# Patent application title: EXTENDED RANGE INTERFEROMETRIC METHODS AND SYSTEMS

##
Inventors:
Vladimir Slastion (Toronto, CA)

IPC8 Class: AG06F1500FI

USPC Class:
702189

Class name: Data processing: measuring, calibrating, or testing measurement system measured signal processing

Publication date: 2011-08-25

Patent application number: 20110208481

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## Abstract:

An interferometer estimates at least one interferometric parameter of one
or more signals emitted from a source. The interferometer includes at
least one phase measurement module for measuring phase differences
between the source signals received at different signal receiving
sensors. At least one coarse estimate of a sought parameter used to
represent the at least one interferometric parameter is generated by
processing the one or more signals received from the source. At least one
fine estimate of the sought parameter is also generated by processing the
at least one coarse sought parameter using the plurality of phase
measurements received from the at least one phase measurement module. The
at least one fine sought parameter represents the at least one
interferometric parameter with greater accuracy than the at least one
coarse sought parameter and over an extended range of values in which the
sought parameter is not unambiguously determinable using only the
plurality of phase measurements.## Claims:

**1.**An interferometer for estimating at least one interferometric parameter of one or more signals received from a source, the interferometer comprising: at least one phase measurement module configured to determine a plurality of phase measurements of the one or more signals received from a source; at least one coarse sought parameter estimator configured to determine at least one coarse sought parameter representing the at least one interferometric parameter by processing the one or more signals received from the source; a fine sought parameter estimator configured to process the at least one coarse sought parameter received from the at least one coarse sought parameter estimator, using the plurality of phase measurements received from the at least one phase measurement module, to determine at least one fine sought parameter representing the at least one interferometric parameter with greater accuracy than the at least one coarse sought parameter.

**2.**The interferometer of claim 1, wherein the fine sought parameter estimator comprises: a combined estimator configured to determine at least one partial sought parameter, representing the interferometric parameter over a narrower range of values than the at least one coarse sought parameter, and at least one noise parameter associated with the plurality of phase measurements by processing the plurality of phase measurements received from the at least one phase measurement module; and at least one partial sought parameter extender configured to calculate the at least one fine sought parameter using the at least one partial sought parameter received from the combined estimator and the at least one coarse sought parameter received from the at least one coarse sought parameter estimator.

**3.**The interferometer of claim 2, wherein the combined estimator is configured to estimate a vector Θ of M partial sought parameters and the at least one noise parameter by processing a vector φ of N phase measurements received into the combined estimator, where N is greater than M, each element of the vector φ of phase measurements defined within one phase cycle, and the vector φ of phase measurements is related to the vector Θ of partial sought parameters by: a vector of N integer numbers k of phase cycles missed in the N phase measurements φ, a vector n of N phase errors associated with the N phase measurements φ, and a matrix A with dimensions N×M comprising M column vectors a

_{i}that are N-dimensional linearly independent vectors of relatively prime numbers.

**4.**The interferometer of claim 3, wherein the vector φ of phase measurements is related to the vector Θ of partial sought parameters according to: φ=AΘ-k+n.

**5.**The interferometer of claim 4, wherein one or more of the at least one partial sought parameter extenders is configured to calculate a corresponding fine sought parameter Θ

_{F}according to: Θ

_{F}=Θ*

_{C}+Θ

_{P}, where Θ

_{P}represents a corresponding partial sought parameter received from the combined estimator, and Θ*

_{C}is calculated by processing a corresponding coarse sought parameter Θ

_{C}, received from the coarse sought parameter estimator, according to: Θ C * = { Θ CR ; if ( Δ Θ C - Θ P ) ≦

**0.**5 Θ CR + 1 ; if ( Δ Θ C - Θ P ) >

**0.**5 Θ CR - 1 ; if ( Δ Θ C - Θ P ) < -

**0.**5 , ##EQU00025## where |X| is an absolute value of X , and where Θ

_{CR}represents an integer component of the corresponding coarse sought parameter Θ

_{C}defined according to: Θ

_{CR}=rnd[Θ

_{C}], where rnd[ . . . ] is a procedure for rounding an element inside the square brackets [ . . . ] to a nearest integer, and where ΔΘ

_{C}represents a residual component of the corresponding coarse sought parameter Θ

_{C}defined according to: ΔΘ

_{C}=rrni{Θ

_{C}}, where rrni{ . . . } is a procedure for calculating a residual of rounding the element inside the braces { . . . } to the nearest integer.

**6.**The interferometer of claim 4, wherein the combined estimator comprises: a first phase measurements converter configured to calculate an M-dimensional vector ψ by processing the vector cp of phase measurements, received from the at least one phase measurement module, according to: ψ=Vφ, where V is a matrix with dimensions M×N that is predefined for the matrix A; a second phase measurements converter configured to calculate an (N-M) dimensional vector δ by processing the vector φ of phase measurements, received from the at least one phase measurement module, according to: δ=Uφ, where U is a matrix with dimensions (N-M)×N that is predefined for the matrix A; and a noise parameters calculator configured to process the vector δ received from the second phase measurements converter to calculate an (N-M) dimensional vector v of noise parameters according to: v=rrni{δ}, where rrni{ . . . } is a procedure for calculating residuals of rounding each element of the vector inside the braces { . . . } to nearest integers.

**7.**The interferometer of claim 6, wherein the combined estimator further comprises a partial sought parameters estimator configured to determine the vector Θ of partial sought parameters by processing the vector ψ, received from the first phase measurements converter, according to: Θ=rrni{ψ}, wherein the vector Θ of partial sought parameters and the at least one noise parameter are outputs of the combined estimator.

**8.**The interferometer of claim 7, wherein the vector v of noise parameters is an output of the combined estimator.

**9.**The interferometer of claim 7, wherein the combined estimator further comprises a common noise parameter estimator configured to calculate a common noise parameter a according to: α = ( i = 1 N - M v i 2 ) 1 / 2 , ##EQU00026## where each v

_{i}is an element of the vector v of noise parameters received from the noise parameters calculator, and wherein the common noise parameter α is an output of the combined estimator.

**10.**The interferometer of claim 7, wherein the combined estimator further comprises a discrete noise parameter estimator having the vector v of noise parameters received from the noise parameters calculator and (N-M)×Z threshold values γ

_{ij}as inputs, the discrete noise parameter estimator configured to calculate: β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ; i = 1 , ( N - M ) , j = 1 , Z , ##EQU00027## where |v

_{i}| is an absolute value of v

_{i}, and wherein the discrete noise parameter estimator is configured to calculate Z elements of a vector ε according to: ε

_{j}=(β

_{1}jvβ

_{2}jv . . . vβ.sub.(N-M)j); j=1, . . . Z, where V is a logical disjunction, and wherein the discrete noise parameter estimator is configured to calculate a discrete noise parameter q according to: q=count[ε], where count[ . . . ] is a procedure for counting a number of logical ones in the binary vector inside the square brackets, and wherein the discrete noise parameter q is an output of the combined estimator.

**11.**The interferometer of claim 7, wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the at least one noise parameter received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**12.**The interferometer of claim 6, wherein the combined estimator further comprises a partial sought parameters estimator configured to determine the vector Θ of partial sought parameters according to: Θ=rrni{Hξ}, where H is a matrix with dimensions M×N that is predefined for the matrix A and for a covariance matrix B that characterizes the vector n of phase errors, and ξ is an N-dimensional vector combination of the vector v of noise parameters received from the noise parameters calculator, and the vector ψ received from the first phase measurements converter, according to: ξ = ( v ψ ) , ##EQU00028## wherein the vector Θ of partial sought parameters is an output of the combined estimator.

**13.**The interferometer of claim 12, wherein the at least one noise parameter is an output of the combined estimator.

**14.**The interferometer of claim 13, wherein the vector v of noise parameters is an output of the combined estimator.

**15.**The interferometer of claim 13, wherein the combined estimator further comprises a common noise parameter estimator configured to calculate a common noise parameter α according to: α = ( i = 1 N - M v i 2 ) 1 / 2 , ##EQU00029## where each v

_{i}is an element of the vector v of noise parameters received from the noise parameters calculator, and wherein the common noise parameter α is an output of the combined estimator.

**16.**The interferometer of claim 13, wherein the combined estimator further comprises a discrete noise parameter estimator having the vector v of noise parameters received from the noise parameters calculator and (N-M)×Z threshold values γ

_{ij}as inputs, the discrete noise parameter estimator configured to calculate: β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ; i = 1 , ( N - M ) , j = 1 , Z , ##EQU00030## where |v

_{i}| is an absolute value of v

_{i}, and wherein the discrete noise parameter estimator is configured to calculate Z elements of a vector ε according to: ε

_{j}=(β

_{1}jvβ

_{2}jv . . . vβ.sub.(N-M)j); j=1, . . . Z, where V is a logical disjunction, and wherein the discrete noise parameter estimator is configured to calculate a discrete noise parameter q according to: q=count[ε], where count[ . . . ] is a procedure for counting a number of logical ones in the binary vector inside the square brackets, and wherein the discrete noise parameter q is an output of the combined estimator.

**17.**The interferometer of claim 13, wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the at least one noise parameter received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**18.**The interferometer of claim 6, wherein the combined estimator further comprises: a noise parameters converter configured to process the vector v of noise parameters received from the noise parameters calculator to calculate an (N-M)-dimensional vector η according to: η=Pv, where P is a matrix with dimensions (N-M)×(N-M) that is predefined for the matrix A and for a covariance matrix B that characterizes the vector n of phase errors; a region shift calculator configured to process the vector η received from the noise parameters converter to calculate an (N-M)-dimensional vector χ* according to: χ * = arg min χ (

**0.**5 ( χ i T P χ i ) + χ i T η ) , ##EQU00031## where each χ

_{i}is an (N-M)-dimensional vector comprising elements of 0 or .+

**-.**1 that is predefined for the matrix A and for the covariance matrix B; a noise parameters corrector configured to process the vector v of noise parameters received from the noise parameters calculator and the vector χ* received from the region shift calculator to calculate an (N-M)-dimensional vector ρ according to: ρ=v+χ*; and a partial sought parameters estimator configured to calculate the vector Θ of the partial sought parameters according to: Θ=rrni{Hτ}, where H is a matrix with dimensions M×N that is predefined for the matrix A and for the covariance matrix B, and τ is an N-dimensional vector combination of the vector ρ received from the noise parameters corrector, and the vector ψ received from the first phase measurements converter, according to: τ = ( ρ ψ ) , ##EQU00032## wherein the vector Θ of partial sought parameters is an output of the combined estimator.

**19.**The interferometer of claim 18, wherein the at least one noise parameter is an output of the combined estimator.

**20.**The interferometer of claim 19, wherein the vector v of noise parameters is an output of the combined estimator.

**21.**The interferometer of claim 19, wherein the combined estimator further comprises a common noise parameter estimator configured to calculate a common noise parameter α according to: α = ( i = 1 N - M v i 2 ) 1 / 2 , ##EQU00033## where each v

_{i}is an element of the vector v of noise parameters received from the noise parameters calculator, and wherein the common noise parameter α is an output of the combined estimator.

**22.**The interferometer of claim 19, wherein the combined estimator further comprises a discrete noise parameter estimator having the vector v of noise parameters received from the noise parameters calculator and (N-M)×Z threshold values γ

_{ij}as inputs, the discrete noise parameter estimator configured to calculate: β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ; i = 1 , ( N - M ) , j = 1 , Z , ##EQU00034## where |v

_{i}| is an absolute value of v

_{i}, and wherein the discrete noise parameter estimator is configured to calculate Z elements of a vector ε according to: ε

_{j}=(β

_{1}jvβ

_{2}jv . . . vβ.sub.(N-M)j); j=1, . . . Z, where V is a logical disjunction, and wherein the discrete noise parameter estimator is configured to calculate a discrete noise parameter q according to: q=count[ε], where count[ . . . ] is a procedure for counting a number of logical ones in the binary vector inside the square brackets, and wherein the discrete noise parameter q is an output of the combined estimator.

**23.**The interferometer of claim 19, wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the at least one noise parameter received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**24.**The interferometer of claim 6, wherein the combined estimator further comprises: a noise parameters converter configured to process the vector v of noise parameters received from the noise parameters calculator to calculate an (N-M)-dimensional vector η according to: η=Pv, where P is a matrix with dimensions (N-M)×(N-M) that is predefined for the matrix A and for a covariance matrix B that characterizes the vector n of phase errors; a region shift calculator configured to process the vector η received from the noise parameters converter to calculate an (N-M)-dimensional vector χ* according to: χ * = arg min χ (

**0.**5 ( χ i T P χ i ) + χ i T η ) , ##EQU00035## where each χ

_{i}is an (N-M)-dimensional vector comprising elements of 0 or .+

**-.**1 that is predefined for the matrix A and for the covariance matrix B; a noise parameters corrector configured to process the vector χ* received from the region shift calculator to calculate an M-dimensional vector f according to: f=Rχ*, where R is a matrix with dimensions M×(N-M) that is predefined for the matrix A and for the covariance matrix B; and a partial sought parameters calculator configured to calculate the vector Θ of partial sought parameters by processing the vector f, received from the noise parameters corrector, according to: Θ=rrni{Hξ+f}, where H is a matrix with dimensions M×N that is predefined for the matrix A and for the covariance matrix B, and ξ is an N-dimensional vector combination of the vector v of noise parameters received from the noise parameters calculator, and the vector ψ received from the first phase measurements converter, according to: ξ = ( v ψ ) , ##EQU00036## wherein the vector Θ of partial sought parameters is an output of the combined estimator.

**25.**The interferometer of claim 24, wherein the at least one noise parameter is an output of the combined estimator.

**26.**The interferometer of claim 25, wherein the vector v of noise parameters is an output of the combined estimator.

**27.**The interferometer of claim 25, wherein the combined estimator further comprises a common noise parameter estimator configured to calculate a common noise parameter a according to: α = ( i = 1 N - M v i 2 ) 1 / 2 , ##EQU00037## where each v

_{i}is an element of the vector v of noise parameters received from the noise parameters calculator, and wherein the common noise parameter α is an output of the combined estimator.

**28.**The interferometer of claim 25, wherein the combined estimator further comprises a discrete noise parameter estimator having the vector v of noise parameters received from the noise parameters calculator and (N-M)×Z threshold values γ

_{ij}as inputs, the discrete noise parameter estimator configured to calculate: β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ; i = 1 , ( N - M ) , j = 1 , Z , ##EQU00038## where |v

_{i}| is an absolute value of v

_{i}, and wherein the discrete noise parameter estimator is configured to calculate Z elements of a vector ε according to: ε

_{j}=(β

_{1}jvβ

_{2}jv . . . vβ.sub.(N-M)j); j=1, . . . Z, where V is a logical disjunction, and wherein the discrete noise parameter estimator is configured to calculate a discrete noise parameter q according to: q=count[ε], where count[ . . . ] is a procedure for counting a number of logical ones in the binary vector inside the square brackets, and wherein the discrete noise parameter q is an output of the combined estimator.

**29.**The interferometer of claim 25, wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the at least one noise parameter received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**30.**The interferometer of claim 6, wherein the combined estimator further comprises: a discrete noise parameter estimator having the vector v of noise parameters received from the noise parameters calculator and (N-M)×Z threshold values γ

_{ij}as inputs, the discrete noise parameter estimator configured to calculate: β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ; i = 1 , ( N - M ) , j = 1 , Z , ##EQU00039## where |v

_{i}| is an absolute value of v

_{i}, and wherein the discrete noise parameter estimator is configured to calculate Z elements of a vector ε according to: ε

_{j}=(β

_{1}jvβ

_{2}jv . . . vβ.sub.(N-M)j); j=1, . . . Z, where V is a logical disjunction, and wherein the discrete noise parameter estimator is configured to calculate a discrete noise parameter q according to: q=count[ε], where count[ . . . ] is a procedure for counting a number of logical ones in the binary vector inside the square brackets; and an adaptive estimator having the discrete noise parameter q received from the discrete noise parameter estimator, the vector v of noise parameters received from the noise parameters calculator, and the vector ψ received from the first phase measurements converter as inputs, the adaptive estimator configured to determine the vector Θ of sought parameters differently based upon the value of the discrete noise parameter q.

**31.**The interferometer of claim 30, wherein the adaptive estimator is configured to determine the vector Θ of partial sought parameters: if q is below or equal to a first threshold, according to: Θ=rrni{ψ}; if q is above the first threshold and below or equal to a second threshold greater than the first threshold, according to: Θ=rrni{Hξ}, where H is a matrix with dimensions M×N that is predefined for the matrix A and for a covariance matrix B that characterizes the vector n of phase errors, and ξ is an N-dimensional vector combination of v and ψ according to: ξ = ( v ψ ) ; ##EQU00040## and if q is above the second threshold, by calculating an (N-M)-dimensional vector η according to: η=Pv, where P is a matrix with dimensions (N-M)×(N-M) that is predefined for the matrix A and for the covariance matrix B, and by further calculating an (N-M)-dimensional vector χ* according to: χ * = arg min χ (

**0.**5 ( χ i T P χ i ) + χ i T η ) , ##EQU00041## where each χ

_{i}is an (N-M)-dimensional vector comprising elements of 0 or .+

**-.**1 that is predefined for the matrix A and for the covariance matrix B, and by further calculating an (N-M)-dimensional vector ρ according to: ρ=v+χ*, and by calculating the vector Θ of partial sought parameters according to: Θ=rrni{Hτ}, where τ is an N-dimensional vector combination of ρ and ψ according to: τ = ( ρ ψ ) , ##EQU00042## wherein the vector Θ of partial sought parameters is an output of the combined estimator.

**32.**The interferometer of claim 31, wherein the discrete noise parameter q is an output of the combined estimator, and wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the discrete noise parameter q received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**33.**The interferometer of claim 30, wherein the adaptive estimator is configured to determine the vector Θ of partial sought parameters: if q is below or equal to a first threshold, according to: Θ=rrni{ψ}; if q is above the first threshold and below or equal to a second threshold greater than the first threshold, according to: Θ=rrni{Hξ}, where H is a matrix with dimensions M×N that is predefined for the matrix A and for a covariance matrix B that characterizes the vector n of phase errors, and ξ is an N-dimensional vector combination of v and ψ according to: ξ = ( v ψ ) ; ##EQU00043## and if q is above the second threshold, by calculating an (N-M)-dimensional vector η according to: η=Pv, where P is a matrix with dimensions (N-M)×(N-M) that is predefined for the matrix A and for the covariance matrix B, and by further calculating an (N-M)-dimensional vector χ* according to: χ * = arg min χ (

**0.**5 ( χ i T P χ i ) + χ i T η ) , ##EQU00044## where each χ

_{i}is an (N-M)-dimensional vector comprising elements of 0 or .+

**-.**1 that is predefined for the matrix A and for the covariance matrix B, and by further calculating an M-dimensional vector f according to: f=Rχ*, where R is a matrix with dimensions M×(N-M) that is predefined for the matrix A and for the covariance matrix B, and by calculating the vector Θ of sought parameters according to: Θ=rrni{Hξ+f}, wherein the vector Θ of partial sought parameters is an output of the combined estimator.

**34.**The interferometer of claim 33, wherein the discrete noise parameter q is an output of the combined estimator, and wherein the fine sought parameter estimator further comprises a postprocessor configured to generate at least one postprocessed fine sought parameter by processing the at least one fine sought parameter received from the at least one partial sought parameter extender, using the discrete noise parameter q received from the combined estimator, to improve an estimate of the at least one interferometric parameter.

**35.**The interferometer of claim 2, wherein the combined estimator is configured to: compare the at least one noise parameter with at least one threshold; and based on a result of the comparison, determine the at least one partial sought parameter by processing the plurality of phase measurements.

**36.**The interferometer of claim 1, wherein the fine sought parameter estimator comprises: a combined estimator configured to determine a vector Θ of M partial sought parameters, representing the at least one interferometric parameter over a narrower range of values than the at least one coarse sought parameter, by processing a vector φ of N phase measurements received into the combined estimator, where N is greater than M, each element of the vector φ of phase measurements defined within one phase cycle, and the vector φ of phase measurements is related to the vector Θ of partial sought parameters according to: φ=AΘ-k+n, where k represents a vector of N integer numbers of phase cycles missed in the N phase measurements φ, n represents a vector of N phase errors associated with the N phase measurements φ, and A represents a matrix with dimensions N×M comprising M column vectors a

_{i}that are N-dimensional linearly independent vectors of relatively prime numbers, the combined estimator comprising: a phase measurements converter configured to calculate an M-dimensional vector ψ by processing the vector φ of phase measurements, received from the at least one phase measurement module, according to: ψ=Vφ, where V is a matrix with dimensions M×N that is predefined for the matrix A; and a partial sought parameters estimator configured to process the vector p received from the phase measurements converter to determine the vector Θ of partial sought parameters according to: Θ=rrni{ψ}, wherein the vector Θ of partial sought parameters is an output of the combined estimator; and at least one partial sought parameter extender configured to calculate the at least one fine sought parameter using the at least one partial sought parameter received from the combined estimator and the at least one coarse sought parameter received from the at least one coarse sought parameter estimator.

**37.**The interferometer of claim 36, wherein one or more of the at least one partial sought parameter extender is configured to calculate a corresponding fine sought parameter Θ

_{F}according to: Θ

_{F}=Θ*

_{C}+Θ

_{P}, where Θ

_{P}represents a corresponding partial sought parameter received from the combined estimator, and Θ*

_{C}is calculated by processing a corresponding coarse sought parameter Θ

_{C}, received from the coarse sought parameter estimator, according to: Θ C * = { Θ CR ; if ( Δ Θ C - Θ P ) ≦

**0.**5 Θ CR + 1 ; if ( Δ Θ C - Θ P ) >

**0.**5 Θ CR - 1 ; if ( Δ Θ C - Θ P ) < -

**0.**5 , ##EQU00045## where |X| is an absolute value of X, and where Θ

_{CR}represents an integer component of the corresponding coarse sought parameter Θ

_{C}defined according to: Θ

_{CR}=rnd[Θ

_{C}], where rnd[ . . . ] is a procedure for rounding an element inside the square brackets [ . . . ] to a nearest integer, and where ΔΘ

_{C}represents a residual component of the corresponding coarse sought parameter Θ

_{C}defined according to: ΔΘ

_{C}=rrni{Θ

_{C}}, where rrni{ . . . } is a procedure for calculating a residual of rounding the element inside the braces { . . . } to the nearest integer.

**38.**A fine sought parameter estimator for use in an interferometer to estimate at least one interferometric parameter, the fine sought parameter estimator comprising a processor configured to: receive a vector φ of N phase measurements and a vector Θ

_{C}of M coarse sought parameters; estimate a vector Θ of M partial sought parameters by processing the vector φ of phase measurements, where N is greater than M, each element of the vector φ of phase measurements defined within one phase cycle, and the vector φ of phase measurements related to the vector Θ of partial sought parameters by: a vector of N integer numbers k of phase cycles missed in the N phase measurements φ, a vector n of N phase errors associated with the N phase measurements φ, and a matrix A with dimensions N×M comprising M column vectors a

_{i}that are N-dimensional linearly independent vectors of relatively prime numbers; and process the vector Θ of M partial sought parameters and the vector Θ

_{C}of M coarse sought parameters to generate a vector Θ

_{F}of M fine sought parameters representing the at least one interferometric parameter with greater accuracy than the vector Θ

_{C}of M coarse sought parameters and over a greater range of values than the vector Θ of M partial sought parameters.

**39.**A method of estimating at least one interferometric parameter of one or more signals from a source, the method comprising: determining a plurality of phase measurements of the one or more signals received from a source; determining at least one coarse sought parameter representing the at least one interferometric parameter by processing the one or more signals received from the source; and processing the at least one coarse sought parameter using the plurality of phase measurements to determine at least one fine sought parameter representing the at least one interferometric parameter with greater accuracy than the at least one coarse sought parameter.

## Description:

**CROSS**-REFERENCE TO RELATED APPLICATIONS

**[0001]**This application claims the benefit of U.S. Provisional Application Ser. No. 61/306,046 filed on Feb. 19, 2010, the entire contents of which are hereby incorporated by reference.

**FIELD**

**[0002]**Embodiments described herein relate generally to location systems and methods for calculating the distance to an object or a location of an object by estimating one or more time parameters or angles of arrival and, more specifically, to interferometric systems and methods for estimating locations on the basis of multiple ambiguous phase measurements.

**INTRODUCTION**

**[0003]**Location systems are used to estimate the location of objects in one-dimensional, two-dimensional or three-dimensional spaces. To provide this functionality, most location systems operate by measuring angles of arrival, or alternatively some time parameters of a signal emitted or reflected by a located object.

**[0004]**Different structures of location systems utilize different methods to estimate object locations. For example, triangulation is a method used to estimate locations based on angles of arrival (AOA). Trilateration is a method used by some location systems to estimate the location of an object by measuring the time of flight (TOF) or time of arrival (TOA) of a signal emitted from that object to several receivers. In a different method, known as multilateration (also known as hyperbolic positioning), the location of an object may be estimated by computing the time difference of arrival (TDOA) of a signal emitted from that object to three or more receivers.

**[0005]**A user of the location system often needs to be able to determine object locations accurately, with high reliability and over wide ranges. The accuracy and reliability with which the object location may be determined in various systems generally depend on how accurate and reliable are the estimates of AOA, TOF, TOA or TDOA. Location systems often work in conditions where noisy signals are received or where the received signals have multipath propagation. Each of these factors may significantly affect the accuracy and reliability of the AOA, TOF, TOA or TDOA estimates. Interferometric estimation of such parameters is often one of the most accurate methods. It can be used for estimating location information with high accuracy, in wide ranges and with generally good quality and reliability.

**SUMMARY**

**[0006]**Some embodiments described herein relate to a combined estimator. In some embodiments, the combined estimator is for use in an interferometric system, which may include one or more direction finding interferometers or one or more interferometric location systems. In some embodiments, the combined estimator comprises a processor. In some embodiments the combined estimators described herein can be implemented in hardware, in software running on microprocessor, ASIC, or in combination of hardware and software. In some such embodiments, the combined estimator estimates a plurality of parameters, which may be referred to as sought parameters, and which can in turn be used to estimate one or more interferometric parameters of a source signal. In some embodiments the combined estimator also estimates noise parameters that may be independent of the one or more interferometric parameters being estimated by the interferometric system.

**[0007]**In some embodiments, the noise parameters are used to determine the quality of associated estimated parameters. In some embodiments, the noise parameters are used to process or filter associated estimated parameters. In some embodiments, if the noise component is above a threshold then the associated estimated parameters are discarded and therefore are not used in the estimation of the one or more interferometric parameters. Alternatively, in some embodiments, if the noise component is above a threshold then the associated estimated parameters are weighted in such a way that reliable estimates take precedence over unreliable estimates. In this way, the estimate of the one or more interferometric parameters may be improved.

**[0008]**Some embodiments described herein relate to an interferometer for determining an interferometric parameter. The interferometer is configured to: determine a plurality of phase measurement values; determine a noise parameter associated with phase measurement values; determine if the noise parameter is above a threshold; if the noise parameter is above the threshold, discard the associated estimated parameters' values; determine the interferometric parameter based on the non-discarded estimated parameters' values.

**[0009]**In some embodiments, the estimated interferometric parameter may be an angle of arrival of a signal. In some embodiments, the estimated interferometric parameter may be a time parameter of a signal that is used in the interferometer to estimate a location of the signal, such as an object that emitted the signal.

**[0010]**In some embodiments, each phase measurement is a phase difference in signals received by one or more signal sensors. In some embodiments, the phase measurement is a phase difference in signals received at two signal sensors. In some embodiments, the phase difference is outputted by a phase detector coupled to receivers that are in turn coupled to the signal sensors.

**[0011]**In some embodiments, a noise parameter is determined, where the noise parameter is indicative of the level of noise. In some embodiments, the noise parameter is a noise component that is independent of the interferometric parameter.

**[0012]**In some embodiments, at least one sought parameter is determined. In some such embodiments, the interferometric parameters are determined from the sought parameters. In some embodiments, the noise parameter associated with sought parameters is determined. In some embodiments, the noise parameters are used to process or filter associated estimated sought parameters. Thus, in some embodiments, If the noise parameter is above a threshold then the associated sought parameters are discarded and are not used in the determination of the interferometric parameters or, alternatively, are adaptively filtered according to the level of the noise parameter.

**[0013]**Some embodiments described herein relate to a method of determining interferometric parameters, the method comprises: determining a plurality of phase measurement values; determining a noise parameter associated with phase measurement values; determining if the noise parameter value is above a threshold; if the noise parameter value is above the threshold, discarding the associated phase measurement values; and determining the interferometric parameters based on the non-discarded phase measurement values.

**[0014]**Some embodiments described herein relate to an interferometer for estimating at least one interferometric parameter of one or more signals received from a source. The interferometer has at least one phase measurement module configured to determine a plurality of phase measurements of the one or more signals received from a source. At least one coarse sought parameter estimator is configured to determine at least one coarse sought parameter representing the at least one interferometric parameter by processing one or more signals received from the source. A fine sought parameter estimator is configured to process the at least one coarse sought parameter, received from the at least one coarse sought parameter estimator, using the plurality of phase measurements received from the at least one phase measurement module to determine at least one fine sought parameter representing the at least one interferometric parameter with greater accuracy than the at least one coarse sought parameter.

**[0015]**In some embodiments, the fine sought parameter estimator comprises a combined estimator configured to determine at least one partial sought parameter, which represents the interferometric parameter over a narrower range of values than the at least one coarse sought parameter. The combined estimator also may determine at least one noise parameter associated with the plurality of phase measurements by processing the plurality of phase measurements. In some embodiments, the fine sought parameter estimator also comprise at least one partial sought parameter extender configured to calculate the at least one fine sought parameter using the partial sought parameter received from the combined estimator and the coarse sought parameter received from the coarse sought parameter estimator.

**[0016]**In some embodiments, the coarse sought parameter estimator generates the coarse estimate of the sought parameter based on a time difference of arrival of the source signal at a pair of signal receiving antennas determined by comparing the magnitude of the received signals against a threshold level. The time difference of arrival is then normalized by an unambiguous time interval in order to determine the coarse estimate of the sought parameter.

**[0017]**In some embodiments, a partial estimate of the sought parameter is also generated to estimate time parameters unambiguously within the unambiguous time interval. The partial estimate of the sought parameter may be generated based on the measured phase differences. Combining the partial and coarse estimates of the sought parameters then yields the fine estimate of the sought parameter with greater accuracy than the coarse estimate and not limited to the same finite range as the partial estimate.

**BRIEF DESCRIPTION OF THE DRAWINGS**

**[0018]**For a better understanding of the embodiments described herein and to show more clearly how they may be carried into effect, reference will now be made, by way of example only, to the accompanying drawings which show at least one example embodiment, and in which:

**[0019]**FIG. 1 illustrates a schematic diagram of various embodiments of an interferometric location system that estimates two spatial coordinates;

**[0020]**FIG. 2 is a graph that illustrates coarse estimation of a sought parameter in various embodiments;

**[0021]**FIG. 3 is a graph that illustrates, for various embodiments, a relationship between measured phase differences φ

_{i}, a coarse sought parameter estimate Θ

_{C}, a partial sought parameter estimate Θ

_{P}, a fine sought parameter estimate Θ

_{F}, and corresponding time parameters for different frequency components f

_{i}of received signals.

**[0022]**FIG. 4A illustrates a schematic diagram of various embodiments of a fine sought parameter estimator for estimating one sought parameter;

**[0023]**FIG. 4B illustrates a schematic diagram of various embodiments of a fine sought parameter estimator for estimating one sought parameter;

**[0024]**FIG. 5A illustrates a schematic diagram of various embodiments of a fine sought parameter estimator for estimating M sought parameters;

**[0025]**FIG. 5B illustrates a schematic diagram of various embodiments of a fine sought parameter estimator for estimating M sought parameters;

**[0026]**FIG. 5C illustrates a schematic diagram of various embodiments of a fine sought parameter estimator for estimating M sought parameters;

**[0027]**FIG. 6 is a diagram illustrating δ, χ, v, and Voronoi regions for various embodiments that have N-M=2;

**[0028]**FIG. 7 is a graph illustrating the relationship between φ, k, a, and n for various embodiments of interferometers estimating a sought parameter Θ on the basis of two phase measurements;

**[0029]**FIG. 8 is a block diagram illustrating various embodiments of a combined estimator;

**[0030]**FIG. 9 is a block diagram illustrating various embodiments of a combined estimator;

**[0031]**FIG. 10 is a diagram illustrating a Voronoi region and three threshold parallelotopes in

^{2}for various embodiments;

**[0032]**FIG. 11 is a block diagram illustrating various embodiments of a discrete noise parameter estimator;

**[0033]**FIG. 12 is a block diagram illustrating various embodiments of a combined estimator;

**[0034]**FIG. 13 is a block diagram illustrating various embodiments of a combined estimator;

**[0035]**FIG. 14 is a block diagram illustrating various embodiments of a combined estimator;

**[0036]**FIG. 15 is a graph that illustrates, for various embodiments, the difference between the probability of correct ambiguity resolution in the calculation of interferometric parameters with and without the rejection of measurements based on the level of noise parameter;

**[0037]**FIG. 16 is a block diagram illustrating various embodiments of a combined estimator;

**[0038]**FIG. 17 is a block diagram illustrating various embodiments of a combined estimator;

**[0039]**FIG. 18 is a block diagram illustrating various embodiments of a combined estimator;

**[0040]**FIG. 19 is a block diagram illustrating various embodiments of a combined estimator;

**[0041]**FIG. 20 is a block diagram illustrating various embodiments of a combined estimator;

**[0042]**FIG. 21 is a block diagram illustrating various embodiments of a combined estimator;

**[0043]**FIG. 22 is a block diagram illustrating various embodiments of a combined estimator;

**[0044]**FIG. 23 is a block diagram illustrating various embodiments of a combined estimator;

**[0045]**FIG. 24 is a block diagram illustrating various embodiments of a combined estimator;

**[0046]**FIG. 25 is a graph that illustrates, for various embodiments, the difference between the probability of correct ambiguity resolution in the calculation of interferometric parameters with and without the rejection of measurements based on the level of noise parameter; and

**[0047]**FIG. 26 is a block diagram illustrating various embodiments of a combined estimator.

**DETAILED DESCRIPTION OF EMBODIMENTS**

**[0048]**The accuracy with which a location system can estimate the location or range of the located signal-emitting object may depend, among other factors, on the accuracy of the time parameter estimation used by the location system. Interferometric phase measurements may be used to achieve very accurate time parameter estimation, which in turn would enable very accurate estimates of the object location. However, interferometric phase measurements of an oscillating signal are often inherently ambiguous, requiring multiple estimations of the same time parameter to resolve the inherent ambiguity. The requirement of multiple time parameter estimations tends to increase overall system complexity, for example, in terms of additional hardware components or additional computing resources.

**[0049]**The embodiments described herein generally relate to interferometric systems and methods that are operable to resolve the inherent ambiguity in time parameter estimation without incurring undue system complexity. Certain of the described embodiments may be applied to radars of different types and configurations, as well as other forms of location and/or navigation systems. Some of the embodiments process ambiguous phase measurements in order to produce estimates of one or more time parameters, as described above, such as time of flight (TOF), time of arrival (TOA) and time difference of arrival (TDOA). Some of the embodiments are also operable to process ambiguous phase measurements in order to estimate of one or more Angles of Arrival (AOA), as is the case of direction finding interferometers. For convenience, reference may be made primarily to interferometric systems and related methods for locating objects by measuring time difference of arrival of one or more signals emitted by the object. Considered interferometric methods are generalized on estimation of several interferometric parameters, which can be used in direction finding interferometers estimating more than one angle of arrival.

**[0050]**Various interferometric systems are known in the art. Many of these interferometric systems are direction finding interferometers that utilize an antenna array in order to estimate the AOA of an incoming signal, which characterizes the direction from the antenna array to the located object. Depending on the application, the antenna array may be a linear antenna array capable of measuring one angle of arrival, a planar antenna array capable of measuring two angles of arrival, or a three-dimensional antenna array capable of measuring more than two different angles of arrival.

**[0051]**Phase interferometers for use in location systems may be implemented using an array of several spatially separated receiving antennas, where the respective location of each antenna in the array is known. In such systems, the time parameter measured is often TDOA, although other time parameters, such as those referenced above, can also be measured instead. In some embodiments, a located object emits pulsed signals with known carrier frequencies that are detected in turn by the receiving antennas. The utilized time parameter (e.g. TDOA) may be estimated as the elapsed time between the beginnings of respective signal pulses received at different antenna pairs in the system.

**[0052]**If the signals emitted by the located object have wide spectral bandwidth, the pulses received at each antenna have a relatively sharp rise to full signal amplitude. The start of each pulse may then be relatively easy to detect with good accuracy. However, not all signals emitted from the located object will have wide spectral bandwidth. In particular, if the signals emitted by the located object do not have a wide spectral bandwidth, the pulses received at each antenna may have a relatively slow rise to full signal amplitude, which can make it difficult to accurately detect the start of each signal pulse. In these cases, accurate estimates of TDOA may be difficult to produce based on pulse arrival times. Therefore, estimating the start of signals received at different antennas may be used in some cases as a course estimate of TDOA. Pulse arrival times may also be used to produce initial or intermediate estimates of TDOA.

**[0053]**To provide a finer estimate of this time parameter, in addition to measuring the start times of the signal pulses received at different antenna, the phase difference between like frequency signals received at different antennas may also be measured. In general, the shorter the wavelength of the signal used for measuring phase difference, the more accurate will be the estimate of a given time parameter, such as TDOA. Reducing the wavelength of the received signals therefore provides one way to improve the accuracy of the time parameter estimate.

**[0054]**However, when measuring the phase difference between two signals received at a pair of antennas, an inherent ambiguity will generally arise if the distance between the pair of receiving antennas is greater than one half wavelength of the received signals. In that case, the actual phase difference between the two received signals can be much more than 360° and, yet, not be fully detected because phase difference is only measurable within a 360° range. Consequently, integer numbers of whole cycles of phase differences can be missed in the measurements of phase. The integer numbers of whole cycles are often reproduced through subsequent processing of the phase measurements in order to provide unambiguous time parameter estimation.

**[0055]**The described embodiments are operable with located objects that emit pulsed signals having multiple different known frequency components (or alternatively multiple different known wavelengths). By measuring multiple different phase differences between like frequency components of the emitted signals received at different receiving antennas, the described embodiments provide for ambiguity resolution and fine estimation of time parameters. The fine estimate of the time parameters may be provided instead of, or in addition to, the course estimate of the time parameter generated using pulse start times, as will be explained in more detail below.

**[0056]**Another difficulty with phase measurements in interferometric location systems is that the multipath of signal propagation can introduce significant phase measurement errors. This effect, together with errors associated with other noise components of the received signals, can significantly decrease the probability of correct ambiguity resolution. For example, ambiguity resolution in interferometric systems can be incorrect if the sum of all phase errors in the phase measurements is above a given threshold level. This limit can vary depending on the particular configuration of the interferometer and is selectable in various embodiments.

**[0057]**If the sum of all phase errors in the phase measurements is above the given threshold level, then the probability of incorrect ambiguity resolution for the corresponding phase measurements is high. This in turn can mean that the result of the phase measurements is unreliable. Accordingly, in various embodiments, phase measurements having a corresponding amount of phase noise that is above the threshold level of noise can be rejected or specifically processed to improve the accuracy of the time parameter estimation.

**[0058]**Moreover, in various embodiments, the level of noise in the phase measurements is used to characterize the quality of the time parameter estimate. In some of the described embodiments, both noise parameters and the estimate of the time parameter are computed concurrently. In some embodiments, the noise parameters are analyzed in order to estimate the degree of phase errors present in the phase measurements and, upon that basis, determine the reliability of the resulting time parameter estimate. In some such embodiments, if a particular estimate or sample of a time parameter is determined to be unreliable, then that particular estimate is discarded and not used in an overall estimate of the time parameter. Discarding unreliable estimates of the time parameter can improve the overall accuracy of the interferometric location system.

**[0059]**Some embodiments described herein relate to an interferometric location system that produces a fine sought parameter estimate Θ

_{F}, that is obtained as a combination of a coarse sought parameter estimate Θ

_{C}and a partial sought parameter estimate Θ

_{P}. The partial sought parameter estimate Θ

_{P}has generally greater accuracy compared to the coarse sought parameter estimate Θ

_{C}, but is also defined within a more limited range of values. In some cases, the coarse sought parameter estimate Θ

_{C}may exist within limits that significantly exceed the limits imposed on the partial sought parameter estimate Θ

_{P}. The fine sought parameter estimate Θ

_{F}may combine both the accuracy of the partial sought parameter estimate Θ

_{P}and the extended range of the coarse sought parameter estimate Θ

_{C}.

**[0060]**Various embodiments described herein relate to an interferometric location system that estimates a partial sought parameter Θ

_{P}, which is related to an interferometric parameter, such as a time parameter estimated by the interferometric location system to determine the position or range of a located object. Each of the partial sought parameter Θ

_{P}and the fine sought parameter estimate Θ

_{F}is related to one or more noise parameters determined after processing N phase differences φ

_{1}, φ

_{2}, . . . , φ

_{N}measured on N signal components received at pairs of spatially separated antennas and having different wavelengths with respect to one another. In some embodiments, the time parameter to which the various sought parameters Θ

_{F}, Θ

_{P}, and Θ

_{C}are related can represent any of TOA, TOF, or TDOA.

**[0061]**While reference may be made primarily to interferometric location systems that estimate a time parameter, the described embodiments may also be suitable for use in direction finding interferometers. Accordingly, in some embodiments, the sought parameter estimates Θ

_{F}, Θ

_{P}, and Θ

_{C}may relate to estimates of angle of arrival.

**[0062]**In some embodiments, the noise parameters computed and used by the interferometric location system are independent of the particular interferometric parameter, e.g. TOF, TOA, TDOA and AOA, which the interferometric location system is configured to estimate. For example, for some interferometric location systems that are made in accordance with the embodiments disclosed herein, the noise parameters are independent of the position of the located object. In some embodiments, the noise parameters characterize the multipath components of the received signals.

**[0063]**Referring now to FIG. 1, there is illustrated a schematic diagram of an interferometric location system 1100 that estimates two spatial coordinates. As illustrated in FIG. 1, the interferometric location system 1100 estimates two spatial coordinates of a located object 1105 by measuring a time parameter, e.g. TDOA, of a signal emitted from the object 1105 and received at a plurality of signal receiving sensors 1110. However, it should be understood that FIG. 1 (as with other figures referenced herein) is merely illustrative and is not limiting in any way of the described embodiments.

**[0064]**For example, while FIG. 1 illustrates embodiments in which antennas are used to implement the signal receiving sensors 1110, other types or configurations of signal receiving sensors may be used in variant embodiments. In general, any suitable signal receiving sensor 1110 can be used including, but not limited to, an antenna, a light detector, an ultrasonic transducer, or some other sensor. In addition, while three signal receiving sensors 1110 are depicted in FIG. 1, any appropriate number of signal sensors may be utilized. In some embodiments, four or more signal receiving sensors 1110 may be used. The particular number of signal receiving sensors 1110 can vary depending on different factors and the application for which the interferometric location system 1100 is used.

**[0065]**The signal receiving sensors 1110 are spatially distributed within a plane in which the object 1105 is located. The respective locations of the signal receiving sensors 1110 are not necessarily fixed, but are generally known at the moment the signal emitted by the located object 1105 is received at each respective signal receiving sensor 1110. In some embodiments, the signal receiving sensors 1110 are fixed (i.e. stationary) and their locations are known. In some embodiments, the signal receiving sensors 1110 are mobile (i.e. transitory), but have tracked or otherwise knowable trajectories from which their respective locations can be continuously determined.

**[0066]**The located object 1105 emits multiple frequency component signals from which at least N frequency components related to each other as relatively prime numbers may be selected. In some embodiments, the signals emitted by the located object 1105 have at least N components with different known frequencies that relate to each other as relatively prime numbers. The known frequencies may not themselves be prime or relatively prime numbers, but instead relate to each other as relatively prime numbers after being divided by a common multiplier. For example, the signal emitted by the located object 1105 can have two component frequencies f

_{1}=6 MHz and f

_{2}=10 MHz. After dividing through by a common multiplier of 2×10

^{6}Hz, the ratio of these two component frequencies is 3 to 5, which are relatively prime numbers. As used throughout the description, the phrase "relating to one another as relatively prime numbers" may have this general meaning.

**[0067]**In some embodiments, the located object 1105 emits signals that have at least N+1 components with known frequencies that can be combined to produce at least N signal components with different known frequencies that relate to each other as N relatively prime numbers. For example, the signal emitted by the located object 1105 can have three component frequencies f

_{1}=1000 MHz, f

_{2}=1030 MHz and f

_{3}=1040 MHz. These three signal components may be combined to yield two components with frequencies f

_{4}=f

_{2}-f

_{1}=30 MHz and f

_{5}=f

_{3}-f

_{1}=40 MHz. After dividing through by a common multiplier of 10

^{7}Hz, the combined component frequencies f

_{4}and f

_{5}are in the ratio 3 to 4, which are relatively prime numbers. Additional aspects of the frequency components, and how they may relate to each other through the common multiplier as relatively prime numbers, will be discussed further below.

**[0068]**In some cases, the located object 1105 concurrently emits multiple signal components from which the at least N signal components related to each other as relatively prime numbers may be directly selected or otherwise obtained by combining signal components. In other cases, the located object 1105 emits the multiple signal components sequentially, for example, in accordance with a frequency hopping protocol.

**[0069]**Depending on the location of the object 1105 and the relative positioning of the signal receiving sensors 1110, the emitted signal can arrive at the signal receiving sensors 1110 at corresponding different times. Accordingly, the time parameters of the received signals, e.g. TDOA, which the interferometric location system 1100 utilizes to estimate the position of the located object 1105 may generally depend also on the location of the object 1105 and the relative positioning of the signal receiving sensors 1110. In addition, the difference in arrival times of the emitted signal at different signal receiving sensors 1110 also results in the received signals having generally different phases relative to one another.

**[0070]**The signals emitted by the located object 1105 are received at the signal receiving sensors 1110, after which the signals pass through corresponding receivers 1120 and signal transmitting channels 1121 to a plurality of extended interferometers 1130. In some embodiments, one receiver 1120 and one signal transmitting channel 1121 is associated with each signal receiving sensor 1110 to pass the received signals to the plurality of extended interferometers 1130. In various embodiments, the signal transmitting channels 1121 can be cables connected between a respective signal receiver 1120 and extended interferometer 1130, although other types of signal transmitting channels 1121 are possible as well.

**[0071]**Each extended interferometer 1130 may be associated with a corresponding pair of signal receiving antennas 1120, so that the interferometric location system 1100 may include X signal receiving sensors 1110 and X-1 extended interferometers 1130. As illustrated in FIG. 1, in some embodiments, the interferometric location system 1100 may include three signal receiving sensors 1110 and correspondingly two extended interferometers 1130. However, to estimate three spatial coordinates, as described above, the interferometric location system 1100 may include four signal receiving sensors 1110 and correspondingly three extended interferometers 1130. In alternative embodiments, four signal receiving sensors 1110 may be used to estimate two spatial coordinates, but with some added redundancy for increased accuracy and resolution.

**[0072]**In some embodiments, each extended interferometer 1130, which is provided with the signals received at a different pair of the signal receiving sensors 1110, can comprise a phase measurement module 1140, a coarse sought parameter estimator 1150, and a fine sought parameter estimator 1160. The respective pair of signals received at the extended interferometers 1130 are passed to both the phase measurement module 1140 and the coarse sought parameter estimator 1150, so that each of the phase measurement module 1140 and the coarse sought parameter estimator 1150 within a given extended interferometer 1130 receives the same pair of signals for processing.

**[0073]**The phase measurement module 1140 measures N phase differences φ

_{i}, 1<=i<=N corresponding to N like frequency components of the signals received, or otherwise obtained by combining the signals received, at different pairs of the signal receiving sensors 1110. However, in various embodiments, the phase measurement module 1140 can also pre-process the N phase differences φ

_{i}including, but not restricted to, averaging, filtering, and decorrelation of phase measurements made on different frequencies. The N phase differences φ

_{i}generated and output by the phase measurements module 1140 are passed as inputs into the fine sought parameter estimator 1160. A coarse sought parameter estimate Θ

_{C}generated and output by the coarse sought parameter estimator 1150 is also passed to the fine sought parameter estimator 1160.

**[0074]**In some embodiments, the phase measurements φ

_{i}, 1<=i<=N, as well as the estimation of coarse sought parameters Θ

_{C}can be organized in different suitable ways, based on the particular configuration or application of the interferometric location system. Thus, in some embodiments, fine sought parameter estimator can be used in various extended interferometers for accurate estimation of AOA, TDOA, TOF, or TOA in different interferometric location systems, where one or more phase measurement modules can be used to determine the phase measurements φ

_{i}, 1<=i<=N corresponding to the particular system. Likewise, one or more coarse sought parameter estimators can be used to determine the coarse sought parameters corresponding to the particular configuration or application of the interferometric location system.

**[0075]**In various embodiments, the fine sought parameter estimator 1160 processes the N phase differences φ

_{i}and the coarse sought parameter estimate Θ

_{C}to produce the fine sought parameter estimate Θ

_{F}. To generate the fine sought parameter estimate Θ

_{F}, the fine sought parameter estimator 1160 calculate a partial sought parameter estimate Θ

_{P}on the basis of the N phase differences φ

_{i}without using the coarse sought parameter estimate Θ

_{C}. The fine sought parameter estimator 1160 can then generate the fine sought parameter estimate Θ

_{F}using the partial sought parameter estimate Θ

_{P}combined with the coarse sought parameter estimate Θ

_{C}.

**[0076]**As explained in more detail below, the partial sought parameter estimate Θ

_{P}corresponds to a time interval t

_{PM}inside of which the interferometric parameter can be detected accurately and unambiguously based on the N phase differences φ

_{i}obtained from the phase measurement module 1140. In some embodiments, the partial sought parameter estimate Θ

_{P}is also restricted to values within the range -0.5≦Θ

_{P}<0.5. The coarse sought parameter estimate Θ

_{C}is not restricted to the same range as the partial sought parameter estimate Θ

_{P}and may be any dimensionless real number representing an estimate of the time parameter, e.g. TDOA, normalized by the value of unambiguous time interval t

_{PM}. In a particular case in which the time parameter falls within the time interval t

_{PM}, the coarse sought parameter estimate Θ

_{C}is valued within the range -0.5≦Θ

_{C}<0.5 and corresponds to the partial sought parameter estimate Θ

_{P}although with less accuracy.

**[0077]**The interferometric location system 1110 also includes a location calculator 1170 coupled to each extended interferometer 1130. Each fine sought parameter estimate Θ

_{F}generated by a corresponding fine sought parameter estimator 1160 is passed as an input to the location calculator 1170, which reconstructs the measured time parameters from the fine sought parameter estimates Θ

_{F}. The location calculator 1170 then may use any suitable method of calculating coordinates for the located object 1105 using the time parameters. For example, possible methods are described in Y. T. Chan and K. C. Ho, Solution and Performance Analysis of Geolocation by TDOA, IEEE Transactions on Aerospace and Electronic Systems, Vol. 29, No. 4, 1993 and in Wade H. Foy, Position-Location Solutions by Taylor-Series Estimation, IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-12, No. 2, pp. 187-193, 1976. As illustrated in FIG. 1, the location calculator 1170 receives two fine sought parameter estimates Θ

_{F1}and Θ

_{F2}and determines two spatial coordinates, e.g., x and y coordinates, of the located object 1105.

**[0078]**In alternative embodiments, the location calculator 1170 calculates three spatial coordinates, e.g. x, y and z coordinates, of the located object 1105. However, it should be appreciated that at least one additional signal receiving sensor 1110, receiver 1120, signal transmitting channel 1121, and extended interferometer 1130 may each be included in the location calculator 1170 to calculate the additional spatial coordinate.

**[0079]**Referring now to FIG. 2, the operation of the coarse sought parameter estimator 1150 in some embodiments is explained using graphs 2100 and 2200. Each of the graphs 2100 and 2200 plots a signal received at one of the signal receiving sensors 1110 (FIG. 1) over time. For example, graph 2100 plots the magnitude S

_{1}(t) as a function of time of a signal 2105 received at one of the signal receiving sensors 1110. Similarly, graph 2200 plots the magnitude S

_{2}(t) of a different signal 2205 received at one other of the signal receiving sensors 1110. As shown in FIG. 2, the two signals 2105 and 2205 have substantially the same frequency and represent the signals received at the inputs of one of the coarse sought parameter estimators 1150 shown in FIG. 1.

**[0080]**Each of the graphs 2100 and 2200 also has defined a threshold value Th, which is at the same level on each graph 2100 and 2200. The time point t

_{1}on graph 2100 represents the first time at which the magnitude S

_{1}(t) of signal 2105 exceeds the threshold value Th, while time point t

_{2}on graph 2200 represents the corresponding first time at which the magnitude S

_{2}(t) of signal 2205 exceeds the threshold value Th.

**[0081]**In various embodiments, the coarse sought parameter estimator 1150 estimates the time difference of arrival between the two signals 2105 and 2205 at respective signal receiving sensors 1110 as the difference between the first times each of the signals 2105 and 2205 exceeds the threshold value Th, i.e. TDOA=t

_{2}-t

_{1}. To generate the coarse sought parameter estimate Θ

_{C}, the coarse sought parameter estimator 1150 then normalizes the time difference of arrival t

_{2}-t

_{1}by the unambiguous time interval t

_{PM}defined above. However, the course sought parameter estimator 1150 may also generate the coarse sought parameter estimate Θ

_{C}differently in alternative embodiments.

**[0082]**It can be seen from FIG. 2 that some error may be associated with the coarse sought parameter estimate Θ

_{C}. For example, the measured time difference of arrival t

_{2}-t

_{1}may vary depending on the respective growth rates and noise levels of each signal 2105 and 2205. The signal 2205 exceeds the threshold value Th at the peak of the second full period, while the signal 2105 does not exceed the threshold value Th until the upslope of the third full period. In some embodiments, the coarse sought parameter estimate Θ

_{C}may incorporate some degree of error, for example, corresponding to an error in the estimate of time difference of arrival, which may exceed one or more cycles of the received signals.

**[0083]**Referring back to FIG. 1, by using the N measured phase differences φ

_{i}, the fine sought parameter estimator 1160 may produce the fine sought parameter estimate Θ

_{F}with more accuracy than the coarse sought parameter estimator 1150 is able to produce the coarse sought parameter Θ

_{C}. In the following description, a relationship between the phase differences φ

_{i}and the partial sought parameter estimate Θ

_{P}will be developed.

**[0084]**The range between the phase center of a signal emitting antenna of the located object 1105 and that of a j

^{th}signal receiving sensor 1110 can be represented as R

_{j}. Analogously, R

_{m}may be used to represent the range between the phase centers of the signal emitting antenna of the located object 1105 and an m

^{th}signal receiving sensor 1110. Then a range difference AR defined for the j

^{th}and m

^{th}signal receiving sensors 1110 can be calculated according to:

**ΔR=R**

_{j}-R

_{m}. (1)

**[0085]**The phase difference φ

_{i}represents a measured phase difference between i

^{th}signals components of the same frequency f

_{i}received at the j

^{th}and m

^{th}signal receiving sensors 1110. In various embodiments, phase measurement module 1140 can measure each phase difference φ

_{i}within the limits -π≦φ

_{i}<π, which is equivalent to -0.5≦φ

_{i}<0.5 using normalized phase values. It will be assumed herein through that the measured phase differences φ

_{i}are normalized.

**[0086]**In some embodiments, the measured phase differences φ

_{i}relate to the range difference ΔR according to:

**Φ i + k i = Δ R λ i + Φ CHi + k CHi + n i , ( 2 ) ##EQU00001##**

**where the measured phase differences**φ

_{i}in equation (2) above may be further defined as:

**φ**

_{i}=φ

_{0}i+n

_{i}. (3)

**In equations**(2) and (3), n

_{i}represents a phase error associated with the given phase difference φ

_{i}, and φ

_{0}1 represents an ideal phase difference that would be measured between i

^{th}signals components if no phase errors were present, i.e., if n

_{i}=0 in equation (3).

**[0087]**Because the phase measurement module 1140 measures the phase differences φ

_{i}only within one cycle, integer multiples k

_{i}of full cycles of the phase differences φ

_{i}may be lost during the phase measurements. In various embodiments, as explained in greater detail below, the integer multiples k

_{i}of full cycles are recovered implicitly in generating the partial sought parameter estimate Θ

_{P}.

**[0088]**The wavelength λ

_{i}and frequency f

_{i}of the i

^{th}signal component are related according to:

**λ i = c f i , ( 4 ) ##EQU00002##**

**where c is a speed of signal propagation of the i**

^{th}signal component. Moreover, in equation (2), phase shifts due to signal propagation from the signal receiving sensors 1110 to the phase measurement module 1140 are accounted for according to:

**φ**

_{CHi}+k

_{CHi}=φ

_{C}Hj+k

_{C}Hj-φ

_{C}Hm-k

_{C}Hm. (5)

**In equation**(5) above, φ

_{CHi}is limited to values in the range -0.5≦φ

_{CHi}<0.5 and represents a partial phase difference of signal phase shifts that result due to propagation of signals received at the j

^{th}and m

^{th}signal receiving sensors 1110, respectively, to the phase measurement module 1140. The value of φ

_{CHi}and integer multiples k

_{CHi}of full cycles of φ

_{CHi}can be estimated and known after calibration of the interferometric location system 1100. Alternatively, these values can be calculated based upon particular locations of the signal receiving sensors 1110 at the moment phase measurements are taken.

**[0089]**To the extent that the values of φ

_{CHi}and integer multiples k

_{CHi}are known or ascertainable, and moreover do not depend on the particular ranges R

_{j}and R

_{m}, the phase measurement module 1140 can compensate for the effects of φ

_{CHi}and k

_{CHi}on the measured phase difference φ

_{i}. In the discussion following below, it will be assumed that the phase measurement module 1140 compensates for the effects of φ

_{CHi}and k

_{CHi}, with the result that the measured phase difference φ

_{i}output by the phase measurement module 1140 does not generally depend on these quantities in at least some embodiments.

**[0090]**Applying the above-stated assumptions, equation (1) can be re-written according to:

**Φ i + k i = Δ R λ i + n i . ( 6 ) ##EQU00003##**

**Taking equation**(4) into consideration, equation (6) can then be re-written as:

**φ**

_{i}+k

_{i}=t

_{P}*f

_{i}+n

_{i}, (7)

**where t**

_{P}is a partial time parameter that is defined as the ratio ΔR/c and represents the time difference of arrival between the signals received at the j

^{th}and m

^{th}signal receiving sensors 1110, respectively. The component frequencies f

_{i}, 1≦i≦N, which relate to each other as relatively prime numbers, may be represented as:

**f**

_{i}=βa

_{i}, (8)

**where**β is a common multiplier shared by each f

_{i}and where a

_{i}, 1≦i≦N, represent the relatively prime numbers through which the component frequencies f

_{i}relate.

**[0091]**By processing the N phase differences φ

_{i}, the partial time parameter t

_{P}may be estimated unambiguously within the limits:

**- t PM 2 ≦ t P < t PM 2 , ( 9 ) ##EQU00004##**

**where t**

_{PM}represents a maximum partial time parameter and is defined as:

**t PM**= Δ R MA X c . ( 10 ) ##EQU00005##

**In equation**(10) above, ΔR

_{MAX}represents a maximum range difference that may be unambiguously estimated based on N phase differences φ

_{i}, which measured between signals received at two different signal sensors 1110. The maximum range difference ΔR

_{MAX}can be calculated as:

**ΔR**

_{MAX}=a

_{i}λ

_{i}, (11)

**for any**1≦i≦N. According to equation (11), each relatively prime number a

_{i}represents the number of range spans, expressed in terms of a corresponding wavelength λ

_{I}, within the maximum range difference ΔR

_{MAX}.

**[0092]**Using equations (4) and (8), the wavelength λ

_{i}can be represented as:

**λ i = c β a i . ( 12 ) ##EQU00006##**

**Substituting equation**(12) into equation (11) also yields:

**Δ R MA X = c β . ( 13 ) ##EQU00007##**

**Correspondingly**, by combining equations (13) and (10), the maximum partial time parameter t

_{PM}can be further calculated as:

**t PM**= 1 β , ( 14 ) ##EQU00008##

**and thereby also is related inversely to the common multiplier**β. According to equation (9), the partial time parameter t

_{P}that is defined in equation (7) can be represented as:

**t**

_{P}=t

_{PM}Θ

_{P}, (15)

**where**Θ

_{P}represents the partial sought parameter estimate and, as noted above, is limited to values within the range -0.5≦Θ

_{P}<0.5.

**[0093]**Referring now to FIG. 3, graphs 3100, 3200 and 3300 are used to explain the relationship between the measured phase differences φ

_{i}, the coarse sought parameter estimate Θ

_{C}, the partial sought parameter estimate Θ

_{P}, the fine sought parameter estimate Θ

_{F}, and the corresponding time parameters for different frequency components f

_{i}of the received signals. Specifically, graph 3100 illustrates time difference of arrival of the received signals on a single axis 3105. Various time parameters, including time parameters corresponding to the coarse sought parameter estimate Θ

_{C}, the partial sought parameter estimate Θ

_{P}, and the fine sought parameter estimate Θ

_{F}, are plotted on the axis 3105 in relation to points on the graphs 3200 and 3300.

**[0094]**Graphs 3200 and 3300 plot the magnitude of different measured phase differences φ

_{i}as a function of the fine sought parameter estimate Θ

_{F}for different frequency components f

_{i}. In particular, curve 3205 on graph 3200 plots φ

_{1}(Θ

_{F}), which represents the relationship between a measured phase difference φ

_{1}and the fine sought parameter estimate Θ

_{F}for a first frequency component f

_{1}. Likewise curve 3305 on graph 3300 plots φ

_{2}(Θ

_{F}), which represents the relationship between a measured phase difference φ

_{2}and the fine sought parameter estimate Θ

_{F}for a second frequency component f

_{2}. It is assumed in curves 3205 and 3305 that n

_{1}=n

_{2}=0, which reflects the assumption of no noise present in the measured phase differences φ

_{i}.

**[0095]**After dividing through by the common multiplier β, the frequency components f

_{1}and f

_{2}are in the ratio of 3 to 4, which are relatively prime numbers. This is seen in FIG. 3 from the fact that curve 3205 undergoes three full cycles of φ

_{1}for each whole integer of Θ

_{F}on the x-axis, while curve 3305 undergoes four full cycles of φ

_{2}for each whole integer of Θ

_{F}on the x-axis. Curves 3205 and 3305 also reflect that a

_{1}=3 and a

_{2}=4 in equation (8). For example, in some embodiments, the corresponding frequency components can have values of f

_{1}=3 MHz and f

_{2}=4 MHz (in which case the common multiplier β would equal 10

^{6}Hz), although other combinations are possible as well.

**[0096]**Graphs 3200 and 3300 also illustrate the relationships between the partial sought parameter estimate Θ

_{P}, the coarse sought parameter estimate Θ

_{C}, and the fine sought parameter Θ

_{F}. According to some embodiments, the partial sought parameter estimate Θ

_{P}is defined within the range -0.5≦Θ

_{P}<0.5 inside of which the partial sought parameter estimate Θ

_{P}and the fine sought parameter Θ

_{F}would be equal. However, if the fine sought parameter estimate Θ

_{F}is outside the range of the partial sought parameter Θ

_{P}, these two values would not equate.

**[0097]**For three different values of the partial sought parameter estimate Θ

_{P}within the range -0.5≦Θ

_{P}<0.5 the measured phase difference φ

_{1}is equal to a given value, denoted by 3210 in graph 3200. Point 3220 represents one such value of the partial sought parameter estimate Θ

_{P}. Similarly, there are four different values of the partial sought parameter estimate Θ

_{P}within the range -0.5≦Θ

_{F}<0.5 at which the measured phase difference φ

_{2}is equal to a given value, denoted by 3310 in graph 3300. Point 3320 represents one such value of the partial sought parameter estimate Θ

_{P}for which this is true. Points 3220 and 3320 together represent the only pairing in which the partial sought parameter estimate Θ

_{P}is the same for each measured phase difference φ

_{i}. Accordingly, this value of the partial sought parameter estimate Θ

_{P}results from the measured phase differences φ

_{1}and φ

_{2}being equal to values 3210 and 3310, respectively.

**[0098]**Moreover, the partial sought parameter estimate Θ

_{P}obtained from values 3210 and 3310 of the measured phase differences φ

_{1}and φ

_{2}can be ambiguously represented at multiple different points on the graphs 3200 and 3300. These points correspond to integer multiples of whole numbers added to the partial sought parameter estimate Θ

_{P}, which are denoted on the x-axis of graphs 3200 and 3300 as Θ

_{P}±1, Θ

_{P}±2, Θ

_{P}±3, etc. The partial sought parameter estimate Θ

_{P}may be ambiguously represented by a multiple of additional values outside of the range -0.5≦Θ

_{P}<0.5 to reflect the fact that the TDOA between received signals, i.e. t

_{2}-t

_{1}, may be greater than the maximum partial time parameter t

_{PM}defined in equation (14). By providing a rough estimate of the fine sought parameter estimate Θ

_{F}, the course sought parameter Θ

_{C}can be used to produce the fine estimate of the sought parameter Θ

_{F}from the partial sought parameter estimate Θ

_{P}and some integer number of full cycles.

**[0099]**Considering all phase differences φ

_{i}measured on N frequency components f

_{i}, and by combining equations (8), (14), and (15) together, equation (7) can be written in vector form according to:

**φ+k=AΘ**

_{P}+n, (16)

**where**φ, k, A and n are each N-dimensional column vectors, with every i

^{th}element in the column vector corresponding to a respective vector component determined for the i

^{th}frequency component of the signal received at the signal receiving sensors 1110. In accordance with various embodiments, the value of the partial sought parameter Θ

_{P}may be estimated, as will be described, by solving equation (16).

**[0100]**Referring now to FIGS. 4A and 4B, the fine sought parameter estimator 1160 is illustrated in more detail according to different embodiments. The fine sought parameter estimator 1160 generates the fine sought parameter estimate Θ

_{F}, or alternatively a post-processed fine sought parameter estimate Θ'

_{F}, by processing the vector φ of N measured phase differences (shown explicitly as φ

_{1}. . . φ

_{N}in FIGS. 4A and 4B) and the coarse sought parameter estimate Θ

_{C}in accordance with the described embodiments.

**[0101]**More specifically, FIG. 4A illustrates a fine sought parameter estimator 1160 that comprises a combined estimator 4100 and a partial sought parameter extender 4110. Based on the vector φ of N measured phase differences, the combined estimator 4100 generates the partial sought parameter estimate Θ

_{P}and may also calculate at least one noise parameter that can be used to improve the quality of the fine sought parameter estimate Θ

_{F}. However, in some embodiments of the fine sought parameter estimator 1160, the combined estimator 4100 outputs the partial sought parameter estimate Θ

_{P}to the partial sought parameter extender 4110 for calculating the fine sought parameter estimate Θ

_{F}, but does not output the at least one noise parameter. In such embodiments, the at least one noise parameter may be used internally within the combined estimator 4100 to improve the accuracy of the partial sought parameter estimate Θ

_{P}, relative to other embodiments of the combined estimator 4100 that compute the partial sought parameter estimate Θ

_{P}differently. These different methods for calculating the partial sought parameter estimate Θ

_{P}are explained in more detail below.

**[0102]**In the alternative embodiments illustrated by in FIG. 4B, the fine parameter estimator 1160 comprises combined estimator 4200, partial sought parameter extender 4110 and postprocessor 4300. The combined estimator 4200 is similar to the combined estimator 4100, but in these alternative embodiments of the fine sought parameter estimator 1160, the combined estimator 4200 outputs both the partial sought parameter estimate Θ

_{P}and the at least one noise parameter. The partial sought parameter estimate Θ

_{P}is provided to the partial sought parameter extender 4110 for calculating the fine sought parameter estimate Θ

_{F}. The at least one noise parameter calculated by the combined estimator 4200 is provided to the postprocessor 4300, together with the fine sought parameter estimate Θ

_{F}.

**[0103]**In some embodiments, the postprocessor 4300 uses the at least one noise parameter received from the combined estimator 4200 to improve the estimate of the fine sought parameter Θ

_{F}via processing or filtering fine sought parameter estimates Θ

_{F}based on the value of at least one noise parameter. Thus, in some embodiments, the postprocessor 4300 uses the at least one noise parameter received from the combined estimator 4200 to discard any fine sought parameter estimates Θ

_{F}that are determined to be unreliable. For example, the fine sought parameter estimates Θ

_{F}may be determined to be unreliable if the level of noise associated with the vector φ of measured phase differences exceeds a threshold noise level. As an alternative to discarding, an adaptive filtering of the fine sought parameter estimates Θ

_{F}can be performed based on the value of the at least one noise parameter. Thus, in some embodiments, the postprocessor may apply a weighting factor to each fine sought parameter estimate Θ

_{F}based on the at least one noise parameter. Fine sought parameter estimates Θ

_{F}generated from less noisy phase differences φ may be weighted more heavily than fine sought parameter estimates Θ

_{F}generated from more noisy phase differences φ. As a still further alternative, the postprocessor 4200 may combine the fine sought parameter estimate Θ

_{F}and the at least one noise parameter into a single vector for processing by some other component of the location system based on the value of the at least one noise parameter.

**[0104]**In various embodiments of the fine sought parameter estimator 1160, of which FIGS. 4A and 4B represent only two exemplary configurations, the partial sought parameter extender 4110 can process the coarse sought parameter estimate Θ

_{C}and the partial sought parameter estimate Θ

_{P}to generate the fine sought parameter estimate Θ

_{F}according to:

**Θ**

_{F}=Θ*

_{C}+Θ

_{P}. (17)

**In equation**(17) above, Θ*

_{C}represents a corrected coarse sought parameter equal to:

**Θ C * = { Θ CR ; if ΔΘ C - Θ P ≦ 0.5 Θ CR + 1 ; if ( ΔΘ C - Θ P ) > 0.5 Θ CR - 1 ; if ( Δ Θ C - Θ P ) < - 0.5 , ( 18 ) ##EQU00009##**

**where**|X| is an absolute value of X. Moreover, Θ

_{CR}and ΔΘ

_{C}introduced in equation (17) are defined according to:

**Θ**

_{CR}=rnd[Θ

_{C}], (19)

**and**

**ΔΘ**

_{C}=rrni{Θ

_{C}}, (20)

**where rnd**[ . . . ] is a procedure of rounding to the nearest integer of an element inside of the square brackets [ . . . ], and where rrni{ . . . } is a procedure of calculating the residual of rounding to the nearest integer of an element inside of the braces { . . . }.

**[0105]**Each fine sought parameter estimator 1160 of a corresponding extended interferometer 1130 (FIG. 1) generates a fine sought parameter estimate Θ

_{F}, or alternatively a post-processed fine sought parameter estimate Θ*

_{F}. The different fine sought parameter estimates Θ

_{F}or post-processed fine sought parameter estimates Θ*

_{F}correspond to different unambiguous time parameters estimated from the signals received at a different pair of signal receiving sensors 1110. Each such estimate Θ

_{F}or Θ*

_{F}is provided as an input to the location calculator 1170 (FIG. 1), which calculates the corresponding unambiguous time parameter t

_{F}, e.g. TDOA, from the fine sought parameter estimate Θ

_{F}according to:

**t**

_{F}=t

_{PM}Θ

_{F}. (21)

**As the fine sought parameter estimate**Θ

_{F}is not restrained to the range -0.5≦Θ

_{P}<0.5 defined for the partial sought parameter Θ

_{P}, and may instead be any real number, the unambiguous time parameter t

_{F}is not limited to values less than the maximum partial time parameter t

_{PM}. Having calculated the unambiguous time parameters t

_{F}using equation (21), the location calculator 1170 then determines spatial coordinates of the located object 1105 using any known method.

**[0106]**In various embodiments, either the combined estimator 4100 (FIG. 4A) or the combined estimator 4200 (FIG. 4B) may calculate the partial sought parameter estimate Θ

_{P}without directly calculating or otherwise estimating the integer numbers k of lost cycles. In various embodiments, the combined estimators 4100 and 4200 also calculate one or more noise parameters that characterize the quality of the partial sought parameter estimate Θ

_{P}. In the case of the combined estimator 4200, which provides the one or more noise parameters as inputs for the postprocessor 4300, the one or more noise parameters may also be used to produce the postprocessed fine sought parameter estimate Θ'

_{F}having improved accuracy. It will be described below how, in various embodiments, the postprocessor 4300 may output the fine sought parameter estimate Θ'

_{F}as a combination of the fine sought parameter estimate Θ

_{F}and at least one noise parameter used as a quality factor to indicate the reliably of the fine sought parameter estimate Θ

_{F}.

**[0107]**In addition to the interferometric location system 1100 (FIG. 1), the method of generating a fine sought parameter estimate Θ

_{F}as a combination of a partial sought parameter estimate Θ

_{P}and a coarse sought parameter estimate Θ

_{C}is also applicable to direction finding interferometers. The aforementioned method is also application to some interferometers that estimate M interferometric parameters based on N measured phase differences when N>M. Direction finding interferometers can include linear, planar, or three-dimensional antenna arrays to estimate one, two, or three Angles of Arrival of a signal from a source. Direction finding interferometers can comprise several receiving antennas, wherein the distances between the different receiving antennas in the direction finding interferometer are known. The lines between phase centers of respective antennas in the direction finding interferometers may be referred to generally as baselines. Phase differences are generally measured between signals received on those baselines to compute AOA.

**[0108]**The direction finding interferometer may comprise a linear antenna array having baselines between respective antennas, which sizes relate to each other as relatively prime numbers. In such implementations, equation (16) is applicable and may be solved to compute the partial sought parameter estimate Θ

_{P}as herein described. However, when applied to estimating AOA in a direction finding interferometer, the various parameters defined in equation (16) may represent different physical quantities as compared to a TDOA interferometric location system. In particular, φ represents a vector of N phase differences measured on the N baselines, the elements of vector k represent numbers of full cycles lost in the phase measurements taken on corresponding baselines, A represents a vector of relatively prime numbers which define corresponding sizes of the N baselines, and the elements of vector n represent phase errors associated with the phase measurements taken on corresponding baselines. The partial sought parameter estimate Θ

_{P}solved using equation (16) represents a cosine or sine of the AOA of the source signals. In this way, equation (16) has applicability to both direction finding interferometers for estimating angle of arrival and interferometric location systems that estimate time parameters, provided the interferometers are suitably configured.

**[0109]**If the N baselines in the antenna array of a direction finding interferometer are organized in two-dimensional space, corresponding phase measurements φ are defined by two angles of arrival. In direction finding interferometers with a three-dimensional antenna array, the phase differences cp by extension may depend on three angles of arrival. In such cases, two or three angles of arrival can be estimated on the basis of the phase measurements φ performed on N baselines. Generally, some interferometers can estimate M partial sought parameters Θ

_{P1}. . . Θ

_{PM}by processing the N phase measurements φ

_{1}. . . φ

_{N}on N measuring scales (N>M), in which case equation (16) may be re-written in vector form according to:

**φ+k=AΘ+n, (22)**

**where**φ, k, and n are N-dimensional column vectors with every i

^{th}element corresponding to an i

^{th}baseline in the direction finding interferometer. Again, the elements of vector k represent numbers of full cycles lost in the phase measurements taken on corresponding baselines and the elements of vector n represent phase errors associated with the phase measurements φ taken on corresponding baselines. In comparison to equation (16), Θ

_{P}now represents an M-dimensional column vector of partial sought parameters Θ

_{P1}. . . Θ

_{PM}, while matrix A has dimensions N×M and is composed of column vectors a

_{i}that are N-dimensional linearly independent vectors of relatively prime numbers, which are defined by the structure of antenna array of the interferometer.

**[0110]**Non-extended direction finding interferometers may be configured to provide very accurate unambiguous estimates of several angles of arrival in restricted angle sectors. In extended direction finding interferometers, additional direction finding components can be implemented to provide coarse estimation of sought parameters Θ

_{C1}. . . Θ

_{CM}to obtain less accurate unambiguous estimates in wider angle sectors. For instance, extended direction finding interferometers can be configured to estimate partial sought parameters Θ

_{P1}and Θ

_{P2}with 0.1° angle accuracy within a 10° angle sectors. Coarse sought parameter estimators included in such interferometers can also calculate coarse sought parameter estimates Θ

_{C1}and Θ

_{C2}with a 2° angle accuracy within 90° angle sectors. Then, by combining the partial sought parameter estimates Θ

_{Pi}with the coarse sought parameters Θ

_{Ci}, the direction finding interferometers can produce fine sought parameter estimates Θ

_{F1}and Θ

_{F2}with a 0.1° angle accuracy within 90° angle sectors.

**[0111]**In direction finding interferometers, each partial sought parameter estimate Θ

_{Pi}, 1≦i<M represents the sine or cosine of angle of arrival. Extended direction finding interferometer will in some cases provide correspondence in dimensions and values between partial sought parameter estimates Θ

_{Pi}and coarse sought parameter estimates Θ

_{Ci}, 1≦i<M. For example, while the partial sought parameter Θ

_{Pi}is limited to the range -0.5≦Θ

_{Pi}<0.5, the coarse sought parameter estimate Θ

_{Ci}may be any real number that is not generally restricted to the same range. But within the limited range -0.5≦Θ

_{Pi}<0.5, the coarse sought parameter estimate Θ

_{Ci}and the partial sought parameter estimate Θ

_{Pi}will correspond to the same physical quantity, e.g. an AOA.

**[0112]**For instance, in some embodiments, the extended direction finding interferometer produces a coarse estimate of AOA in a wide angle sector and a partial sought parameter estimate Θ

_{Pi}corresponding to a narrow angle sector. In that case, the coarse sought parameter estimator can obtain the coarse sought parameter estimate Θ

_{Ci}as coarse estimate of AOA divided by the size of the narrow angle sector in degrees. If such correspondence is achieved, then M partial sought parameter extenders 4110 can be used for producing M fine sought parameter estimates Θ

_{Fi}, 1≦i<M. The M fine sought parameter estimates Θ

_{Fi}will have the same accuracy as the corresponding partial sought parameter estimates Θ

_{Pi}, but will represent AOA in the wide-angle sector in which the coarse estimate of angle of arrival is defined. However, it should be appreciated that in different embodiments of direction finding interferometers, the coarse sought parameter estimates Θ

_{Ci}can be defined differently, provided correspondence in maintained between Θ

_{Pi}and Θ

_{Ci}in terms of both dimension and value.

**[0113]**In some embodiments, extended phase interferometer 1130 (FIG. 1) is applicable for time parameter estimation and/or for angle of arrival estimation in different interferometric location systems. In different interferometric location systems, each phase measurement module 1140 and coarse sought parameter estimator 1150 may be embodied differently, while the fine sought parameter estimator 1160 may be embodied the same in each interferometric location system.

**[0114]**In addition, it should be appreciated that the various elements defined in equation (22) are not restricted only to representing time parameters or angles of arrival. In some embodiments, still other interferometric systems not explicitly described herein may be designed to estimate one or more different interferometric parameters by representing the one or more interferometric parameters using a vector Θ of sought parameters and solving equation (22). Regardless of the physical meaning of the one or more interferometric parameters, if represented by a vector Θ of sought parameters, equation (22) may be solved as described in more detail below to estimate the one or more interferometric parameters.

**[0115]**Referring now to FIGS. 5A, 5B and 5C, a fine sought parameter estimator 5000 for estimating M fine sought parameters in an interferometric location system is illustrated according to different embodiments. The fine sought parameter estimators 1160 illustrated in FIGS. 4A and 4B represent embodiments of particular cases of the fine sought parameter estimator 5000 for interferometric location systems that estimate only a single fine sought parameter Θ

_{F}.

**[0116]**In some embodiments of the interferometric location system 1100, more than one fine sought parameter estimate Θ

_{Fi}may be calculated by the extended interferometer 1130 by processing the N phase measurements φ

_{1}. . . φ

_{N}. In such embodiments, more than one coarse sought parameter estimates Θ

_{Ci}may also be calculated by a corresponding number of coarse sought parameter estimators 1150 (FIG. 1). More than one partial sought parameter extenders 4110 or 5110 may also then be included in the fine sought parameter estimator 5000 and, consequently, in each extended interferometer 1130 to calculate the more than one fine sought parameter estimates Θ

_{Fi}.

**[0117]**As seen in FIG. 5A, combined estimator 5100 calculates M partial sought parameter estimates Θ

_{Pi}, 1≦i<M by processing N measured phase differences φ

_{1}. . . φ

_{N}. The M partial sought parameter estimates Θ

_{Pi}are provided to a corresponding number of partial sought parameter extenders 4110. Each of the partial sought parameter extenders 4110 also receives a corresponding one of M coarse sought parameter estimates Θ

_{Ci}, and generates one of M fine sought parameter estimates Θ

_{Fi}based on the received partial sought parameter estimate Θ

_{Pi}and coarse sought parameter estimate Θ

_{Ci}. The M fine sought parameter estimates Θ

_{F1}. . . Θ

_{FM}are output from the fine parameter estimator 5000.

**[0118]**In FIG. 5B, the fine sought parameter estimator 5000 includes a combined estimator 5200 instead of the combined estimator 5100 shown in FIG. 5A and further includes postprocessor 5300. In addition to the M partial sought parameter estimates Θ

_{Pi}, the combined estimator 5200 calculates and outputs one or more noise parameters generated based on the N measured phase differences φ

_{1}. . . φ

_{N}. Each of the partial sought parameter extenders 4110 receives a corresponding one of M coarse sought parameter estimates Θ

_{Ci}, and generates one of the M fine sought parameter estimates Θ

_{Fi}based on the one of M partial sought parameter estimate Θ

_{Pi}, received from the combined estimator 5200, and coarse sought parameter estimate Θ

_{Ci}. The one or more noise parameters are provided to the postprocessor 5300 together with the M fine sought parameter estimates Θ

_{Fi}produced by the M partial sought parameter extenders 4110. The postprocessor 5300 generates M post-processed fine sought parameter estimates Θ'

_{F}.

**[0119]**As seen in FIG. 5C, combined estimator 5400 calculates M partial sought parameter estimates Θ'

_{Pi}by processing N measured phase differences φ

_{1}. . . φ

_{N}. In some embodiments, the combined estimator 5400 calculates at least one noise parameter, processes the at least one noise parameter to compare the at least one noise parameter with at least one noise threshold. Based on the result of the comparison, the combined estimator 5400 calculates pre-processed partial sought parameter estimates Θ'

_{Pi}, 1≦i<M using the at least one noise parameter and the N measured phase differences φ

_{1}. . . φ

_{N}. In some embodiments, if at least one noise parameter exceeds at least one noise threshold, the combined estimator 5400 discards the corresponding pre-processed partial sought parameter estimates Θ'

_{Pi}generated based on the N measured phase differences φ

_{1}. . . φ

_{N}. Alternatively, in some embodiments, if at least one noise parameter exceeds at least one noise threshold, the combined estimator 5400 does not calculate pre-processed partial sought parameter estimates Θ'

_{Pi}.

**[0120]**The M pre-processed partial sought parameter estimates Θ'

_{Pi}are provided to a corresponding number of partial sought parameter extenders 5110 included in the combined estimator 5400. Each of the partial sought parameter extenders 5110 also receives a corresponding one of M coarse sought parameter estimates Θ

_{Ci}, and generates one of the M fine sought parameter estimates Θ'

_{Fi}based on the received partial sought parameter estimate Θ'

_{Pi}and coarse sought parameter estimate Θ

_{Ci}. The M fine sought parameter estimates Θ'

_{F1}, . . . , Θ'

_{FM}are output from the fine parameter estimator 5000.

**[0121]**In some embodiments, the postprocessor 5300 (FIG. 5B) combines the at least one noise parameter with each of the fine sought parameter estimates Θ

_{Fi}and outputs the combination as the pre-processed fine sought parameter estimates Θ'

_{Fi}. For example, each of the pre-processed fine sought parameter estimates Θ'

_{Fi}may be represented by a 16-bit digital word, in which 12 bits represent the value of the fine sought parameter estimate Θ

_{Fi}and the remaining 4 bits are allocated to the at least one noise parameter.

**[0122]**Similarly, in some embodiments, the combined estimator 5400 (FIG. 5C) may output the pre-processed partial sought parameter estimates Θ'

_{Pi}as a combination of the at least one noise parameter and the partial sought parameters estimates Θ

_{Pi}. In such embodiments, each partial sought parameter extender 5110 (FIG. 5C) may then extend the partial sought parameter estimate Θ

_{Pi}to a fine sought parameter estimate Θ

_{Fi}, and thereby generate the pre-processed fine sought parameter estimates Θ'

_{Fi}as a combination of the fine sought parameter estimate Θ

_{Fi}and the at least one noise parameter (taken from the pre-processed partial sought parameter estimate Θ'

_{Pi}). As another example, each of the pre-processed partial sought parameter estimates Θ'

_{Pi}may be represented by a 16-bit digital word, in which 12 bits represent the value of the partial sought parameter estimate Θ

_{Pi}and the remaining 4 bits are allocated to the at least one noise parameter. Then each of the pre-processed fine sought parameter estimates Θ'

_{Fi}may be represented by a 24-bit digital word, in which 20 bits represent the value of the fine sought parameter estimate Θ

_{Fi}and the remaining 4 bits are allocated to the at least one noise parameter.

**[0123]**As will be appreciated, the combined estimators 4100 and 4200 shown in FIGS. 4A and 4B may, respectively, represent a particular implementation the combined estimators 5100 and 5200 shown in FIGS. 5A and 5B for the case of generating only a single fine sought parameter estimate Θ

_{F}.

**[0124]**How the combined estimators 4100 and 4200 calculate one partial sought parameter estimate Θ

_{P}(or alternatively the combined estimators 5100, 5200 and 5400 estimate the vector Θ of M partial sought parameters) and the one or more noise parameters, as well as how the post-processor 4300 calculates one post-processed fine sought parameter estimate Θ'

_{F}(or alternatively the postprocessor 5300 calculates the M post-processed fine sought parameter estimates Θ'

_{F1}. . . Θ'

_{FM}, or the preprocessor within combined estimator 5400 calculates the M pre-processed combined partial sought parameter estimates Θ'

_{Pi}, 1≦i<M) are now discussed.

**[0125]**Equation (22) may be solved to determine the vector Θ of partial sought parameters on the assumption that vector n is a Gaussian random vector with covariance matrix B. Then a maximum likelihood estimate of the vector Θ of partial sought parameters can be found as the estimate that maximizes the likelihood function:

**W**( Θ , k | Φ ) = T * exp ( - 1 2 ( Φ + k - A Θ ) T B - 1 ( Φ + k - A Θ ) ) , ( 23 ) ##EQU00010##

**where T is a multiplier that depends on the covariance matrix B**.

**[0126]**For a fixed vector k, the quadratic form in equation (23) is minimized if:

**Θ=(A**

^{TB}

^{-1}A)

^{-1}A

^{TB}

^{-1}(φ+k). (24)

**[0127]**The vector k can be found by minimizing the following quadratic form:

**k**= ar g min k ( ( Φ + k ) T C ( Φ + k ) ) , ( 25 ) ##EQU00011##

**where C is a matrix defined by vector A and matrix B according to**:

**C**=B

^{-1}-B

^{-1}A(A

^{TB}

^{-1}A)

^{-1}A

^{TB}

^{-1}. (26)

**[0128]**Each of the described interferometers of an interferometer has a specific set of vectors k that shall be considered in equation (25). From this set, N-M linearly independent vectors k

_{1}, . . . , k

_{N}-M can be chosen in the way that they provide N-M lowest values of

**d**

_{i}=k

_{i}

^{T}Ck

_{i}(27)

**Those vectors found from equation**(27) can be combined in matrix K, which has dimensions N×(N-M), according to:

**K**=(k

_{1},k

_{2}, . . . , k

_{N}-M). (28)

**[0129]**Characteristic matrix S with dimensions N×N can be obtained by combining matrices K and A as follows:

**S**=(KA). (29)

**Matrix S is used in various embodiments of the methods described herein**in the effective estimation of the vector Θ of sought parameters and noise parameters. Matrix S has a property that det(S)=±1.

**[0130]**Equation (24) can be rewritten as:

**Θ=(A**

^{TB}

^{-1}A)

^{-1}A

^{TB}

^{-1}SS

^{-1}(φ+k), (30)

**or equivalently as**:

**Θ=HS**

^{-1}(φ+k), (31)

**where H is a matrix defined by matrices A and B as**:

**H**=(A

^{TB}

^{-1}A)

^{-1}A

^{TB}

^{-1}S (32)

**[0131]**In turn, matrix S

^{-1}can be partitioned into two matrices:

**S**- 1 = ( U V ) ( 33 ) ##EQU00012##

**where U is a matrix comprised of the first**(N-M) row vectors of S

^{-1}according to:

**U**= ( S 1 - 1 S 2 - 1 S N - M - 1 ) , ( 34 ) ##EQU00013##

**and where V is a matrix comprised of the last M row vectors of S**

^{-1}according to:

**V**= ( S N - M + 1 - 1 S N - 1 ) . ( 35 ) ##EQU00014##

**[0132]**Accordingly, S

^{-1}φ can be partitioned into a vector δ given by:

**δ=Uφ, (36)**

**and a vector**ψ given by:

**ψ=Vφ. (37)**

**[0133]**Any N-dimensional vector k in equation (30) can be represented as a linear combination of the column-vectors from matrix S according to:

**k**=e

_{1}k

_{1}+e

_{2}k

_{2}+ . . . +e.sub.(N-M)k.sub.(N-M)+e.sub.(N-M+1)a

_{1}+ . . . +e

_{N}a

_{M}, (38)

**where each of the e**

_{i}in equation (38) are integers. Also, as will be appreciated:

**S**

^{-1}S=SS

^{-1}=I. (39)

**[0134]**Taking into consideration equations (29), (31), (37), (38) and (39), the part of equation (31) can be written as:

**V**( Φ + k ) = ψ + ( e ( N - M + 1 ) e N ) . ( 40 ) ##EQU00015##

**[0135]**Matrix H can be partitioned into two matrices as:

**H**=(RI), (41)

**where R is an M**×(N-M)-dimensional matrix of real numbers, and I is the M×M-dimensional identity matrix.

**[0136]**If there are no phase errors in the measurements (n=0), conducted by the interferometer, or alternatively if phase errors are small, and k is a vector that minimizes the quadratic form in equation (25), it can be assumed that:

**S**- 1 ( Φ + k ) = ( O V ( Φ + k ) ) , ( 42 ) ##EQU00016##

**where O is the**(N-M)-dimensional zero vector. According to equations (31), (40), (41), and (42), the vector Θ of sought parameters equals to:

**Θ = ψ + ( e ( N - M + 1 ) e N ) . ( 43 ) ##EQU00017##**

**[0137]**The elements of the vector Θ of sought parameters are bounded by the limits: -0.5≦Θ

_{i}<0.5. Thus, e

_{j}in equation (43) can be eliminated and equation (43) can be rewritten as:

**Θ=ψ-rnd[ψ], (44)**

**where rnd**[ . . . ] is a procedure of rounding to the nearest integer every element of a vector inside of the square brackets [ . . . ]. Equation (44) can also be rewritten as:

**Θ=rrni{ψ}, (45)**

**where rrni**{ . . . } is a procedure of calculating the residual of rounding to the nearest integer every element of a vector inside of the braces { . . . }.

**[0138]**The accuracy of Θ calculated according to equation (45) can be very sensitive to the level of phase errors. Accordingly, in various embodiments, the level of phase errors, or the noise parameters, which are related to the level of phase errors, are utilized as "quality parameters" or parameters that characterize the quality of Θ. In various embodiments, noise parameters are estimated through the use of matrix U. Equations (29), (34) and (39) indicate that U projects φ and k in a space orthogonal to the column vectors of A. Vectors δ, expressed in equation (36), and χ, where:

**χ=Uk, (46)**

**are**(N-M)-dimensional vectors in space orthogonal to A. Any χ is a point of a lattice in . The quadratic form in equation (25) describes Voronoi regions with χ being the center.

**[0139]**Reference is now made to FIG. 6, which illustrates δ, χ, and Voronoi regions 6311 for N-M=2. The maximum likelihood estimation of k according to equation (25) implies finding (-k), that projection U(-k) is a center of Voronoi region χ with δ inside of this Voronoi region. Thus, if the k that minimizes equation (25) is known, then

**U**(φ+k)=δ-χ. (47)

**[0140]**Moreover, the center of the Voronoi region 6311 that is closest to δ can be approximately estimated as:

**χ=rnd[δ]. (48)**

**In various embodiments**, the rounding region 6312 is used instead of Voronoi region 6311, and equation (47) can be written as:

**v**=rrni{δ}. (49)

**[0141]**Considering the ideal case when there are no phase errors, implying n=0, then: φ=φ

_{0}, φ+k=AΘ, and δ=χ

_{j}in for any Θ. Consequently, if vector v≠0, it is a projection of an N-dimensional error vector n on orthogonal to A. Any N-dimensional vector n can be represented as a sum of components lying in where column vectors a

_{i}from matrix A are allocated, and components in that are orthogonal to A. The procedure of projecting n onto excludes components allocated in from the result of the projection, and it leaves components in that are the elements of v. Thus, vector v is defined by phase errors only, and in some embodiments it is used in the estimation of noise parameters along with estimation of Θ.

**[0142]**Reference is now made to FIG. 7, which illustrates the relationship between φ, k, a, and n for various embodiments of interferometers that comprise a linear antenna array with two baselines. Vector n is represented as a sum of two components 7413 and 7414. Component 7413 is allocated in the line of a. Component 7414 can be calculated as v, shown in equation (49). The two dimensional vector v in for N-M=2 is shown in FIG. 6. In various embodiments, the elements of v are sent to a postprocessor (e.g., 5300 in FIG. 5B) as noise parameters.

**[0143]**Reference is now made to FIG. 8, which is a block diagram illustrating various embodiments of a combined estimator 8516 that calculates Θ and the elements of v. In various embodiments, phase measurements converter module 8517 processes the input vector φ, and calculates δ and ψ through the use of equations (36) and (37). In some embodiments, instead of one phase measurements converter module 8517, two phase measurement converter modules can be used alternatively, with one of the phase measurement converters processing the input vector φ to calculate δ through the use of equation (36), and the other of the phase measurement converters processing the input vector φ to calculate ψ through the use of equation (37). In addition, in various embodiments, partial sought parameters estimator module 8519 utilizes equation (45) to calculate Θ. Noise parameters calculator module 8518 performs equation (49) and calculates noise parameters v. In various embodiments, these noise parameters v and partial sought parameters Θ are outputs of the combined estimator 8516. In some embodiments noise parameters v are sent from the combined estimator 8516 to a postprocessor (e.g., 5300 in FIG. 5B).

**[0144]**In some embodiments, the whole vector v is not inputted into the postprocessor. In some such embodiments, the combined estimator can output a noise parameter, which in some embodiments is calculated as the length of vector v. This parameter α is related to the length of noise vector n and in various embodiments is used as a parameter that indicates how noisy is the estimate of Θ. The noise parameter a can be calculated according to:

**α = ( i = 1 N - M v i 2 ) 1 2 . ( 50 ) ##EQU00018##**

**[0145]**Reference is now made to FIG. 9, which is a block diagram illustrating various embodiments of a combined estimator 9516 that calculates α along with Θ. In various embodiments, phase measurements converter module 8517 processes the input vector φ and calculates S

^{-1}φ. As above, and herein throughout, phase measurements converter module 8517 may in some cases be implemented as two separate phase measurement converter modules. In various embodiments, noise parameters calculator module 8518 calculates v according to equation (49) and partial sought parameters estimator module 8519 calculates Θ according to equation (45). In some embodiments, common noise parameter estimator 9520 calculates noise parameter α according to equation (50). In various embodiments, α is sent to a postprocessor (e.g., 5300 in FIG. 5B) and, in some such embodiments, the postprocessor utilizes the magnitude of α as a criterion for the acceptance of the associated fine sought parameters values. Similarly, in various embodiments, α is sent to a preprocessor included in a combined estimator (e.g., 5400 in FIG. 5C) and, in some such embodiments, the preprocessor utilizes the magnitude of α as a criterion for the acceptance of the associated partial sought parameters values. Thus, in some embodiments, if the magnitude of α exceeds a threshold, then the associated partial or fine sought parameters values are discarded, adaptively filtered according to the noise parameter or specifically processed, as described above and herein throughout, for the postprocessor 4300 (FIG. 4B).

**[0146]**In some embodiments, alternative methods are used to estimate a noise parameter. For example, in some embodiments, a noise parameter is estimated by detecting whether or not v is out of the (N-M) dimensional parallelotope with center at χ (48), and with sizes defined by thresholds 0≦γ

_{ij}<0.5. Reference is again made to FIG. 6. Parallelotope 6313 is the parallelotope in for χ=0, that corresponds to an embodiment when N-M=2. Rounding regions 6312 corresponds to the rounding procedure expressed in equation (48). Vector v illustrated in FIG. 6 is shown inside of a rounding region 6312. In various embodiments, every i

^{th}element of V is compared with corresponding threshold γ

_{ij}to detect if v is out of j

^{th}parallelotope 6313. Several parallelotopes can be used to detect or to quantify how far vector v is from the center of rounding region 6312. For example, FIG. 10 illustrates the case with three threshold parallelotopes in These are examples only and in some embodiments any appropriate number of parallelotopes can be used.

**[0147]**A vector of Z noise parameters ε can be obtained by comparing v

_{i}with Z thresholds corresponding to Z parallelotopes, as in the following:

**j**= ( β 1 j β 2 j β ( N - M ) j ) ; j = 1 , Z ( 51 ) β ij = { 1 , v i ≧ γ ij 0 , v i < γ ij ( 52 ) ##EQU00019##

**where v in equation**(51) is a logical disjunction, and |v

_{i}| in equation (52) is an absolute value of v

_{i}. Noise parameter q can be calculated according to:

**q**=count[ε], (53)

**where count**[ . . . ] is a procedure of counting number of elements of the binary vector in the square brackets that are a logical "1", obtained as shown for example in equation (52). If every, γ

_{ij}<γ

_{i}(j+1), then q shows the number of largest parallelotope with v outside of it. Thus noise parameter q shows how far vector v is from the center of rounding region 6312.

**[0148]**Reference is now made to FIG. 11, which illustrates various embodiments of a discrete noise parameter estimator 11624 that calculates q in accordance with equation (53). Each i

^{th}comparison module 11621 compares the magnitude of |v

_{i}| with γ

_{ij}and calculates β

_{ij}according to equation (52). Logical disjunction module 11622 performs the logical disjunction procedure according to equation (51). Counting module 11623 counts discrete noise parameter q according to equation (53). FIG. 10 illustrates the relationship between v

_{i}, γ

_{ij}, β

_{ij}, ε

_{j}, q, and rounding region 6312 for various embodiments of an interferometer with N-M=2.

**[0149]**Reference is now made to FIG. 12, which is a block diagram that illustrates various embodiments of a combined estimator 12516 that calculates Θ according to equation (45) and q according to equation (53). In various embodiments, phase measurements converter module 8517 processes the input vector φ and calculates S

^{-1}φ. In various embodiments, noise parameters calculator module 8518 calculates v according to equation (49) and partial sought parameters estimator module 8519 calculates Θ according to equation (45). In some embodiments, discrete noise parameter estimator 11624 calculates discrete noise parameter q according to equation (53). In some embodiments, q is sent to a postprocessor (e.g., 5300 in FIG. 5B), and in some such embodiments, the postprocessor utilizes the magnitude of q as a criterion for the acceptance of the associated partial or fine sought parameters values. Thus, in some embodiments, if the magnitude of q exceeds a threshold, then the associated partial or fine sought parameters values are discarded. In various embodiments, the combined estimator 12516 and discrete noise parameter estimator 11624 have (N-M)*Z inputs of threshold γ

_{ij}values. In some embodiments, the magnitudes of those threshold values are set to be constant. In various other embodiments, these threshold values can be variable. In some embodiments, the threshold γ

_{ij}values are generated internally by the combined estimator 12516.

**[0150]**In various embodiments, both vectors v and ψ are utilized during the estimation of Θ, according to:

**Θ=rrni{Hξ}, (54)**

**where**ξ is a vector combination of vectors v and ψ as follows:

**ξ = ( v ψ ) . ( 55 ) ##EQU00020##**

**[0151]**In various embodiments, the accuracy of Θ calculated according to equation (54) is less sensitive to the phase errors than the accuracy of Θ when calculated according to equation (45).

**[0152]**Reference is now made to FIG. 13, which is a block diagram illustrating various embodiments of a combined estimator 13516 that calculates a along with Θ. In various embodiments, phase measurements converter module 8517 processes the input vector φ and calculates S

^{-1}φ. In various embodiments, noise parameters calculator module 8518 calculates v according to equation (49) and a second type sought parameters estimator module 13700 calculates Θ according to equation (54). In some embodiments, common noise parameter estimator 9520 calculates common noise parameter a according to equation (50). In some embodiments, the noise parameter α output by common noise parameter estimator 9520 and the values of Θ output by second type sought parameters estimator module 13700 are outputs of the combined estimator 13516.

**[0153]**In various embodiments, a is sent to a postprocessor (e.g., 5300 in FIG. 5B) and in some such embodiments the postprocessor utilizes the magnitude of α as a quality parameter or as a criterion for the acceptance of the associated partial or fine sought parameters values. Thus in some embodiments, if the magnitude of a exceeds a threshold, then the associated partial or fine sought parameters values are discarded.

**[0154]**In various embodiments, the ambiguity of the phase measurement is resolved correctly and Θ is calculated without abnormal errors when equation (54) is utilized, and corresponding δ is inside of the right rounding region 6312, as illustrated by the dashed lines, in FIG. 6. Vectors δ (6314), χ

_{1}(6315), and v (6316) in FIG. 6 illustrate the correct ambiguity resolution if v is in the rounding region 6312 with χ

_{1}in the center, and k projected into χ

_{1}would give the correct Θ according to equation (24) for n=0. An incorrect ambiguity resolution decision can occur if δ is supposed to be rounded to χ

_{i}, but due to a high level of phase errors is rounded to χ

_{i}+χ

_{j}instead. In such a situation Θ might be calculated with abnormally high errors. For instance, consider the case where, for some angle of arrival, δ is supposed to be rounded to χ

_{2}(indicated by reference indicium 6317), if the level of phase errors is high, δ may be rounded to χ

_{1}(indicated by reference indicium 6315) instead. This can result in abnormally high errors in the estimation of Θ. In various embodiments, the decision is made that if v is close to a rounding region border, then there is a relatively high probability that it was calculated with an incorrect ambiguity resolution. Accordingly, in some embodiments, the corresponding Θ estimate calculated using equation (54) with such a value for the v vector can be considered as unreliable in such embodiments and associated with v partial or fine sought parameters are rejected in the postprocessor. In various embodiments, this kind of rejection increases the probability of the correct ambiguity resolution. Thus, in various embodiments, the magnitude of one or more noise parameters, such as for example but not limited to, α or q are considered to be a criterion for a decision as to whether or not to reject partial or fine sought parameters estimates. Parameter a shows the length of v. However, it does not inform about the position of v regarding the borders of rounding region 6312. Parameter q indicates how close v is to the border of the rounding region 6312 and, accordingly, in some embodiments, q is a more convenient criterion for rejection in postprocessing.

**[0155]**Reference is next made to FIG. 14, which is a block diagram illustrating various embodiments of combined estimator 14516 that calculates Θ according to equation (54) and q according to equation (53). In various embodiments, phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. In various embodiments, noise parameters calculator module 8518 calculates v according to equation (49), second type sought parameters estimator module 13700 calculates Θ according to equation (54), and discrete noise parameter estimator 11624 calculates q according to equation (53). In various embodiments, combined estimator 14516 and discrete noise parameter estimator 11624 have (N-M)*Z inputs of threshold γ

_{ij}values. In some embodiments, the magnitudes of these threshold values can be set to be constants. In various other embodiments, these threshold values can be variable and can be adjusted as desired. In some embodiments, the q output of discrete noise parameter estimator 11624 and the Θ outputs of second type sought parameters estimator module 13700 are outputs of the combined estimator 14516.

**[0156]**Reference is now made to FIG. 15, which is a graph that illustrates, for various embodiments, the difference between the probability of correct ambiguity resolution in the calculation of one partial sought parameter Θ

_{P}according to equation (54) without rejection and with rejection on q=1 if only one threshold parallelotope 6313 with γ

_{11}=γ

_{2}1=0.4 is considered in the discrete noise parameter estimator 11624. The probability of correct ambiguity resolution has been estimated after 10000 trials in a simulation of the combined estimator 14516 for a TDOA estimating interferometer designed to locate an object emitting three signal components which frequencies relate to each other as relatively prime numbers defining vector A in (16) as:

**A**= ( 7 3 2 ) . ( 56 ) ##EQU00021##

**[0157]**As can be seen from FIG. 15, in some embodiments, the rejection of unreliable Θ

_{F}samples in the postprocessor allows for up to a 10% increase in the probability of correct ambiguity resolution for the particular conditions listed above.

**[0158]**Reference is now made to FIG. 16, which is a block diagram that illustrates various embodiments of a combined estimator 16516 that calculates Θ according to equation (54) and outputs a vector of noise parameters along with Θ. As FIG. 16 indicates, in some embodiments, the interferometers may utilize the whole vector v for postprocessing. Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 determines v in accordance with equation (49). Second type sought parameters estimator module 13700 calculates Θ according to equation (54). In some embodiments, the vector of noise parameters v output by noise parameters calculator module 8518 and the Θ values output by second type sought parameters estimator module 13700 are outputs of the combined estimator 16516.

**[0159]**In various embodiments, the use of equation (54) can be suboptimal, because it determines whether the vector v is inside of rounding region 6312 as opposed to whether the vector v is inside of Voronoi region 6311. Referring back to FIG. 6, it can be seen that rounding region 6312 does not completely correspond to the Voronoi region 6311, which is defined by the quadratic form in equation (25). In particular, it is possible for a 6 vector to be inside rounding region 6312 but to be outside of the corresponding Voronoi region 6311 and vice versa. In addition, Voronoi region 6311 can have up to 2(2

^{N}-M-1) sides, while the corresponding rounding region 6312 has 2(N-M) sides. Accordingly, the larger the number (N-M) is, the greater the difference between a Voronoi region 6311 and the corresponding rounding region 6312 tends to be.

**[0160]**In various embodiments, as a result of the lack of complete correspondence between the Voronoi region 6311 and the rounding region 6312, some samples of δ calculated by equation (36) and processed according to equations (49) and (54) produce the sought parameters with abnormally high errors due to incorrect ambiguity resolution. This can be illustrated with vector δ

_{1}in FIG. 6. According to equation (54), δ

_{1}will be rounded to (χ

_{1}-χ

_{2}) and v

_{1}will be used for calculation of Θ. However, k obtained according to the maximum likelihood in equation (25) corresponds to (-χ

_{2}); δ

_{1}is inside of Voronoi region with center at (-χ

_{2}) and v

_{2}should be used for correct calculation of Θ. In various embodiments, the optimal determination using equation (25) can be significantly simplified with the use of vector v determined according equation (49). Equation (25) corresponds to:

**χ * = arg min χ ( ( v + χ i ) T P ( v + χ i ) ) , ( 57 ) ##EQU00022##**

**where P is a matrix defined as**:

**P**=K

^{TCK}, (58)

**and where**χ

_{i}are vectors which form Voronoi region 6311 with center at χ=0. Equation (57) corresponds to:

**χ * = arg min χ ( 0.5 ( χ i T P χ i ) + χ i T η ) , ( 59 ) ##EQU00023##**

**where**η is a vector defined as:

**η=Pv. (60)**

**[0161]**Voronoi region 6311 can have up to 2(2

^{N}-M-1) sides. Vectors χ

_{i}, defining these sides and χ=0 shall be considered in equation (59). Therefore, the number of χ

_{i}to estimate them in equation (59) is not more than (2

^{N}-M+1-1). Such χ

_{i}has only 0 and ±1 in its elements and, therefore, every χ

_{i}

^{T}η equation (59) is a linear combination of corresponding elements of η. As far as set of χ

_{i}forming Voronoi region 6311 for particular matrix A are predefined, it also predefines the set of linear combinations of corresponding elements of η to be considered in equation (59). The magnitudes of 0.5(χ

_{i}

^{TP}χ

_{i}) are predefined constants, which do not depend on the phase measurements. In various embodiments, these conditions make a combined estimator designed based on the minimization procedure according to equation (59) more effective and efficient than a combined estimator that is designed around a computational procedure that is based on equation (25), especially given that equation (59), while more efficient given the above conditions, is nonetheless, in terms of the final estimate that is produced in the end, equivalent to equation (25).

**[0162]**After the searching of χ* according to equation (59) is performed, the vector Θ of sought parameters can be estimated according to:

**Θ=rrni{Hτ}, (61)**

**where**τ is a vector combination of ρ and ψ according to:

**τ = ( ρ ψ ) , ( 62 ) ##EQU00024##**

**and where**ρ is given by:

**ρ=v+χ*. (63)**

**[0163]**Alternatively the vector Θ of sought parameters can be estimated according to:

**Θ=rrni{Hξ+f}, (64)**

**where f is a vector given by**:

**f**=Rχ*, (65)

**and where R is a part of matrix Has defined in equation**(41).

**[0164]**Reference is next made to FIG. 17, which is a block diagram illustrating various embodiments of combined estimator 17516 that calculates a maximum likelihood estimate of Θ in accordance with equation (61). Phase measurements converter module 8517 processes the input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region (VR) shift calculator module 17720 calculates χ* according to equation (59). Noise parameters corrector module 17730 calculates ρ according to equation (63). Second type sought parameters estimator module 13700 calculates Θ according to equation (61). In various embodiments, the outputs of second type sought parameters estimator module 13700 are the outputs of combined estimator 17516.

**[0165]**Reference is next made to FIG. 18, which is a block diagram illustrating various embodiments of combined estimator 18516 that calculates a maximum likelihood estimate of Θ in accordance with equation (64). Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Second type noise parameters corrector module 18730 calculates f according to equation (65). Third type sought parameters estimator module 18700 calculates Θ according to equation (64). In various embodiments, the outputs of third type sought parameters estimator module 18700 are the outputs of combined estimator 18516.

**[0166]**In various embodiments, given that equations (61) or (64) completely correspond to the maximum likelihood principle of estimation of Θ, the probability of correct ambiguity resolution for an combined estimator that is designed based on the use of either of these equations is greater than the probability of correct ambiguity resolution for a combined estimator that is designed based on the use of equation (54). For example, FIG. 25 and FIG. 15 are graphs illustrating the difference between those algorithms for matrix A defined in equation (56).

**[0167]**Reference is next made to FIG. 19, which is a block diagram illustrating various embodiments of combined estimator 19516 that calculates a maximum likelihood estimate of Θ in accordance with equation (61), and also outputs the vector of noise parameters v along with Θ. Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Noise parameters corrector module 17730 calculates ρ according to equation (63). Second type sought parameters estimator module 13700 calculates Θ according to equation (61). In various embodiments, the vector of noise parameters v output by noise parameters calculator module 8518 and the values of Θ output by second type sought parameters estimator module 13700 are the outputs of combined estimator 19516.

**[0168]**Reference is next made to FIG. 20, which is a block diagram illustrating various embodiments of combined estimator 20516 that calculates a maximum likelihood estimate of Θ in accordance with equation (64), and also outputs the vector of noise parameters v along with Θ. Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Second type noise parameters corrector module 18730 calculates f according to equation (65). Third type sought parameters estimator module 18700 calculates Θ according to equation (64). In various embodiments, the vector of noise parameters v output by noise parameters calculator module 8518 and the values of Θ output by third type sought parameters estimator module 18700 are the outputs of combined estimator 20516.

**[0169]**Reference is next made to FIG. 21, which is a block diagram illustrating various embodiments of combined estimator 21516 that calculates a maximum likelihood estimate of Θ in accordance with equation (61) and common noise parameter α according to equation (50). Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Noise parameters corrector module 17730 calculates ρ according to equation (63). Second type sought parameters estimator module 13700 calculates Θ according to equation (61). Common noise parameter estimator 9520 calculates a according to equation (50). In various embodiments, the common noise parameter a output by common noise parameter estimator 9520 and the values of Θ output by second type sought parameters estimator module 13700 are outputs of combined estimator 21516.

**[0170]**Reference is now made to FIG. 22, which is a block diagram illustrating various embodiments of combined estimator 22516 that calculates a maximum likelihood estimate of Θ in accordance with equation (64) and common noise parameter a according to equation (50). Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Second type noise parameters corrector module 18730 calculates f according to (65). Third type sought parameters estimator module 18700 calculates Θ according to equation (54). Common noise parameter estimator 9520 calculates a according to equation (50). In various embodiments, the common noise parameter α output by common noise parameter estimator 9520 and the values of Θ output by third type sought parameters estimator module 18700 are outputs of combined estimator 22516.

**[0171]**Reference is next made to FIG. 23, which is a block diagram illustrating various embodiments of combined estimator 23516 that calculates a maximum likelihood estimate of Θ in accordance with equation (61) and discrete noise parameter q according to equation (53). Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Noise parameters corrector module 17730 calculates ρ according to equation (63). Second type sought parameters estimator module 13700 calculates Θ according to equation (61). Discrete noise parameter estimator 11624 calculates q according to equation (53). Combined estimator 23516 and discrete noise parameter estimator 11624 have (N-M)*Z inputs of threshold γ

_{ij}values. In some embodiments, the magnitudes of those threshold values are set to constant. In various other embodiments, these threshold values can be variable. In various embodiments, the discrete noise parameter q output by discrete noise parameter estimator 11624 and the values of Θ output by second type sought parameters estimator module 13700 are outputs of combined estimator 23516.

**[0172]**Reference is now made to FIG. 24, which is a block diagram illustrating various embodiments of combined estimator 24516 that calculates a maximum likelihood estimate of Θ in accordance with equation (64) and discrete noise parameter q according to equation (53). Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). Noise parameters converter module 17710 calculates η according to equation (60). Voronoi Region shift calculator module 17720 calculates χ* according to equation (59). Second type noise parameters corrector module 18730 calculates f according to (65). Third type sought parameters estimator module 18700 calculates Θ according to equation (64). In various embodiments, discrete noise parameter estimator 11624 calculates q according to equation (53). Combined estimator 24516 and discrete noise parameter estimator 11624 have (N-M)*Z inputs of threshold γ

_{ij}values. The magnitudes of those threshold values can be set to constant, or they can be variable. In various embodiments, the discrete noise parameter q output by discrete noise parameter estimator 11624 and the values of Θ output by third type sought parameters estimator module 18700 are outputs of combined estimator 24516.

**[0173]**Reference is again made to FIG. 25, which is a graph that illustrates, in various embodiments, the difference between the probability of correct ambiguity resolution in the calculation of partial sought parameter Θ

_{P}according to equations (61) or (64) without rejection and with rejection on q=1, if only one threshold parallelotope 6313 with γ

_{11}=γ

_{2}1=0.45 is considered in the discrete noise parameter estimator 11624. The probability of correct ambiguity resolution has been estimated after 10000 trials in a simulation of the combined estimators 23516 and 24516 for a TDOA estimating interferometer designed to locate an object emitting three signal components which frequencies relate to each other as relatively prime numbers defining vector A in (16) as it is shown in equation (56). As can be seen from FIG. 25, in various embodiments, the rejection of unreliable Θ

_{F}samples in a postprocessor (e.g., 5300 in FIG. 5B) allows up to 5% increasing the probability of correct ambiguity resolution for the particular conditions listed above.

**[0174]**Some embodiments and some applications may require a high level of Θ accuracy, very high probability of correct ambiguity resolution, and high interferometer throughput. Accordingly, in some embodiments, the combined estimator can work in an adaptive manner to reduce the amount of computation required and thereby also reduce the amount of time required. In particular, in some embodiments, the combined estimator makes a decision regarding the level of noise and which algorithm is most suitable given the level of noise. In some embodiments, the least computationally intensive algorithm or the equation that is most efficient but still applicable given the level of noise is selected. In other embodiments, any of the applicable equations are selected.

**[0175]**For example, in some embodiments, the discrete noise parameter q can be calculated and a determination of position of v with respect to 2 threshold parallelotopes in If v is inside of the smallest parallelotope and if q=0, then Θ can be estimated according to equation (45). However, if v is outside of the smallest parallelotope, but is inside of the second parallelotope and if q=1, then Θ can be estimated according to equation (54). Also, if v is out of the biggest parallelotope and if q=2, then Θ can be estimated according to equation (61) or (64).

**[0176]**Alternatively, assuming a larger number of parallelotopes is defined, if v is inside of a range of the smallest parallelotopes, so that q is below or equal to a first threshold value (i.e., q≦T

_{1}), then Θ can be estimated according to equation (45). However, if v is outside of the range of smallest parallelotopes, but is inside of a range of intermediate parallelotopes, so that q is below or equal to a second threshold value larger than the first threshold value (i.e., T

_{1}<q≦T

_{2}), then Θ can be estimated according to equation (54). Also, if v is outside of the range of intermediate parallelotopes, so that q is larger than the second threshold value (i.e., T

_{2}<q), then Θ can be estimated according to equation (61) or (64).

**[0177]**Reference is now made to FIG. 26, which is a block diagram that illustrates various embodiments of combined estimator 26516 that calculates Θ in different manners depending on the magnitude of discrete noise parameter q. Phase measurements converter module 8517 processes input vector φ and calculates S

^{-1}φ. Noise parameters calculator module 8518 calculates v according to equation (49). In various embodiments, discrete noise parameter estimator 11624 calculates q according to equation (53). Adaptive estimator 26800 calculates Θ based on the magnitude of q. If q≦T

_{1}, corresponding to the first range of values, adaptive estimator 26800 calculates Θ according to equation (45). If T

_{1}<q≦T

_{2}, corresponding to the second range of values larger than the first range, adaptive estimator 26800 calculates Θ according to equation (54). If T

_{2}<q, corresponding to the third range of values larger than the second range, adaptive estimator 26800 calculates Θ according to equation (61) or (64). Combined estimator 26516 and discrete noise parameter estimator 11624 have (N-M)*Z inputs of threshold γ

_{ij}values. In some embodiments, the magnitudes of those threshold values are set to constant. In various other embodiments, these threshold values can be variable. In various embodiments, the discrete noise parameter q output by discrete noise parameter estimator 11624 and the values of Θ output by adaptive estimator module 26800 are outputs of combined estimator 26516.

**[0178]**The various embodiments of interferometers described herein can be implemented in hardware, in software running on microprocessor, ASIC, or in combination of hardware and software.

**[0179]**Various systems, apparatus and methods have been described according to example embodiments of the invention, including at least one example of each claimed embodiment. None of the above-described embodiments is limiting in any way, and the claimed embodiments may cover systems, apparatus and methods, as well as aspects thereof, which were not explicitly described above. The claimed embodiments are not limited to systems, apparatus and methods having all of the features of any one example system, apparatus or method described above, or to common features shared by two or more of the systems, apparatus and methods described above. It is possible that a system, apparatus, or method described above does not directly relate to a claimed embodiment of the invention.

**[0180]**While the above description provides example embodiments, it will be appreciated that some features and/or functions of the described embodiments may be susceptible to modification without departing from the scope or operating principles of the described embodiments. What has been described above is intended to be non-limiting and merely illustrative of the invention, the scope of which is defined only by the claims appended hereto.

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