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Method

Mathematical basis

A transient electromagnetic signal can be represented as:⁵

y ( t ) = x ( t ) + n ( t ) ≈ ∑ i = 1 M R i e s i t + n ( t ) ; 0 ≤ t ≤ T , {\displaystyle y(t)=x(t)+n(t)\approx \sum _{i=1}^{M}R_{i}e^{s_{i}t}+n(t);0\leq t\leq T,}

where

y ( t ) {\displaystyle y(t)} is the observed time-domain signal, n ( t ) {\displaystyle n(t)} is the signal noise, x ( t ) {\displaystyle x(t)} is the actual signal, R i {\displaystyle R_{i}} are the residues ( R i {\displaystyle R_{i}} ), s i {\displaystyle s_{i}} are the poles of the system, defined as s i = − α i + j ω i {\displaystyle s_{i}=-\alpha _{i}+j\omega _{i}} , z i = e ( − α i + j ω i ) T s {\displaystyle z_{i}=e^{(-\alpha _{i}+j\omega _{i})T_{s}}} by the identities of Z-transform, α i {\displaystyle \alpha _{i}} are the damping factors and ω i {\displaystyle \omega _{i}} are the angular frequencies.

The same sequence, sampled by a period of T s {\displaystyle T_{s}} , can be written as the following:

y [ k T s ] = x [ k T s ] + n [ k T s ] ≈ ∑ i = 1 M R i z i k + n [ k T s ] ; k = 0 , . . . , N − 1 ; i = 1 , 2 , . . . , M {\displaystyle y[kT_{s}]=x[kT_{s}]+n[kT_{s}]\approx \sum _{i=1}^{M}R_{i}z_{i}^{k}+n[kT_{s}];k=0,...,N-1;i=1,2,...,M} ,

Generalized pencil-of-function estimates the optimal M {\displaystyle M} and z i {\displaystyle z_{i}} 's.⁶

Noise-free analysis

For the noiseless case, two ( N − L ) × L {\displaystyle (N-L)\times L} matrices, Y 1 {\displaystyle Y_{1}} and Y 2 {\displaystyle Y_{2}} , are produced:⁷

[ Y 1 ] = [ x ( 0 ) x ( 1 ) ⋯ x ( L − 1 ) x ( 1 ) x ( 2 ) ⋯ x ( L ) ⋮ ⋮ ⋱ ⋮ x ( N − L − 1 ) x ( N − L ) ⋯ x ( N − 2 ) ] ( N − L ) × L ; {\displaystyle [Y_{1}]={\begin{bmatrix}x(0)&x(1)&\cdots &x(L-1)\\x(1)&x(2)&\cdots &x(L)\\\vdots &\vdots &\ddots &\vdots \\x(N-L-1)&x(N-L)&\cdots &x(N-2)\end{bmatrix}}_{(N-L)\times L};}   [ Y 2 ] = [ x ( 1 ) x ( 2 ) ⋯ x ( L ) x ( 2 ) x ( 3 ) ⋯ x ( L + 1 ) ⋮ ⋮ ⋱ ⋮ x ( N − L ) x ( N − L + 1 ) ⋯ x ( N − 1 ) ] ( N − L ) × L {\displaystyle [Y_{2}]={\begin{bmatrix}x(1)&x(2)&\cdots &x(L)\\x(2)&x(3)&\cdots &x(L+1)\\\vdots &\vdots &\ddots &\vdots \\x(N-L)&x(N-L+1)&\cdots &x(N-1)\end{bmatrix}}_{(N-L)\times L}}

where L {\displaystyle L} is defined as the pencil parameter. Y 1 {\displaystyle Y_{1}} and Y 2 {\displaystyle Y_{2}} can be decomposed into the following matrices:⁸

[ Y 1 ] = [ Z 1 ] [ B ] [ Z 2 ] {\displaystyle [Y_{1}]=[Z_{1}][B][Z_{2}]} [ Y 2 ] = [ Z 1 ] [ B ] [ Z 0 ] [ Z 2 ] {\displaystyle [Y_{2}]=[Z_{1}][B][Z_{0}][Z_{2}]}

where

[ Z 1 ] = [ 1 1 ⋯ 1 z 1 z 2 ⋯ z M ⋮ ⋮ ⋱ ⋮ z 1 ( N − L − 1 ) z 2 ( N − L − 1 ) ⋯ z M ( N − L − 1 ) ] ( N − L ) × M ; {\displaystyle [Z_{1}]={\begin{bmatrix}1&1&\cdots &1\\z_{1}&z_{2}&\cdots &z_{M}\\\vdots &\vdots &\ddots &\vdots \\z_{1}^{(N-L-1)}&z_{2}^{(N-L-1)}&\cdots &z_{M}^{(N-L-1)}\end{bmatrix}}_{(N-L)\times M};}   [ Z 2 ] = [ 1 z 1 ⋯ z 1 L − 1 1 z 2 ⋯ z 2 L − 1 ⋮ ⋮ ⋱ ⋮ 1 z M ⋯ z M L − 1 ] M × L {\displaystyle [Z_{2}]={\begin{bmatrix}1&z_{1}&\cdots &z_{1}^{L-1}\\1&z_{2}&\cdots &z_{2}^{L-1}\\\vdots &\vdots &\ddots &\vdots \\1&z_{M}&\cdots &z_{M}^{L-1}\end{bmatrix}}_{M\times L}}

[ Z 0 ] {\textstyle [Z_{0}]} and [ B ] {\textstyle [B]} are M × M {\textstyle M\times M} diagonal matrices with sequentially-placed z i {\textstyle z_{i}} and R i {\textstyle R_{i}} values, respectively.⁹

If M ≤ L ≤ N − M {\textstyle M\leq L\leq N-M} , the generalized eigenvalues of the matrix pencil

[ Y 2 ] − λ [ Y 1 ] = [ Z 1 ] [ B ] ( [ Z 0 ] − λ [ I ] ) [ Z 2 ] {\displaystyle [Y_{2}]-\lambda [Y_{1}]=[Z_{1}][B]([Z_{0}]-\lambda [I])[Z_{2}]}

yield the poles of the system, which are λ = z i {\displaystyle \lambda =z_{i}} . Then, the generalized eigenvectors p i {\displaystyle p_{i}} can be obtained by the following identities:¹⁰

[ Y 1 ] + [ Y 1 ] p i = p i ; {\displaystyle [Y_{1}]^{+}[Y_{1}]p_{i}=p_{i};}      i = 1 , . . . , M {\displaystyle i=1,...,M} [ Y 1 ] + [ Y 2 ] p i = z i p i ; {\displaystyle [Y_{1}]^{+}[Y_{2}]p_{i}=z_{i}p_{i};}      i = 1 , . . . , M {\displaystyle i=1,...,M}

where the + {\displaystyle ^{+}} denotes the Moore–Penrose inverse, also known as the pseudo-inverse. Singular value decomposition can be employed to compute the pseudo-inverse.

Noise filtering

If noise is present in the system, [ Y 1 ] {\textstyle [Y_{1}]} and [ Y 2 ] {\textstyle [Y_{2}]} are combined in a general data matrix, [ Y ] {\textstyle [Y]} :¹¹

[ Y ] = [ y ( 0 ) y ( 1 ) ⋯ y ( L ) y ( 1 ) y ( 2 ) ⋯ y ( L + 1 ) ⋮ ⋮ ⋱ ⋮ y ( N − L − 1 ) y ( N − L ) ⋯ y ( N − 1 ) ] ( N − L ) × ( L + 1 ) {\displaystyle [Y]={\begin{bmatrix}y(0)&y(1)&\cdots &y(L)\\y(1)&y(2)&\cdots &y(L+1)\\\vdots &\vdots &\ddots &\vdots \\y(N-L-1)&y(N-L)&\cdots &y(N-1)\end{bmatrix}}_{(N-L)\times (L+1)}}

where y {\displaystyle y} is the noisy data. For efficient filtering, L is chosen between N 3 {\textstyle {\frac {N}{3}}} and N 2 {\textstyle {\frac {N}{2}}} . A singular value decomposition on [ Y ] {\textstyle [Y]} yields:

[ Y ] = [ U ] [ Σ ] [ V ] H {\displaystyle [Y]=[U][\Sigma ][V]^{H}}

In this decomposition, [ U ] {\textstyle [U]} and [ V ] {\textstyle [V]} are unitary matrices with respective eigenvectors [ Y ] [ Y ] H {\textstyle [Y][Y]^{H}} and [ Y ] H [ Y ] {\textstyle [Y]^{H}[Y]} and [ Σ ] {\textstyle [\Sigma ]} is a diagonal matrix with singular values of [ Y ] {\textstyle [Y]} . Superscript H {\textstyle H} denotes the conjugate transpose.¹²¹³

Then the parameter M {\textstyle M} is chosen for filtering. Singular values after M {\textstyle M} , which are below the filtering threshold, are set to zero; for an arbitrary singular value σ c {\textstyle \sigma _{c}} , the threshold is denoted by the following formula:¹⁴

σ c σ m a x = 10 − p {\displaystyle {\frac {\sigma _{c}}{\sigma _{max}}}=10^{-p}} ,

σ m a x {\textstyle \sigma _{max}} and p are the maximum singular value and significant decimal digits, respectively. For a data with significant digits accurate up to p, singular values below 10 − p {\textstyle 10^{-p}} are considered noise.¹⁵

[ V 1 ′ ] {\textstyle [V_{1}']} and [ V 2 ′ ] {\textstyle [V_{2}']} are obtained through removing the last and first row and column of the filtered matrix [ V ′ ] {\textstyle [V']} , respectively; M {\textstyle M} columns of [ Σ ] {\textstyle [\Sigma ]} represent [ Σ ′ ] {\textstyle [\Sigma ']} . Filtered [ Y 1 ] {\textstyle [Y_{1}]} and [ Y 2 ] {\textstyle [Y_{2}]} matrices are obtained as:¹⁶

[ Y 1 ] = [ U ] [ Σ ′ ] [ V 1 ′ ] H {\displaystyle [Y_{1}]=[U][\Sigma '][V_{1}']^{H}} [ Y 2 ] = [ U ] [ Σ ′ ] [ V 2 ′ ] H {\displaystyle [Y_{2}]=[U][\Sigma '][V_{2}']^{H}}

Prefiltering can be used to combat noise and enhance signal-to-noise ratio (SNR).¹⁷ Band-pass matrix pencil (BPMP) method is a modification of the GPOF method via FIR or IIR band-pass filters.¹⁸¹⁹

GPOF can handle up to 25 dB SNR. For GPOF, as well as for BPMP, variance of the estimates approximately reaches Cramér–Rao bound.²⁰²¹²²

Calculation of residues

Residues of the complex poles are obtained through the least squares problem:²³

[ y ( 0 ) y ( 1 ) ⋮ y ( N − 1 ) ] = [ 1 1 ⋯ 1 z 1 z 2 ⋯ z M ⋮ ⋮ ⋱ ⋮ z 1 N − 1 z 2 N − 1 ⋯ z M N − 1 ] [ R 1 R 2 ⋮ R M ] {\displaystyle {\begin{bmatrix}y(0)\\y(1)\\\vdots \\y(N-1)\end{bmatrix}}={\begin{bmatrix}1&1&\cdots &1\\z_{1}&z_{2}&\cdots &z_{M}\\\vdots &\vdots &\ddots &\vdots \\z_{1}^{N-1}&z_{2}^{N-1}&\cdots &z_{M}^{N-1}\end{bmatrix}}{\begin{bmatrix}R_{1}\\R_{2}\\\vdots \\R_{M}\end{bmatrix}}}

Applications

The method is generally used for the closed-form evaluation of Sommerfeld integrals in discrete complex image method for method of moments applications, where the spectral Green's function is approximated as a sum of complex exponentials.²⁴²⁵ Additionally, the method is used in antenna analysis, S-parameter-estimation in microwave integrated circuits, wave propagation analysis, moving target indication, radar signal processing,²⁶²⁷²⁸ and series acceleration in electromagnetic problems.²⁹

See also

• Estimation of signal parameters via rotational invariance techniques
• Generalized eigenvalue problem
• Matrix pencil
• MUSIC (algorithm)
• Prony's method

References

1. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

2. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

3. Sarkar, T.; Nebat, J.; Weiner, D.; Jain, V. (November 1980). "Suboptimal approximation/identification of transient waveforms from electromagnetic systems by pencil-of-function method". IEEE Transactions on Antennas and Propagation. 28 (6): 928–933. Bibcode:1980ITAP...28..928S. doi:10.1109/TAP.1980.1142411. /wiki/Tapan_Kumar_Sarkar ↩

4. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

5. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

6. Hua, Y.; Sarkar, T. K. (May 1990). "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (5): 814–824. doi:10.1109/29.56027. /wiki/Tapan_Kumar_Sarkar ↩

7. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

8. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

9. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

10. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

11. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

12. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

13. Hua, Y.; Sarkar, T. K. (May 1990). "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (5): 814–824. doi:10.1109/29.56027. /wiki/Tapan_Kumar_Sarkar ↩

14. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

15. Hua, Y.; Sarkar, T. K. (May 1990). "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (5): 814–824. doi:10.1109/29.56027. /wiki/Tapan_Kumar_Sarkar ↩

16. Hua, Y.; Sarkar, T. K. (May 1990). "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (5): 814–824. doi:10.1109/29.56027. /wiki/Tapan_Kumar_Sarkar ↩

17. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

18. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

19. Hu, Fengduo; Sarkar, T. K.; Hua, Yingbo (January 1993). "Utilization of Bandpass Filtering for the Matrix Pencil Method". IEEE Transactions on Signal Processing. 41 (1): 442–446. Bibcode:1993ITSP...41..442H. doi:10.1109/TSP.1993.193174. /wiki/Tapan_Kumar_Sarkar ↩

20. Hua, Y.; Sarkar, T. K. (February 1989). "Generalized pencil-of-function method for extracting poles of an EM system from its transient response". IEEE Transactions on Antennas and Propagation. 37 (2): 229–234. Bibcode:1989ITAP...37..229H. doi:10.1109/8.18710. /wiki/Tapan_Kumar_Sarkar ↩

21. Hu, Fengduo; Sarkar, T. K.; Hua, Yingbo (January 1993). "Utilization of Bandpass Filtering for the Matrix Pencil Method". IEEE Transactions on Signal Processing. 41 (1): 442–446. Bibcode:1993ITSP...41..442H. doi:10.1109/TSP.1993.193174. /wiki/Tapan_Kumar_Sarkar ↩

22. Hua, Y.; Sarkar, T. K. (May 1990). "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise". IEEE Transactions on Acoustics, Speech, and Signal Processing. 38 (5): 814–824. doi:10.1109/29.56027. /wiki/Tapan_Kumar_Sarkar ↩

23. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

24. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

25. Dural, G.; Aksun, M. I. (July 1995). "Closed-form Green's functions for general sources and stratified media". IEEE Transactions on Microwave Theory and Techniques. 43 (7): 1545–1552. Bibcode:1995ITMTT..43.1545D. doi:10.1109/22.392913. hdl:11693/10756. /wiki/%C4%B0r%C5%9Fadi_Aksun ↩

26. Sarkar, T. K.; Pereira, O. (February 1995). "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials". IEEE Antennas and Propagation Magazine. 37 (1): 48–55. Bibcode:1995IAPM...37...48S. doi:10.1109/74.370583. /wiki/Tapan_Kumar_Sarkar ↩

27. Kahrizi, M.; Sarkar, T. K.; Maricevic, Z. A. (January 1994). "Analysis of a wide radiating slot in the ground plane of a microstrip line". IEEE Transactions on Microwave Theory and Techniques. 41 (1): 29–37. doi:10.1109/22.210226. /wiki/Tapan_Kumar_Sarkar ↩

28. Hua, Y. (January 1994). "High resolution imaging of continuously moving object using stepped frequency radar". Signal Processing. 35 (1): 33–40. Bibcode:1994SigPr..35...33H. doi:10.1016/0165-1684(94)90188-0. https://escholarship.org/uc/item/5vv1k6fs ↩

29. Karabulut, E. Pınar; Ertürk, Vakur B.; Alatan, Lale; Karan, S.; Alişan, Burak; Aksun, M. I. (2016). "A novel approach for the efficient computation of 1-D and 2-D summations". IEEE Transactions on Antennas and Propagation. 64 (3): 1014–1022. Bibcode:2016ITAP...64.1014K. doi:10.1109/TAP.2016.2521860. hdl:11693/36512. /wiki/%C4%B0r%C5%9Fadi_Aksun ↩