Generalized Pencil Of Function Method

Transcription

1 Generalized Pencil Of Function Method In recent times, the methodology of approximating a function by a sum of complex exponentials has found applications in many areas of electromagnetics For example in antenna pattern synthesis [] and in the extraction of the s-parameters of microwave integrated circuits [2], in the computation of input impedance of electrically wide slot antennas [3] and in the efficient evaluation of the Sommerfeld integral [4], that is our goal Two of the most popular linear methods used to estimate the parameters of a sum of complex exponentials are the polynomial method, ie Prony method, and the matrix pencil method (called also GPOF)[5] This method differs from the well known Prony method because it finds the poles by solving a generalized eigenvalue problem instead of the conventional two-step process where the first step involves the solution of a matrix equation, and the second step entails finding the roots of a polynomial, as is required by Prony method Formulation Let us consider a generic signal y(t) y(t) = x(t) + n(t) R i e sit + n(t) () where n(t) is the possible noise of the system We sample y(t) at kt s, where k = 0,, N and T s is the sampling period obtaing where y(kt s ) = x(kt s ) + n(kt s ) R i zi k + n(kt s ) (2) z i = e s it s = e (α i+jω i )T s (3)

2 2 where i =, 2, M Let us consider first a noiseless signal y(t) sampling this signal we obtain with y(t) = x(t) R i e s it y(kt s ) = x(kt s ) (4) R i zi k (5) z i = e s it s = e (α i+jω i )T s (6) where i =, 2, M Our goal is to estimate M, s i, R i that approximate the function y(t) We define two matrices (N L) L Y, Introducing y(0) y() y(l ) y() y(2) y(l) Y y(n L ) y(n L) y(n 2) y() y(2) y(l) y(2) y(3) y(l + ) y(n L) y(n L + ) y(n ) Z R R 2 0 R 0 0 R m z z 2 0 Z z m z z 2 z M z N L z2 N L z N L M (N L) M (N L) L (N L) L (7) (8) (9) (0) ()

3 3 Y, can be written as Z 2 z z L z 2 z2 L z M z L M M L (2) Y = Z R Z 2 (3) Now let us consider the matrix pencil = Z R Z 0 Z 2 (4) λy = Z R {Z 0 λi} Z 2 (5) where I is a M M identity matrix We can demostrate that if M L N M the rank of λy is M; however, if λ = z i i =, 2, M the i th row of Z 0 λi is zero and the rank of λy is reduced to M Moreover, the poles are the generalized eigenvalues of the matrix pencil, so the problem of solving for z i can be cast as an ordinary eigenvalue problem Let us define the Moore Penrose pseudo inverse matrix of Y as Y + = {Y H Y } Y H (6) where the superscript h indicates the conjugate transpose Then, ] [ ] [ λy = Y + Z R Z 0 λi Z 2 (7) Y + = [ ] ] Y H Y Y H [Y Y H Y 2 Y H λy (8) [ ] = Y H Y Y H λi (9) = Y + Y λi (20) 2 Thus the parameters z i can be determined by solving the generalized eigenvalue problem of Y + Y λi (2) 2 In presence of noise, (M L) we define the matrix Y as a combination of Y, y 0 y y L y y 2 y L Y (22) y N L y N L y N 2 (N L) (L+)

4 4 We carry out a Singular Value Decomposition of the matrix Y Y (N L) (L+) = U Σ V H (23) where U, V are unitary matrices with dimensions (N L) (N L) e (L + ) (L + ), respectively given by the eigenvalues of Y Y H and Y H Y While Σ is a (N L) (L + ) diagonal matrix composed of the singular values of Y : Σ = diag(σ, σ 2,, σ p ); p = min{n L, L+} (24) where σ σ 2 σ p 0 Now we can estimate the M parameter This is done comparing the largest singular value σ max with the other σ c and defining a threshold q This is linked to the significant decimal digits in the data Therefore, we consider only the singular values that are σ c σ max 0 q (25) The other singular values are treated as zero, since we consider them only given by noise We can now construct a filtered matrix V so that it contains only the M dominant right singular vectors of V so that Y is given by Y = U Σ V H (26) Σ is containg only the M dominant singular values of Σ We have defined Y and that are constructed deleting respectively the last and the firts row of Y Y = U Σ V H = U Σ V H 2 we can demostrate that the eigenvalues of {Y + λi } (27) (28) (29) are the same as the ones of {V H } + {V 2 H } + λi (30) Once the M and z i are known we can solve the matrix equation by least square to calculate R i

5 5 y 0 y y N = z z 2 z m z 2 z2 2 zm 2 z N z2 N zm N R R 2 R M (3) Prony-Type Method For sake of completeness, we describe also the Prony method,even if its computationally less efficient ans also unstable in case of noisy signal We consider once again the same generic signal y(t) Given M complex number z i, with i =, 2, M, there exist only one complex number a i C with i =, 2, M, so that z i are solutions of a k z k = 0 a 0 (32) finding the coefficient a i is equivalent to determine the poles z i Moreover, we can demostrate that if p(λ) = a k λ k = 0 (33) where p(z i ) 0 i =, 2, M If y n = R i zi n (34) for n = 0,,, N then, if L m N with M L y m k a k = 0 (35) Considering substituting y m k a k = y m k = R i z m k (36) ( M ) R i zi m z k i a k = R i zi m z k i a k = 0 (37) }{{} p(z i ) 0 (38)

6 6 We can write ( 35) in matrix form m (l, N ) Y a y 0 y y L y L y y 2 y L y L+ y N L y N 2 y N Y a = Y y a a 0 a L a L a 0 }{{} = 0 (39) (40) since a 0 = Y a = Y a + y = 0 so, = Y a = y where Y has dimension of L [(N ) L] In order to generalize the method we have to write the polynomial form of the signal so that we can distinguish the roots of the signal to those due to noise This solution, called minimum-norm solution is given by a = Y + y (4) where Y + is once again the Moore-Penrose pseudo-inverse matrix In case of noisy signal the Moore-Penrose pseudo-inverse matrix Y + is replaced by a truncated rank- M pseudo-inverse which is formed by the first M largest singular values The choice of M is the same as in the Matrix pencil method

7 Bibliography [] E K Miller, G J Burke, Using model based parameter estimation to increase the physical interpretability and numerical efficiency of computational electromagnetics Computer physics Communications, n 68, 99, pp43-75 [2] T K Sarkar, Z A Maricevic, M Kahrizi, An accurate dembedding procedure for characterizing discontinuities, International Journal of Microwave and MIllimeter wave Computer aided Engineering, vol 2, Feb 992, pp [3] M Kahrizi, T K Sarkar, Z A Maricevic, Analysis of a wide radiating slot in the ground plane of a microstrip line, IEEE Transactions on Microwave Theory and Techniques, Vol 4, n, Jan 993, pp29-37 [4] R M Shubair, YL Chow, Efficient computation of the periodic Green s function in layered dielectric media, Microwave Theory and Techniques, IEEE Transactions on, Vol 4, n 3, March 993, pp [5] Y Hua, T K Sarkar, Generalized Pencil-of-Function Method for Extracting Poles of an EM System from Its Transient Response, IEEE Trans Antennas Propagat, vol 37, no 2, pp , Feb989 7