Radio Science Bulletin 313 - June 2005 - URSI

Next, construct the ‘filtered’ matrix [ V ′],[ V ] [ v v ]′ = 1 , ! , , (7)Mso that it contains only the M dominant right singularvectors of [ V ]; that is, the L+ 1-Mright singular vectorsfrom M + 1 to L + 1 corresponding to the small singularvalues, are discarded. Therefore,[ Y ] [ U][ ′][ V′]H1 1[ Y2] [ U][ ′][ V2′]where [ V 1′ ] is obtained from [ 1]of[ V 1′ ], [ 2]and [ ′]= Σ , (8)H= Σ , (9)V ′ by deleting the last rowV ′ is obtained by removing the first row of[ V 1′ ],Σ is obtained from the M columns of [] Σcorresponding to the M dominant singular values. The polesof the signals are given by the nonzero eigenvalues of{[ ]H +V1 } [ V2]H′ ′ , (10)which are the same as the eigenvalues ofH{ }H[ V ] [ V ]2 1+′ ′ . (11)The + superscript denotes the pseudo-inverse of a matrix[14].zm = m + j m TSareknown for 1 ≤m≤ M , the residues R m are solved from thefollowing least-squares problemOnce the M poles exp( α ω )⎡ y()0 ⎤⎡ 1 1 " 1 ⎤⎡ R1⎤⎢ ⎥⎢⎥y()1 z1 z2 z⎢M R⎥⎢ ⎥⎢" ⎥ 2= ⎢ ⎥.⎢⎢ ⎥# ⎥⎢ # # # ⎥⎢ # ⎥⎢ ⎥ ⎢⎣y( N −1)⎥⎦ N−1 N−1 N−1Rz M1 z2z⎢⎣ ⎥⎢ ⎦⎣" M ⎥⎦(12)The sampling time T S is typically assumed to be unity in adigital system. However, for real data, one needs to specifya value for T S . The damping coefficients αmand frequenciesωmin the continuous s domain are related to the discretepoles z m in the z domain by sm = αm + jωmz = exp sT .and ( )S3. State-Space Approach 1 (SS1)Consider a stable, linear, time-invariant, discretetimesystem of degree n described by the minimal realizationx( k + ) = [ A] x() k + [ B] u()k() = [ ] ⎡ () ⎤,⎡⎣ 1 ⎤⎦ ⎡⎣ ⎤⎦,y k C ⎣x k ⎦(13)where ⎡⎣x()k ⎤⎦is the length-n state vector at discrete time kand [ A ] is the n× nstate transition matrix. Generally, theinput u and output y can be vectors, the well knownmultiple-input multiple-output (MIMO) problem. However,for the sake of brevity, u()k and y()k are scalars in thisdiscussion. The poles of the system are then the eigenvaluesof the state-space matrix[ A ]. In addition, [ B]and [ C]areauxiliary matrices of dimension n × 1 and1× n , respectively.Therefore, for a single-input single-output system, thescalar Markov parameters { h()k }, which essentially arethe samples of the impulse response of the system, are givenbyk−1= (14)hk ( ) [ C][ A] [ B].{ ⎤}Define the sequence of Hankel matrices ⎡Mq( k,l)(equation 15 - see below)where the index q is a nonnegative integer and K× L is thedimension of M q . The elements of each M q represent theimpulse response of the system [7,8].For a minimal realization of the matrices[ A ], [ B ],and [ C ] of degree n and N time samples ( N ≥ 1 ), the rthorderreduced model is obtained by using the followingprocedure. First, generate the square Hankel matrix⎡⎣M1 ( N + 1, N + 1)⎤⎦ and perform an SVD of this matrix to⎣⎦q,h() q h( q+ 1) " h( q+ l) " h( q+L)( + 1) ( + 2) " ( + 1+ ) " ( + 1+)⎡⎤⎢⎥⎢h q h q h q l h q L⎥⎢ # # # # # # ⎥⎡⎣Mq( k,l)⎤=⎦ ⎢ ⎥⎢h( q+ k) h( q+ k + 1) " h( q+ k+ l) " h( q+ k+L)⎥⎢ # # # # # # ⎥⎢⎥⎢⎣h( q+ K) h( q+ K + 1) " h( q+ K + l) " h( q+ K + L)⎥⎦(15)TheRadio Science Bulletin No 313 (June, 2005) 29