J. L. Brewer, \Kronecker products and matrix calculus in systems theory", IEEE Trans. on Circuits and Systems, vol. 25, No. 9, pp. 772-781  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a[k] a. The superscript (p) stands for polynomial . This is achieved in the following way: All product pairs of elements of a and h are combined in a N P 1 vector c = h a = h 1 a 1 ; h 1 a 2 ; h 1 a P ; h 2 a 1 ; hN 1 a P ] T ; 45) where denotes the Kronecker product [13]. With the same notation for the adaptive coecients c[k] h[k] a[k] 46) 14 we can de ne a joint system distance for the polynomial preprocessor and the FIR lter D (p) k] 10 log 10 jjc c[k]jj 2 2 jjcjj 2 2 : 47) Unlike D h [k] D (p) k] can never be equal to ERLE, ....

J. L. Brewer, \Kronecker products and matrix calculus in systems theory", IEEE Trans. on Circuits and Systems, vol. 25, No. 9, pp. 772-781

....T a(t; q) 8.4) We may also define, in terms of (8.3) n 2 q q T o a(t; q) q A(t; q) 2 R mn Thetam ; 8.5) AB) q = A q B i I m Omega A j B q 2 R mn Thetal ; 8. 6) where I m is an m Theta m identity matrix, and q is an m Gammacomponent column vector; see [10]. Baker Rihan 20 8.2 The second order sensitivity coefficients for (2.1) We here present the variational equations for the sensitivity coefficients, r ij , of second order for the model (2.1) They comprise a set of coupled neutral delay differential equations: d dt r 11 (t; p) s 1 (t; p) ....

J.W. Brewer, Kronecker Products and Matrix Calculus in System Theory, IEEE Tranc. on Circuits and Systems 25(9) (1978) 772-781.

.... Define the Toeplitz correlation matrix R = R 0;0 and the cross correlation vector r = r 0;0 , where r k;n and R k;n are defined in (22) and (23) Then it follows from (50) that E[hh H ] R Omega Gamma and E[hh H (0) r Omega Gamma, where Omega denotes the Kronecker product [26]. Thus, 49) can be equivalently expressed as G opt = Q Gamma1=2 i Q 1=2 (R Omega Gamma)Q 1=2 oe 2 p I j Gamma1 Q 1=2 (r Omega Gamma) Q Gamma1=2 i R Omega (P 1=2 GammaP 1=2 ) oe 2 p I j Gamma1 Q 1=2 (r Omega Gamma) 51) where the second equality ....

J. W. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Trans. Circ. and Syst.,

....C in the M Theta N c matrix UC = u c1 u c2 : u c Nc ] and the eigenvectors of R s in the N Theta N s matrix U S = u s1 u s2 : u s Ns ] The matrix U S defines the temporal KL transform for the signal source s(k) k = 1; 2; N . Using elementary properties of Kronecker products [13], the eigenvectors of Q s in (12) are given by u i = u c j Omega u s k (17) for i = 1; 2; N c N s , j = 1; 2; N c , k = 1; 2; N s . We can also write the stacked matrix of eigenvectors for Q s in Kronecker form as U = U C Omega U S ; 18) revealing that the ....

J. W. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Transactions on Circuits and Systems, vol. CAS-25, no. 9, pp. 772--776, September 1978.

....Y n is an aperiodic, irreducible, homogeneous Markov chain on the states S = Q N j=1 S j with transition matrix P and invariant measure given by P = P 1 Omega Delta Delta Delta Omega PN (3.4) 1 Omega Delta Delta Delta Omega N (3. 5) 15 where Omega denotes the Kronecker product [2, 18]. Define U n (i) P N j=1 A j n (i j ) Gamma c with i = i 1 ; i N ) 2 S, and let Phi( e Gamma c Gamma Psi 1 ( Omega Delta Delta Delta Omega Psi N ( Delta (3.6) where Psi j ( diag (E [exp( A j n (i) i 2 S j ) Thus, by using the identities ....

....) 2 S, and let Phi( e Gamma c Gamma Psi 1 ( Omega Delta Delta Delta Omega Psi N ( Delta (3. 6) where Psi j ( diag (E [exp( A j n (i) i 2 S j ) Thus, by using the identities (A Omega B) T = A T Omega B T and (A Omega B) C Omega D) AC) Omega (BD) cf. [2]) the matrix H( P T Phi( is given by e Gamma c (P 1 Omega Delta Delta Delta Omega PN ) T Gamma Psi 1 ( Omega Delta Delta Delta Omega Psi N ( Delta = e Gamma c Gamma P T 1 Psi 1 ( Delta Omega Delta Delta Delta Omega Gamma P T N Psi N ....

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J. W. Brewer, "Kronecker Products and Matrix Calculus in System Theory", IEEE Trans. on Circuit and Syst., 25, No. 9, pp. 772-781, 1978.

....circular Gaussian random vector x with mean m and covariance matrix R. The statistical expectation operator is written as E[ and the Euclidean norm of vector x is denoted as kxk. The symbol Omega denotes Kronecker product and vec(A) is formed by stacking the columns of matrix A into a vector [3]. 2. POLARIZATION SIGNALING SCHEME h v u h v u SOURCE COORDINATE SENSOR COORDINATE SCATTERERS Figure 1: Polarization signaling coordinate system. Consider a transmit and receive scheme as depicted in Figure 1. The transmitter utilizes 2 perpendicular antennas as the vector ....

.... be written as r(t) b [H1 Delta Delta Delta HL ] z C 2 4 m(t Gamma 1) m(t Gamma L) 3 5 z f(t) n(t) b Gamma f T (t) Omega I3 Delta vec(C) z c n(t) 5) The second equality follows from the identity vec(AXB) B T Omega A)vec(X) [3]. The coherent test statistic Z corresponds to matched filtering and MRC. Using (5) we can represent it as Z = c H Z T 0 Gamma f T (t) Omega I3 Delta H r(t) dt (6) where c is an estimate of the channel coefficient vector c. Numerous existing channel estimation techniques may be ....

J. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Trans. Circ. System, vol. CAS-25, pp. 772--781, Sept. 1978.

....methods to implement the optimal processor in the partially coherent case. It will be shown how this method naturally leads to banded and subarray beamforming techniques. Note from (3. 8) that the signal s autocorrelation function across the array can be written as the following Kronecker product [15]: P s = C Omega R s ; 6.1) where C is an M Theta M matrix which contains the decorrelation coefficients from (3.2) that is, the (i; j)th element of C is given by c ij . Similarly, given the assumptions made in Chapter 3, the optimal processing matrix in (3.6) can be expressed as A opt = ....

J. W. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Transactions on Circuits and Systems, vol. CAS-25, no. 9, pp. 772--776, September 1978.

....n( 1 B ) Delta Delta Delta ; n( M Gamma 1 B ) 8) Let r def = vec(R) and h k def = vec(H k ) be the RM Gammadimensional vectorized received signal and channel coefficients, respectively. Here, vec(X) denotes a vector formed by stacking the columns of matrix X into a vector [25]. Using the identity vec(AXB) B T Omega A)vec(X) where Omega denotes Kronecker matrix product [25] we can write r as r = b 1 B 1 h 1 z desired signal K X k=2 b k p ae k B k h k z MAI n : 9) The columns of B k def = Q k Omega A k form the basis for the received ....

....= vec(H k ) be the RM Gammadimensional vectorized received signal and channel coefficients, respectively. Here, vec(X) denotes a vector formed by stacking the columns of matrix X into a vector [25] Using the identity vec(AXB) B T Omega A)vec(X) where Omega denotes Kronecker matrix product [25], we can write r as r = b 1 B 1 h 1 z desired signal K X k=2 b k p ae k B k h k z MAI n : 9) The columns of B k def = Q k Omega A k form the basis for the received signal for user k with dimension N k = P k Gamma P Gamma k 1) Theta (L k 1) RM . Note ....

[Article contains additional citation context not shown here]

J. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Trans. Circ. System, vol. CAS-25, pp. 772--781, Sept. 1978.

....0 [u(t) t] u T (t) fi fi fi fi u(t) u (t) F 0 [u(t) t] u 1 . F 0 [u(t) t] u 2 . Delta Delta Delta . F 0 [u(t) t] u n fi fi fi fi u(t) u (t) 3. 58) Here we are using Vetter s notation [136] and [137] for the calculus of matrices (see also Brewer [18] for an overview of matrix calculus) The operation Omega represents the Kronecker product for matrices, which for any n Theta m matrix A and p Theta q matrix B is defined as A Omega B j 0 B B B B a 11 B a 12 B : a 1mB a 21 B a 22 B : a 2mB . a n1 B a n2 B : ....

Brewer, J.W., 1978: Kronecker products and matrix calculus in system theory. IEEE Trans. Circuits and Systems, 25, 772--781.

....ratio test (GLRT) 7] in which an estimate of the parameters is formed and used to obtain the optimal test statistic. The GLRT based test statistic is given by LG = max ( ff; L ( ff) Y ) 20) Using the Kronecker forms in (13) and (14) and elementary properties of the Kronecker product [12], the optimal test statistic is given by LG = max ( ff; hC Gamma1 n C s C Gamma1 n Omega R Gamma1 n R ( ff) s R Gamma1 n Y ; Y i: 21) Upon expanding the inner product in (21) we obtain LG = max ( ff; M X i=1 M X j=1 a ij hR Gamma1 n R ( ff) s R ....

J. W. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Transactions on Circuits and Systems, vol. CAS-25, no. 9, pp. 772--776, September 1978. 25

....of C in the M ThetaN c matrix UC = uc 1 uc 2 : uc N c ] and the eigenvectors of Rs in the N Theta Ns matrix US = us 1 us 2 : us Ns ] The matrix US defines the temporal KL transform for the signal source s(k) k = 1; 2; N . Using elementary properties of Kronecker products [10], the eigenvectors of Qs in (10) are given by u i = uc j Omega us k (15) for i = 1; 2; NcNs , j = 1; 2; Nc , k = 1; 2; Ns . We can also write the stacked matrix of eigenvectors for Qs in 1 In general Nc = M except in the case of a perfect signal coherence between ....

J. W. Brewer, "Kronecker products and matrix calculus in system theory," IEEE Transactions on Circuits and Systems, vol. CAS-25, no. 9, pp. 772--776, September 1978.

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Brewer, J. W., "Kronecker products and matrix calculus in system theory", IEEE Trans. Circuits and Systems, CAS25 (9), 772-781 (1978).

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