Journal ArticleAbstract  This paper is concerned with the blind identification of bilinear systems excited by higher-order white noise. Unlike prior work that restricted the bilinear system model to simple forms and required the excitation to be Gaussian distributed, the results of this paper are applicable to a more general class of bilinear systems and for the case when the excitation is non-Gaussian. We describe an estimation procedure for the computation of the system parameters using output cumulants of order less than four

Kalouptsidis, N.

Mathews, V. John

English

The University of Utah: J. Willard Marriott Digital Library

BLIND IDENTIFICATION OF BILINEAR SYSTEMS N. K alouptsidis P. K oukoulas V. J. Mathews Department of Informatics University of Athens Panepistimiopolis 15771 Athens, GREECE DSP Lab., Intracom S.A 19.5 Km Markopoulou Ave. 190 02 Peania Athens, GREECE Electrical Eng. Dept. University of Utah Salt Lake City, UT 84112 USA ABSTRACT This paper is concerned with the blind identification of bilinear systems excited by higher-order white noise. Un-like prior work that restricted the bilinear system model to simple forms and required the excitation to be Gaussian dis-tributed, the results of this paper are applicable to a more general class of bilinear systems and for the case when the excitation is non-Gaussian. We describe an estimation pro-cedure for the computation of the system parameters using output cumulants of order less than four. 1. INTRODUCTION We consider bilinear systems of the form Ka Kb yen) L a(i)y(n - i) + L b(i)u(n - i) i=1 i=O Kcu Key + L L c(i,j)y(n - i)u(n - j) + v(n) (1) j=1 i=j where yen) is the output of the system, u(n) its input and v(n) the measurement noise. We assume that b(O) = 1, c(Keu, Keu) =J:. 0, Keu ~ Kb, and C, the matrix of coeffi-cients c(i,j), is lower triangular. We assume also that the measurement noise v(n) belongs to a Gaussian white pro-cess with zero mean value and is independent of the input, and that the input signal u(n) is a higher-order white pro-cess that has zero mean value and is not necessarily Gaus-sian distributed. The cumulants of u(n) are given by cum[u(n), u(n -lI),' .. , u(n - lk-d] = /'ko(h,···, lk-d (2) where 0(l1, ... ,lk-d is the (k -1) dimensional unit impulse signal and /'k denotes the signal intensity of order k. The coefficients a(i), b(i) and c(i,j) are such that yen) is a sta-tionary process. Sufficient conditions for the stationarity of bilinear processes are derived in [1, 4, 5]. The blind iden-tification problem addressed in this paper is formulated as follows. Given the system orders K a , Kb, Keu and Key and the output statistics, determine the system parameters a(i), 0-7803-7041-41011$10.00 ©2001 IEEE 3501 b(i) and c(i,j), as well as the higher-order signal intensities /,2, /'3, /,4 and /'5 associated with the input signal u(n). Closed form expressions that relate measurable statis-tics of the output signal to the parameters that must be es-timated are available only for a very restricted class of non-linear system models [6, 7]. Consequently, the most com-mon approach to estimating the parameters of the model is to resort to some form of numerical search algorithm that operates in an iterative manner [8]. This paper describes a set of equations that relate parameters of the system in (1) to cumulants of the output signal up to order four, and then provides a direct method for estimating the parame-ters based on these relationships. The rest of this paper is organized as follows. The next section describes the blind estimation procedure. The es-timation is performed in a sequential manner. The coeffi-cients a(i) are estimated first, and the estimated parame-ter values are employed to find the remaining parameters. Because of space limitations, we only describe the method-ology employed in our approach. Complete derivations will be presented in [3]. The concluding remarks are made in Section 3. 2. BLIND ESTIMATION OF THE PARAMETERS 2.1. Estimation of a(i) The first stage of the blind estimation algorithm involves the computation of the a(i) parameters through the out-put covariance c~2)(I) = cum[y(n), yen -I)]. It is relatively straightforward to show that the covariance function of the output signal satisfies the following results for sufficiently large values of the lag: Proposition 1. Let L > K = max(Kb, Keu). Then, the output covariance sequence satisfies the autoregressive model Ka c~2)(L) = L a(i)c~2)(L - i). (3) i=l Proposition 1 states that the output covariance sequence behaves in a manner that is identical to the covariance func-tion of an autoregressive signal for sufficiently large values of the lag I. This property enables the computation of the a( i) parameters with the aid of the second-order statistics and a linear system Toeplitz solver such as a variant of the Levinson algorithm [2]. Estimation of the c( i, j) coefficients relies on the third and fourth order cumulant sequences (4) and The behavior of the cumulants of order 3 or 4 when all arguments 11, 12 and b are large is easy to predict. Similar to the results in Proposition 1, we can show that C~3) (h, 12 ) and C~4) (h, 12, b) satisfy the same autoregressive model: Ka c~3)(Ll,L2) = La(i)c~3)(Ll-i,L2 -i)j i=l (6) and Ka c~4)(Ll,L2,L3) = La(i)c~4)(Ll-i,L2 -i,L3 -i)j i=l 2.2. Estimation of the Last Column of C To estimate the remaining parameters b( i) and c( i, j) re-quires the derivation of relationships that relate the cumu-lants for smaller lag values through these coefficients. To derive such relationships in a manageable fashion we hold one of the arguments at a large value and analyze the cumu-lants as the remaining lag values fall in the range 0 ::; 1 ::; K. The following result allows the estimation of the last column of the coefficient matrix C. Proposition 2. Let Ka D2(1) = - L a(i)c~2)(1 - i) a(O) = -1, (8) i=O Ka D3(l1, 12) = - L a(i)c~3)(ll - i, 12 - i) (9) i=O and Ka D4(ll, 12 , 13 ) = - L a(i)c~4)(ll - i, 12 - i, b - i). (10) i=O Then Key D3(K,L) =,2 L c(i,Kcu)C~2)(L - i) (11) i=Kcu 3502 and Key 2,2 L ·c(i, Keu)C~3) (K - i, L - i) i=Kcu + ,3 D 3 (K, L). ,2 (12) The quantities Drn (-) can be computed from the out-put cumulants since the parameters a(i) have already been computed. Application of (11) for Key - Kcu + 1 different values of L enables us to determine the last column of C scaled by the input variance ,2. Furthermore, evaluation of (12) at a value of L such that D 3 (K, L) f. 0, enables us to determine the ratio ,3/,2. 2.3. Estimation of b(K) We analyze the statistics of the output signal sequentially, starting with lag values of K, and then descending to smaller lag values. The following expressions provide useful infor-mation for lag values equal to K. Key D2(K) = ,2b(K) + fh2 L c(i, K) + ,3c(K, K) (13) i=K Key ,3b(K) + y,3 L c(i, K) + ,4c(K, K) i=K Key +2,2 L c(i, K)C~2)(K - i) (14) i=K Key ,4b(K) +y,4 L c(i, K) + ,5c(K, K) + i=K Key +3,2 L c(i, K)C~3\K - i, K - i) i=K Key +3,3 L c(i, K)C~2) (K - i) (15) i=K In the above expressions y denotes the mean value of yen). Equation (13) yields '2b(K). Equations (14) and (15) pro-vide ,4h2 and '5h2. 2.4. Estimation of Other Parameters The remaining calculations require the definition of the fol-lowing parameters: K S3(m, L) = - L a(l- m)D3(1, L), l=m (16) and K S4(m, K, L) = - L a(l-m)D4(1, K, L), l::;m::;K-l. l=rn (17) The estimation of the remaining columns of C relies on expressions of the following form: Key S4(K -1,K, L) + a(I)--y2 L c(i, K)C~3)(K - i, L - i) i==K Key -'YZ I: c(i, K)C~3) (K - 1 - i, L - i) = Key 2: [D2(K)c~3)(1 - i, L - K + 1 - i) + (D3(K,K)-'Y2 %c(n'K)c~2)(K-n») X xc~2)(L-K+l-i)+ Key "(2 L c(n, K) X n=K X c14 )(n - K + 1 - i, 1 - i, L - K + 1 - i) + Key +'Y3 2: c(n, K)c~3)(n - K + 1 - i, L - K + 1 - i) Key +'Y4 2: c(n, K)c~2)(L - n)J(i - 1) + n:;:.:K Key 'Y2 L c(n, K)c~2) (L - n)c~2) (1 - i) + n=K Key 1113 L c(n,K)c~2)(L - n) + n==K Key +13 2: c(n, K)c~3) (K - n, L - n)J(i - 1) + Key 1 Y'Y2 ~ c(n, K)C~3)(K - n, L - n) c(i, 1) + +12c(K -1, f( - l)c~3)(1, L - K + 1) + Key L ("(2 c(i, f( -1) + b(1)--y2c(i, K» c~3)(K - i, L - i) + i=K The above equation evaluated for different choices of L leads to a linear system of equations with the following groups of unknowns: 1. The first column of C, c(i, 1) 2. The term 'Y2c(K - 1, K - 1) 3. A linear combination of the last two columns of C: 'Y2c(i, K - 1) + b(1h2c(i, K). 4. Thetermb(1)~+"(2c(I,1). We estimate the four unknown groups of parameters by solving the system of linear equations resulting from (18). In group 4 we have one linear equation with unknowns bel) and 12, since 'Y3/'Y2 and c(l, 1) are already available from prior calculations. A second equation for b(l) and 12 is obtained using an expression similar to (18) for S3(K -1, K) and given by Key S3(K - 1, K) + a(lh2 I: c(i, K)C~2)(K - i) -Key key 1'22: c(i, K) L c(n, l)c;3)(i - K + 1 - n, 1 - n) -i=K n=l Key key "(3 L c(i, K) L c(n, 1)c~2)(i - K + 1 - n) -i=K n=l Key "(22: c(i, K)C~2) (K - 1 - i) -i=K Key i=l Key 14c(K, Klfj L c(i, 1) -12c(K -1, K - l)c~2)(I) -i=l Key - L ("(2 c(i, K - 1) + b(lh2c(i, K» c~2)(K - i) i=K Key -c(l, 1)':3 L c(i, K)C~2) (K - i) -i=K Key key -fj L c(i, 1h2 2: c(n, K)c~2)(K - n) i=l n=k (Key ) -c(l,lh4 b(K) + 11 t; c(i, K) -Key ( Key ) - ~ c(i, 1hz b(K) + fj ~ c(n, Kl C~2) (1 - i) -(Key ) Key Y13 b(K) + Y t; c(i, K) ~ c(n, 1) = bel) ['Y4C(K, K) + 'Y3 (b(K) + 17 ~ c(i, K») 1 + 12 [3'Y3C(K, K)c(l, 1) + c(l, 1h2 (bCK) + fj ~ c(i, K») 1 (19) The determinant of the above system is c(l, 1 )c(K, K) (,,(4 _ 3;:) 3503 Provided that c(l, 1) =I- 0, c(K, K) =I- ° and 1'4 =I- 3"t5h2, b(l) and 1'2 are uniquely determined. It is shown in [3] that the above assumption can be rei axed if we use cumulants of order 5. Having determined bel) and 1'2 we return to groups 2 and 3 and compute the K - 1 columns of C. Finally b(K - 1) is obtained from 5 2(K - 1) as b(K - 1)'2 = 52(K - 1) -1'3c(K - 1, K - 1) - Y1'2c(K - 1, K - 1) Key -y L (r2c(i, K - 1) + b(1),2c(i, K)) i=K Key -D2(K)fj L c(i, 1) i=l Key key -1'22::: c(i, K) 2::: c(n, 1)c~2)(i - K + 1 - n) i=K n=! -c(l,l) [1'3 (b(K)+Y~C(i'K))+1'4C(K'K)l -b(l) (1'2b(K) + 1'3c(K, K)) -1'2/2c(K, K)c(l, 1).(20) In summary, (18)-(20) lead to the estimation of the first column of C, c(i, 1), the (K -l)th column of C, c(i, K -1), b( 1), b( K - 1) and 1'2. The remaining parameters can be de-termined in a similar manner through successive evaluation of 54(1, K, L), 5 3(1, K) and 5 2(1) for 1= K - 2, K - 3,···. 3. CONCLUDING REMARKS This paper has dealt with the blind identification of bilinear systems from measurements of the output signals. The pa-rameters are determined via a sequence of linear systems in-volving cumulants up to order four. Unlike prior work that restricted the bilinear system model to simple forms and re-quired the excitation to be Gaussian distributed [6, 7], the results of this paper are applicable to a more general class of bilinear systems and for the case when the excitation is non-Gaussian. 4. REFERENCES [1] T. Bose and M. Q. Chen, "BIBO Stability of the Dis-crete Bilinear Systems," Digital Signal Processing: A Review Journal, Vol. 5, No.3, 1995. [2] N. Kalouptsidis, "Signal Processing Systems: Theory and Design", John Wiley, 1997. [3] N. Kalouptsidis, P. Koukoulas and V. J. Mathews, "Blind Identification of Bilinear Input Output Sys-terns", under preparation. [4] S. Kotsios and N. Kalouptsidis, "BIBO Stability Crite-ria for a Certain Class of Discrete Nonlinear Systems," Int. J. Control, vol. 58, No.3, pp. 707-730, September 1993. [5] J. Lee and V. J. Mathews, "A Stability Theorem for Bilinear Systems," IEEE Trans. on Signal Processing, vol. SP-41, No.7, pp. 1871-1873, July 1994. 3504 [6] V. J. Mathews and T. K. Moon, "Parameter Estima-tion for a Bilinear Time Series Model," Proc. IEEE Int. ConI Acoust. Speech and Signal Proc., pp. 3513-3516, Toronto, May 1991. [7] S. A. Sesay and T. Subba Rao, "Difference Equations for Higher Order Moments and Cumulants for Bilin-ear Time Series Model BP(p,O,p,I)," J. Time Series Analysis, vol. 12, No.2, pp. 159-177, February 1991. [8] T. Subba Rao, "On the Theory of Bilinear Time Series Models," J. Royal Soc. Ser. B, vol. 43, pp. 244-255, 1981. 

Blind identification of bilinear systems

https://collections.lib.utah.edu/dl_files/fc/4f/fc4f817b74154cda3ed2e73e18b54e9d8a2f60df.pdf

Blind identification of bilinear systems

Abstract

Similar works

Full text

Available Versions

The University of Utah: J. Willard Marriott Digital Library