ARMA SPECTRAL ESTIMATION BY AN ADAPTIVE IIR FILTER. by JIANDE CHEN, JOOS VANDEWALLE, and BART DE MOOR4

Transcription

1 560 R. BRU AND J. VITbRIA REFERENCES Y. H. Au-Yeung and Y. T. Poon, 3X3 orthostochastic matrices and the convexity of generalized numerical ranges, Linear Algebra Appl. 27:69-79 (1979). N. Bebiano, Some variations on the concept of c-numerical range, Portugal. Math. 43: ( ). N. Bebiano, Some analogies between the c-numerical range and a certain variation on this concept, Linear Algebra Appl. 81:47-54 (1986). C. K. Li, T. Y. Tam, and N. K. Tsing, The generalized spectral radius and spectral norm, Linear and Multilinear Algebra 16: (1984). C. K. Li, The c-spectral, c-radial and c-convex matrices, Linear and Multilinear Algebra 20:5-15 (1986). M. E. Miranda, On the trace of the product and the determinant of the sum of complex matrices with prescribed singular values, in Proceedings of Encuentro It&ma&ma1 de Algebra Lineal y Aplicaciones, Vitoria-Gasteiz, Spain, 1983, pp G. N. de Oliveira, Normal matrices (research problem), Linear and Multilinear Algebra 12: (1982). ARMA SPECTRAL ESTIMATION BY AN ADAPTIVE IIR FILTER by JIANDE CHEN, JOOS VANDEWALLE, and BART DE MOOR4 1. Zntroduction Spectral estimation is a frequently encountered problem in digital signal processing. Conventional methods of spectral estimation, such as in [l, 21, are based on fast-fourier-transform (FFT) techniques. They are generally efficient in computation and produce reasonable results for stationary signals. These FFT-based approaches, however, have several disadvantages as follows: (a) The limitation of the ability to distinguish the spectral responses of two or more signals. The frequency resolution in Hz is roughly the reciprocal of the time interval in seconds over which sampled data are available. (b) The problem of leakage in frequency domain, i.e., the energy in the main lobe of a spectral response leaks into the sidelobes, masking adjacent spectral responses that are present. This is due to the implicit windowing of the data when processing. Careful selection of windows can reduce the sidelobe leakage, but always at the expense of reduced resolution. 4Katholieke Universiteit L..euven, Lab. ESAT, Dept. E. E., Kard. Mercierlaan 94, B-3030 Leuven, Belgium; Te. (32)(16)22@331, Telex elekul.

2 CONFERENCE REPORT 561 (c) The limitation on tracking time-varying spectra due to the block processing of the FFT. In order to avoid the problems resulting from the FFT, modem spectral estimation techniques have been proposed [3, 41. The basic principles and results of many modem methods are shown in an excellent survey paper [5]. Among them autoregressive moving-average (ARMA) methods are more generalized and produce better results than the others. The ARMA model assumes that a time series v, can be modeled as the output of a p-pole and q-zero filter excited by white noise: Y,= - 5 akyn-k+ 5 cknn-k7 k=l k=o (1) where n, is white noise and c, = 1. Once the parameters of model are identified, the spectral estimate of the time series calculated: the ARMA y, can be U2At11+~;&eXp( -jwkat)12 PV(ti)= (l+cp,,a,exp( -jokht)\ (2) where a2 is the variance of the white noise and - 1/(2At) < f< l/(2 At). Many ARMA parameter estimation techniques have been proposed, which usually involve many matrix computations and iterative optimization techniques. They are normally not practical for real-time processing. In the following sections a suboptimal method with considerably less computations is proposed. 2. Spectral Estimation by an Adaptive ZIR Filter The structure of an adaptive IIR filter which was first proposed in [3] is shown in Figure 1. y, is an input time series, and f is the estimate of yn by the adaptive filter. The set a, and ck are the feedforward and feedback coefficients of the filter, respectively. They are iteratively adjusted according to the least-mean-square (LMS) algorithm to be derived in the following. Using vector notation and assuming that the time series y,, is wide-sense stationary and has zero mean, the LMS algorithm can be derived in analogy

3 562 R. BRU AND J. VITORIA Y n J+ A Yn FIG. 1. The structure of an adaptive IIR filter. with that in [6]: p q Yn = - L aknyn-k + L cknnn-k (3) k=l k=l (4) (5) where Y,: = [Yn-l' Yn-2"'" Yn-p],

4 CONFERENCE REPORT 563 Let E [.] denote the expectation operation, and replace An' C n by A, C for simplicity. Then the mean square error E[ e;] can be expressed as: where The solution for the adaptive filter coefficients that minimize the mean square error is obtained by setting the gradient vector with respect to the filter parameters equal to zero: V'A[ E(e;)] = 2RA - 2P +2QC = 0, A = R-1(P - QC), (7) V'c[E( e;)] = 2R'C - 2P' +2QA = 0, (8) The Wiener solution for the filter parameters can be obtained if all the second-order statistics are known. In most real applications, however, these statistics are not known a priori. The steepest-descent method is applied for the adaptation of the filter parameters: According to the LMS algorithm [7, 8] the above unknown matrices are replaced by instantaneous estimates of their values, i.e., R is estimated by

5 564 R. BRU AND J. VITORIA YnY,;, R' by e n e P ~ by, YnYn, P' by Ynen, and Q by Y n e Finally, ~. the LMS algorithm for the adaptation of the filter coefficients is obtained as follows: (11) (12) where J.tl' J.t2 are constants which control the convergence speed of the algorithm. These values must be chosen less than the reciprocal of the total input energy to the adaptive filter. Small values result in better performance at the expense of slower convergence. The determination of the order of the ARM A model, i.e., the values of p and q, can be found in the literature [9, 10]. Now comparing the expression of the filter output (3) with the ARMA model in Equation (1), we find that the filter output is indeed an ARMA estimation of the input time series Y n if the residual error en is a white process. The whiteness of the en has been proved in [7]. Therefore, once the algorithm converges, the power-spectrum estimate can be calculated from the filter parameters by Equation (2). 3. Results For investigating the performance of the method proposed in the previous section, computer simulations have been conducted in comparison with a conventional FFT-based periodogram method. In the first simulation the input time series Y n consists of two sinusoids in white noise. The ratio of signal (two sinusoids) to noise equals 17 db. The frequencies of these two sinusoids are 0.1 and 0.2 Hz, respectively. For the proposed algorithm the lengths of filter feedforward and feedback coefficients, p, q, are chosen to be 30 and 20. The results are shown in Figure 2, where the vertical axis represents power and horizontal frequency. Curve 1 is the spectral estimate by the periodogram method, and curve 2 is that by the proposed method. Comparing these two results, one can see that the proposed method produces higher resolution, i.e., the two peaks in curve 2 are much narrower than those in curve 1. The second simulation is designed to investigate the ability to distinguish two very close spectral lines. The input time series has the same composition as the previous simulation, but the frequencies of the two sinusoids are equal

6 CONFERENCE REPORT 565 ~. a E + e l -2.0E+01 -'\ 0E+12l1 o OOE E-12l1 2 12l0E-12l1 3.0I2lE-':ll. '\ I2lI2lE-01 FIG. 2. Curve 1: spectral estimation by periodogram; curve 2: by the proposed method. Results of spectral estimation. Real frequencies: il = 0.1 Hz, h = 0.2 Hz. to 0.1 and 0.11 Hz, respectively. Figure 3 shows the results. The spectral estimate by the periodogram method is shown as curve 1, from which one can see that two spectral responses are merged into one. The estimation by the proposed method is shown in curve 2, where two different spectral responses are well recognizable. 4. Conclusion In this paper an ARMA spectral estimation by an adaptive IIR filter is described. The LMS algorithm for the adaptation of the filter coefficients is derived. The proposed adaptive IIR filter produces an ARM A estimation of the input time series, and the whole system functions as a whitening filter. Once the algorithm converges, the spectral estimate of the input time series can be calculated from the filter coefficients. The proposed method has been compared with the FFT-based periodogram method by computer simulations. It has advantages of producing higher resolution and of resulting in higher ability to distinguish close spectral lines. It is simple and has low computational complexity, so it can be used for real-time processing. It is adaptive and so can be used for tracking the

7 566 R. BRU AND J. VIT6RIA 0.0JiE E-EL 2,00E E-01 t. WE-01 FIG. 3. Results of spectral estimation. Real frequencies: fi = 0.1 Hz, f, = 0.11 Hz. Curve 1: spectral estimation by periodogram; curve 2: by the proposed method. instantaneous frequency of the time series. It produces better results than autoregressive methods with the same complexity of computation, such as the method in [4], because it is based on the ARMA model. Since the convergence of the LMS algorithm is relatively slow, it is not efficient for application to a time series with only a few data available. The proposed method has been applied to biomedical signal processing with success [ 111. Recently a recursive least-squares algorithm has been developed. REFERENCES 1 2 R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover, New York, P. D. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograrns, ZEEE Trans. Audio Ekctroacoust. AU-15:70-73 (1967).

8 CONFERENCE REPORT B. Friedlander, A recursive maximum likelihood algorithm for ARMA line enhancement, 1EEE Trans. Acoust. Speech Signal Process. ASSP-30: (1982). L. J. Griffiths, Rapid measurement of digital instantaneous frequency, IEEE Trans. Acoust. Speech Signal Process. ASSP-23~ (1975). S. M. Kay and S. L. Marple, Jr., Spectrum analysis-a modem perspective, Proc. IEEE 69: (1981). P. L. Feintuch, An adaptive recursive LMS filter, Proc. IEEE 64: (1976). B. Widrow, Adaptive filter, in Aspects of Network and System Theory (R. E. Kalman and N. DeClaris, Eds.), Holt, Rinehart and Winston, New York, B. Widrow et al., Adaptive noise canceling: Principles and applications, Proc. IEEE 63: (1975). Y. T. Chan and J. C. Wood, A new order determination technique for ARMA process, IEEE Trans. Acoust. Speech Signd Process. ASSP32: (1984). J. J. Fuchs, ARMA order estimation via matrix perturbation theory, IEEE Trans. Automat. Control AC-32: (1987). J. Chen, J. Vandewalle, W. Sansen, G. Vantrappen, and J. Janssens, On-line spectral analysis of electrogastrographical signals, submitted to European Conference for Signal Processing, France, SOME PROPERTIES OF THE RIESZ PROJECTOR FOR REGULAR MATRIX POLYNOMIALS by N. CHONDROS and J. MAROULAS5 1. Preliminmies L&-t L(X) = A,$ + A,X A,, (1) be an m X m matrix polynomial such that the leading coefficient A, is not necessarily the identity or even invertible. When the spectrum o(l)= {h~c:detl(x)=o} of L(X) is not the empty set, L(X) is said to be regular, and when L(0) = I, 5Department of Mathematics, National Technical University, Zografou Campus, Athens, Greece.