SYSTEM RECOSTRUCTIO FROM SELECTED HOS REGIOS Haralambos Pozidis and Athina P. Petropulu Electrical and Computer Engineering Department Drexel University, Philadelphia, PA 94 Tel. (25) 895-2358 Fax. (25) 895-695 pozidis@cbis.ece.drexel.edu, athina@artemis.ece.drexel.edu Abstract System reconstruction from arbitrarily selected slices of the n-th order output spectrum is considered. We establish that unique identication of the impulse response of a system can be performed, up to a scalar and a circular shift, based on any two horizontal slices of the discretized n-th order output spectrum, (n 3), as long as the distance between the slices and the grid size satisfy a simple condition. For the special case of real systems, one slice suces for reconstruction. The ability to select the slices to be used for reconstruction enables one to avoid regions of the n-th order spectrum where the estimation variance is high, or where the ideal bispectrum is expected to be zero, as in the case of bandlimited systems. We propose a mechanism for selecting slices that result in improved system estimates. We also demonstrate via simulations the superiority, in terms of estimation bias and variance, of the proposed method over existing approaches in the case of bandlimited systems.. Introduction System reconstruction has been a very active eld of research in the recent years. Higher-order spectra (HOS) have been applied successfully to the problem of system reconstruction, mainly because of their ability to preserve the true system phase, and their robustness to additive Gaussian noise of unknown covariance. System reconstruction methods can be divided into two main categories: parametric and nonparametric. Parametric methods t a specic model to the output observations and use the output statistics to identify the model parameters. Sensitivity to model order mismatch is their main disadvantage. This work was supported by SF under grant MIP-9553227. on-parametric methods reconstruct the system by recovering its Fourier phase and magnitude. In this paper we consider non-parametric system reconstruction methods. on-parametric methods can be divided into two main sub-categories, those that utilize the whole bispectrum information ([4], [], [6], [8], [5]), and those that require bispectrum slices ([3], [2]). Methods that use xed bispectrum slices cannot be applied for the reconstruction of bandlimited systems, since, depending on the system, the ideal bispectrum along these slices can be zero. Moreover, in the presence of noise and nite data records, bispectrum estimates along xed slices can exhibit high estimation variance, and since single slices are used, there is no averaging mechanism to reduce estimation errors. Several bispectrum slices within the bispectrum principal domain have been used in a method proposed in [], and averaging was performed over the frequency response sample estimates. However, as the sample number decreases, the number of independent realizations of the corresponding frequency response sample over which averaging can be performed decreases too. As a result, low frequency samples of the frequency response exhibit considerable estimation variance in the presence of noise or data of nite lengths. Errors in low frequency samples can propagate to the remaining samples, since the method is iterative in nature. However, the method performs well in the case of wideband systems. The methods in [4] and [8] can also be viewed as methods that use several bispectrum slices. According to these methods, the non-redundant bispectrum is used to form a linear system of equations, which can be solved for the unknown frequency response samples. Although the system of equations is overdetermined and a solution could be obtained even if some slices were discarded, there is no mechanism for selecting the \best" slices to use, nor is there any guarantee that certain bispectrum regions can be avoided.
In this paper, we consider the possibility of system reconstruction from arbitrarily selected higher-order spectra slices of the system output. We rst establish that unique identication of the impulse response of an arbitrary system can be performed, up to a scalar and a circular shift, based on any two horizontal slices of the output discretized n-th order spectrum, of any order n 3, of the system, as long as the distance between the slices and the grid size satisfy a simple condition. When the system is real, one slice suces for system reconstruction. We then propose a method for system reconstruction based on a pair of selected HOS slices of the system output. In [7] it was shown that the obtained system estimates are asymptotically unbiased and consistent. We propose a mechanism to select the slices that will result in improved system estimates. We demonstrate via simulation examples the superiority of the proposed method over existing approaches, for the case of bandlimited systems. 2. System reconstruction from any pair of horizontal slices of the HOS of the system output We dene a horizontal slice of the n-th order spectrum Cn(!; x!2;... ;! n ) of a signal x(n) as the onedimensional sequence that arises when we x all indices!2;... ;! n to certain real numbers, and allow! to take all possible values in (?; ]. Throughout the paper, the term \slice" will be used instead of \horizontal slice". The distance between two slices C x n(!; ;... ; n?) and C x n(!; ;... ; n?) is dened as the l -norm of the vector?, i.e., k? k = P n? i= j i? i j, where = [;... ; n?] T and = [;... ; n?] T. In this paper we consider reconstruction from thirdorder spectra. A generalization of the results to the n-th order spectra case can be found in [7]. Consider a stationary process x(n) given by: x(n) = e(n) h(n) + w(n); () where e(n) is an i.i.d. non-gaussian process with zero mean and nite n-th order cumulant n e 6=, for n 2; w(n) is a stationary zero-mean Gaussian process of unknown covariance which is assumed independent of e(n); h(n) is the impulse response of an exponentially stable, generally mixed-phase, complex LTI system which has to be estimated from the output x(n). It is initially assumed that h(n) does not have zeros on the unit circle, however this assumption is relaxed later. The frequency-domain bispectrum of x(n) is given by C x 3 (!;!2) = e 3 H(! )H(!2)H(?!?!2); (2) with H(!) denoting the frequency response of the system. The following proposition holds: Proposition For the process x(n) described by eq. (), h(n) is always identiable, within a (complex) constant and a circular shift, from any two slices of the discretized output bispectrum, i.e, C3 x(2 k; 2 l ) and C3 x ( 2 k; 2 l 2), k = ;... ;?, if and only if and r = jl?l2j are relatively prime integers. If h(n) is real, then it is identiable, within a constant and a circular shift, based on a single slice of the discretized output bispectrum, i.e., C3 x(2 k; 2 l), if and only if and r = 2l are coprime integers. The proof of this proposition can be found in [7]. 3. The reconstruction method By evaluating (2) at discrete frequencies! = 2 k, k 2 [;...;? ], we obtain the discrete bispectrum of x(n), i.e., C3 x (k; l) = 3 e H(k)H(l)H(?k? l): (3) Let us consider two slices of the discrete bispectrum at distance r, i.e. slices (: :l) and (:; l + r), with l arbitrarily chosen, i.e., C3 x(k; l) = e 3H(k)H(l)H(?k? l) C3 x (k; l + r) = 3 e H(k)H(l + r)h(?k? l? r)(4) Taking natural logarithms of both sides in (4) and subtracting we get: log C3 x(k; l)? log Cx 3 (k; l + r) = H(l) log H(l + r) + log H(?k? l) H(?k? l? r) Substituting m =?k? l in (5) we get: (5) log H(m) = log H(m? r) + log H(l + r)? log H(l) + log C x 3 (?m? l; l)? log C x 3 (?m? l; l + r); (6) where it can be shown that: log H(l + r)? log H(l) =? X [log C 3 x (k; l + r)? log C3 x (k; l)] : (7) k=
From (6) and (7) we would be able to calculate the frequency response of the system recursively, provided that the initial conditions flog H(k); k = ; ;...; r? g were known a priori. However, a solution can still be obtained, without the need of any a priori information. Let h l = [log H();... ; log H(? )] T be the (? ) vector of the unknown samples of the logarithm of the frequency response of the system. By substituting s = m? r in (6), and letting s take the values ; ;...;? 2, we form the linear system of equations: Ah l = c; (8) where c is a (? ) vector of bispectrum values along the slices (:; l) and (:; l + r), with c i = log C x 3 (?i?r? l; l)?log Cx 3 (?i?r? l; l + r) + c l;r (9) where i = ; ;...;? 2, and c l;r = log H(l + r)? log H(l): () Matrix A is a sparse matrix with special structure; it is bidiagonal if r = and tridiagonal otherwise, and contains 's and (?)'s. It can be proved, [7], that matrix A is nonsingular if and only if and r are relatively prime integers and that if A is nonsingular, then det A =. In this method, the logarithm of a pair of bispectrum slices is used to recover the logarithm of the frequency response of a system. Although the phase of the bispectrum appears implicitly in the expressions, only the principal argument of the bispectrum is actually needed. To see that, let ~ (k; l) denote the principal argument of the phase sample (k; l). Then it holds: (k; l) = ~ (k; l) + 2I(k; l); () where I(k; l) is an integer function of k; l. The solution of (8), when c is computed based on ~ (k; l), becomes: ~h l =A? c + j2i (l) + j 2 I 2(l) =h l + adj(a) j2i (l) + j 2 det(a) I 2(l) (2) where I (l); I 2 (l) are vectors of integer values. However, since det(a) =, the reconstructed frequency response of the system will only dier from the true one by a complex exponential factor of the form e jm 2 I(l), where I(l) is an integer function of l, which corresponds to a circular shift of the impulse response of the system. Therefore, h(n) is reconstructed within a multiplicative scalar and a circular shift, as stated previously. Throughout the derivation of the method it has been assumed that the system h(n) does not have zeros on the unit circle, since then H(k) = for some k. However, if C3 x (k; l) = for some (k; l), then we can change the spacing between samples, or equivalently re-estimate the bispectrum in a dierent grid of frequency points to surpass that problem, as suggested in [] and [8]. The procedure outlined in this Section is valid for any pair of bispectrum slices, subject to the condition given in Proposition. Therefore, by using different pairs of slices, we can average the reconstructed systems in the time-domain (after scaling and shifting them appropriately), thus reducing the eects of noise and nite data lengths in the estimation of cumulants. 4. Simulations In this Section we demonstrate the performance of the proposed method and compare it to the methods of [] (BLW) and [8] (RG) for the reconstruction of bandlimited systems. The BLW and RG methods use several bispectrum slices within the bispectrum principal domain and are applicable to the reconstruction of bandlimited systems. Since the RG method was developed for the reconstruction of real systems, comparisons are performed for a real system. We implemented all methods using the bispectrum of the system output for simplicity. We generated the input process e(n) as an i.i.d. sequence with zero mean and nonzero skewness, and added zero-mean, Gaussian noise to the output of the system, at various signalto-noise ratios (SRs). The bispectrum of the output signal was estimated using the indirect method. Data of length L were segmented into non-overlapping records of length 256 symbols. The third-order cumulants of each record were estimated in a square grid of (2M? ) (2M? ) lags, and then averaged over all L 256 records. Finally, a two-dimensional FFT was applied on the averaged cumulants to obtain the discrete bispectrum of the output process. Although the system considered was real-valued, we used two slices for the reconstruction procedure instead of only one slice (see Proposition ), since an FFT size of = 64, a power of 2, was used, in order to speed up computations. The reconstruction procedure was repeated using several dierent pairs of slices, and the estimated systems were averaged in the time-domain, in order to reduce noise eects. The comparison method BLW determines the frequency response of the unknown system except for a linear-phase term, e jk(), where (!) is the phase response. Although this is not a drawback, it creates representation problems for the time-domain recovered signal h(n) if () is not an integer, since it then cor-
responds to a non-integer time-delay. To circumvent that problem, we supplied the correct value of () to the BLW algorithm. We generated a bandpass system as: regions in the reconstruction procedure is responsible for poor performance. On the other hand, selection of regions with higher signal information only, as in the proposed method, leads to better results. h(t) = :77 jtj cos(2:49 t)+:8(:65) jtj sin(2:38 t+ 5 ) where?4:5 t 3 seconds. A discrete-time signal h(n) of length 6 symbols was generated by sampling h(t) every.5 seconds. Since the true length of the signal h(n) is assumed unknown, we used M = 2 in the computation of third-order cumulants. Although any pair of slices, as soon as it satises the identiability conditions, should provide identical results, it was found in practice that some slices provide better results than others. Let us dene the term \frequency content" as Z? j C3 x (!; l) j d! Based on our experience with simulations, we found that the use of bispectrum slices with low frequency content resulted in poor performance; the opposite was also found to be true. In order to select the slices for better reconstruction, we run dierent simulations and computed the frequency content of each slice at each run. The average frequency content over all runs, is shown in Fig. where the shaded area indicates standard deviation. It can be seen that slices -8 exhibit a consistently higher frequency content than all others. Then we run simulations of the proposed method, using slices 3-4 (corresponding to \good" slices) and 23-24 (corresponding to \bad" slices), and the results are shown in Fig. 2 for L = 248 output samples and SR equal to and db. Clearly, the use of slices 3-4 produces superior results, both in bias and variance, compared to those of slices 23-24. ext, we run simulations of the proposed method, using averaging over slices -7 (in pairs of consecutive slices), and of methods BLW and RG. The results are shown in Figs. 3 and 4 for SR of and db respectively. In both gures, graphs, and correspond to L = 24, while, (e) and (f) to L = 248 output samples used. The graphs on the left, center and right correspond to the proposed, BLW and RG methods respectively. It can be seen that method RG performs better than BLW both in terms of bias and variance, but both methods are outperformed by the proposed one. This can be attributed to the fact that the actual system h(n) is bandpass, therefore its output bispectrum contains regions of low magnitude, where the useful signal information is signicantly corrupted by noise. The inclusion of such 25 2 5 5 FREQUECY COTET 5 5 2 25 3 SLICE IDEX Figure. Frequency content for slices - 33 of the output bispectrum of the system. Circles and solid line represent the average over simulations, while shaded area indicates sample standard deviation. 5. Conclusions A non-parametric method for system reconstruction based on HOS slices was presented. It was shown that an arbitrary system can be uniquely identied based on any two slices of the output discretized n-th order spectrum, with n 3, as long as the distance between the slices and the grid size satisfy a simple condition. By using the logarithm of two n-th order spectrum slices, the logarithm of the frequency response of the system was obtained as the solution of a linear system of equations. A mechanism to select slices that result in improved system estimates was proposed. Simulation examples conrmed the superiority of the proposed method as compared to existing schemes, when tested on bandlimited systems. The exibility of the proposed method in selecting the slices to be used for reconstruction could allow one to avoid regions of the n-th order spectrum where the useful system information is limited or distorted by noise.
.5 SR=Inf, L=248.5.5.5 2 3.5 SR=, L=248.5.5 SLICES : 34.5 2 3.5.5.5 SR=Inf, L=248.5.5.5 2 3.5 SR=, L=248.5.5 SLICES : 23 24.5 2 3 Figure 2. Results of the proposed method using slices 3-4 and 23-24 of the output bispectrum with SR db ( and ), and SR db ( and ), and L = 248 samples. Actual system is in solid lines, the average over estimates in dash-dotted lines, and shaded area indicates standard deviation..5 SR=Inf, L=24.5.5 2 3.5 SR=Inf, L=248.5 Proposed Method.5 2 3.5 SR=Inf, L=24.5.5.5 2 3.5 SR=Inf, L=248.5.5 BLW Method (e).5 2 3.5 SR=Inf, L=24.5.5.5 2 3.5 SR=Inf, L=248.5.5 RG Method (f).5 2 3 Figure 3. Comparison of the proposed, BLW and RG methods for L = 24 (, and ), and L = 248 output samples (, (e) and (f)), and SR= db. Actual system is in solid lines, the average over estimates in dash-dotted lines, and shaded area indicates standard deviation..5 SR=, L=24.5.5.5 2 3.5 SR=, L=248.5.5 Proposed Method.5 2 3.5 SR=, L=24.5.5.5 2 3.5 SR=, L=248.5.5 BLW Method (e).5 2 3.5 SR=, L=24.5.5.5 2 3.5 SR=, L=248.5.5 RG Method (f).5 2 3 Figure 4. Comparison of the proposed, BLW and RG methods for L = 24 (, and ), and L = 248 output samples (, (e) and (f)), and SR= db. Actual system is in solid lines, the average over estimates in dash-dotted lines, and shaded area indicates standard deviation. References [] H. Bartelt, A.W. Lohmann and B. Wirnitzer, \Phase and amplitude recovery from bispectra", Applied Optics, vol. 23, no. 8, pp. 32-329, Sept. 984. [2] S.A. Dianat and M.R. Raghuveer, \Fast Algorithms for Phase and Magnitude Reconstruction from Bispectra", Opt. Eng., vol. 29, pp. 54-52, May 99. [3] K.S. Lii and M. Rosenblatt, \Deconvolution and Estimation of Transfer Function Phase and Coecients for ongaussian Linear Processes", The Annals of Statistics, vol., pp. 95-28, 982. [4] T. Matsuoka and T.J. Ulrych, \Phase Estimation Using the Bispectrum", Proc. IEEE, vol. 72, pp. 43-4, Oct. 984. [5] R. Pan and C.L. ikias, \The Complex Cepstrum of Higher Order Cumulants and onminimum Phase System Identication", IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 86-25, Feb. 988. [6] A.P. Petropulu and C.L. ikias, \Signal Reconstruction from the Phase of the Bispectrum", IEEE Trans. Acoust., Speech, Signal Processing, vol. 4, pp. 6-6, March 992. [7] H. Pozidis, and A.P. Petropulu, \System Reconstruction Based on Selected Regions of the Discretized HOS", IEEE Trans. Sig. Proc., submitted in 997. [8] M. Rangoussi and G.B. Giannakis, \FIR Modeling Using Log-Bispectra: Weighted Least-Squares Algorithms and Performance Analysis", IEEE Trans. Circ. and Sys., vol. 38, no. 3, pp. 28-296, Mar. 99.