Simplified Design of Low-Delay Oversampled NPR GDFT Filterbanks
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Simplified Design of Low-Delay Oversampled NPR GDFT Filterbanks ¨ 1 Bogdan Dumitrescu,1, 2 Robert Bregovi´c,1 and Tapio Saramaki 1 Institute
of Signal Processing, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland of Automatic Control and Computers, “Politehnica” University of Bucharest, 060032 Bucharest, Romania
2 Department
Received 01 September 2004; Revised 17 April 2005; Accepted 18 April 2005 We propose an efficient algorithm for designing the prototype filters of oversampled, near-perfect reconstruction (NPR), GDFT modulated filterbanks (FB) with arbitrary delay. We describe simplified conditions for imposing NPR, posed on the frequency response of the distortion transfer function and on the stopband attenuation of the prototype filters. Given the analysis prototype, we show that the minimization of the stopband energy of the synthesis prototype, subject to the simplified NPR constraints, can be expressed as a convex optimization problem. Our algorithm consists of initialization with the prototype of a near-orthogonal FB— which can also be designed via convex optimization—and then successive optimization of the synthesis and analysis prototypes. We give design examples, discuss the properties of the obtained FBs, and present synthetic echo control experiments. The presented results show that, for a given delay, our algorithm produces FBs with significantly better properties than the near-orthogonal FBs. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
Recent studies [1–4] on subband adaptive filtering have shown that good performances and flexibility are obtained by using oversampled nearly prefect reconstruction filterbanks. Moreover, a low implementation complexity is ensured by uniform filterbanks whose filters are obtained by (complex) modulation from a single prototype; the filters have complex coefficients, but the prototype is real. In this paper, we give an efficient design algorithm for such filterbanks. A general filterbank (FB) structure is presented in Figure 1. The FB is oversampled when the down-sampling factor R is smaller than the number of channels M. The subband signals xk [n], k = 0 : M − 1, are processed by adaptive filtering or other type of algorithms; however, since here we are interested in a general design method, we ignore the subband processing and assume that yk [n] = xk [n]. We assume that all analysis filters Hk (z) and synthesis filters Fk (z), k = 0 : M − 1, have complex coefficients. Ideally, the passband of a filter has a width of 2π/M; more precisely, the passband of Hk (z) or Fk (z) covers the interval [2kπ/M, 2(k + 1)π/M], as shown in Figure 2(b). If the input signal is real, then we can take M even, and only the first M/2 channels of the FB are necessary. In this case, the FB processes the frequencies between 0 and π; those from −π to 0 are discarded; no information is lost, but the subband signals are complex. For such an FB, the output y[n] is twice the real part of the sum of synthesis bank outputs.
(Real subband s
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