Noniterative Design of 2-Channel FIR Orthogonal Filters
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Research Article Noniterative Design of 2-Channel FIR Orthogonal Filters ´ M. Elena Dom´ınguez Jimenez TACA Research Group, ETSI Industriales, Universidad Polit´ecnica de Madrid, 28006 Madrid, Spain Received 1 January 2006; Revised 12 June 2006; Accepted 26 August 2006 Recommended by Gerald Schuller This paper addresses the problem of obtaining an explicit expression of all real FIR paraunitary filters. In this work, we present a general parameterization of 2-channel FIR orthogonal filters. Unlike other approaches which make use of a lattice structure, we show that our technique designs any orthogonal filter directly, with no need of iteration procedures. Moreover, in order to design an L-tap 2-channel paraunitary filterbank, it suffices to choose L/2 independent parameters, and introduce them in a simple expression which provides the filter coefficients directly. Some examples illustrate how this new approach can be used for designing filters with certain desired properties. Further conditions can be eventually imposed on the parameters so as to design filters for specific applications. Copyright © 2007 M. Elena Dom´ınguez Jim´enez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
INTRODUCTION
Filterbanks are widely used in all signal processing areas. In the 2-channel case, the filterbank decomposes any signal into its lowpass and highpass components; this is achieved by convolution with a lowpass filter h and a highpass filter g. When using finite impulse response (FIR) filters, the implementation is even more direct. In particular, for signal compression applications, orthogonal subband transforms are desired; hence, paraunitary filterbanks are required. Thus, some design techniques have been addressed in the literature. Moreover, the appearance of the wavelet theory gave a new insight into the filter bank theory, and also provided new methods for the design of real FIR paraunitary filters. Despite the wide number of publications, we have focused on the most well-known results, which are contained in [1–5]. We can merge these main approaches into three groups. (a) Methods based on spectral factorizations. Commonly used in wavelet theory [1, 3], they are based on the following characterization: a real L-tap filter h = (h1 , h2 , . . . , hL ) is paraunitary if and only if its transfer function H(z) = Ln=1 hn z1−n verifies H(z)2 + H(−z)2 = 2,
|z| = 1.
(1)
Hence, it suffices to find the power spectral P(z) = |H(z)|2 which satisfies P(z) + P(−z) = 2, and then factors it as P(z) = |H(z)|2 = H(z)H(z−1 ) so as to get the real filter coefficients. Thus, it is necessary to compute roots of a 2L − 1 degree polynomial P; but the main drawback is that the corresponding iterative algorithms generally become numerically unstable for long filters. (b) Lattice filters design. Instead of computing a polynomial, this approach designs the polyphase matr
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