Abstract
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.
Highlights
Filterbanks are widely used in all signal processing areas
The appearance of the wavelet theory gave a new insight into the filter bank theory, and provided new methods for the design of real finite impulse response (FIR) paraunitary filters
We have presented a novel characterization of real paraunitary FIR filterbanks
Summary
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. Any paraunitary real L-tap FIR causal filter is obtained through iteration, because its polyphase matrix can be factorized as. In order to design a paraunitary filter of length L, we need a total amount of L/2 parameters in [−1, 1], and L/2 signs. This algorithm behaves well numerically, but it is difficult to apply when imposing extra desired properties upon the filter. The paper is organized as follows: in Section 2 we derive the explicit expression of all real 2-channel FIR orthogonal filterbanks. By reversing the order of its rows and columns we obtain its transpose
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