Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain

A maximally decimated filter bank system (with possibly unequal decimation ratios in the subbands) can be regarded as a generalization of the short-time Fourier transformer. In fact, it is known that such a “filter bank transformer” is closely related to the wavelet transformation. A natural question that arises when we conceptually pass from the traditional Fourier transformer to the filter bank transformer is: what happens to the convolution theorem, i.e., is there an analog of the convolution theorem in the world of “filter bank transforms”? In this paper we address the question first for uniform decimation and then generalize it to the nonuniform case. The result takes a particularly simple and useful form for paraunitary or orthonormal filter banks. It shows how we can convolve two signals x ( n ) and g ( n ) by directly convolving the subband signals of a paraunitary filter bank and adding the results. The advantage of the method is that we can quantize in the subbands based on the signal variance and other perceptual considerations, as in traditional subband coding. As a result, for a fixed bit rate, the result of convolution is much more accurate than direct convolution. That is, we obtain a coding gain over direct convolution. We will derive expressions for optimal bit allocation and optimal coding gain for such paraunitary convolvers. As a special case, if we take one of the two signals to be the delta function (e.g., g ( n ) = 6(n)) , we can recover the well-known bit allocation and coding gain formulas of traditional subband coding. The derivations also show that these formulas are valid regardless of the filter quality, as long as orthonormality is not violated. A special case similar to orthogonal transform coding is also considered and good convolutional coding gains for speech are demonstrated, with the use of the DCT matrix. Finally, the result is extended to the case of nonuniform biorthonormal filter banks; with the incorporation of an additional trick, the convolution theorems in this case become as simple as for the orthonormal case.