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In t his case, both the analysis and synthesis perfect reconstruction filters have the same length N. Assuming a stationary input signal x(n), the error component in t he reconstructed signal i:(n) is cyclostationary with period M, due to the presence of the interpolators in t he synthesis stage. T herefore, the objective is to minimize t he average power I MSE = M trace(E[ee']), (4) x- x is the error betwe€n the reconstructed signal vector (2) and where e = an appropriately defined original signal vector.

Vi(O) ], with v;(n) the noise component in subband i. Interestingly enough, when quantization noise is added to t he subband coefficients, the analysis fil ters that minimize the MSE between the original and reconstructed signals are no longer given by t he perfect reconstruction filter pair. In this paper, we are interested in designing the optimal reconstruction filters, given the analysis filters and a statistical model for the input and the quantization noise. 2 Determining the optimal filters In the following, we will consider the case of paraunitary analysis filters, but t he discussion can be easily adapted to other classes of filters.

In coding applications, the subband coefficients y;(n) are actually quantized before the synthesis stage. The quantization operation can be modeled 21 by adding the quantization error to the subband coefficients, as shown in Fig. 1 for the case of a two-subband system . Thus , equation (1) has to be modified as x=Fz , (3) where % = y+ v and tI = [vo, ... , VM _I )f , V i = [vi(- k), ... ,Vi(O) ], with v;(n) the noise component in subband i. Interestingly enough, when quantization noise is added to t he subband coefficients, the analysis fil ters that minimize the MSE between the original and reconstructed signals are no longer given by t he perfect reconstruction filter pair.