Sufficient Condition for Perfect Reconstruction

Above, we found that, for any integer which divides , a sufficient condition for perfect reconstruction is

and the output signal is then

More generally, we allow any nonzero scaling and any additional delay:

where is any constant and is any nonnegative integer. In this case, the output signal is

Thus, given any polyphase matrix , we can attempt to compute : If it is stable, we can use it to build a perfect-reconstruction filter bank. However, if is FIR, will typically be IIR. In §10.5 below, we will look at paraunitary filter banks, for which is FIR and paraunitary whenever is.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.