With MM uniformly spaced sub-bands, the sub-band width is 2πM 2 M radians, implying that the sub-band signal can be downsampled by a factor MM (but not more than MM) without loss of information. This is referred to as a "critically sampled" filterbank. This maximal level of downsampling is advantageous when storing or further processing the sub-band signals. With critical sampling, the total number of downsampled sub-band output samples equals the total number of input samples. Assuming lossless sub-band processing, the critically-sampled synthesis/analysis procedure is illustrated in Figure 1:

Recall that one of our goals in filter design is to ensure that yn≈xn-d y n x n d for some integer delay dd. From the block diagram above, one can see that imperfect analysis filtering will contribute aliasing errors to the sub-band signals. This aliasing distortion will degrade yn y n if it is not cancelled by the synthesis filterbank. Though ideal brick-wall filters H k z H k z and G k z G k z could easily provide perfect reconstruction (i.e., yn=xn-d y n x n d ), they would be unimplementable due to their doubly-infinite impulse responses. Thus, we are interested in the design of causal FIR filters that give near-perfect reconstruction or, if possible, perfect reconstruction.

There are two principle approaches to the design of filterbanks:

1. Classical: Approximate ideal brick wall filters to ensure good sub-band isolation (i.e., frequency selectivity) and accept (a hopefully small amount of) aliasing and thus reconstruction error.
2. Modern: Constrain the filters to give perfect (or near-perfect) reconstruction and hope for good sub-band isolation.