2.1 Uniform filterbanks

Uniform filterbanks

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

Recall that one of our goals in filter design is to ensure that $y(n)\approx x(n-d)$ for some integer delay $d$ . From the block diagram above , one can see that imperfect analysis filtering will contribute aliasing errorsto the sub-band signals. This aliasing distortion will degrade $y(n)$ if it is not cancelled by the synthesis filterbank. Thoughideal brick-wall filters $H_k(z)$ and $G_k(z)$ could easily provide perfect reconstruction ( i.e. , $y(n)=x(n-d)$ ), they would be unimplementable due to their doubly-infinite impulse responses. Thus, we are interestedin 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:

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