Perfect Reconstruction Cosine Modulated Filter Banks

By changing the phases $\phi_k$ , the pseudo-QMF filter bank can yield perfect reconstruction:

$$\phi_k \eqsp \left(k+\frac{1}{2}\right)\left(L+1\right)\frac{\pi}{2}$$ (12.101)

where $L$ is the length of the polyphase filter ($M=LN$ ).

If $M=2N$ , then this is the oddly stacked Princen-Bradley filter bank and the analysis filters are related by cosine modulations of the lowpass prototype:

$$f_k(n) \eqsp h(n)\hbox{cos}\left[\left(n+\frac{N+1}{2}\right)\left(k+\frac{1}{2}\right)\frac{\pi}{N}\right],\quad k=0,\ldots,N-1$$ (12.102)

However, the length of the filters $M$ can be any even multiple of $N$ :

$$M\eqsp LN, \quad (L/2) \in \cal{Z}$$ (12.103)

The parameter $L$ is called the overlapping factor. These filter banks are also referred to as extended lapped transforms, when $K \ge 2$ [159].