0.8 Filter banks and transmultiplexers  (Page 12/23)

The frequency responses in [link] can be obtained by shifting an the response of an ideal lowpass filter (supportedin $[-\frac{\pi }{2M},\frac{\pi }{2M}]$ ) by $(i+\frac{1}{2})\frac{\pi }{M}$ , $i\in \left\{0,,,...,,,M,-,1\right\}$ . This can be achieved by modulating with a cosine (or sine) with appropriate frequency and arbitrary phase. However, some choicesof phase may be incompatible with perfect reconstruction. A general choice of phase (and hence modulation, that covers all modulated filter banks of this type) is givenby the following definition of the analysis and synthesis filters:

Here $\alpha$ is an integer parameter called the modulation phase. Now one can substitute these forms for the filters in [link] to explicit get PR constraints on the prototype filters $h$ and $g$ . This is a straightforward algebraic exercise, since the summation over $i$ in [link] is a trigonometric sum that can be easily computed. It turns out thatthe PR conditions depend only on the parity of the modulation phase $\alpha$ . Hence without loss of generality, we choose $\alpha \in \left\{M,-,1,,,M,-,2\right\}$ —other choices being incorporated as a preshift into the prototype filters $h$ and $g$ .

Thus there are two types of MFBs depending on the choice of modulation phase:

The PR constraints on $h$ and $g$ are quite messy to write down without more notational machinery. But the basic nature of the constraintscan be easily understood pictorially. Let the $M$ polyphase components of $h$ and $g$ respectively be partitioned into pairs as suggested in [link] . Each polyphase pair from $h$ and an associated polyphase pair $g$ (i.e., those four sequences) satisfy the PR conditions for a two-channel filter bank. In other words, these subsequences couldbe used as analysis and synthesis filters respectively in a two-channel PR filter bank.As seen in [link] , some polyphase components are not paired. The constraints on these sequences that PR imposes will be explicitlydescribed soon. Meanwhile, notice that the PR constraints on the coefficients are decoupled into roughly $M/2$ independent sets of constraints (since there are roughly $M/2$ PR pairs in [link] ). To quantify this, define $J$ :

In other words, the MFB PR constraint decomposes into a set of $J$ two-channel PR constraints and a few additional conditions on the unpaired polyphasecomponents of $h$ and $g$ .

We first define $M$ polyphase components of the analysis and synthesis prototype filters, viz., $P_l\left(z\right)$ and $Q_l\left(z\right)$ respectively. We split these sequences further into their even and odd components to give $P_{l,0}\left(z\right)$ , $P_{l,1}\left(z\right)$ , $Q_{l,0}\left(z\right)$ and $Q_{l,1}\left(z\right)$ respectively. More precisely, let

and let

with $\mathcal{Q}\left(z\right)$ defined similarly. Let $\mathcal{I}$ be the $2\times 2$ identity matrix.

Theorem 47 (Modulated Filter Banks PR Theorem) An modulated filter bank (Type 1 or Type 2) (as defined in [link] and [link] ) is PR iff for $l\in \mathcal{R}\left(J\right)$

and furthermore if $\alpha$ is even $P_{\frac{\alpha }{2},0}\left(z\right)Q_{\frac{\alpha }{2},0}\left(z\right)=\frac{1}{M}$ . In the Type 2 case, we further require $P_{M-1}\left(z\right)Q_{M-1}\left(z\right)=\frac{2}{M}$ .