0.8 Filter banks and transmultiplexers  (Page 14/23)

In the Type 1 case [link] , [link] ,

and therefore

In the Type 2 case [link] ,

and hence

Linear phase filter banks

In some applications. it is desirable to have filter banks with linear-phase filters [link] . The linear-phase constraint (like the modulation constraint studied earlier) reduces the numberof free parameters in the design of a filter bank. Unitary linear phase filter banks have been studied recently [link] , [link] . In this section we develop algebraic characterizations of certain typesof linear filter banks that can be used as a starting point for designing such filter banks.

In this section, we assume that the desired frequency responses are as in [link] . For simplicity we also assume that the number of channels, $M$ , is an even integer and that the filters are FIR. It should be possible to extend the results that follow to the case when $M$ is an odd integer in a straightforward manner.

Consider an $M$ -channel FIR filter bank with filters whose passbands approximate ideal filters.Several transformations relate the $M$ ideal filter responses. We have already seen one example where all the ideal filters are obtained bymodulation of a prototype filter. We now look at other types of transformations that relate the filters. Specifically,the ideal frequency response of $h_{M-1-i}$ can be obtained by shifting the response of the $h_i$ by $\pi$ . This either corresponds to the restriction that

or to the restriction that

where $N$ is the filter length and for polynomial $H\left(z\right)$ , $H^R\left(z\right)$ denotes its reflection polynomial (i.e. the polynomial with coefficients in the reversedorder). The former will be called pairwise-shift (or PS) symmetry (it is also known as pairwise-mirror image symmetry [link] ) , while the latter will be called pairwise-conjugated-shift (or PCS) symmetry (also known as pairwise-symmetry [link] ). Both these symmetries relate pairs offilters in the filter bank. Another type of symmetry occurs when the filters themselves are symmetric or linear-phase. The only type of linear-phase symmetry we will consider isof the form

where the filters are all of fixed length $N$ , and the symmetry is about $\frac{N-1}{2}$ . For an $M$ -channel linear-phase filter bank (with $M$ an even integer), $M/2$ filters each are even-symmetric and odd-symmetric respectively [link] .

We now look at the structural restrictions on $H_p\left(z\right)$ , the polyphase component matrix of the analysis bank that these three types of symmetries impose.Let $J$ denote the exchange matrix with ones on the antidiagonal. Postmultiplying a matrix $A$ by $J$ is equivalent to reversing the order of the columns of $A$ , and premultiplying is equivalent to reversing the order of the rows of $A$ . Let $V$ denote the sign-alternating matrix, the diagonal matrix of alternating $\pm 1$ 's. Postmultiplying by $V$ , alternates the signs of the columns of $A$ , while premultiplying alternates the signs of the rows of $A$ . The polyphase components of $H\left(z\right)$ are related to the polyphase components of $H^R\left(z\right)$ by reflection and reversal of the ordering of the components. Indeed, if $H\left(z\right)$ is of length $Mm$ , and ${\displaystyle H\left(z\right)=\sum _{l=0}^{M-1}{z}^{-l}{H}_l\left(z^M\right)}$ , then,