0.8 Filter banks and transmultiplexers  (Page 2/23)

1. Perfect Reconstruction (i.e., $y\left(n\right)=x\left(n\right)$ ).
2. Usefulness . Clearly this depends on the application. For the subband coding application, the filter frequency responses mightapproximate the ideal responses in [link] . In other applications the filters may have to satisfy other constraints orapproximate other frequency responses.

If the signals and filters are multidimensional in [link] , we have the multidimensional filter bank design problem.

Transmultiplexer

A transmultiplexer is a structure that combines a collection of signals into a single signal at a higher rate; i.e., it is the dual of a filterbank. If the combined signal depends linearly on the constituent signal, we have a linear transmultiplexer. Transmultiplexers were originally studied in the context of converting time-domain-multiplexed (TDM) signalsinto frequency domain multiplexed (FDM) signals with the goal of converting back to time-domain-multiplexed signals at some later point.A key point to remember is that the constituent signals should be recoverable from the combined signal. [link] shows the structure of a transmultiplexer. The input signals $y_i\left(n\right)$ were upsampled, filtered, and combined (by a synthesis bank of filters)to give a composite signal $d\left(n\right)$ . The signal $d\left(n\right)$ can be filtered (by an analysis bank of filters) and downsampled to give a set of signals $x_i\left(n\right)$ . The goal in transmultiplexer design is a choice of filters that ensures perfect reconstruction (i.e., for all $i$ , $x_i\left(n\right)=y_i\left(n\right)$ ). This imposes bilinear constraints on the synthesis and analysis filters. Also,the upsampling factor must be at least the number of constituent input signals, say $L$ . Moreover, in classical TDM-FDM conversion the analysis and synthesis filters must approximate the ideal frequency responses in [link] . If the input signals, analysis filters and synthesis filters are multidimensional, we havea multidimensional transmultiplexer.

Perfect reconstruction—a closer look

We now take a closer look at the set of bilinear constraints on the analysis and synthesis filters of a filter bank and/or transmultiplexerthat ensures perfect reconstruction (PR). Assume that there are $L$ analysis filters and $L$ synthesis filters and that downsampling/upsampling is by some integer $M$ . These constraints, broadly speaking, canbe viewed in three useful ways, each applicable in specific situations.

1. Direct characterization - which is useful in wavelet theory (to characterize orthonormality and frame properties),in the study of a powerful class of filter banks (modulated filter banks), etc.
2. Matrix characterization - which is useful in the study of time-varying filter banks.
3. z-transform-domain (or polyphase-representation) characterization - which is useful in the design and implementation of (unitary) filter banks and wavelets.

Direct characterization of pr

We will first consider the direct characterization of PR, which, for both filter banks and transmultiplexers, follows from an elementary superpositionargument.

Theorem 38 A filter bank is PR if and only if, for all integers $n_1$ and $n_2$ ,