Chapter 6 Filter Banks

Filter banks are arrangements of low pass, bandpass, and highpass filters used for the spectral decomposition and composition of signals. They play an important role in many modern signal processing applications such as audio and image coding. The reason for their popularity is the fact that they easily allow the extraction of spectral components of a signal while providing very efficient implementations. Since most filter banks involve various sampling rates, they are also referred to as multirate systems. To give an example, Figure 6.1 shows an M-channel filter bank. The input signal is decomposed into M socalled subb and signalsby applying M analysis filters with different passbands. Thus, each of the subband signals carries information on the input signal in a particular frequency band. The blocks with arrows pointing downwards in Figure 6.1 indicate downsampling (subsampling) by factor N, and the blocks with arrows pointing upwards indicate upsampling by N. Subsampling by N means that only every Nth sample is taken. This operation serves to reduce or eliminate redundancies in the M subband signals. Upsampling by N means the insertion of N 1 consecutive zeros between the samples. This allows us to recover the original sampling rate. The upsamplers are followed by filters which replace the inserted zeros with meaningful values. In the case M = N we speak of critical subsampling, because this is the maximum downsampling factor for which perfect reconstruction can be achieved. Perfect reconstruction means that the output signal is a copy of the input signal with no further distortion than a time shift and amplitude scaling.