ABSTRACT
An important class of digital filter system is the multirate filter bank. In this chapter we shall only be considering the one-dimensional (1-D) type of filter bank, as might be applied to typical signals that evolve with time, such as audio waveforms and communications signals. The reason that filter banks are important is that we can often achieve useful functionality by
separating signals into various different frequency bands and applying different processing to each
band. Typical examples of this are the widely used MP3 digital audio compression systems. In order to achieve a low coded data file size for digital music tracks, the MP3 coding standard specifies that the audio signal should be split into many frequency bands and that separate adaptive quantization should be applied to each band. The quantization is designed to take maximum advantage of the noise masking properties of the human auditory system, such that frequency bands containing substantial audio energy are quantized quite coarsely (because the quantizing noise gets masked by the signal here), whereas bands with low levels of audio are quantized more finely (since the masking is only effective at frequencies close to those containing most of the audio energy). Filter banks can operate with filtered outputs being sampled at the same rate as the input signal.
However, with many filters operating in parallel, this can lead to an unacceptably large amount of output data being generated. It is therefore sensible to subsample the outputs of the filters in a filter bank so that the total output data rate from all the filters is similar to that of the input. Such filters are called multirate filters, and the complete system is a multirate filter bank. When signals are subsampled, aliasing can occur and cause degradation of signal quality, but, with careful design, aliasing effects can be eliminated in multirate filter banks as long as the total output data rate is no less than the input rate. Multirate filters can be implemented with much less computational cost than the equivalent full rate filters, and, in the case of compression systems, they generate much less data to be coded by the adaptive quantizers. An alternative way to view the advantages of filter banks is that, with careful design, they can
encourage signal sparsity, i.e., most of the energy of an input signal can be concentrated in a small proportion of the output samples from the filter bank. Sparsity has been shown to be a key element in successfully performing many signal processing tasks, such as compression, denoising, signal separation, and other enhancement techniques. Sparsity can be achieved if, at any given time, the input signal can be well approximated by a weighted sum of the impulse responses from just a few of the filters in the filter bank. This occurs when the filters are matched to typical components of the signal. Digital filter banks have been actively studied since the 1960s. However their use achieved a consid-
erable boost with the development of wavelet theory in the 1980s. The theory of wavelet transforms was developed principally by French and Belgian mathematicians, notably A. Grossman, J. Morlet, Y. Meyer, I. Daubechies, and S. Mallat, and efficient implementation of the wavelet transform is usually achieved with multirate filter banks. The two topics are now firmly linked and are of great importance for signal analysis and compression. The discrete wavelet transform (DWT) may be used to analyze a wide variety of signals, particularly
those that combine long low-frequency events with much shorter high-frequency events (e.g., transients). It has perhaps achieved its greatest success with images. Although these are two-dimensional (2-D) signals, the 1-D filters considered in this chapter are still highly relevant since 2-D wavelet transforms are usually achieved using separable 1-D processing along the rows and then down the columns of the image (or vice versa). In this chapter we shall be introducing some of the basic ideas behind 1-D filter banks, and then will
concentrate much of our coverage on the 2-band multirate filter bank, which is the workhorse of the DWT. In the final sections we shall extend the discussion to M-band (M > 2) filter banks and also to Hilbert pairs of filter banks, which lead to the dual-tree complex wavelet transform. Most of the following discussions in this chapter will assume that the filters are finite-impulse-response
(FIR) filters and that all samples of the input signal are available in the memory of the signal processing hardware. Hence causality of the filters is not an issue and filter taps corresponding to negative delays (positive powers of z in the z-transform) pose no implementation problems. In this situation, the most natural way to design filters is for zero overall delay (the zero-phase condition). If these assumptions are not valid (e.g., in the case of a continuously evolving audio signal), then
appropriate delays can usually be inserted so that the filters are still implementable. Only in the case of recursive infinite-impulse-response (IIR) filters is the causality issue a real potential problem; and so more care is required on this issue in Section 24.7. Elsewhere we shall ignore problems of causality.
