1-D multirate filter banks

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 considerable 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