There exist many applications in modern signal processing where it is advantageous to separate a signal into different frequency ranges called sub-bands. The spectrum might be partitioned in the uniform manner illustrated in Figure 1, where the sub-band width Δ k =2πM Δ k 2 M is identical for each sub-band and the band centers are uniformly spaced at intervals of 2πM 2 M .

Alternatively, the sub-bands might have a logarithmic spacing like that shown in Figure 2.

For most of our discussion, we will focus on uniformly spaced sub-bands.

The separation into sub-band components is intended to make further processing more convenient. Some of the most popular applications for sub-band decomposition are audio and video source coding (with the goal of efficient storage and/or transmission).

Figure 3 illustrates the use of sub-band processing in MPEG audio coding. There a psychoacoustic model is used to decide how much quantization error can be tolerated in each sub-band while remaining below the hearing threshold of a human listener. In the sub-bands that can tolerate more error, less bits are used for coding. The quantized sub-band signals can then be decoded and recombined to reconstruct (an approximate version of) the input signal. Such processing allows, on average, a 12-to-1 reduction in bit rate while still maintaining "CD quality" audio. The psychoacoustic model takes into account the spectral masking phenomenon of the human ear, which says that high energy in one spectral region will limit the ear's ability to hear details in nearby spectral regions. Therefore, when the energy in one sub-band is high, nearby sub-bands can be coded with less bits without degrading the perceived quality of the audio signal. The MPEG standard specifies 32-channels of sub-band filtering. Some psychoacoustic models also take into account "temporal masking" properties of the human ear, which say that a loud burst of sound will temporarily overload the ear for short time durations, making it possible to hide quantization noise in the time interval after a loud sound burst.

In typical applications, non-trivial signal processing takes place between the bank of analysis filters and the bank of synthesis filters, as shown in Figure 4. We will focus, however, on filterbank design rather than on the processing that occurs between the filterbanks.

Our goals in filter design are:

1. Good sub-band frequency separation (i.e., good "frequency selectivity").
2. Good reconstruction (i.e., yn≈xn-d y n x n d for some integer delay dd) when the sub-band processing is lossless.

The first goal is driven by the assumption that the sub-band processing works best when it is given access to cleanly separated sub-band signals, while the second goal is motivated by the idea that the sub-band filtering should not limit the reconstruction performance when the sub-band processing (e.g., the coding/decoding) is lossless or nearly lossless.