Filters

In the previous chapter we saw that a delay network can have a non-uniform
frequency response--a gain that varies as a function of
frequency. Delay networks also typically change the phase of incoming signals
variably depending on frequency. When the delay times used are very short,
the most important properties of a delay network become its frequency and
phase response. A delay network that is designed specifically for its
frequency or phase response is called a
*filter*.

In block diagrams, filters are shown as in Figure 8.1 (part a). The curve shown within the block gives a qualitative representation of the filter's frequency response. The frequency response may vary with time, and depending on the design of the filter, one or more controls (or additional audio inputs) might be used to change it.

Suppose, following the procedure of Section 7.3, we put
a unit-amplitude, complex-valued sinusoid with angular frequency into a filter. We
expect to get out a sinusoid of the same frequency and some amplitude, which
depends on . This gives us a complex-valued function , which is called the
*transfer function*
of the filter.

The frequency response is the gain as a function of the frequency . It is is equal to the magnitude of the transfer function. A filter's frequency response is customarily graphed as in Figure 8.1 (part b). An incoming unit-amplitude sinusoid of frequency comes out of the filter with magnitude .

It is sometimes also useful to know the phase response of the filter, equal to . For a fixed frequency , the filter's output phase will be radians ahead of its input phase.

The design and use of filters is a huge subject, because the wide range of uses a filter might be put to suggests a wide variety of filter design processes. In some applications a filter must exactly follow a prescribed frequency response, in others it is important to minimize computation time, in others the phase response is important, and in still others the filter must behave well when its parameters change quickly with time.

- Taxonomy of filters

- Elementary filters
- Elementary non-recirculating filter
- Non-recirculating filter, second form
- Elementary recirculating filter
- Compound filters
- Real outputs from complex filters
- Two recirculating filters for the price of one

- Designing filters
- One-pole low-pass filter
- One-pole, one-zero high-pass filter
- Shelving filter
- Band-pass filter
- Peaking and stop-band filter
- Butterworth filters
- Stretching the unit circle with rational functions
- Butterworth band-pass filter
- Time-varying coefficients
- Impulse responses of recirculating filters
- All-pass filters

- Applications

- Examples
- Prefabricated low-, high-, and band-pass filters
- Prefabricated time-varying band-pass filter
- Envelope followers
- Single sideband modulation
- Using elementary filters directly: shelving and peaking
- Making and using all-pass filters

- Exercises