Linear Filter - Frequency Response

Frequency Response

The frequency response or transfer function of a filter can be obtained if the impulse response is known, or directly through analysis using Laplace transforms, or in discrete-time systems the Z-transform. The frequency response also includes the phase as a function of frequency, however in many cases the phase response is of little or no interest. FIR filters can be made to have zero phase, but with IIR filters that is generally impossible. With most IIR transfer functions there are related transfer functions having a frequency response with the same magnitude but a different phase; in most case the so-called minimum phase transfer function is preferred.

Filters in the time domain are most often requested to follow a specified frequency response. Then a mathematical procedure is used to find a filter transfer function which can be realized (within some constraints) and which approximates the desired response to within some criterion. Common filter response specifications are described as follows:

• A low-pass filter passes low frequencies while blocking higher frequencies.
• A high-pass filter passes high frequencies.
• A band-pass filter passes a band (range) of frequencies.
• A band-stop filter passes high and low frequencies outside of a specified band.
• A notch filter has a null response at a particular frequency. This function may be combined with one of the above responses.
• An all-pass filter passes all frequencies equally well, but alters the phase relationship among them.
• An equalization filter is not designed to fully pass or block any frequency, but instead to gradually vary the amplitude response as a function of frequency: filters used as pre-emphasis filters, equalizers, or tone controls are good examples.