Group Delay and Phase Delay (Digital Filter Design Toolkit)

For a filter with a frequency response of H(e^jω), the phase delay response is defined by the following equation:

The group delay response is defined as the negative derivative of the phase response ω, as shown in the following equation:

Both the group delay and phase delay are in samples.

For a generalized linear phase filter with arg[H(e^jω)] = –ω+β, the group delay is represented by the following equation:

The phase delay is represented by the following equation:

You can represent the phase delay as the time delay in samples experienced by each frequency component of the input signal. The filter is represented by the following illustration:

The filter H(e^jω) shifts all frequency components by a phase β and then filters the signal with a new filter H_new(e^jω) that has a phase of –ω. You can interpret the group delay as the time delay in samples experienced by each frequency component through the new filter H_new(e^jω).

Linear phase filters are characterized by a constant group delay. The deviation of the group delay from a constant value within the passband indicates the degree of nonlinearity in the phase. Use the group delay to analyze the linearity of a filter.