Group Delay and Phase Delay

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Updated2025-10-10
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This section covers the formulas used to calculate and the relationships between the group delay response and the phase delay response of a filter.

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

τ_{ρ} = - \frac{\arg [H (e^{j ω})]}{ω}

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

τ_{g} (ω) = - \frac{d}{d ω} \arg [H (e^{j ω})]

Both the group delay and phase delay are in samples.

For a generalized linear phase filter with

\arg [H (e^{j ω})] = - α ω + β

the group delay is represented by the following equation:

τ_{g} (ω) = - \frac{d}{d ω} \arg [H (e^{j ω})] = α

The phase delay is represented by the following equation:

τ_{ρ} = - \frac{\arg [H (e^{j ω})]}{ω} = α - \frac{β}{ω}

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 equation:

x (n) \to e^{j β} \to H_{n e w} (e^{j ω}) \to y (n)

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 –αω. 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.

LabVIEW Digital Filter Design Toolkit User Manual

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Group Delay and Phase Delay