Group delay and phase delay
In signal processing, group delay and phase delay are functions that describe in different ways the delay times experienced by a signal’s various sinuoidal frequency components as they pass through a linear time-invariant (LTI) system (such as a microphone, coaxial cable, amplifier, loudspeaker, communications system, ethernet cable, digital filter, or analog filter).
Fourier analysis reveals how signals in time can alternatively be expressed as the sum of sinusoidal frequency components, each based on the trigonometric function
The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a frequency component of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.
A varying phase response as a function of frequency, from which group delay and phase delay can be calculated, typically occurs in devices such as microphones, amplifiers, loudspeakers, magnetic recorders, headphones, coaxial cables, and antialiasing filters.
[2] All frequency components of a signal are delayed when passed through such devices, or when propagating through space or a medium, such as air or water.
The phase delay property in general does not give useful information if the device input is a modulated signal.
The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system.
The simplest use case for group delay is illustrated in Figure 1 which shows a conceptual modulation system, which is itself an LTI system with a baseband output that is ideally an accurate copy of the baseband signal input.
In Figure 1, the outer system phase delay is the meaningful performance metric.
Linear time-invariant system § Fourier and Laplace transforms expresses this relationship as: where
Suppose that such a system is driven by a wave packet formed by a sinusoid multiplied by an amplitude envelope
This condition can be expressed mathematically as: Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated[clarification needed] as: Here
driven by a complex sinusoid of unit amplitude, the output is where the phase shift
is The phase of a 1st-order low-pass filter formed by a RC circuit with cutoff frequency
Filters will have negative group delay over frequency ranges where its phase response is positively-sloped.
If a signal is band-limited within some maximum frequency B, then it is predictable to a small degree (within time periods smaller than 1⁄B).
[9] Circuits with negative group delay (e.g., Figure 2) are possible, though causality is not violated.
[2][13] It is therefore important to know the threshold of audibility of group delay with respect to frequency,[14][15][16] especially if the audio chain is supposed to provide high fidelity reproduction.
[17] Flanagan, Moore and Stone conclude that at 1, 2 and 4 kHz, a group delay of about 1.6 ms is audible with headphones in a non-reverberant condition.
[18] Other experimental results suggest that when the group delay in the frequency range from 300 Hz to 1 kHz is below 1.0 ms, it is inaudible.
Leach[19] introduced the concept of differential time-delay distortion, defined as the difference between the phase delay and the group delay: An ideal system should exhibit zero or negligible differential time-delay distortion.
[19] It is possible to use digital signal processing techniques to correct the group delay distortion that arises due to the use of crossover networks in multi-way loudspeaker systems.
[20] This involves considerable computational modeling of loudspeaker systems in order to successfully apply delay equalization,[21] using the Parks-McClellan FIR equiripple filter design algorithm.
It is often desirable for the group delay to be constant across all frequencies; otherwise there is temporal smearing of the signal.
, it therefore follows that a constant group delay can be achieved if the transfer function of the device or medium has a linear phase response (i.e.,
TTD is an important characteristic of lossless and low-loss, dispersion free, transmission lines.
Telegrapher's equations § Lossless transmission reveals that signals propagate through them at a speed of
, such as that typically found in the definition of filter designs, may have the group delay determined by taking advantage of the phase relation,
of an ideal linear phase response would be expected to have a value of 0 across the frequency range of interest (such as the pass band of a filter), while the
of a real-world approximately linear phase response may deviate from 0 by a small finite amount across the frequency range of interest.
