In signal processing, group delay and phase delay are functions that describe in different ways the delay times experienced by a signal's various sinusoidal 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).
These delays are sometimes frequency dependent, which means that different sinusoid frequency components experience different time delays. As a result, the signal's waveform experiences distortion as it passes through the system. This distortion can cause problems such as poor fidelity in analog video and analog audio, or a high bit-error rate in a digital bit stream.
Background Frequency components of a signal
Fourier analysis reveals how signals in time can alternatively be expressed as the sum of sinusoidal frequency components, each based on the trigonometric function with a fixed amplitude and phase and no beginning and no end.
Linear time-invariant systems process each sinusoidal component independently; the property of linearity means they satisfy the superposition principle.
Introduction 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. 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.
While a phase response describes phase shift in angular units (such as degrees or radians), the phase delay is in units of time and equals the negative of the phase shift at each frequency divided by the value of that frequency. Group delay is the negative derivative of phase shift with respect to frequency.
Phase delay A linear time-invariant system or device has a phase response property and a phase delay property, where one can be calculated exactly from the other. Phase delay directly measures the device or system time delay of individual sinusoidal frequency components. If the phase delay function at any given frequency—within a frequency range of interest—has the same constant of proportionality between the phase at a selected frequency and the selected frequency itself, the system/device will have the ideal of a flat phase delay property, a.k.a. linear phase. Since phase delay is a function of frequency giving time delay, a departure from the flatness of its function graph can reveal time delay differences among the signal's various sinusoidal frequency components, in which case those differences will contribute to signal distortion, which is manifested as the output signal waveform shape being different from that of the input signal.
The phase delay property in general does not give useful information if the device input is a modulated signal. For that, group delay must be used.
Group delay The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system. For a modulation signal (passband signal), the information carried by the signal is carried exclusively in the wave envelope. Group delay therefore operates only with the frequency components derived from the envelope.
Basic modulation system
A device's group delay can be exactly calculated from the device's phase response, but not the other way around. 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. This system as a whole is referred to here as the outer LTI system/device, which contains an inner (red block) LTI system/device. As is often the case for a radio system, the inner red LTI system in Fig 1 can represent two LTI systems in cascade, for example an amplifier driving a transmitting antenna at the sending end and the other an antenna and amplifier at the receiving end.
Amplitude Modulation Amplitude modulation creates the passband signal by shifting the baseband frequency components to a much higher frequency range. Although the frequencies are different, the passband signal carries the same information as the baseband signal. The demodulator does the inverse, shifting the passband frequencies back down to the original baseband frequency range. Ideally, the output (baseband) signal is a time delayed version of the input (baseband) signal where the waveform shape of the output is identical to that of the input.
In Figure 1, the outer system phase delay is the meaningful performance metric. For amplitude modulation, the inner red LTI device group delay becomes the outer LTI device phase delay. If the inner red device group delay is completely flat in the frequency range of interest, the outer device will have the ideal of a phase delay that is also completely flat, where the contribution of distortion due to the outer LTI device's phase response—determined entirely by the inner device's possibly different phase response—is eliminated. In that case, the group delay of the inner red device and the phase delay of the outer device give the same time delay figure for the signal as a whole, from the baseband input to the baseband output. It is significant to note that it is possible for the inner (red) device to have a very non-flat phase delay (but flat group delay), while the outer device has the ideal of a perfectly flat phase delay. This is fortunate because in LTI device design, a flat group delay is easier to achieve than a flat phase delay.
Angle Modulation In an angle-modulation system—such as with frequency modulation (FM) or phase modulation (PM)—the (FM or PM) passband signal applied to an LTI system input can be analyzed as two separate passband signals, an in-phase (I) amplitude modulation AM passband signal and a quadrature-phase (Q) amplitude modulation AM passband signal, where their sum exactly reconstructs the original angle-modulation (FM or PM) passband signal. While the (FM/PM) passband signal is not amplitude modulation, and therefore has no apparent outer envelope, the I and Q passband signals do indeed each have their own separate amplitude modulation envelopes. (However, unlike with regular amplitude modulation, the I and Q envelopes do not resemble the wave shape of the baseband signals, even though 100 percent of the baseband signal is represented in a complex manner by those I and Q envelopes.) So, for each of the I and Q passband signals, a flat group delay ensures that neither the I pass band envelope nor the Q passband envelope will have wave shape distortion, so when the I passband signal and the Q passband signal are added back together, the sum is the original FM/PM passband signal, which will also be unaltered.
Theory According to LTI system theory (used in control theory and digital or analog signal processing), the output signal of an LTI system can be determined by convolving the time-domain impulse response of the LTI system with the input signal . expresses this relationship as:
where denotes the convolution operation, and are the Laplace transforms of the input and impulse response , respectively, is the complex frequency, and is the inverse Laplace transform. is called the transfer function of the LTI system and, like the impulse response , fully defines the input-output characteristics of the LTI system. This convolution can be evaluated by using the integral expression in the time domain, or (according to the rightmost expression) by using multiplication in the Laplace domain and then applying the inverse transform to return to time domain.
LTI system response to wave packet Suppose that such a system is driven by a wave packet formed by a sinusoid multiplied by an amplitude envelope , so the input can be expressed in the following form:
Also suppose that the envelope is slowly changing relative to the sinusoid's frequency . 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 as:
Here is the group delay and is the phase delay, and they are given by the expressions below (and potentially are functions of the angular frequency ). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, . The envelope of the sinusoid is delayed in time by the group delay, .
Mathematical definition of group delay and phase delay The group delay, , and phase delay, , are (potentially) frequency-dependent and can be computed from the unwrapped phase shift . The phase delay at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:
The group delay at each frequency equals the negative of the slope (i.e. the derivative with respect to frequency) of the phase at that frequency:
In a linear phase system (with non-inverting gain), both and are constant (i.e., independent of ) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely ) is negative, with magnitude increasing linearly with frequency .
LTI system response to complex sinusoid More generally, it can be shown that for an LTI system with transfer function driven by a complex sinusoid of unit amplitude,
the output is
where the phase shift is
1st order low- or high-pass RC filter example The phase of a 1st-order low-pass filter formed by a RC circuit with cutoff frequency is:
Similarly, the phase for a 1st-order RC high-pass filter is:
Taking the negative derivative with respect to for either this low-pass or high-pass filter yields the same group delay of:
For frequencies significantly lower than the cutoff frequency, the phase response is approximately linear (arctan for small inputs can be approximated as a line), so the group delay simplifies to a constant value of:
Similarly, right at the cutoff frequency,
As frequencies get even larger, the group delay decreases with the inverse square of the frequency and approaches zero as frequency approaches infinity.
Negative group delay
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 ). A filter whose group delay is negative over that signal's entire frequency range is able to use the signal's predictability to provide an illusion of a non-causal time advance. However, if the signal contains an unpredictable event (such as an abrupt change which makes the signal's spectrum exceed its band-limit), then the illusion breaks down. Circuits with negative group delay (e.g., Figure 2) are possible, though causality is not violated.
Negative group delay filters can be made in both digital and analog domains. Applications include compensating for the inherent delay of low-pass filters, to create zero phase filters, which can be used to quickly detect changes in the trends of sensor data or stock prices.
Group delay in audio Group delay has some importance in the audio field and especially in the sound reproduction field.
Threshold of audibility