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 ${\displaystyle \sin(x)}$ 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.

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 in the steady state.^[3] 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 linear phase, which results in a constant group delay.^[1] 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 various signal 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.

Group delay

While phase delay describes the system's response to steady state sinusoidal components, group delay describes the response to amplitude modulated sinusoids.

The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system.^[4][5] 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

Figure 1: Outer and Inner LTI Devices

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.

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 ${\displaystyle \displaystyle A_{\text{env}}(t)>0}$, so the input ${\displaystyle \displaystyle x(t)}$ can be expressed in the following form:

${\displaystyle x(t)=A_{\text{env}}(t)\cos(\omega t+\theta )\,.}$

Also suppose that the envelope ${\displaystyle \displaystyle A_{\text{env}}(t)}$ is slowly changing relative to the sinusoid's frequency ${\displaystyle \displaystyle \omega }$. This condition can be expressed mathematically as:

${\displaystyle \left|{\frac {d}{dt}}\log {\big (}A_{\text{env}}(t){\big )}\right|\ll \omega \ .}$

Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated as:

${\displaystyle y(t)={\big |}H(i\omega ){\big |}\ A_{\text{env}}(t-\tau _{g})\cos {\big (}\omega (t-\tau _{\phi })+\theta {\big )}\;.}$

Here ${\displaystyle \displaystyle \tau _{g}}$ is the group delay and ${\displaystyle \displaystyle \tau _{\phi }}$ is the phase delay, and they are given by the expressions below (and potentially are functions of the angular frequency ${\displaystyle \displaystyle \omega }$). 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, ${\displaystyle \displaystyle \tau _{\phi }}$. The envelope of the sinusoid is delayed in time by the group delay, ${\displaystyle \displaystyle \tau _{g}}$.

Mathematical definition of group delay and phase delay

The group delay, ${\displaystyle \displaystyle \tau _{g}}$, and phase delay, ${\displaystyle \displaystyle \tau _{\phi }}$, are (potentially) frequency-dependent^[6] and can be computed from the unwrapped phase shift ${\displaystyle \displaystyle \phi (\omega )}$. The phase delay at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:

${\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}\,.}$

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:^[7]

${\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}\,.}$

In a linear phase system (with non-inverting gain), both ${\displaystyle \displaystyle \tau _{g}}$ and ${\displaystyle \displaystyle \tau _{\phi }}$ are constant (i.e., independent of ${\displaystyle \displaystyle \omega }$) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely ${\displaystyle \displaystyle -\omega \tau _{\phi }}$) is negative, with magnitude increasing linearly with frequency ${\displaystyle \displaystyle \omega }$.

LTI system response to complex sinusoid

More generally, it can be shown that for an LTI system with transfer function ${\displaystyle \displaystyle H(s)}$ driven by a complex sinusoid of unit amplitude,

${\displaystyle x(t)=e^{i\omega t}\ }$

the output is

${\displaystyle {\begin{aligned}y(t)&=H(i\omega )\ e^{i\omega t}\ \\&=\left({\big |}H(i\omega ){\big |}e^{i\phi (\omega )}\right)\ e^{i\omega t}\ \\&={\big |}H(i\omega ){\big |}\ e^{i\left(\omega t+\phi (\omega )\right)}\ \\\end{aligned}}\ }$

where the phase shift ${\displaystyle \displaystyle \phi }$ is

${\displaystyle \phi (\omega )\ {\stackrel {\mathrm {def} }{=}}\ \arg \left\{H(i\omega )\right\}\;.}$

Negative group delay

• Figure 2: Negative group delay filter circuit
• 
  Circuit with negative group delay of ${\displaystyle \displaystyle \tau _{g}}$ = −RC = −1 ms for frequencies much lower than 1⁄RC = 1 kHz.
• 
  LTspice AC simulation of ${\displaystyle \displaystyle \tau _{g}}$ from 1 Hz (${\displaystyle \displaystyle \tau _{g}}$ ≅ −1 ms) to 10 kHz (${\displaystyle \displaystyle \tau _{g}}$ ≅ 0 ms).
• 
  Transient simulation of an input (green) wave whose output (red) is ahead by 1 ms, but with instability when the input turns on and off.

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). 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.^[8] Circuits with negative group delay (e.g., Figure 2) are possible, though causality is not violated.^[9]

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.^[10]