physics

Group delay and phase delay

Group delay and phase delay is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Group delay and phase delay rather than just read about it. In short: 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…

Key takeaways

Group delay and phase delay belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Group delay and phase delay to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Group delay and phase delay from memory before moving on to harder problems.

Reference excerpt

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 sin ⁡ ( x ) {\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.

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.

… excerpt ends here. Continue reading the full article.

Illustrations

Group delay and phase delay illustration

Worked examples

Example 1 — a first encounter with Group delay and phase delay

Start with the simplest possible case. Write down what Group delay and phase delay claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Group delay and phase delay before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Group delay and phase delay ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Group delay and phase delay

In research: Group delay and phase delay appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Group delay and phase delay in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Group delay and phase delay is common in secondary-school and first-year university syllabi. It links to neighbouring topics Electrical engineering, Optics, Signal processing, so understanding it makes those chapters shorter.
In everyday life: Look for Group delay and phase delay outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Group delay and phase delay in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Group delay and phase delay means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Group delay and phase delay out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Group delay and phase delay in simple terms?

Why does Group delay and phase delay matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Group delay and phase delay?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Group delay and phase delay.