Group delay and phase delay

Among the group Duration (English group delay) is the time it takes for an excitation signal in the form of a wave group to go through an LTI system such as an electronic filter or a transmission channel. A wave group has the appearance of a wave packet or a wave train.

Illustration and meaning

The right figure shows a wave group, before and after passing through a LTI system (hereinafter referred to as " system " ), such as an electronic filter or a transmission line. The so-called envelope is shown as a blue dashed line, the carrier frequency in purple and the group delay between input and output signal denoted by. The group delay is correlated with the temporal offset of the envelope.

The transit time of a wave group with a system depends on the characteristics of the system and the carrier frequency of the wave group. The group delay time often has a nonlinear dependence of the frequency. Is the group delay time, however constant, then the processing time of various wave groups at different frequencies are the same. That is, when passing through the system to view all the same delay wave groups, and therefore the relative position of the groups to one another is maintained. For this reason, no " melts " a broadband signal when passing such a system. Based on the optical physics virtually no dispersion occurs.

Has a system of a linear phase characteristic, it has a constant group delay.

A constant group delay is desirable, for example in case of transmission routes, so that the phase response of a useful signal to be transmitted is retained in the transfer if possible. The transfer function of a coaxial cable has, for example, to a good approximation a linear phase response and thus a constant group delay.

Definition

The system is a transfer function with magnitude frequency response and phase response is based, so that applies:

Respectively. :

The group delay is given by from the phase transition:

For a given transfer function in Laplacebildbereich the group delay can be calculated directly ( without going through the phase diagram):

Mathematical Description

For the derivation of the group delay time studying the response of a system to excitation with a signal in the form of a wave group. The excitation signal should therefore not be as well as the response of the system are real, complex. The wave group can write as a product of a real envelope with a real harmonic fixed frequency (carrier frequency):

The corresponding amplitude density spectrum is

The magnitude frequency response of the envelope is intended to be restricted to a sufficiently small frequency range and sufficiently quickly disappear outside this range. Sufficiently small and fast in this context means that the magnitude frequency response of the transfer function and the phase curve in the fields and can be sufficiently well approximated by linear functions.

For systems with real impulse response of the magnitude frequency response is an even function in. In the areas around therefore approximately true:

For systems with real impulse response of the phase is an odd function in. In the areas around therefore approximately true:

The constants and are obtained by Taylor expansion of the phase transition. Specifically applies.

By applying the transfer function in the frequency range of the wave group and by inverse Fourier transform under the assumptions made, at the output of the system response:

It can be seen on the envelope argument that with the maturity of the group is associated. With the definition of the group delay

Identifies you. Thus the envelope of the group is delayed by exactly one group delay time after passing through the system.

The group delay time is different from the phase delay time. In this example, the phase delay is:

Examples

First order low pass

It is the group delay of a low-pass filter (linear continuous first-order system ) can be specified. This system can be implemented for example as an RC element. From the frequency response

Yields the phase transition to:

This is a filter-dependent constant. The group delay for this low-pass filter is given by:

Acoustics

A frequency-independent as possible group delay is also in electro-acoustics, especially for a natural sound, of importance. Many components of an audio reproduction chain, such as the loudspeaker crossover (crossover ), alter the group delay of the signal. But the architecture of the listening room has a significant impact here. Acoustic resonators can also affect the group delay as components of loudspeaker systems. It is important to know the perception threshold for audibility of group delay changes as a function of frequency, especially when the audio chain is designed for hi-fi reproduction. Tables can be found in Blauert and Laws: