# Group delay and phase delay

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All signal components are delayed when passing through a device such as an amplifier or a loudspeaker. The signal delay can be (and often is) different for different frequencies. The delay variation means that signals consisting of different frequency components suffer delay (or time) distortion. A small delay variation is usually not a problem, but larger delays can cause trouble such as poor fidelity and intersymbol interference. High speed modems use adaptive equalizers to compensate for group delay.

## Contents

### Introduction

Group delay is a useful measure of time distortion, and is calculated by differentiating the insertion phase response of the device under test (DUT) versus frequency. Another way to say this is that group delay is a measure of the slope of the transmission phase response. The linear portion of the phase response is converted to a constant value (representing the average signal-transit time) and deviations from linear phase are transformed into deviations from constant group delay. The variations in group delay cause signal distortion, just as deviations from linear phase cause distortion. Group delay is just another way to look at linear phase distortion.

In LTI system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, $\displaystyle x(t)$, to output signal, $\displaystyle y(t)$, of an LTI system is governed by:

Or, in the frequency domain,

where

and

Here $\displaystyle h(t)$ is the time domain impulse response of the LTI system and $\displaystyle X(s)$, $\displaystyle Y(s)$, $\displaystyle H(s)$, are the Laplace transforms of $\displaystyle x(t)$, $\displaystyle y(t)$, and $\displaystyle h(t)$, respectively. $\displaystyle H(s)$ is called the transfer function of the LTI system and, as does the impulse response, $\displaystyle h(t)$, fully defines the input-output characteristics of the LTI system.