What Is Group Delay In Digital Filters?

Group Delay

A more commonly encountered representation of filter phase response is called the group delay, divers past

$\displaystyle \zbox {D(\omega) \isdefs - \frac{d}{d\omega} \Theta(\omega).} \qquad\hbox{(Group Delay)}$

For linear phase responses, i.eastward.,

$\Theta(\omega) = -\alpha\omega$ for some constant $\alpha$ , the group delay and the phase delay are identical, and each may be interpreted equally time filibuster (equal to $\alpha$ samples when

$\omega\in[-\pi,\pi]$ ). If the phase response is nonlinear, then the relative phases of the sinusoidal signal components are generally altered past the filter. A nonlinear phase response commonly causes a ``smearing'' of assail transients such as in percussive sounds. Another term for this type of phase distortion is phase dispersion. This tin can be seen below in §7.6.five.

An example of a linear phase response is that of the simplest lowpass filter, $\Theta(\omega) = -\omega T/2 \,\,\Rightarrow\,\, P(\omega)=D(\omega)=T/2$ . Thus, both the phase delay and the group delay of the simplest lowpass filter are equal to one-half a sample at every frequency.

For whatsoever reasonably smooth stage role, the group delay $D(\omega)$ may exist interpreted as the time delay of the amplitude envelope of a sinusoid at frequency $\omega$ [63]. The bandwidth of the aamplitude envelope in this interpretation must exist restricted to a frequency interval over which the phase response is approximately linear. We derive this effect in the next subsection.

Thus, the proper name ``group filibuster'' for $D(\omega)$ refers to the fact that it specifies the delay experienced past a narrow-band ``grouping'' of sinusoidal components which take frequencies within a narrow frequency interval about $\omega$ . The width of this interval is limited to that over which $D(\omega)$ is approximately constant.

Derivation of Group Delay as Modulation Delay

Suppose we write a narrowband signal centered at frequency $\omega_c$ equally

$\displaystyle x(n) = a_m(n) e^{j\omega_c n} \protect$

(8.6)

where $\omega_c$ is defined as the carrier frequency (in radians per sample), and is some ``lowpass'' amplitude modulation signal. The modulation can be complex-valued to represent either phase or amplitude modulation or both. By ``lowpass,'' we mean that the spectrum of is concentrated near dc, i.due east.,

$\displaystyle a_m(n) \isdefs \frac{1}{2\pi} \int_{-\pi}^{\pi} A_m(\omega)e^{j\... ... \frac{1}{2\pi} \int_{-\epsilon}^{\epsilon} A_m(\omega)e^{j\omega n} d\omega,$

for some

$\left\vert\epsilon\right\vert\ll\pi$ . The modulation bandwidth is thus bounded by

$2\epsilon\ll\pi$ .

Using the above frequency-domain expansion of , can be written as

$\displaystyle x(n) \eqsp a_m(n) e^{j\omega_c n} \eqsp \left[\frac{1}{2\pi} \int_{-\epsilon}^{\epsilon} A_m(\omega)e^{j\omega n} d\omega\right] e^{j\omega_c n},$

which we may view as a scaled superposition of sinusoidal components of the grade

$\displaystyle x_\omega(n) \isdefs A_m(\omega)e^{j\omega n} e^{j\omega_c n} = A_m(\omega)e^{j(\omega+\omega_c) n}$

with $\omega$ near 0. Let u.s.a. at present pass the frequency component

$x_\omega(n)$ through an LTI filter having frequency response

$\displaystyle H(e^{j\omega}) = G(\omega) e^{j\Theta(\omega)}$

to go

$\displaystyle y_\omega(n) = \left[G(\omega_c+\omega)A_m(\omega)\right] e^{j[(\omega_c +\omega) n + \Theta(\omega_c+\omega)]}. \protect$

(8.7)

Assuming the stage response

$\Theta(\omega)$ is approximately linear over the narrow frequency interval

$\omega\in[\omega_c-\epsilon,\omega_c+\epsilon]$ , we can write

$\displaystyle \Theta(\omega_c+\omega)\;\approx\; \Theta(\omega_c) + \Theta^\prime(\omega_c)\omega \isdefs \Theta(\omega_c) - D(\omega_c)\omega,$

where

$D(\omega_c)$ is the filter group delay at $\omega_c$ . Making this substitution in Eq. $\,$ (vii.7) gives

$\begin{eqnarray*} y_\omega(n) &=& \left[G(\omega_c+\omega)A_m(\omega)\right] e^... ...\right] e^{j\omega[n-D(\omega_c)]} e^{j\omega_c[n-P(\omega_c)]}, \end{eqnarray*}$

where nosotros also used the definition of phase delay, $P(\omega_c) = -\Theta(\omega_c)/\omega_c$ , in the terminal step. In this expression we can already see that the carrier sinusoid is delayed by the stage delay, while the amplitude-envelope frequency-component is delayed by the grouping filibuster. Integrating over $\omega$ to recombine the sinusoidal components (i.eastward., using a Fourier superposition integral for ) gives

$\begin{eqnarray*} y(n) &=& \frac{1}{2\pi}\int_{\omega} y_\omega(n) d\omega \\ &... ...)]}\\ &=& a^f[n-D(\omega_c)] \cdot e^{j\omega_c[n-P(\omega_c)]} \end{eqnarray*}$

where denotes a zero-phase filtering of the amplitude envelope by $G(\omega+\omega_c)$ . We see that the amplitude modulation is delayed by $D(\omega_c)$ while the carrier moving ridge is delayed by $P(\omega_c)$ .

We accept shown that, for narrowband signals expressed every bit in Eq. $\,$ (seven.half dozen) as a modulation envelope times a sinusoidal carrier, the carrier wave is delayed by the filter stage delay, while the modulation is delayed by the filter group delay, provided that the filter phase response is approximately linear over the narrowband frequency interval.

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Group Delay Examples in Matlab
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