Phase and group delays
Whenever a signal is transmitted through a dispersive, that is frequency-selective system, such as a communication channel, some delay is introduced into the output signal in relation to the input signal. The delay is determined by the phase response of the system. For the convenience of presentation, let φ(w) = arg{H(jw)} denote the phase response of a dispersive communication channel, where H(jw ) is the frequency response of the channel. Suppose that a sinusoidal signal is transmitted through the channel at a frequency wc. The signal received at the channel output lags the transmitted signal by φ(wc) radians. The time delay corresponding to this phase lag is called the phase delay of the channel.
Though, we must keep in mind that the phase delay is not necessarily the true signal delay. This follows from the fact that a sinusoidal signal has infinite duration, with each cycle exactly like the preceding one. Such a signal does not convey information, except for the fact that it is just there to speak of it. It would therefore be incorrect to deduce from the preceding reasoning that the phase delay is the true signal delay. In reality, as we know, information can be transmitted through a channel by only applying some form of modulation to a carrier.
Suppose that the modulation frequency w0 is smaller compared with the carrier frequency wc, which implies that the side frequencies w1, and w2, are close together, with wc between them. This modulated signal is known as a narrowband signal. Then we may approximate the phase response in the vicity of w=wc, by the two-term Taylor series expansion.
While solving the equation for the Taylor series expansion we will find a time delay incurred by the message signal, or the envelope of the modulated signal. The time delay is called the envelope delay or group delay. Therefore, the group delay is defined as the negative of the derivative of the phase response of the channel with respect to w, evaluated at carrier frequency wc.
In general, we find that when a modulated signal is transmitted through a communication channel, there are two different types of delays to be considered:
- The carrier or phase delay
- The envelope or group delay.
The group delay is the true signal delay.
So now that we know that the group delay is the true signal delay when a modulated signal is transmiited through a communication channel, we will ask ourselves about the practical importance of group delay.
As we have seen previously, group delay strictly applies to modulated signals that are narrowband; that is, the bandwidth of the message signal is small compared with the carrier frequency. It is only when this condition is satisfied that we would have justified in using the two-term approximation for the phase response.
There are many cases where the narrowband assumption is not true because the message bandwidth is comparable to the carrier frequency. In these situations, the group delay is formulated as a frequency-dependent parameter.
When a wideband modulated signal is transmitted through a dispersive channel, the frequency components of the message signal are delayed by different amounts at the channel output. Consequently, the message signal undergoes a form of linear distortion known as delay distortion.
To reconstruct a faithful version of the original message signal in the receiver, we use a delay equalizer. This equalizer has to be designed in such a way that when it is connected in a cascade with the channel, the overall group delay is constant.
One practical example would be to consider the ubiquitous telephone channel, the useful frequency band of which extends from about 0.1 to 3.1 kHz. Over this band of frequencies, the magnitude response of the band is considered to be essentially constant, so that there is little amplitude distortion. On the other hand, the group delay of the channel is highly dependent on frequency. As far as telephonic communication is concerned the variation of group delay in the channel with frequency is of no real consequence because our ears are relatively insensitive to delay distortion. However, in the case when wideband data are transmitted over a telephone channel. For example, for a data rate of 4 kilobits per second. So, in the telephone channel, the group delay varies from zero to several milliseconds. Accordingly, delay distortion is extremely harmful to wideband data transmission over a telephone channel. In such an application delay equalization is essential for satisfactory operation.
To summarize phase and group delay, we can say the group delay is the true signal; and is of paramount importance when it comes to the application of wideband modulated signal. Both the phase and group delay are defined in terms of the phase response of a channel over which a modulated signal is transmitted.
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