# Pulse Modulation - Principles of Communication

## What is Pulse Modulation?

Until know, we have debated about continuous-wave modulation. Currently it’s time for discrete signals. The Pulse modulation techniques, deals with discrete signals. Let us understand how to convert a continuous signal into a discrete one. The process called Sampling helps us with this.

## Sampling

The procedure of changing continuous time signals into corresponding discrete time signals, can be termed as Sampling. A definite instant of data is repeatedly sampled in the sampling process.

The resulting figure specifies a continuous-time signal x(t) and a sampled signal xs(t). When x(t) is multiplied by a periodic impulse train, the sampled signal xs(t) is obtained. A sampling signal is a periodic train of pulses, having unit amplitude, sampled at equal intervals of time Ts, which is called as the Sampling time. This data is transmitted at the time instants Ts and the carrier signal is transmitted at the remaining time.

## Sampling Rate

To discretize the signals, the gap amongst the samples should be fixed. That gap can be termed as the sampling period Ts.

$Sampling\phantom{\rule{mediummathspace}{0ex}}Frequency=\frac{1}{{T}_{s}}={f}_{s}$

Where,

Ts = the sampling time
fs = the sampling frequency or sampling rate

## Sampling Theorem

While seeing the sampling rate, a significant point concerning how much the rate has to be, should be measured. The rate of sampling should be such that the data in the message signal should neither be lost nor it should get over-lapped.

The sampling theorem states that, “a signal can be precisely replicated if it is sampled at the rate fs which is greater than or equal to twice the maximum frequency W.”

To put it in modest words, for the effective reproduction of the original signal, the sampling rate should be twice the highest frequency.

Which means,

${f}_{s}\ge 2W$Where,

fs = the sampling frequency
W is the highest frequency

This rate of sampling is called as Nyquist rate.

The sampling theorem, which is also called as Nyquist theorem, conveys the theory of adequate sample rate in terms of bandwidth for the class of functions that are bandlimited.

For the continuous-time signal x(t), the band-limited signal in frequency domain, can be signified as shown in the resulting figure. If the signal is tested beyond the Nyquist rate, the original signal can be improved. The resulting figure clarifies a signal, if tested at a higher rate than 2w in the frequency domain. If the same signal is tested at a rate less than 2w, then the sampled signal would look like the resulting figure. We can detect from the beyond pattern that the over-lapping of information is done, which indicates to mixing up and loss of information. This annoying phenomenon of over-lapping is called as Aliasing.

Aliasing can be mentioned to as “the phenomenon of a high-frequency constituent in the spectrum of a signal, taking on the identity of a lower-frequency component in the spectrum of its sampled version.”

Therefore, the sampling of the signal is selected to be at the Nyquist rate, as was stated in the sampling theorem. If the sampling rate is equal to twice the highest frequency (2W).

That means,

${f}_{s}=2W$

Where,

fs = the sampling frequency
W is the highest frequency The result will be as presented in the above figure. The information is substituted without any loss. Therefore, this is a good sampling rate.

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