# Pulse modulation

Principles of communication
2020-11-20 02:10:33
# Pulse modulation

So far we have discussed continuous wave modulation. Now is the time for discrete signals. ** pulse modulation ** techniques deal with discrete signals. Let's see how to convert a continuous signal to a discrete signal. The process called Sampling helps us with this.

## Sampling

The process of converting continuous time signals to equivalent discrete time signals can be called ** Sampling **. A certain instant of data is continuously sampled in the sampling process.

The following figure shows a continuous time signal ** x (t) ** and a sampled signal ** x **_{ s } (t) . When ** x (t) ** is multiplied by a periodic pulse train, the sampled signal ** x **_{ s } (t) is obtained.

A ** the sampling signal ** is a periodic train of pulses, having a ** unit d 'amplitude **, sampled at equal time intervals **T**_{s} , which is called as ** sampling time **. This data is transmitted at the **T**_{s} time instants and the carrier signal is transmitted at the remaining time.

## Sampling frequency

To discretize the signals, the difference between the samples must be corrected. This difference can be termed as ** sampling period ** ** T **_{ s } .

$$ Sampling: Frequency = frac {1} {T_s} = f_s $$

Where,

** T **_{ s } = the sampling time

** f **_{ s } = the sampling rate or sampling rate

## Sampling theorem

While considering the sampling rate sampling, an important point concerning the value of the rate must be considered. The ** sampling rate ** should be such as the dataof the message signal are neither lost nor exceeded.

The ** sampling theorem ** states that "a signal can be exa reproduced correctly if it is sampled at the frequency ** f **_{ s } which is greater than or equal to twice the maximum frequency W. ”

Put it in simpler words, for efficient reproduction of the original signal, the frequency of The sampling should be twice the higher frequency.

Which means,

$$ f_s geq 2W $$

Where,

** f **_{ s } = the sample rate

** W ** is the highest frequency

This sampling rate is called ** Nyquist rate **.

The sampling theorem, also called ** Nyquist theorem **, provides the theory of a sufficient sampling frequency in terms of bandwidth for the class of functions which arelimited band.

For the continuous time signal ** x (t) **, the band limited signal in the frequency domain, can be represented as shown in the following figure.

If the signal is sampled above the Nyquist rate, the original signal can be recovered. The following figure explains a signal, if it is sampled at a rate greater than 2w in the frequency domain.

If the same signal is sampled at a frequency lower than 2w, then the sampled signal will look like the following figure.

It can be seen from the above diagram that the overlap of information is done, which leads to mixing and loss of information . This unwanted phenomenon of overlap is called ** Aliasing **.

Aliasing can be called "the phenomenon of a high frequency component in the spectrum of a signal, assuming the identity of a low frequency component in the spectrum of its sampled version. "

Therefore, the signal sampling is chosen to be freeNyquist quence, as indicated in the sampling theorem. If the sampling rate is twice the higher frequency (2W).

This means,

$$ f_s = 2W $$

Where,

** f **_{ s } = the sample rate

** W ** is the highest frequency

The result will be as shown in the figure above. The information is replaced without any loss. So this is a good sample rate.