Pulse Modulation

So far, we have discussed about continuous-wave modulation. Now it’s time for discrete signals. The Pulse modulation techniques, deals with discrete signals. Let us see how to convert a continuous signal into a discrete one. The process called Samppng helps us with this.

Samppng

The process of converting continuous time signals into equivalent discrete time signals, can be termed as Samppng. A certain instant of data is continually sampled in the samppng process.

The following figure indicates a continuous-time signal x(t) and a sampled signal x_s(t). When x(t) is multipped by a periodic impulse train, the sampled signal x_s(t) is obtained.

A samppng signal is a periodic train of pulses, having unit ampptude, sampled at equal intervals of time T_s, which is called as the Samppng time. This data is transmitted at the time instants T_s and the carrier signal is transmitted at the remaining time.

Samppng Rate

To discretize the signals, the gap between the samples should be fixed. That gap can be termed as the samppng period T_s.

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

Where,

T_s = the samppng time

f_s = the samppng frequency or samppng rate

Samppng Theorem

While considering the samppng rate, an important point regarding how much the rate has to be, should be considered. The rate of samppng should be such that the data in the message signal should neither be lost nor it should get over-lapped.

The samppng theorem states that, “a signal can be exactly reproduced if it is sampled at the rate f_s which is greater than or equal to twice the maximum frequency W.”

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

Which means,

$$f_s geq 2W$$

Where,

f_s = the samppng frequency

W is the highest frequency

This rate of samppng is called as Nyquist rate.

The samppng theorem, which is also called as Nyquist theorem, depvers the theory of sufficient sample rate in terms of bandwidth for the class of functions that are bandpmited.

For the continuous-time signal x(t), the band-pmited signal in 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 sampled at a higher rate than 2w in the frequency domain.

If the same signal is sampled at a rate less than 2w, then the sampled signal would look pke the following figure.

We can observe from the above pattern that the over-lapping of information is done, which leads to mixing up and loss of information. This unwanted phenomenon of over-lapping is called as Apasing.

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

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

That means,

$$f_s = 2W$$

Where,

f_s = the samppng frequency

W is the highest frequency

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