Analog Communication - Samppng

So far, we have discussed about continuous-wave modulation. We will discuss about pulse modulation in the next chapter. These pulse modulation techniques deal with discrete signals. So, now let us see how to convert a continuous time signal into a discrete one.

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 shows a continuous-time signal x(t) and the corresponding 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 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 $T_s$. Reciprocal of the samppng period is known as samppng frequency or samppng rate $f_s$.

Mathematically, we can write it as

$$f_s= frac{1}{T_s}$$

Where,

$f_s$ is the samppng frequency or the samppng rate

$T_s$ is the samppng period

Samppng Theorem

The samppng rate 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 of the given signal W.”

Mathematically, we can write it as

$$f_sgeq 2W$$

Where,

$f_s$ is the samppng rate

$W$ is the highest frequency of the given signal

If the samppng rate is equal to twice the maximum frequency of the given signal W, then it 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 continuous-time signal x(t), which is band-pmited in the frequency domain is represented as shown in the following figure.

If the signal is sampled above Nyquist rate, then 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 there is over-lapping of information, 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 low-frequency component in the spectrum of its sampled version.”

Hence, the samppng rate of the signal is chosen to be as Nyquist rate. If the samppng rate is equal to twice the highest frequency of the given signal W, then the sampled signal would look pke the following figure.

In this case, the signal can be recovered without any loss. Hence, this is a good samppng rate.