# Sampling Theorem | Sampling In PCM | Digital Communication

The sampling theorem, also known as the Nyquist-Shannon sampling theorem, is a fundamental concept in signal processing and digital communication.

Sampling is a process of converting a continuous-time analog signal into discrete-time signal. This conversion is done with the help of sampler.

Sampling is the second step in Pulse Code Modulation (PCM) technique where we convert analog signal into digital signal. In PCM after sampling amplitude of the discrete-time signal is quantized with the help of quantizer and then this quantized signal is coded to the binary sequence using encoder, allow the computer to process the signal, then transmitted to the far end receiver then it is converted back to the original signal.

## Need for Sampling

Most of the real-life signals, such as audio, video, temperature etc., are continuous in nature and represented as analog signals. However, for efficient processing, storage, and transmission, these analog signals need to be converted into digital form. Sampling make possible this conversion by taking a limited number of discrete data points from the continuous signal.

Sampling convert continuous-time, continuous amplitude signal into discrete-time continuous amplitude signal.

Analog signals are more susceptible to noise and external interference, which can degrade signal quality. Through sampling and subsequent digital processing, noise can be filtered out or reduced, resulting in cleaner and more reliable data.

## What is the Sampling Theorem?

The sampling theorem, also known as the Nyquist-Shannon theorem, it is a fundamental principle that guides the process of converting analog signals to digital form.

The sampling theorem states that “if a signal is sampled at regular time intervals (ts), then the sequence of samples can be reconstructed or recreate the original signal. when the sampling rate is at least twice the highest frequency component present in the signal.”

Mathematically,

if a massage signal (continuous signal ) is denoted as m(t), and its highest frequency component is denoted as fm, then the sampling theorem states that to accurately reconstruct m(t) from its samples, the sampling frequency (or rate), denoted as fs, must be greater than or equal to twice the highest frequency component:

fs ≥ 2 * fm

When this condition is met, the original signal can be reconstructed from its samples using techniques like interpolation or various digital signal processing methods. The process of converting a continuous signal into discrete samples is known as “sampling” or “digitization.”

## What is the Nyquist Rate?

The Nyquist rate, also known as the Nyquist frequency or Nyquist limit, is the minimum sampling rate required to accurately represent a continuous signal in a discrete form without introducing distortion or aliasing. It is a fundamental concept in signal processing and digital communication.

The Nyquist rate is defined as follows:

“The Nyquist rate is equal to twice the highest frequency component present in a continuous signal.”

Mathematically,

if the highest frequency component in a signal is denoted as fm, then the Nyquist rate, denoted as fs (the minimum sampling frequency), is given by:

fs = 2 * fm

If the sampling rate is less than the Nyquist rate, aliasing can occur, where higher-frequency components of the signal “overlap” into lower-frequency ranges, making it impossible to accurately reconstruct the original signal from the samples.

### What is sampling?

Sampling is a process of converting a continuous-time analog signal into discrete-time signal. This conversion is done with the help of sampler.

### Why do we need to sample signals in digital communication?

Digital communication systems operate with discrete signals, and most information sources generate continuous analog signals. Sampling allows us to represent these analog signals in a digital format, making it easier to process, transmit, and store them.

### Statement of sampling theorem

The sampling theorem states that “if a signal is sampled at regular time intervals (ts), then the sequence of samples can be reconstructed or recreate the original signal. when the sampling rate is at least twice the highest frequency component present in the signal.”
Mathematically,
if a massage signal (continuous signal ) is denoted as m(t), and its highest frequency component is denoted as fm, then the sampling theorem states that to accurately reconstruct m(t) from its samples, the sampling frequency (or rate), denoted as fs, must be greater than or equal to twice the highest frequency component:
fs ≥ 2 * fm

### what is Nyquist rate?

The Nyquist-Shannon sampling theorem states that to accurately represent a continuous signal, the sampling rate must be at least twice the highest frequency component in the signal (Nyquist rate). It’s crucial to avoid aliasing and information loss during sampling.

### How do I choose the right sampling rate for a signal?

Determine the highest frequency component (f_max) in your signal and choose a sampling rate (fs) that is at least twice f_max (fs ≥ 2 * f_max) to satisfy the Nyquist theorem.