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Table of contents
1.
Introduction
2.
What is the Sampling Theorem?
3.
What is the Need of Sampling?
4.
Nyquist Criterion for Sampling
5.
Sampling Theorem for Low Pass Signals
6.
Proof of Sampling Theorem
7.
Types of Sampling 
7.1.
Impulse Sampling
7.2.
Natural Sampling
7.3.
Flat Top Sampling
8.
Practical Implications of Sampling Theorem
9.
Applications of Sampling Theorem
10.
What is the Aliasing Effect in Sampling?
11.
Frequently Asked Questions
11.1.
What is meant by the sampling theorem?
11.2.
What is the formula for sampling theory?
11.3.
What is sampling theory in mathematics?
11.4.
What is Nyquist formula?
12.
Conclusion
Last Updated: Apr 18, 2024
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Sampling Theorem

Author Nitika
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23 Jul, 2024 @ 01:30 PM

Introduction

The sampling theorem plays an important role in digital signals and processing along with advanced control operations. In this article, we will explore the depth of the sampling theorem, with its types and applications.

sampling theorem

What is the Sampling Theorem?

The sampling theorem simply converts continuous-time signals into discrete time signals.

The statement for sampling theorem can be given in two forms-

  • A band-limited signal of finite energy, which has no frequency component higher than fm Hz, is completely described by its sample values at uniform intervals less than or equal to 1/2fm second apart.
     
  • A band-limited signal of finite energy, which has no frequency components higher than fm Hz, may be completely recovered from the knowledge of its samples taken at the rate of 2fm samples per second.

 

Combining the above two forms, the sampling theorem states that: 

A continuous-time signal may be completely represented in its samples and recovered back if the sampling frequency is fs≥ 2fm. Here fs is the sampling frequency, and fm is the maximum frequency present in the signal.

The sampling frequency is Fs = 1Ts

For the signal to be sampled, it needs to go through the sampling process where a sufficient number of samples of the signal are taken to represent it entirely in its samples. The number of samples depends on the maximum signal frequency present in the signal.

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What is the Need of Sampling?

Sampling is an essential element in the world of electronics, which is why sampling is needed in various steps. We will learn where and how sampling is used in further sections of this article.

Sampling is needed for the conversion of continuous-time signals in areas where the input signals are discrete-time signals. Most of the Digital Signal Processing methods use sampling for the same. Also, the continuous-time signals are more challenging in terms of transmission. Hence sampling comes into play to convert them into discrete time signals.

Sampling is also needed in noise reduction techniques in channels.

In addition to that, the conversion of analog signals ad digital signals increases the capability between systems of that kind. It also helps in interfacing as well.

Nyquist Criterion for Sampling

When the sampling frequency equals twice the input signal,

Fs =2Fm

It is said to be the Nyquist Rate for sampling.

 

When the sampling frequency is less than twice the input frequency, 

Fs <2Fm

It is called an Aliasing effect.

Sampling Theorem for Low Pass Signals

The Sampling Theorem states that the sampling frequency must be at least twice the highest frequency present in the signal to reconstruct a continuous low-pass signal from its samples accurately. In other words, if we want to capture all the essential information of a signal, we need to sample it at a rate greater than or equal to twice its highest frequency component. This prevents "aliasing," where high-frequency content appears as lower frequencies, distorting the signal during reconstruction. Following the Sampling Theorem ensures we can faithfully represent and recover the original signal from its sampled version without losing essential details.

Proof of Sampling Theorem

Let us assume

x(t) is a continuous-time signal 

and,

x(t) is 0 for |ω|>ωm

Proof of Sampling Theorem

The above image shows x(t) is subjected to sapling by multiplying it with an impulse train δ(t)

The output obtained by the multiplier will be a discrete-time signal y(t) (sampled) 

 

Sampled signal

 equation-1

The trigonometric Fourier series representation of δ(t) is

equation-2

Where, 

an and bn sample

Substituting values in ……(2)

Theta(n) value

Substituting values in ……(1)

y(t) value

Taking Fourier transform on both sides,

Final result

Where n=0,±1,±2,..

Types of Sampling 

Sampling can be done in various ways but there are three general types of sampling techniques:

  • Impulse sampling
     
  • Natural sampling
     
  • Flat Top sampling
     

Let us learn about these types of sampling in detail:

Impulse Sampling

Also known as instantaneous or impulse training, this method has the sampling signal as a periodic impulse train. The area of each impulse in the sampled signal equals the instantaneous value of the input signal x(t). As a result, this method is unsuitable for transmission purposes.

Natural Sampling

Also called practical sampling, this method has the sampling signal as a pulse train. The top of each pulse in the sampled signal retains the shape of the input signal x(t) during the pulse interval. The structure of the top of the pulse at the receiver end is very difficult to determine.

Flat Top Sampling

In this sampling type, the sampling signal is also a pulse train. The top of each pulse in the sampled signal remains constant and is equal to the instantaneous value of the input signal 𝑥(𝑛) at the start of the samples.

Practical Implications of Sampling Theorem

The sampling theorem has its implications in the following areas:

  1. Discrete-Time Processing of Continuous-Time Signal
    The sampling formula and the reconstruction technique are used for the discrete-time processing of continuous-time signals. When applying a linear time-invariant filter to a band-limited signal, the output is a signal with the same band limit.
    This makes use of digital computing power and flexibility to be used in continuous-time signal processing.
     
  2. Psychoacoustics
    Sampling also has its uses in Psychoacoustics, like digital devices that use sampling rates associated with the frequency range of human vocalizations and auditory sensitivity. 

Also, discarding frequencies greater than or equal to 4 kHz by the use of an anti-aliasing filter is essential to avoid aliasing in telephone systems. If no filter is added, it could have a negative impact on the 

Applications of Sampling Theorem

Here are a few of the applications of the sampling theorem:

  • Used for maintaining sound quality in music recordings
     
  • Used in the conversion of analog to discrete forms
     
  • Used in speech and pattern recognition systems.
     
  • Used in Modems
     
  • Used in sensor data evaluation systems
     
  • Used in Radar and radio navigation systems
     
  • Used in digital watermarking, biometric identification systems, and surveillance systems.

What is the Aliasing Effect in Sampling?

The Aliasing Effect in sampling occurs when a continuous signal is not adequately sampled, resulting in distorted or misleading representations during reconstruction. If the sampling rate is too low and doesn't meet the requirements of the Sampling Theorem, high-frequency components get incorrectly represented as lower frequencies, leading to overlapping signals. This causes information loss and ambiguity in the sampled data, making it challenging to reconstruct the original signal accurately. Aliasing is a crucial consideration in signal processing, and avoiding it by adhering to the Nyquist-Shannon sampling criterion ensures faithful reproduction of the original signal from its samples.

Frequently Asked Questions

What is meant by the sampling theorem?

The Sampling Theorem, also known as Nyquist-Shannon Sampling Theorem, states that when we want to convert a continuous analog signal into a digital form (sampling), we must sample it at a rate at least twice the highest frequency in the analog signal. 

What is the formula for sampling theory?

If the continuous-time low-pass signal x(t) has a band limit such that 

For ω≥ωmax, 𝑥(ω)=0 

is seen through the samples it provides.

What is sampling theory in mathematics?

In mathematics, sampling theory deals with the ideas and procedures for choosing a sample, or subset of data points, from a larger population in order to draw conclusions about the features of the population with a given degree of accuracy and confidence. 

What is Nyquist formula?

The Nyquist formula states that the minimum sampling rate needed to accurately represent a signal is twice the highest frequency present in the signal.

Conclusion

In this blog, we have thoroughly discussed on sampling theorem. We have discussed the need for the sampling theorem, its types, and applications. 

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