# What is Sampling Theorem?

By Aina Parasher|Updated : August 26th, 2022

By using the sampling theorem, we convert the continuous-time analog signal into a discrete-time signal, then the amplitude of the discrete-time signal is quantized and coded to the binary sequence to allow the computer to process the signal, then transmitted to the far end receiver then it is converted back to the original signal.

In this way, sampling has a fair share of importance in the development of digital processing and advanced control operations. In this article we will discuss the sampling theorem, to understand the sampling theorem, we must have a basic knowledge of what is sampling in signal processing, and along with that, we will discuss the Nyquist sampling theorem and the significance of sampling.

## What is the Sampling Theorem?

The sampling theorem states that a continuous-time signal needs to be uniformly sampled at a minimum rate in order to recover or reconstruct the original signal.

### Sampling Theorem Statement

The sampling theorem states that if a signal is sampled at regular intervals, then the sequence of samples can be reconstructed to recreate the original signal.

Most of the signals that we use frequently in communication and control operations are analog signals. Even though it is easier to handle analog signals in a simple operation like a simple telephone system where we convert the sound signal into its analogous electrical signal in the form of voltage, where the amplitude of the voice signal reflects in the voltage of the electrical signal and transmitted at the velocity equal to that of the light. But if they are to be sent to longer distances, the strength of the signal decreases and leads to an increase in noise levels. Hence modern telephonic systems have employed the concept of digital processing of the voice, in which the signal is converted into digital code that can be handled by a computer. If we wish to process the signal with digital equipment like a computer it involves various processes called sampling, quantizing, and coding.

## Sampling Theorem Formula

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

𝑥(ω)=0 for ω≥ωmax

is represented in the form of its samples.

Then x(t) can be recovered in its original form if the sampling frequency is greater than or equal to twice the maximum frequency of the message signal x(t).

If ωs≥2ω𝑚𝑎𝑥 (Nyquist sampling rate condition);

x(nTs) = x(t), n=0, ±1, ±2, ±3, ……

Here Ts is the sampling period (sec/sample).

The Nyquist sampling rate condition can also be written as 𝑓𝑠=1𝑇𝑠≥𝜔𝑚𝑎𝑥𝜋

Here fsis the sampling frequency (sample/second).

• If the Nyquist sampling rate condition is satisfied, then the original signal x(t) can be recovered by passing the sampled signal through an ideal low pass filter with the frequency response H(ω)=Ts; when -ωs/2<ω<ωs/2and equal to zero elsewhere.

## Important Aspects of Sampling Theorem

In the process of sampling an analog signal, one must choose an extremely small value for the sampling period to make sure that there is no significant difference between the analog signal and discrete signal in the perspective of information contained as well as visual appearance. In such representation, in discrete form, some redundant values may come across, but we can spare them without losing the information in the original signal. In case we opt for a large value of sampling it may help us in data compression, but it comes with the significant risk of losing the information in the original signal.

Hence choosing the appropriate value of sampling is a vital measure in recovering the information from the message signal, the sampling theory and some other results introduced by Nyquist and Shannon constitute the bridge between analog and digital signals.

## What are the Types of Sampling?

To convert the continuous-time analog signal into the appropriate discrete signal the initial step is to consider the samples of the given analog signal x(t) at uniform times 𝑡=𝑛𝑇𝑠

x(nTs) = x(t)|t=nTs; n is an integer

Here Ts is the sampling period.

We can do the sampling in different ways like

• Pulse amplitude modulation (PAM), and
• Ideal impulse sampling.

### Pulse Amplitude Sampling

Pulse amplitude sampling is a basic approach in digital communication. In this method of sampling, the message signal is modulated with a pulse train to acquire a sequence of narrow pulses having a similar amplitude to the continuous-time signal in the pulse.

Thus, the pulse amplitude modulation is the multiplication of the continuous-time signal x(t) by a periodic signal p(t) that consists of the pulses of the width W, amplitude 1𝑊, and period 𝑇𝑠. The discrete form of the original signal xPAM(t) with a small pulse width W, and the amplitude of the pulses being x(mTs), then

xPAM(t) = x(t)p(t) ≈ 1W∑x (mTs)[u(t-mTs)-u(t-mTs-W)]

As p(t) is a periodic signal we can represent it by Fourier series as p(t)=∑Pkejkω0tk; ω0=2ΠTs

Here Pk is the Fourier series coefficient.

Hence the pulse amplitude modulated signal can be expressed as xPAM(t) = ∑Pkx(t)ejkw0tk

The Fourier transform of the signal is xPAM (ω)=∑PkX(ω-kω0)k

### Ideal Impulse Sampling

This sampling is called ideal sampling because the impulse signal has zero width and captures the signal value at any instant as per our need.

If the pulse width of the periodic pulse train p(t) is much smaller than the sampling period(Ts), then p(t) can be replaced by a periodic impulse train δTs(t), with a period of Ts. This development will considerably simplify the analysis and makes the results easier to grasp.

The sampling function is δTs(t)=∑δ(t-nTs)n

The sampled signal is given by xs(t)=∑x(nTs)δ(t-nTs)n

If X(ω) is the Fourier transform of the signal x(t) then the Fourier signal of xs(t) is Xs(ω)=1Ts∑x(ω-kωs)k

In the frequency domain, the spectrum of the sampled signal contains the copies of the spectrum of the continuous-time signal x(t) repeating at the regular intervals of 𝜔𝑠(sampling frequency. Practically impulse functions are not available hence we use the 𝑠𝑖𝑛𝑐 function in sampling applications.

If x(t) is a band-limited signal with a lowpass spectrum of finite support, that is x(ω)=0 for ω>ωmax as shown below.

Case (1): ωs≥2ωmax

In this case, the spectrum of the sampled signal consists of shifted non-overlapping versions of (1Ts)X(ω). It is possible by obeying Nyquist’s sampling rate condition. In this case, we can recover X(ω) or x(t) from Xs(ω) or Xs(t).

Case(2): ωs<2ωmax

In this case, the spectra of X(ω) are overlapped as the sampling rate is less than that of the Nyquist rate, hence it is not possible to recover the original signal from sampled signal. Due to overlapping of the spectra some frequency components of the original signal will acquire a different frequency, this phenomenon is called frequency aliasing.

## What is the Aliasing Effect in Sampling?

If the sampling frequency is less than Nyquist’s standard frequency (ωs<2ωmax), then high-frequency components of the spectra take the identity of low-frequency components, this phenomenon is known as aliasing. Due to this, it is not possible to reconstruct the original continuous-time signal that has undergone sampling.

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## FAQs on Sampling Theorem

• The sampling theorem states that it is not possible to recover the original signal from the sampled one if a continuous-time signal is sampled with a frequency less than twice the maximum frequency of the original signal or message signal. Hence the sampling frequency ωs must obey the below condition that is known as the Nyquist sampling rate condition.

ωs≥2ω𝑚𝑎𝑥

• In the era of digital processing, it is necessary to process the signals using computers and various other digital equipment. But we cannot do this directly with continuous-time analog signals, we must convert them into the data that can be handled by these digital devices and after the processing, it will be converted back to an original analog signal. For example, in the mobile communication, the analog signal that is the human voice is converted into a discrete-time signal by using proper sampling frequency, then it will undergo quantizing and coding to process the signal, then it can be transmitted to the longer distances at least possible noise to the other end where it is converted back into the analog signal that is the human voice.

• In pulse amplitude modulation a periodic pulse train is used as a multiplier. The analog signal is converted into a discrete-time pulse train that nearly replicates the original signal by choosing the suitable value of sampling frequency. Pulse amplitude modulation is the base for all the digital modulation techniques that we are using.

• In this method of sampling discrete-time impulse train of suitable frequency will be used as the multiplier. The analog signal is converted into the discrete-time signal by multiplication with the impulse train, then the sampled signal will be represented in impulses of different amplitudes. This is known as ideal sampling because the impulse signal has no width, and it can capture any instant on a continuous-time analog signal. We use this method in analytical applications only, practically it is not possible to generate an impulse signal with zero width.

• If a continuous-time signal is sampled at a frequency less than the Nyquist sampling rate (ωs<2ωmax), then it is not possible to recover the original signal from the sampled one, because the high-frequency components of the spectra take the identity of the low-frequency components. We can avoid this effect by selecting the sampling rate greater than or equal to the Nyquist standard rate.

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