Sampling of Signal

Ankush Mahapatra
7 min readMay 29, 2021

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Any signal has three properties which are voltage or amplitude, frequency and phase. The signals are represented only in an analog form where in the digital form of technology is not available. Analog signals are continuous in time and have different voltage levels for different periods of the signal. The main drawback of this is, the amplitude keeps on changing along with the period of the signal. This can be overcome by the digital form of signal representation. The conversion of an analog form of signal into digital form can be done using the sampling technique. Sampling is a process of breakage of continuous signal into discrete signal. The output of this technique represents the discrete version of its analog signal. In this blog, you can find what is the sampling theorem, definition, applications, and its types.

What is the Sampling Theorem?

Any continuous or an analog signal can be represented in the digital version in the form of samples. These samples are also called discrete points. In sampling theorem, the input signal is in an analog form of signal and the second input signal is a sampling signal, which is a pulse train signal and each pulse is equidistant with a period of ‘Ts’. This sampling signal frequency should be more than twice that of the input analog signal frequency. If this condition satisfies, the analog signal can be perfectly represented in discrete form, otherwise, the analog signal may be losing its amplitude values for certain time intervals. How many times the sampling frequency is more than the input analog signal frequency, in the same way, the sampled signal is going to be a perfect discrete form of signal. And these types of discrete signals do well in the reconstruction process for recovering the original signal.

Input signal frequency is denoted by ‘Fm’ and sampling signal frequency is denoted by ‘Fs’.

The output sample signal is represented by the samples. These samples are maintained with a gap and these gaps are termed as sample period or sampling interval (Ts). The reciprocal of the sampling period is known as ‘sampling frequency’ or ‘sampling rate’. The number of samples represented in the sampled signal is indicated by this sampling rate.

Sampling frequency Fs=1/Ts

sampling-block-diagram

The Sampling theorem states that “continuous form of a time-variant signal can be represented in the discrete form of a signal with help of samples and the sampled (discrete) signal can be recovered to original form when the sampling signal frequency Fs having the greater frequency value than or equal to the input signal frequency Fm.”

Fs >= 2Fm

Thus, If the sampling frequency (Fs) equals twice the input signal frequency (Fm), then such a condition is called the ‘Nyquist Criteria’ for sampling.

When the sampling frequency equals twice the input signal frequency it is known as ‘Nyquist rate’.

Fs = 2Fm

If the sampling frequency (Fs) is less than twice the input signal frequency (Fm), such criteria is called an ‘Aliasing effect’.

Fs < 2Fm

So, there are three conditions that are possible from the sampling frequency criteria. They are the sampling, Nyquist and aliasing states.

Nyquist Sampling Theorem

In the sampling process, while converting the analog signal to a discrete version, the chosen sampling signal is a prime factor. And what are the reasons due to which we get distortions in the sampling output while conversion of analog to discrete? Such questions can be answered by the ‘Nyquist sampling theorem’.

Nyquist sampling theorem states that, the sampling signal frequency (Fs) should be double the input signal’s highest frequency component to get distortion less output signal.

Fs = 2Fm

The Nyquist sampling theorem is named after the famous scientist ‘Harry Nyquist’.

Sampling Output Waveforms

The sampling process requires two input signals. The first input signal is an analog signal whereas the other input is sampling pulse or equidistant pulse train signal. And the output signal which is the sampled signal comes from the multiplier block. The sampling process output waveforms are shown below.

Sampling-output-waveforms

Proof of Sampling Theorem

Let us consider a continuous time signal x(t). The spectrum of x(t) is a band limited to Fm Hz i.e. the spectrum of x(t) is zero for |ω|>ωm.

Sampling of this input signal x(t) can be obtained by multiplying x(t) with an impulse train δ(t) of period Ts. The output of multiplier is a discrete signal which is called a sampled signal which is represented by y(t) in the following diagrams…

Here, you can observe that the sampled signal takes the period of impulse. This process of sampling can be explained via. following mathematical expression:

Sampled signal y(t) = x(t).δ(t) ……(1)

The trigonometric Fourier series representation of δ(t) is given by

which we consider as equation ……(2)
Where,

Substitute above values in equation 2.

Substitute δ(t) in equation 1.

Take the Fourier transform on both sides…

To reconstruct x(t), you must recover input signal spectrum X(ω) from sampled signal spectrum Y(ω), which is possible only when there is no overlapping between the cycles of Y(ω).

Possibility of sampled frequency spectrum with different conditions are given by the following diagrams:

Aliasing Effect

It is the effect in which overlapping of a frequency component takes place at the frequency higher than Nyquist rate i.e. fs > 2fm. Loss of signal may occur due to aliasing effect. We can say that aliasing is the phenomenon in which a high frequency component in the frequency spectrum of a signal takes the identity of a lower frequency component in the same spectrum of the sampled signal.

Aliasing is generally avoided by applying low pass filters or anti aliasing filters (AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitable reconstruction filter should then be used when restoring the sampled signal to the continuous domain or converting a signal from a lower to a higher sampling rate.

Sampling Techniques

There are basically three types of Sampling techniques, namely:
1. Natural Sampling
2. Flat top Sampling
3. Ideal Sampling

1. Natural Sampling

Natural Sampling is a practical method of sampling in which pulses have finite width equal to ‘T’. Sampling is done in accordance with the carrier signal which is digital in nature.

Here, you multiply input signal which is x(t) to pulse train as shown below…

Natural Sampling

2. Flat Top Sampling

Flat top sampling is like natural sampling i.e. it is practical in nature.

In comparison to natural sampling flat top sampling can be easily obtained. During transmission, noise is introduced at top of the transmission pulse which can be easily removed if the pulse is in the form of flat top. Here, the top of the samples are flat i.e. they have constant amplitude. Hence, it is called as flat top sampling. In this sampling technique, the top of the samples is equal to the instantaneous value of the message signal x(t) at the start of the sampling process. Sample and hold circuit are used in this type of sampling.

Flat Top Sampling

3. Ideal Sampling

Ideal Sampling is also known as Instantaneous sampling or Impulse Sampling. A train of impulse is used as a carrier signal for ideal sampling. In this sampling technique the sampling function is a train of impulses and the principle used is known as multiplication principle.

Impulse sampling can be performed by multiplying input signal x(t) with impulse train of period ‘T’. Here, the amplitude of impulse changes with respect to amplitude of input

signal x(t). The output of sampler is given by

Ideal Sampling

Applications

Sampling theorem consists of several applications, some of which are listed below. They are

  • For maintaining sound quality in music recordings.
  • Sampling process is applicable in the conversion of analog to discrete form.
  • Used in Speech Recognition systems and pattern recognition systems.
  • In sensor data evaluation systems.
  • Modulation and demodulation systems.
  • Biometric identification systems, digital watermarking and surveillance systems.
  • Radio and radar navigation system sampling is applicable.

In this blog, we learned about sampling, sampling theorem, Nyquist sampling theorem, Aliasing effect, applications of sampling and three types of sampling.

Contributors

Yuvraj Kathar
Swadesh Kelkar
Akshay Khare
Rohan Kulkarni
Ankush Mahapatra

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