Sampling Theorem Statement and Its Applications

A signal has three properties like voltage or amplitude, frequency, phase. The signals are represented only in an analog form where the digital form of technology is not available. Analog signals are continues in time and difference in voltage levels for different periods of the signal. Here, 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. Here conversion of an analog form of the signal into digital form can be done using the sampling technique. The output of this technique represents the discrete version of its analog signal. Here in this article, you can find what is sampling theorem, definition, applications, and its types.

What is the Sampling Theorem?

A continues signal or an analog signal can be represented in the digital version in the form of samples. Here, these samples are also called as 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 equidistance with a period of “Ts”. This sampling signal frequency should be more than twice of the input analog signal frequency. If this condition satisfies, analog signal perfectly represented in discrete form else 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 are well performed in the reconstruction process for recovering the original signal.

sampling-block-diagram
sampling-block-diagram

Sampling Theorem Definition

The sampling theorem can be defined as the conversion of an analog signal into a discrete form by taking the sampling frequency as twice the input analog signal frequency. Input signal frequency denoted by Fm and sampling signal frequency denoted by Fs.

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

Sampling frequency Fs=1/Ts

Sampling Theorem Statement

Sampling theorem states that “continues 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

If the sampling frequency (Fs) equals twice the input signal frequency (Fm), then such a condition is called the Nyquist Criteria for sampling. When sampling frequency equals twice the input signal frequency is known as “Nyquist rate”.

Fs=2Fm

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

Fs<2Fm

So, there are three conditions are possible from the sampling frequency criteria. They are sampling, Nyquist and aliasing states. Now we will see the Nyquist sampling theorem.

Nyquist Sampling Theorem

In the sampling process, while converting the analog signal to a discrete version, the chosen sampling signal is the most important factor. And what are the reasons to get distortions in the sampling output while conversion of analog to discrete? These types of questions can be answered by the “Nyquist sampling theorem”.

Nyquist sampling theorem states that the sampling signal frequency should be double the input signal’s highest frequency component to get distortion less output signal. As per the scientist’s name, Harry Nyquist this is named as Nyquist sampling theorem.

Fs=2Fm

Sampling Output Waveforms

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

Sampling-output-waveforms
Sampling-output-waveforms

Shannon Sampling Theorem

The sampling theorem is one of the efficient techniques in the communication concepts for converting the analog signal into discrete and digital form. Later the advances in digital computers Claude Shannon, an American mathematician implemented this sampling concept in digital communications for converting the analog to digital form. The sampling theorem is a very important concept in communications and this technique should follow the Nyquist criteria for avoiding the aliasing effect.

Applications of Sampling Theorem

There are few applications of sampling theorem are listed below. They are

  • To maintain sound quality in music recordings.
  • Sampling process applicable in the conversion of analog to discrete form.
  • Speech recognition systems and pattern recognition systems.
  • Modulation and demodulation systems
  • In sensor data evaluation systems
  • Radar and radio navigation system sampling is applicable.
  • Digital watermarking and biometric identification systems, surveillance systems.

Sampling Theorem for Low Pass Signals

The low pass signals having the low range frequency and whenever this type of low-frequency signals need to convert to discrete then the sampling frequency should be double than these low-frequency signals to avoid the distortion in the output discrete signal. By following this condition, the sampling signal does not overlap and this sampled signal can be reconstructed to its original form.

  • Bandlimited signal xa(t)
  • Fourier signal representation of xa(t) for reconstruction Xa(F)

Proof of Sampling Theorem

The sampling theorem states that the representation of an analog signal in a discrete version can be possible with the help of samples. The input signals which are participating in this process are analog signal and sample pulse train sequence.

Input analog signal is s(t) 1

The sample pulse train is

sample-pulse-train
sample-pulse-train

The spectrum of an input analog signal is,

Input signal spectrum
Input signal spectrum

Fourier series representation of the sample pulse train is

fourier-series-representation-of-sample-pulse
Fourier-series-representation-of-sample-pulse

The spectrum of the sample output signal is,

spectrum-of-the-sample-output-signal
spectrum-of-the-sample-output-signal

When these pulse train sequences are multiples with the analog signal we will get the sampled output signal which is indicated by here as g(t).

sampled-output-signal
sampled-output-signal

When the signal related to equation 3 passes from the LPF, only Fm to –Fm signal only passed to the output side and the remaining signal is going to be eliminated. Because LPF is assigned to the cut off frequency which is equal to the input analog signal frequency value. In this way at one side analog signal going to converted to discrete and recovered to its original position simply passing from a low pass filter.

Thus, this is all about an overview of the sampling theorem. Here is a question for you, what is the Nyquist rate?

 



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