## 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≥ω_{max}

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 ω_{s}≥2ω_{𝑚𝑎𝑥} (Nyquist sampling rate condition);

x(nT_{s}) = x(t), n=0, ±1, ±2, ±3, ……

Here T_{s} is the sampling period (sec/sample).

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

Here f_{s}is 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)=T
_{s}; when -ω_{s}/2<ω<ω_{s}/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(nT_{s}) = x(t)|_{t=nT}_{s}; n is an integer

Here T_{s} 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 x_{PAM}(t) with a small pulse width W, and the amplitude of the pulses being x(mT_{s}), then

x_{PAM}(t) = x(t)p(t) ≈ 1W∑x (mT_{s})[u(t-mT_{s})-u(t-mT_{s}-W)]

As p(t) is a periodic signal we can represent it by Fourier series as p(t)=∑Pkejkω_{0}tk; ω_{0}=2ΠT_{s}

Here P_{k} is the Fourier series coefficient.

Hence the pulse amplitude modulated signal can be expressed as x_{PAM}(t) = ∑P_{k}x(t)ejkw_{0}tk

The Fourier transform of the signal is xPAM (ω)=∑P_{k}X(ω-kω_{0})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(T_{s}), then p(t) can be replaced by a periodic impulse train δT_{s}(t), with a period of T_{s}. This development will considerably simplify the analysis and makes the results easier to grasp.

The sampling function is δT_{s}(t)=∑δ(t-nT_{s})n

The sampled signal is given by x_{s}(t)=∑x(nT_{s})δ(t-nT_{s})n

If X(ω) is the Fourier transform of the signal x(t) then the Fourier signal of x_{s}(t) is X_{s}(ω)=1T_{s}∑x(ω-kω_{s})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>ω_{max} as shown below.

**Case (1): ω _{s}≥2ω_{max}**

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

**Case(2): ω _{s}<2ω_{max}**

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 (ω_{s}<2ω_{max}), 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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