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The sampling theorem is used to transform a continuous-time analog signal into a discrete-time signal, after which the signal’s amplitude is quantized and coded to a binary sequence so that the computer can process it. The signal is then transmitted to the receiver at the other end, where it is then converted back to its original form.
Sampling thus plays a significant role in the advancement of digital processing and sophisticated control operations. In this article, the sampling theorem will be discussed. To understand the sampling theorem, it is important to have a basic understanding of what sampling in signal processing entails. We will also talk about the Nyquist sampling theorem and the importance of sampling.
What is the Sampling Theorem?
The sampling theorem states that in order to recover or reconstruct the original signal, a continuous-time signal must be uniformly sampled at a minimum rate.
Sampling Theorem Statement
According to the sampling theorem, if a signal is sampled at regular intervals, the original signal can be recreated from the sequence of samples.
Analog signals make up the majority of the signals we frequently use in communication and control operations. The amplitude of the voice signal reflects in the voltage of the electrical signal, which is transmitted at the speed of light, even though it is simpler 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. However, when they are transmitted over greater distances, the signal strength weakens, which raises the 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.
If the sampling frequency is greater than or equal to twice the maximum frequency of the message signal x(t), then x(t) can be recovered in its original form.
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
To ensure that there is no discernible difference between an analog signal and a discrete signal in terms of the information contained and the visual appearance, one must choose an extremely small value for the sampling period when sampling an analog signal. Some redundant values may appear in such a representation in discrete form,. But we can omit them without losing the information in the original signal. If we choose a high sampling value, it might aid in data compression. But carries a sizable risk of losing information from the original signal.
The sampling theory and some other results proposed by Nyquist and Shannon serve as the link between analog and digital signals. Because selecting the right value of sampling is crucial to recovering the information from the message signal.
What are the Types of Sampling?
Consider the samples of the given analog signal x(t) at uniform times t=nTs as the first step in converting the continuous-time analog signal into the appropriate discrete signal.
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
A fundamental method of digital communication is pulse amplitude sampling. This sampling technique involves modulating the message signal with a pulse train to capture a series of short pulses with an amplitude that is close 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
Because the impulse signal has zero width and captures the signal value at any instant as needed,. This sampling is referred to as ideal sampling. A periodic impulse train δTs(t) with a period of Ts can replace the periodic pulse train p(t). If the pulse width is much smaller than the sampling period (Ts). This new development will greatly simplify the analysis and make it simpler to understand the outcomes.
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
The spectrum of the sampled signal in the frequency domain is made up of copies of the spectrum of the continuous-time signal x(t),. Which repeats at regular intervals of 𝜔s(sampling frequency). Due to the practical lack of impulse functions, 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).
Since the sampling rate in this situation is lower than the Nyquist rate. And the spectra of X(ω) are overlapped, it is not possible to reconstruct the original signal from the sampled signal. Frequency aliasing is the term for the phenomenon. Where some frequency components of the original signal acquire a different frequency as a result of the overlapping of the spectra.
What is the Aliasing Effect in Sampling?
Aliasing is a phenomenon that occurs when high-frequency components of a spectrum assume the identity of low-frequency components when the sampling frequency is less than Nyquist’s standard frequency (ωs<2ωmax). The original continuous-time signal that has undergone sampling cannot be recreated as a result.