Sampling Theorem | What is Sampling Theorem

What is the Sampling Theorem?

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/2 and equal to zero elsewhere.

What are the Important Aspects of Sampling?

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=nTs ; 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)=∑Pkejkω₀tk; ω₀=2ΠT_s

Here P_k is the Fourier series coefficient.

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

The Fourier transform of the signal is xPAM (ω)=∑P_kX(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(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 the 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?

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.