Sampling Theorem

Sampling Theorem

sampling of the signals is the fundamental operation in signal-processing.

A continuous time signal is first converted to discrete-time signal by sampling process.

The sufficient number of samples of the signal must be taken so that the original signal is represented in its samples completely. Also, it should be possible to recover or reconstruct the original signal completely from its samples.

The number of samples to be taken depends on maximum signal frequency present in the signal.

Sampling theorem gives the complete idea about the sampling of signals.

Different types of samples are also taken like ideal samples, natural samples and flat-top samples.

Let us discuss the sampling theorem first and then we shall discuss different types of sampling processes.

The statement of sampling theorem can be given in two parts as:

(i) A band-limited signal of finite energy, which has no frequency-component higher than f_m Hz, is completely described by its sample values at uniform intervals less than or equal to 1/ 2f_m second apart.

(ii) A band-limited signal of finite energy, which has no frequency components higher than f_m Hz, may be completely recovered from the knowledge of its samples taken at the rate of  2f_m samples per second.

The first part represents the representation of the signal in its samples and minimum sampling rate required to represent a continuous-time signal into its samples.

The second part of the theorem represents reconstruction of the original signal from its samples. It gives sampling rate required for satisfactory reconstruction of signal from its samples.

Combining the two parts, the sampling theorem may be stated as under:

‘‘A continuous-time signal may be completely represented in its samples and recovered back if the sampling frequency is f_s≥ 2f_m. Here f_s is the sampling frequency and f_m is the maximum frequency present in the signal’’.

Proof of Sampling Theorem

To prove the sampling theorem, we need to show that a signal whose spectrum is band-limited to f_m Hz, can be reconstructed exactly without any error from its samples taken uniformly at a rate f_s > 2 f_m Hz.

Let us consider a continuous time signal x(t) whose spectrum is band-limited to f_m Hz. This means that the signal x(t) has no frequency components beyond f_m Hz.

Therefore, X(ω) is zero for |ω| > ω_m, i.e.,

Where ω_m = 2πf_m

Fig.1 (a) shows this continuous-time signal x(t).

Fig.1 (a) A continuous-time signal, (b) Spectrum of continuous-time signal, (c) Impulse train as sampling function, (d) Multiplier, (e) Sampled signal, (f) Spectrum of sampled signal.

Let X(ω) represents its Fourier transform or frequency spectrum as shown in fig.1(b).

Sampling of x(t) at a rate of f_s Hz (i.e.,f_s samples per second) can be achieved by multiplying x(t) by an impulse train δ_Ts(t). The impulse train δ_Ts(t) consists of unit impulses repeating periodically every T_s seconds, where T_s = 1/f_s.

Fig.1(c) shows this impulse train. This multiplication results in the sampled signal g(t) shown in fig.1(d).

This sampled signal consists of impulses spaced every T_s seconds (the sampling interval). The resulting or sampled signal may be written as under:

…………….. (1)

Again, since the impulse train δ_Ts(t) is a periodic signal of period T_s, it may be expressed as a Fourier series. The Fourier series expansion of impulse-train δ_Ts(t) may be expressed as under:

……………. (2)

Here,

Putting the values of δ_Ts(t) from equation (2) in equation (1), the sampled signal can be written as,

…………….. (3)

Now, to obtain G(ω), the Fourier transformation of g(t), we will have to take the Fourier transform of right hand side.

Thus, Fourier transform of x(t) is X(w).

Fourier transform of 2x(t) cos ω_st is [X(ω – ω_s) + X(ω + ω_s)].

Fourier transform of 2x(t) cos 2ω_st is [X(ω – 2ω_s) + X (ω + 2ω_s)] and so on.

Therefore, on taking Fourier transformation, the equation (3) becomes,.

……………………. (4)

Or,

………….. (5)

From equations (4) and (5), it is quite obvious that the spectrum G(ω) consists of X(ω) repeating periodically with period ω_s = 2π / T_s rad/sec. or f_s = 1 / T_s Hz as shown in fig.1 (f).

Now, if have to reconstruct x(t) from g(t), we must be able to recover X(ω) from G(ω). This is possible if there is no overlap between successive cycles of G(ω). Fig.1 (f) shows that this requires,

But, the sampling interval T_s = 1 / f_s

Hence,

Therefore, as long as the sampling frequency f_s is greater than twice the maximum signal frequency f_m (signal bandwidth, f_m), G(ω) will consist of non-overlapping repetitions of X(ω). If this is true, fig.1 (f) shows that x(t) can be recovered from its samples g(t) by passing the sampled signal g(t) through an ideal low-pass filter (LPF) of bandwidth f_m Hz. This proves the sampling theorem.