Sampling theorem for Low pass signals

SAMPLING:

A message signal may originate from a digital or analog source. If the message signal is analog in nature, then it has to be converted into digital form before it can transmit by digital means. The process by which the continuous-time signal is converted into a discrete–time signal is called Sampling. Sampling operation is performed in accordance with the sampling theorem.

SAMPLING THEOREM FOR LOW-PASS SIGNALS:-

Statement: - “If a band –limited signal g(t) contains no frequency components for ׀f׀ > W, then it is completely described by instantaneous values g(kTs) uniformly spaced in time with period Ts ≤ 1/2W. If the sampling rate, fs is equal to the Nyquist rate or greater (fs ≥ 2W), the signal g(t) can be exactly reconstructed.

Proof:- Part - I If a signal x(t) does not contain any frequency component beyond W Hz, thenthe signal is completely described by its instantaneous uniform samples with sampling interval (or period ) of Ts< 1/(2W) sec.

Part – II The signal x(t) can be accurately reconstructed (recovered) from the set of uniform instantaneous samples by passing the samples sequentially through an ideal (brick-wall) lowpass filter with bandwidth B, where W ≤ B <fs – W and fs = 1/(Ts).

If x(t) represents a continuous-time signal, the equivalent set of instantaneous uniform samples {x(nTs)} may be represented as,

where x(nTs) = x(t)⎢t =nTs , δ(t) is a unit pulse singularity function and „n‟ is an integer.The continuous-time signal x(t) is multiplied by an (ideal) impulse train to obtain {x(nTs)} and can be rewritten as,

Now, let X(f) denote the Fourier Transform F(T) of x(t), i.e.

Now, from the theory of Fourier Transform, we know that the F.T of Σ δ(t- nTs), the impulse train in time domain, is an impulse train in frequency domain:

If Xs(f) denotes the Fourier transform of the energy signal xs(t), we can write using Eq. (1.2.4) and the convolution property:

Xs(f) = X(f)* F{Σ δ(t- nTs)}

X(f)*[fs.Σ δ(f- nfs)]

= fs.X(f)*Σ δ(f- nfs)