Sampling Theorem

Sampling of signals is the fundamental operation in signal processing, a Continuous Time (CT) signal can be converted into a Discrete Time (DT) signal using Sampling process. Sampling is required since the advancement in both signals and systems which are digitized i.e, Digital systems operates only on digital signals only.

Sampling Theorem:-

A CT signal is first converted into DT signal by Sampling process. The sufficient number of samples must be taken so that the original signal is represented in it’s samples completely, and also the signal is represented from it’s samples, these two conditions representation and reconstruction depends on the sampling process ‘fs‘ Hz.

Sampling theorem can be given into two parts

i. A band limited signal of finite energy, which has no frequency component higher than ‘fm‘ Hz, is completely described by it’s sample values at uniform intervals less than (or) equal to 1/2fm seconds apart.

i.e, T_{s}\leq \frac{1}{2f_{m}}  Seconds.

ii. A Band limited signal of finite energy, which has no frequency component higher than fm Hz may be completely recovered from the knowledge of it’s samples if samples are taken at the rate of 2fm samples/second.

i.e, f_{s}\geq 2f_{m} Hz.

Statement:- A Continuous Time signal can be completely represented in it’s samples and recovered from it’s samples if the sampling frequency f_{s}\geq 2f_{m}Hz.

where f_{s} is the sampling frequency.

f_{m} is the highest frequency present in the original signal / Band width of the signal.

proof of Sampling theorem:-

Let us consider a CT signal x(t), which is a band limited to f_{m} Hz as shown

To prove Sampling theorem, it should be shown a signal whose spectrum is band limited to fm Hz can be reconstructed exactly without any error from it’s samples taken uniformly at a rate of f_{s}> 2f_{m} Hz. 

The circuit shows the sampler 

Now sampling of x(t) at a rate of fs may be achieved by multiplying x(t) with a train of impulses  \delta T_{s}(t) with a period ‘Ts‘ seconds.

 

The sampling signal is an ideal (or) instantaneous signal. This is also known as ideal (or) instantaneous sampling.

g(t)=x(t)\delta T_{s}(t)

As \delta T_{s}(t) is a periodic impulse train it can be expressed in it’s Fourier Series expansion as follows

Exponential Fourier Series is

\delta T_{s}(t) = \sum_{n=-\infty }^{\infty }F_{n}e^{jnw_{s}t}

F_{n}= \frac{1}{T_{s}}\int_{\frac{-T_{s}]}{2}}^{\frac{T_{s}}{2}}\delta T_{s}(t)e^{-jn\omega _{s}t}dt

F_{n}=\frac{1}{T_{s}}

F_{n}=f_{s}

∴ Exponential Fourier Series is \delta T_{s}(t)=\sum_{n=-\infty }^{\infty }f_{s}e^{jn\omega _{s}t}

now the sampled signal g(t) = x(t).\delta T_{s}(t)

g(t)=x(t)\sum_{n= -\infty }^{\infty }f_{s}e^{jn\omega _{s}t}

g(t)=\sum_{n= -\infty }^{\infty }f_{s}x(t)e^{jn\omega _{s}t}

By finding Fourier Transform of g(t) is G(f) 

G(f)=\sum_{n= -\infty }^{\infty }f_{s}X(f-nf_{s})

Now the frequency spectrum of the sampled signal G(f) is of the form

From G(f) spectrum the original spectrum of X(f) has been shifted to different center frequencies

i.e, when n=0  center frequency is 0.

                    n=1  center frequency is fs

                    n=-1 center frequency is -fs etc

Some important conclusions from frequency spectrum of sampled signal:-

  1. The spectrum of sampled signal G(f)/G(w) will repeat periodically if f_{s}> 2f_{m} without any overlapping.
  2. G(f) is extending up to infinity and the Band width is infinity as well, out of G(f) , X(f) need to be recovered , which is band limited to fm Hz.
  3. X(f) is centered at f=0 and has fm as the highest frequency, X(f) may be recovered by passing it through a Loe Pass filter with cutoff frequency approximately equals to fm  Hz.
  4. to reconstruct x(t) from g(t) the condition that must be satisfied is  f_{s}\geq 2f_{m}.

 

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Author: Lakshmi Prasanna

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.