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                          Sampling theorem and proof 

Statement: A continuous time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to the twice the highest frequency component of message signal. i. e.
fs2fm.

Proof: Consider a continuous time signal x(t). The spectrum of x(t) is a band limited to fm Hz i.e. the spectrum of x(t) is zero for |ω|>ωm.
Sampling of input signal x(t) can be obtained by multiplying x(t) with an impulse train δ(t) of period Ts. The output of multiplier is a discrete signal called sampled signal which is represented with y(t) in the following diagrams:
Signal Sampling
Here, you can observe that the sampled signal takes the period of impulse. The process of sampling can be explained by the following mathematical expression:
The trigonometric Fourier series representation of Î´(t) is given by
Where 
Substitute above values in equation 2.
Substitute δ(t) in equation 1.
y(t)=x(t).δ(t)
Take Fourier transform on both sides.
To reconstruct x(t), you must recover input signal spectrum X(ω) from sampled signal spectrum Y(ω), which is possible when there is no overlapping between the cycles of Y(ω).
Possibility of sampled frequency spectrum with different conditions is given by the following diagrams:
Sampling

Aliasing Effect

    Aliasing a phenomena in which high frequency components are interference with each other  to due to inadequate sampling frequency

                                   Ex- fs <2fm  

The overlapped region in case of under sampling represents aliasing effect, which can be removed by
  • considering fs >2fm
  • By using anti aliasing filters.

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