Electronics and communication

Sampling theorem and proof

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

f_{s} \geq 2 f_{m} .

Proof: Consider a continuous time signal x(t). The spectrum of x(t) is a band limited to f_m Hz i.e. the spectrum of x(t) is zero for |ω|>ω_m.

Sampling of input signal x(t) can be obtained by multiplying x(t) with an impulse train δ(t) of period T_s. The output of multiplier is a discrete signal called sampled signal which is represented with y(t) in the following diagrams:

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:

Sampled signal y (t) = x (t) . δ (t) . . . . . . (1)

The trigonometric Fourier series representation of

δ

(t) is given by

δ (t) = a_{0} + Σ_{n = 1}^{\infty} (a_{n} \cos n ω_{s} t + b_{n} \sin n ω_{s} t) . . . . . . (2)

Where

a_{0} = \frac{1}{T_{s}} \int_{\frac{- T}{2}}^{\frac{T}{2}} δ (t) d t = \frac{1}{T_{s}} δ (0) = \frac{1}{T_{s}}

a_{n} = \frac{2}{T_{s}} \int_{\frac{- T}{2}}^{\frac{T}{2}} δ (t) \cos n ω_{s} d t = \frac{2}{T_{2}} δ (0) \cos n ω_{s} 0 = \frac{2}{T}

b_{n} = \frac{2}{T_{s}} \int_{\frac{- T}{2}}^{\frac{T}{2}} δ (t) \sin n ω_{s} t d t = \frac{2}{T_{s}} δ (0) \sin n ω_{s} 0 = 0

Substitute above values in equation 2.

∴ δ (t) = \frac{1}{T_{s}} + Σ_{n = 1}^{\infty} (\frac{2}{T_{s}} \cos n ω_{s} t + 0)

Substitute δ(t) in equation 1.

\to y (t) = x (t) . δ (t)

= x (t) [\frac{1}{T_{s}} + Σ_{n = 1}^{\infty} (\frac{2}{T_{s}} \cos n ω_{s} t)]

= \frac{1}{T_{s}} [x (t) + 2 Σ_{n = 1}^{\infty} (\cos n ω_{s} t) x (t)]

y (t) = \frac{1}{T_{s}} [x (t) + 2 \cos ω_{s} t . x (t) + 2 \cos 2 ω_{s} t . x (t) + 2 \cos 3 ω_{s} t . x (t) . . . . . .]

Take Fourier transform on both sides.

Y (ω) = \frac{1}{T_{s}} [X (ω) + X (ω - ω_{s}) + X (ω + ω_{s}) + X (ω - 2 ω_{s}) + X (ω + 2 ω_{s}) + . . .]

∴ Y (ω) = \frac{1}{T_{s}} Σ_{n = - \infty}^{\infty} X (ω - n ω_{s}) w h e r e n = 0, \pm 1, \pm 2, . . .

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:

Aliasing Effect

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

Ex- f_s <2f_m

The overlapped region in case of under sampling represents aliasing effect, which can be removed by

considering f_s >2f_m
By using anti aliasing filters.

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