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Signals Sampling Theorem
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.
fs ≥ 2fm .
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:
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) = a0 + Σ (an cos nωs t + bn sin nωs t) . . . . .
n=1
. (2)
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Where a0
1 2 1 1
= ∫ −T
δ(t)dt = δ(0) =
Ts Ts Ts
2
2 2 2
an = ∫ −T
δ(t) cos nωs dt = δ(0) cos
Ts T2
2
2
nωs 0 =
T
2 2 2
bn = ∫ −T
δ(t) sin nωs t dt = δ(0) sin
Ts Ts
2
nωs 0 = 0
Substitute above values in equation 2.
∴ δ(t) =
1
Ts
+ Σ
∞
n=1
(
2
Ts
cos nωs t + 0)
Substitute δ(t) in equation 1.
→ y(t) = x(t). δ(t)
1 ∞ 2
= x(t)[ + Σ ( cos nωs t)]
Ts n=1 Ts
1 ∞
= [x(t) + 2Σ (cos nωs t)x(t)]
Ts n=1
1
y(t) = [x(t) + 2 cos ωs t. x(t) + 2 cos 2ωs t
Ts
. x(t) + 2 cos 3ωs t. x(t) . . . . . . ]
Take Fourier transform on both sides.
1
Y (ω) = [X(ω) + X(ω − ωs ) + X(ω + ωs )
Ts
+ X(ω − 2ωs ) + X(ω + 2ωs )+ . . . ]
∴ Y (ω) =
1
Ts
∞
Σn=−∞ X(ω − nωs ) where n = 0,
±1, ±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:
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Aliasing Effect
The overlapped region in case of under sampling represents aliasing effect, which can be removed by
considering fs >2fm
By using anti aliasing filters.
