Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Frequency Domain Analysis
N −1
⎛ 2p
kn
x[
n]
= ∑ a exp
⎜
k
j
k = 0 ⎝ N
N −1
⎧ 1 ⎫
= ∑ ⎨ X ( kW
0
) ⎬exp
k = 0 ⎩ N ⎭
N −1
⎧W0
⎫
= ∑ ⎨ X ( kW
0
) ⎬exp
k = 0 ⎩ 2p
⎭
=
1
2p
N −1
∑
k = 0
X ( kW
0
)exp
⎞
⎟
⎠
( − jk W n)
0
( − jk W n)
0
( − jk W0n) W0
With
N → ∞ , W0 → dW
and k → W , the above inverse Fourier transform can be expressed as
W 0
1 2p
x[
n]
= ∫ X ( W)exp
d
2p
0
( jWn) W
(4.5)
Example 4.3 Find the Fourier transform for the unit impulse signal x[ n]
= d [ n]
.
From the definition Eq.(4.4),
∑ ∞
n=
−∞
( − jWn)
X ( W)
= d[
n]exp
= 1
If the unit impulse is shifted,
x[
n]
= d [ n −1]
X ( W)
=
∑ ∞
n=
−∞
d [ n −1]exp
X ( W ) = 1, ∠X
( W)
= −W
.
( − jWn) = exp( − jW)
Example 4.4 Find the Fourier transform of the rectangle signal
1
x[ n]
= n +
5
X ( W)
=
=
∞
∑
n=−∞
0.2
{ d [ n − 2] + d [ n −1] + d [ n] + d [ n + 1] + d [ 2]
}
∞
∑
n=−∞
x[
n]exp
( − jWn)
{ d [ n − 2] + d [ n −1] + d [ n] + d [ n + 1] + d [ n + 2]
} exp( − jWn)
i.e.
47
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