Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Frequency Domain Analysis
4.3 Properties of Fourier transform
The linearity is held by the Fourier transform, i.e.:
If x [ n X ( W)
and x [ n X ( W)
1
] ↔ 1
2
] ↔ 2
Then Ax [ n]
Bx [ n ↔ AX ( Ω) + BX ( Ω)
1
+
2
]
1
2
where ↔ denotes the pair between the Fourier transform and inverse Fourier transform.
The time-shifting property is also held, i.e.:
If x [ n]
↔ X ( W)
Then x[ n − n ] ↔ X ( W) ( − jWn
)
0
exp
0
The Fourier transform of a convolution in time-domain is a product of Fourier transforms in the frequency domain, i.e.:
If x [ n X ( W)
and [ n X ( W)
1
] ↔ 1
Then 1
[ n]*
x2[
n]
↔ X
1( W) X
2
( W)
x .
x ,
2
] ↔ 2
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