Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Frequency Domain Analysis
Table 4.1: Fourier Transform Properties and Pairs
Signal x [n]
Fourier Transform X (Ω)
Property
ax [ n]
+ by[
n]
aX ( Ω)
+ bY ( Ω)
Linearity
x n − n ]
X ( Ω )exp( − jΩn
)
Time-shift
[
0
x[ n]
∗ y[
n]
X ( Ω)
Y ( Ω)
Convolution
x [ n]
y[
n]
X ( Ω) * Y ( Ω)
Modulation
δ [n]
1
δ [ n − n0
]
exp( − jΩn 0
)
u [n]
−1
( ) ∞ 1−
exp( − Ω)
+
= −∞
j πδ ( Ω − 2πk
)
a n −
u[ n]
a < 1
( 1−
a exp( − jΩ)
) 1
x[ n]
= 1 n ≤ m
sin{ ( m + 1/ 2)
Ω}
x[
n]
= 0 n > m
sin( Ω / 2)
k
0
Example 4.6: Find the frequency response from the impulse responses:
a) h[n]=0.2 {d[n-2]+ d[n-1]+ d[n]+ d[n+1]+ d[n+2]}
Taking the Fourier transform, the frequency response is
H ( W)
=
∑ ∞
n=
−∞
h[
n]exp
( − jWn)
= 0.2{exp( − j2W)
+ exp( − jW)
+ 1+
exp( jW)
+ exp( j2W)}
= 0.2{1 + 2cosW + 2cos 2W}
H ( W)
= 0.21+
2 cos W + 2 cos 2W
and the phase Φ H
( W) = 0 .
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