Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Frequency Domain Analysis
In general cases, a system can be described by the difference equation
N
k=
0
a y[
n − k]
=
k
M
k=
0
b x[
n − k]
k
Take Fourier transform for both sides,
N
k=
0
a
k
exp( − jkΩ)
Y ( Ω)
=
M
k=
0
b exp( − jkΩ)
X ( Ω)
k
or
Y ( Ω)
N
k = 0
a
k
exp( − jkΩ)
= X ( Ω)
M
k = 0
b
k
exp( − jkΩ)
Therefore, the frequency response can be obtained by
Y ( W)
H ( W)
= =
X ( W)
M
∑
k=
0
N
∑
k=
0
b exp( − jk W)
Example 4.7: A digital high-pass filter is described by
a
k
k
exp( − jk W)
(4.8)
y[n]= - 0.8 y[n-1] + x[n]- x[n-1]
Find the frequency response and sketch its magnitude and phase over the range 0