Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
2 3 4
h[
n]
= 0,1, α,
α , α , α ... (5.8)
If the impulse response decays with time then the system will return to its initial state before disturbance. Therefore the
stability requires α < 1 in Eq.(5.8) as when n → ∞ makes h [ ∞]
→ 0 . i.e. in the z-plane, the real pole z = α must
lie on the real axis and inside the unit circle.
Consider a transfer function with conjugate imaginary poles:
H ( z)
=
Y ( z)
X ( z)
=
1
( z − jα )( z + jα )
(5.9)
Rearranging,
2
2
z Y ( z)
+ α Y ( z)
=
X ( z)
Or
2
Y ( z)
+ α z
−2
Y ( z)
= z
−2
X ( z)
Taking the inverse z-transform of both sides yields the difference equation
2
y[
n]
+ α y[
n − 2] = x[
n − 2]
or
2
y[
n]
= −α y[
n − 2] + x[
n − 2]
(5.10)
2
From the above we use h[
n]
= −α h[
n − 2] + d [ n − 2]
to evaluate its impulse response, it can be obtained as
2 4 6 8
h[
n]
= 0,0,1,0, −α ,0, α ,0, −α
,0, α ,...
(5.11)
It is clear that the stability condition also requires α < 1, in order to achieve that when n → ∞ , h [ ∞]
→ 0 .
In the case of expression in polar co-ordinates, consider a conjugate pole pair,
H ( z)
=
Y ( z)
X ( z)
=
1
2
2
( z − r exp( jθ
))( z − r exp( − jθ
)) z − 2rz
cosθ
+ r
=
1
(5.12)
where r is the radius, q is the angle on the complex plane. The difference equation can be obtained in the same way as
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