Digital Signal Processing Reference
In-Depth Information
Example 2.20.
Compute the z-transform coefficients of an IIR having poles at 0
.
9
j
and
−
0
.
9
j
; check
your results using the MathScript function
poly
.
The coefficients are
conv([1, -(0 + 0.9j)],[1, -(0 - 0.9j)]) = [1,0,0.81]
and the
z
-transform is
0
z
−
1
0
.
81
z
−
2
)
Y (z)/X(z)
=
1
/(
1
+
+
This can be checked using the function
poly
which converts a vector of roots into polynomial
coefficients:
a = poly([0.9*j,-0.9*j]) = [1,0,0.81]
Example 2.21.
The poles of a certain causal LTI system are at 0
.
9 and
±
0
.
9
j
. There are three zeros at
−
1. Using Eq. (2.8), determine the magnitude of this system's response at DC. Use the function poly to
obtain the polynomial form of the z-transform, and evaluate it to check your work.
We note that with three zeros,
M
= 4 and with three poles,
N
=4.Toget
b
0
, we must obtain the
polynomial coefficients for the numerator of the
z
-transform. Note that the ROC is for
|
z
|
>
0
.
9.
zzs = [-1,-1,-1]; pls = [0.9, 0.9*j,-0.9*j]; Denom = poly(pls);
N = length(pls) +1; M = length(zzs) +1; Num = poly(zzs);
b0 = Num(1);z=1;NProd=1;forNCtr=1:1:M-1;
NProd = NProd*(abs(z - zzs(NCtr))); end
DProd = 1; for DCtr = 1:1:N-1;
DProd = DProd*(abs(z - pls(DCtr))); end
MagResp = abs(b0)*abs(zˆ(N-M))*NProd/DProd
AltMag = abs(sum(Num.*(z.ˆ(0:-1:-(M-1))))/sum(Denom.* (z.ˆ(0:-1:-(N-1)))))
2.3.9 POLE LOCATION AND STABILITY
The impulse response of a single pole IIR is [1,p,p
2
,p
3
..
.]. It can be seen that if
<
1, then a geometrically
convergent series results. A bounded signal (one whose sample values are all finite) will not produce an
unbounded output. If, however,
|
p
|
>
1 then the impulse response does not decay away, and a bounded
input signal can produce an output that grows without bound. For the borderline case when
|
p
|
|
| =
p
1, the
unit impulse sequence as input produces a unit step (
u
[
n
]
) as the corresponding output sequence. If the
input signal is
u
, then the output will grow without bound.
The difference equation for the single-pole IIR
[
n
]
y
[
n
]=
x
[
n
]+
py
[
n
−
1
]
can be used to observe the relationship between pole magnitude, stability, and pole location in the complex
plane by choosing different values of
p
and then processing a test signal
x
[
n
]
such as the unit impulse
[1,0,0 ...] or the unit step [1,1,1 ...]. For
|
p
| ≥
1 and the unit impulse as
x
[
n
]
, the output
y
[
n
]
continues
growing without further input.
