Digital Signal Processing Reference
In-Depth Information
For an LTI system to be stable, all of its poles must have magnitude less than 1.0.
Example 2.22.
Determine the pole magnitudes of the LTI system represented by the
z
-transform
1
H(z)
=
1
.
9
z
−
1
0
.
95
z
−
2
1
−
+
We run the code
pmags = abs(roots([1,-1.9, 0.95]))
which yields
pmags
= [0.9746, 0.9746], meaning that the system is stable.
In summary
•
The poles of a transfer function must all have magnitudes less than 1.0 in order for the
corresponding system to have a stable, bounded response to a bounded input signal (i.e.,
BIBO).
•
Stable poles, when graphed in the
z
-domain (i.e., the complex plane), all lie inside the unit
circle.
Example 2.23.
For an IIR having a single real pole
p
, plot the first 45 samples of the impulse and unit
step responses for the following values of
p
:
(a) 0.9
(b) 1.0
(c) 1.01
We can use this code to generate the impulse and step responses:
p = 0.9; ImpResp = filter(1,[1,-p],[1,zeros(1,44)])
StepResp = filter(1,[1,-p],[ones(1,95)])
Figure 2.5 shows the results for each of the three pole values. Note that for the one stable case (p
= 0.9), the impulse response decays to zero, and the step response converges to a finite value.
You can experiment with pole location for single or complex conjugate pairs of poles and the
corresponding/resultant system impulse response by calling the VI
DemoDragPolesImpRespVI .
Figure 2.6 shows an example of the VI
DemoDragPolesImpRespVI
using a pair of complex poles
having an approximate magnitude of 0.9 and angles of approximately
±
π/
4 radians.
A MATLAB script that performs a similar function is
ML
_
DragPoleZero
which, when called on the Command Line, opens up a GUI that allows you to select a single or complex
conjugate pair of poles or zeros and move the cursor around the
z
-plane to select the pole(s) or zero(s).
The magnitude and phase response of the
z
-transform and the real and imaginary parts of the impulse
response are dynamically displayed as you move the cursor in the
z
-plane. For poles of magnitude 1.0
