Digital Signal Processing Reference
In-Depth Information
p = 0.9
p = 0.9
1
10
5
0.5
0
0
0
10
20
30
40
0
10
20
30
40
(a) Sample Index n
(b) Sample Index n
60
p = 1.0
p = 1.0
1
40
0.5
20
0
0
0
10
20
30
40
0
10
20
30
40
(c) Sample Index n
(d) Sample Index n
2
p = 1.01
p = 1.01
60
40
1
20
0
0
0
10
20
30
40
0
10
20
30
40
(e) Sample Index n
(f) Sample Index n
Figure 2.5:
(a) Impulse response for a single pole IIR, p = 0.9; (b) Step response for same; (c) Impulse
response for a single pole IIR, p = 1.0; (d) Step response for same; (e) Impulse response for a single pole
IIR, p = 1.01; (f ) Step response for same.
or greater, instead of using the
z
-transform to determine the frequency response (which is not possible
since values of
z
along the unit circle are not in the ROC when the poles have magnitude 1.0 or greater),
the DFT of a finite length of the test impulse response is computed. While this is not a true frequency
response (none exists for systems having unstable poles since the DTFT does not converge), it gives
an idea of how rapidly the system response to a unity-magnitude complex exponential grows as pole
magnitude increases beyond 1.0.
2.4
CONVERSION FROM Z-DOMAIN TO TIME DOMAIN
There are a number of methods for computing the values of, or determining an algebraic expression for,
the time domain sequence that underlies a given
z
-transform. Here we give a brief summary of some of
these methods:
