Digital Signal Processing Reference
In-Depth Information
40
35
30
25
20
15
10
5
0
−1
−0.5
0
0.5
1
Frequency, Units of
π
Figure 2.12:
The magnitude of
H(z)
=(1+
z
−
1
)/(1 - 0.95
z
−
1
) evaluated at about 500 points along the
unit circle from -
π
to
π
radians.
2.6.3 FINITE IMPULSE RESPONSE (FIR)
The
z
-transform of a finite sequence
x
[
n
]
is
z
0
z
−
1
z
−
2
z
−
(N
−
1
)
x
[
0
]
+
x
[
1
]
+
x
[
2
]
+
... x
[
N
−
1
]
For a finite causal sequence, the
ROC
(Region Of Convergence) is everywhere in the
z
-plane
except the origin (
z
=
0). Choosing
z
somewhere on the unit circle, we have
e
jω
=
z
We can test the response of, say, an 11-sample FIR with one-cycle correlators (equivalent to those
used for Bin 1 of an 11-point DFT, for example) generated by the
z
-power sequence shown in plot (a) of
Fig. 2.13. To test the response of the same 11-sample FIR with, say, 1.1 cycle correlators, we would use
z
= exp
(j
2
πk/N)
, where
k
=1.1.
b
0
z
0
b
1
z
−
1
b
2
z
−
2
...b
10
z
−
10
b(z)
=
+
+
+
• The
z
-transform evaluated at a value of
z
lying on the unit circle at a frequency corresponding to a
DTFT frequency produces the same numerical result as the DTFT for that particular frequency.
We can also evaluate the
z
-transform at many more (a theoretically unlimited number) values of
z
on the unit circle, corresponding to any sample of the DTFT.
