Biomedical Engineering Reference
In-Depth Information
In most applications, the
transform is somewhat easier to work with than the DFT
because it does not require the use of complex numbers directly. The
z-
transform plays
a similar role for digital signals as the Laplace transform does for the analysis of continu-
ous signals.
If a discrete sequence
z-
x
(
k
) is represented by
x k
, the (one-sided)
z-
transform of the discrete
sequence is expressed by
1
0 x k z k ¼ x 0 þ x 1 z 1
þ x 2 z 2
þ x k z k
X ð z Þ¼
þ
K
ð
11
:
21
Þ
k ¼
Note that the
z-
transform can be obtained directly from the DFT by allowing
N !1
and
2p m
N in Eq. (11.18). In most practical applications, sampled biological
signals are represented by a data sequence with
z ¼ e j
replacing
N
samples so the
z-
transform is estimated
for
transforms and their inverse transforms can be
found in most digital signal processing textbooks.
After a continuous signal has been sampled into a discrete sequence, its
k ¼
0
... N
-1 only. Tables of common
z-
transform is
found quite easily. Since the data sequence of a sampled signal is represented as
z-
x ¼½ x ð
0
Þ
,
x ð T Þ
,
x ð
2
T Þ
,
...
,
x ð kT Þ
ð
11
:
22
Þ
its
z-
transform is obtained by applying Eq. (11.21) to its samples
Þþ x ð T Þ z 1
T Þ z 2
þ ...þ x ð kT Þ z k
X ð z Þ¼ x ð
0
þ x ð
2
ð
11
:
23
Þ
where
is the sampling period or sampling interval.
A sampled signal is a data sequence with each sample separated from its neighboring
samples by precisely one sampling period. In the
T
transform, the value of the multi-
z-
z k have an intuitive graphical
plier,
(
), is the value of the data sample. The terms
x
kT
explanation. The power,
, corresponds to the number of sampling periods following
the start of the sampling process at time zero;
k
z k
can therefore be thought of as a
“shift operator” that delays the sample by exactly
k
sampling periods or
kT
.Thevariable
z 1 , for instance, represents a time separation of one period,
T
, following the start of the
signal at time zero. In Eq. (11.18),
)
is the value of the sampled data that was obtained after the first sampling period.
The
z
(0)isthevalueofthesampleddataat
t ¼
0, and
x
(
T
transform is an important method for describing the sampling process of an
analog signal.
z-
EXAMPLE PROBLEM 11.14
The discrete unit impulse function is represented as the sequence
x ¼
[1, 0, 0, 0,
...
, 0]. Find the
z-
transform of this sequence.
Solution
z 1
z 2
z k ¼
X
ð z Þ¼
1
þ
0
þ
0
þ ...þ
0
1
þ
0
þ
0
þ ...þ
0
¼
1
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