Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 11.15
An A/D converter is used to convert a recorded signal of the electrical activity inside a nerve
into a digital signal. The first five samples of the biological signal are [
60.0,
49.0,
36.0,
23.0,
transform of this data sequence? How many sample periods after the
start of the sampling process was the data sample -23.0 recorded?
14.0] mV. What is the
z-
Solution
z
1
z
2
z
3
z
4
Y
ð
z
Þ¼
60
:
0
49
:
0
36
:
0
23
:
0
14
:
0
The value of the negative exponent of the
23.0 mV
z-
term is 3. Therefore, the data sample with
the value of
23.0 was recorded 3 sampling periods after the start of sampling.
11.5.8 Properties of the z-Transform
The
transform obeys many of the same rules and properties that we've already shown
for the Fourier transform. These properties can significantly simplify the process of evalu-
ating
z-
z-
transforms for complex signals. The following are some of the properties of the
z-
transform. Note the close similarity to the properties for Eqs. (11.11), (11.12), and (11.14).
Let
x
1
(
k
) and
x
2
(
k
) be two digital signals with corresponding
z-
transforms
X
1
(
z
) and
X
2
(
z
).
Linearity:
The
z-
transform is a linear operator. For any constants
a
1
and
a
2
,
1
ð
k
Þ
z
k
¼
a
Z
f
a
x
ð
k
Þþ
a
x
ð
k
Þg ¼
0
½
a
x
ð
k
Þþ
a
x
X
ð
z
Þþ
a
X
ð
z
Þ
ð
11
:
24
Þ
1
1
2
2
1
1
2
2
1
1
2
2
k
¼
Delay:
Let
x
1
(
k
-
n
) be the original signal that is delayed by
n
samples. The
z
-transform of the
delayed signal is
1
1
ð
k
n
Þ
z
k
¼
ð
k
Þ
z
ð
k
þ
n
Þ
¼
z
n
X
Z
f
x
ð
k
n
Þg ¼
ð
z
Þ
ð
11
:
25
Þ
0
x
0
x
1
1
1
1
k
¼
k
¼
z
n
As described previously, note that the operator
represents a shift of
n
samples or
precisely
nT
seconds.
Convolution:
Let
x
(
k
) be the discrete convolution between
x
1
(
k
) and
x
2
(
k
),
x
ð
k
Þ¼
x
1
ð
k
Þ
*
x
2
ð
k
Þ
X
(
z
), the
z-
transform of
x
(
k
), is calculated as
X
ð
z
Þ¼
Z
f
x
ð
k
Þg ¼
X
ð
z
Þ
X
ð
z
Þ
ð
11
:
26
Þ
1
2
As with the Fourier transform, this result demonstrates that convolution between two
sequences is performed by simple multiplication in the
z-
domain.
