605
CHAPTER
33
X
(
s
)
’
m
4
t
’&
4
x
(
t
)
e
&
st
dt
The zTransform
Just as analog filters are designed using the Laplace transform, recursive digital filters are
developed with a parallel technique called the ztransform.
The overall strategy of these two
transforms is the same:
probe the impulse response with sinusoids and exponentials to find the
system's poles and zeros.
The Laplace transform deals with differential equations, the sdomain,
and the splane.
Correspondingly, the ztransform deals with difference equations, the zdomain,
and the zplane.
However, the two techniques are not a mirror image of each other; the splane
is arranged in a rectangular coordinate system, while the zplane uses a polar format.
Recursive
digital filters are often designed by starting with one of the classic analog filters, such as the
Butterworth, Chebyshev, or elliptic.
A series of mathematical conversions are then used to obtain
the desired digital filter.
The ztransform provides the
framework for this mathematics.
The
Chebyshev filter design program presented in Chapter 20 uses this approach, and is discussed in
detail in this chapter.
The Nature of the zDomain
To reinforce that the Laplace and ztransforms are parallel techniques, we will
start with the Laplace transform and show how it can be changed into the z
transform.
From the last chapter, the Laplace transform is defined by the
relationship between the time domain and sdomain signals:
where
and
are the time domain and sdomain representation of the
x
(
t
)
X
(
s
)
signal, respectively.
As discussed in the last chapter, this equation analyzes the
time domain signal in terms of sine and cosine waves that have an
exponentially changing amplitude.
This can be understood by replacing the
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606
X
(
F
,
T
)
’
m
4
t
’&
4
x
(
t
)
e
&
F
t
e
&
j
T
t
dt
X
(
F
,
T
)
’
j
4
n
4
x
[
n
]
e
&
F
n
e
&
j
T
n
y
[
n
]
’
e
&
F
n
y
[
n
]
’
r
n
or
complex variable,
s
, with its equivalent expression,
.
Using this alternate
F
%
j
T
notation, the Laplace transform becomes:
If we are only concerned with
real
time domain signals (the usual case), the top
and bottom halves of the splane are mirror images of each other, and the term,
, reduces to simple cosine and sine waves. This equation identifies each
e
&
j
T
t
location
in the splane by the two parameters,
F
and
T
.
The
value
at each
location is a complex number, consisting of a real part and an imaginary part.
To find the real part, the time domain signal is multiplied by a cosine wave
with a frequency of
T
, and an amplitude that changes exponentially
according to the decay parameter,
F
.
The value of the real part of
X
(
F
,
T
)
is then equal to the integral of the resulting waveform.
The value of the
imaginary part of
is found in a similar way, except
using a sine
X
(
F
,
T
)
wave.
If this doesn't sound very familiar, you need to review the previous
chapter before continuing.
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 Summer '09
 Digital Signal Processing, Lowpass filter, recursion coefficients

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