Digital Signal Processing Reference
In-Depth Information
1.3.2 LAPLACE FAMILY (TIME-VARYING-MAGNITUDE CORRELATORS)
Laplace Transform (LT)
The LT is defined as
∞
∞
x(t)e
−
st
dt
x(t)e
−
σt
e
−
jωt
dt
£
(s)
=
=
−∞
−∞
The LT is the standard frequency transform for use with continuous time domain signals and
systems. The parameter
s
represents the complex number
σ
+
jω
, with
σ
, a real number, being a damping
coefficient, and
jω
, an imaginary number, representing frequency. Both
σ
and
ω
run from negative infinity
to positive infinity. The correlators generated by
e
−
st
are complex exponentials having amplitudes that
decay, grow, or retain unity-amplitude over time, depending on the value of
σ
. By varying both
σ
and
ω
,
the poles and zeros of the signal or system can be identified. Results are graphed in the
s
-Domain (the
complex plane), using rectangular coordinates with
σ
along the horizontal axis, and
jω
along the vertical
axis. The magnitude of the transform can be plotted along a third dimension in a 3-D plot if desired, but,
more commonly, a 2-D plot is employed showing only the locations of poles and zeros.
The LT is a reversible transform, and can be used to solve differential equations, such as those
representing circuits having inductance and capacitance, in the frequency domain. The time domain
solution is then obtained by using the Inverse LT. The LT is used extensively for circuit analysis and
representation in the continuous domain. We'll see later in the topic that certain well known or classical
IIRs (Butterworth, Chebyshev, etc) have been extensively developed in the continuous domain using
LTs, and that the Laplace filter parameters can be converted to the digital domain to create an equivalent
digital IIR.
Note that the FT results when
σ
= 0 in the Laplace transform. That is to say, when the damping
coefficient is zero, the Laplace correlators are constant, unity-amplitude complex exponentials just as
those of the FT. Information plotted along the Imaginary axis in the
s
- or Laplace domain is equivalent,
then, to the FT.
z-Transform (
z-T)
The
z
-transform (
z
-T) is a discrete time form of the Laplace Transform. For those readers not familiar
with the LT, study of the
z
-T can prove helpful since many Laplace properties and transforms of common
signals are analogous to those associated with the
z
-T. The
z
-T is defined as
∞
z
−
n
X(z)
=
x
[
n
]
n
=−∞
The
z
-T converts a number sequence into an algebraic expression in
z
, and, in the reverse or inverse
z
-T, converts an algebraic expression in
z
into a sequence of numbers. The
z
-T is essentially a discrete-time
version of the Laplace transform. The correlators are discrete-time complex exponentials with amplitudes
that grow, shrink, or stay the same according to the value of
z
(a complex number) at which the transform
is evaluated. Values of
z
having magnitudes
<
1 result in a correlation of the signal with a decaying
(discrete) complex exponential, evaluation with
z
having a magnitude equal to 1 results in a Fourier-like
response, and evaluation with
z
having a magnitude greater than 1 results in a correlation of the signal with
a growing discrete complex exponential. Results can be plotted in the
z
-Domain, which is the complex
plane using polar coordinates of
r
and
θ
, where
θ
corresponds to normalized radian frequency and
r
to a
damping factor. The unit circle in the
z
-domain corresponds to the imaginary axis in the
s
-Domain; the
left hand
s
-plane marks a region of stable pole values in the
s
-plane that corresponds to the area inside
