International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Discrete Time Signals: Relation of Z Transform and Laplace
Transform
N.D.Narkhede1,Dr.J.N.Salunke2
1. Associate Professor J.T.Mahajan college of Engineering,Faizpur,Jalgaon (India)
2. Director of School of Mathematical Sciences SRTM University Nanded (India)
Abstract: This paper briefly explain z-transform
and compare it with Laplace transform. It deals
with a review of what z-transform plays role in the
analysis of discrete time signal & LTI system as
Laplace transform plays in the analysis of
continuous time signals & LTI systems. Also it
briefly explain difference equation (not differential
equation). An introduction to Z and Laplace
transform their relation is the topic of this paper. A
pictorial representation of the region of
convergence has been sketched and relation is
discussed. This paper begins with the derivation of
the z-transform from the Laplace transform of a
discrete-time signal.
Key Words: Laplace Transform, Z-Transform,
Discrete time signal, Difference equation, Sampled
signals etc
1 Introduction
With the explosion of digital communication &
media, the need for methods to analyze & process
digital data is becoming more important now.
According to Professor Todd K. Moon, Ztransform like the Laplace transform is an
indispensable mathematical tool/technique for the
design, analysis and monitoring of stability of the
systems. A working knowledge of Z-transform is
essential to the study of discrete time signals &
systems. The z-transform is the discrete-time
counter-part of the Laplace transform and a
generalization of the Fourier transform of a
sampled signal. A useful aspect of the Laplace
transform & the Z-transform are representation of a
system in terms of the locations of the poles and
the zeros of the system transfer function in
complex plane [1, 2]. Z-transform converts a
discrete signals
in to a function
of an
arbitrary complex valued variable z. Like Laplace
transform the z-transform allows insight into
transient behavior, the steady state behavior, and
the stability of discrete-time systems. This paper
begins with the definition of the derivation of the ztransform from the Laplace transform of a discretetime signal. A useful aspect of the Laplace and the
z-transforms are there presentation of a system in
terms of the locations of the poles and the zeros of
the system transfer function in a complex plane [1]
ISSN: 2231-5381
2 Brief comparison between z-transformation
and Laplace transform & Derivation of the ztransform
Z-transform is widely used in linear systems that
are described by difference equation. In sampled
systems inputs & outputs are related by difference
equation & Z-transform techniques are used to
analyze such systems. A difference equation is an
equation which expresses a relation between an
independent variable & the successive values of the
dependent variable or the successive differences of
the dependent variable. This represents a Linear
Time Invariant (LTI) systems & obeys all its usual
properties. Therefore difference equation is to
discrete signal processing, what the differential
equation is to analogue signal processing. The ztransform is the discrete-time counter part of the
Laplace transform. In this section we derive the ztransform from Laplace transform X(s), of a
continuous time signal x(t), is given by the integral
(1)
Where the complex variable s= + jw, and the
lower limit of t=0- allows the possibility that the
signal x[t] may include an impulse.[1,5]. The
inverse Laplace transform is defined by
(2)
Where
is selected so that X[s] is analytic [no
singularities] for s > . The z-transform can be
derived from Eq. [1] by sampling the continuoustime input signal x[t]. for a sampled signal x[mTs],
normally denoted as x[m] assuming the sampling
period Ts=1, the Laplace transform Eq. [1]
becomes
(3)
Substituting the variable es in Eq. [3] with the
variable z we obtain the one-sided z-transform
equation
(4)
The two-sided z-transform is defined as
(5)
A similar relationship exists between the Laplace
transform and the Fourier transform of a
continuous time signal. The Laplace transform is a
one-sided transform with the lower limit of
integration at t=0- , whereas the Fourier transform
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
[1,2] is a two sided transform with the lower limit
of integration at t= - . However for a one-sided
signal, which is zero-valued for t<0-, the limits of
integration for the Laplace and the Fourier
transforms are identical. In that case if the variable
s in the Laplace transform is replaced with the
frequency variable j2 f then the Laplace integral
becomes the Fourier integral. Hence for a onesided signal, the Fourier transform is a special case
of the Laplace transform corresponding to s=j2 f
and
.
right of the s-plane,
>1, is mapped
onto the outside of the unit circle this is the region
of unstable signals and systems. The jw – axis, with
or r=
=1, is itself mapped onto the unit
circle line. Hence the Cartesian co-ordinates used
in s-plane for continuous time signals Fig. 1, is
mapped into a polar representation in the z-plane
for discrete- time signals Fig. 1.(b).
3 The s-plane, z-Plane and The Unit Circle
The frequency variables of the Laplace transform
s= + jw, and the z-transform z=rejw are complex
variables with real and imaginary parts and can be
visualized in a two dimensional plane. The s-plane
of the Laplace transform and the z-plane of ztransform. In the s-plane the vertical jw – axis is
the frequency axis, and the horizontal - axis gives
the exponential rate of decay, or the rate of growth,
of the amplitude of the complex sinusoid as also
shown in fig. 1 shown fig. (1) a.
Since the z-transform is an infinite power series, it
exists only for those values of the variable z for
which the series convergence [ROC] of X[z] is the
set of all the values of z for which X[z] attains a
finite computable value. [1,2,3].
To find the value of z for which the series
converges, we use the ratio test or the root test
states that a series of complex number
with limit
(6)
The inside of unit circle
corresponds to σ<0 part
of s-plane
Im-axis
jw
The jw axis of the s-plane is
the location of the fourier
basis function
0
outside the unit circle
corresponds to σ>0
part of s-plane
3π σ
e
2π
φ
π
Re-axis
4 The Region of Convergence [ROC]
Converges absolutely if A<1 and diverges if A>1
the series may or may not converge.
The root test state that if
(7)
Then the series converges absolutely if A<1, and
diverges if A>1, and may converge or diverge if
A=1. More generally, the series converges
absolutely if
(8)
Where
The unit circle
σ axis
corresponds to the jw
of s-plane
-π
denotes the greatest limit points of
and diverges
(9)
If we apply the root test in equation [4] we obtain
the convergence condition
-2π
Fig 1(b) z-plane
-3π
(10)
Fig 1(a) s-plane
When a signal is sampled in the time domain its
Laplace transform, and hence the s-plane, become
periodic with respect to the jw-axis. This is
illustrated by the periodic horizontal dashes lines in
Fig. 1. Periodic processes can be conveniently
represented using a circular polar diagram such as
the z-plane and its associated unit circle. Now
imagine bending the jw-axis of the s-plane of the
sampled signal of Fig. 1 in the direction of the left
hand side half of the s-plane to form a circle such
that the points and
meet. The resulting circle
is called the z-plane. The area to the left of the splane, i.e.for
or r = <1, is mapped into the
area inside the unit circle, this is the region of
stable causal signals and systems. The area to the
ISSN: 2231-5381
Where R is known as the radius of convergence for
the series. Therefore the series will converge
absolutely for all points in the centered at the origin
( with the possible exception of the pint at infinity).
This region is called the region of convergence
[ROC].
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Im-axis
X(m)
R.O.C.
β
-m
m=
Fig 2(a), two 0sided discrete
signal
,
This infinite power series converges if
.
ISSN: 2231-5381
Re-axis
m
Example: Determine the z-transform the region of
convergence. For the signal
Solution: By definition z-transformation
α
Fig 2(b) ROC,
Conclusion:
1. The work has thoroughly focused on z-transform
& the beauties of z-transform like Laplace
transform.
2. In sampled system Z.T. plays unique role. It
allow designers to analyze the system & predict
performance and to think different terms like
frequency response, digital filters & its use in
digital signal processing (DSP).
References:
[1] EECS 451 Digital Signal Processing and Analysis, lecture
notes J. Fessler
[2] Corriggan D. (2012) Difference Equation & digital filters
Retrieved
May
10,
2015
from
http:
//www.mee.tcd.ie/~corrigad/3cl/DSP.|_2012_students.pdf.
[3] Smith S.W. (1997). The scientist & Engineer’s guide to
digital signal processing. Retrieved may 08, 15 from
http://www.analog.com/media/en/techincal-documentation/dspbook/dsp-book ch 19pdf.
[4] Digital Signal Processing (principles, algorithms and
applications ) by john. G. Prokies, Dimitris g. manolakis
ISBN.9780131873742
[5] Principles of signal processing and linear system
(international version) by b. P. lathi ISBN:3:978-0-19-80622288 Published in India by oxford university Press @ 2009.
[6] Sunetra S. Adsad, Mona V. Dekate (2015), Relation of ztransform & Laplace transform in Discrete Time Signal,
International Research Journal of Engineering & Technology,
Vol. 2, Issue 2, May 2015.
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