Signal and Systems Prof. H. Sameti Chapter 9: Laplace Transform Motivation and Definition of the (Bilateral) Laplace Transform Examples of Laplace Transforms and Their Regions of Convergence (ROCs) Properties of ROCs Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System Geometric Evaluation of Laplace Transforms and Frequency Responses
53
Embed
Signal and Systems Prof. H. Sameti Chapter 9: Laplace Transform Motivatio n and Definition of the (Bilateral) Laplace Transform Examples of Laplace.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Signal and SystemsProf. H. Sameti
Chapter 9: Laplace Transform
Motivation and Definition of the (Bilateral) Laplace Transform Examples of Laplace Transforms and Their Regions of Convergence
(ROCs) Properties of ROCs Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System Geometric Evaluation of Laplace Transforms and Frequency Responses
Book Chapter#: Section#
2
Motivation for the Laplace Transform
CT Fourier transform enables us to do a lot of things, e.g.• Analyze frequency response of LTI systems • Sampling• Modulation
Why do we need yet another transform? One view of Laplace Transform is as an extension of the
Fourier transform to allow analysis of broader class of signals and systems
In particular, Fourier transform cannot handle large (and important) classes of signals and unstable systems, i.e. when
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
3
Motivation for the Laplace Transform (continued)
In many applications, we do need to deal with unstable systems, e.g.• Stabilizing an inverted pendulum• Stabilizing an airplane or space shuttle• Instability is desired in some applications, e.g. oscillators and lasers
How do we analyze such signals/systems? Recall from Lecture #5, eigenfunction property of LTI systems:
is an eigenfunction of any LTI system can be complex in general
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
4
The (Bilateral) Laplace Transform
s = + σ j is a complex variable – Now we explore the full ωrange of
Basic ideas:
1. A critical issue in dealing with Laplace transform is convergence:—X(s) generally exists only for some values of s, located in what is called the region of convergence(ROC): so that
2. If = is in the ROC (i.e. = 0), then σ
Computer Engineering Department, Signal and Systems
absolute integrability needed
absolute integrability condition
Book Chapter#: Section#
5
Example #1: (a – an arbitrary real or complex number)
This converges only if Re(s+a) > 0, i.e. Re(s) > -Re(a)
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
6
Example #2:
This converges only if Re(s+a) < 0, i.e. Re(s) < -Re(a) Same as (s), but different ROC Key Point (and key difference from FT): Need both X(s)
and ROC to uniquely determine x(t). No such an issue for FT.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
7
Graphical Visualization of the ROC Example1: Example2:
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
8
Rational Transforms Many (but by no means all) Laplace transforms of interest to us are
rational functions of s (e.g., Examples #1 and #2; in general, impulse responses of LTI systems described by LCCDEs), where
X(s) = N(s)/D(s), N(s),D(s) – polynomials in s
Roots of N(s)= zeros of X(s)
Roots of D(s)= poles of X(s)
Any x(t) consisting of a linear combination of complex exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and #2) has a rational Laplace transform.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
9
Example #3
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
10
Laplace Transforms and ROCs Some signals do not have Laplace Transforms (have no ROC) for all t since for all
for all t for all X(s) is defined only in ROC; we don’t allow impulses in LTs
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
11
Properties of the ROCThe ROC can take on only a small number of
different forms1. 1) The ROC consists of a collection of lines
parallel to the j -axis in the ω s-plane (i.e. the ROC only depends on ).Why? σdepends only on
2. If X(s) is rational, then the ROC does not contain any poles. Why?
Poles are places where D(s) = 0 ⇒ X(s) = N(s)/D(s) = ∞ Not convergent.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
12
More Properties If x(t) is of finite duration and is absolutely integrable, then the ROC is
the entire s-plane.
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
13
ROC Properties that Depend on Which Side You Are On - I
If x(t) is right-sided (i.e. if it is zero before some time), and if Re(s) = is in the ROC, then all values of s for which Re(s) > are also in the ROC.
Computer Engineering Department, Signal and Systems
ROC is a right half plane (RHP)
Book Chapter#: Section#
14
ROC Properties that Depend on Which Side You Are On -II
If x(t) is left-sided (i.e. if it is zero after some time), and if Re(s) = is in the ROC, then all values of s for which Re(s) < are also in the ROC.
Computer Engineering Department, Signal and Systems
ROC is a left half plane (LHP)
Book Chapter#: Section#
15
Still More ROC Properties If x(t) is two-sided and if the line Re(s) = is in the ROC,
then the ROC consists of a strip in the s-plane
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
16
Example:
Computer Engineering Department, Signal and Systems
Intuition?
Okay: multiply by constant () and will be integrable
Looks bad: no will dampen both sides
Book Chapter#: Section#
17
Example (continued):
Overlap if , with ROC:
What if b < 0? No overlap No Laplace Transform⇒ ⇒
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
18
Properties, Properties If X(s) is rational, then its ROC is bounded by poles or extends to
infinity. In addition, no poles of X(s) are contained in the ROC. Suppose X(s) is rational, then
a) If x(t) is right-sided, the ROC is to the right of the rightmost pole.b) If x(t) is left-sided, the ROC is to the left of the leftmost pole.
If ROC of X(s) includes the j -axis, then FT of x(t) exists.ω
Computer Engineering Department, Signal and Systems
Book Chapter#: Section#
19
Example: Three possible ROCs
x(t) is right-sided ROC: III No x(t) is left-sided ROC: I Nox(t) extends for all time ROC: II Yes
Computer Engineering Department, Signal and Systems
Fourier Transform exists?
20
The z-Transform
• Last time:
• Unit circle (r = 1) in the ROC ⇒DTFT exists
• Rational transforms correspond to signals that are linear combinations of DT exponentials
Computer Engineering Department, Signal and Systems
-depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)
nxZznxzXnxn
n
)(
n
nj rnxrez ||at which ROC
( )jX e
Book Chapter10: Section 1
21
Some Intuition on the Relation between ZT and LT
Computer Engineering Department, Signal and Systems
The (Bilateral) z-Transform
Can think of z-transform as DTversion of Laplace transform with
( ) ( ) ( ) { ( )}stx t X s x t e dt L x t
TenTx nsT
n nxT
)()(lim
0
n
nsT
TenxT )(lim
0
}{)( nxzznxzXnxn
n
sTez
Let t=nT
Book Chapter10: Section 1
22
More intuition on ZT-LT, s-plane - z-plane relationship
LHP in s-plane, Re(s) < 0 |⇒ z| = | esT| < 1, inside the |z| = 1 circle.
Special case, Re(s) = -∞ ⇔|z| = 0. RHP in s-plane, Re(s) > 0 |⇒ z| = | esT| > 1, outside the |z| = 1 circle.
Special case, Re(s) = +∞ ⇔|z| = ∞. A vertical line in s-plane, Re(s) = constant⇔| esT| = constant, a circle in z-plane.
Computer Engineering Department, Signal and Systems