1. University of Khartoum -Signals and Systems- Lecture 11
بسم الله الرحمن الرحيم
University of Khartoum -Signals and Systems- Lecture 11
2015
University of Khartoum
Department of Electrical and Electronic Engineering
Third Year
Signals and Systems
Lecture 15: Laplace Transform
Dr. Iman AbuelMaaly Abdelrahman
Course Specifications

2. Outline Signal Transforms Laplace Transform Region of Convergence
Pole-Zero Plot
Exercises
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3. Signals Transform DSP DSP Signal in time domain x(t)
Continuous-Time signals
Discrete-Time signals
Signal in time domain
x(t)
Signal in time domain
x[n]
Signal in frequency domain
- Fourier Series Ck
- Fourier Transform X(jω)
Signal in frequency domain
- Fourier Series Ck
Fourier Transform X(ejω)
DSP
Signal in Laplace domai Laplace Transform X(s)
Signal in Z- domain
- Z- Transform X(Z)
DSP

4. The Laplace Transform
Assume s is any complex number of form: s =  + j 
That is, s is not purely imaginary and it can also have real values.
X(s)|s=j=X(j).
X(s) is called the Laplace transform of x(t)
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5. Laplace Transform
In general, for a signal x(t):
Is the bilateral Laplace transform, and
Is the unilateral Laplace transform.

6. Laplace Transform ( S-Plane)
The xy-axis plane, where x-axis is the real axis and y-axis is the imaginary axis, is called the s-plane.
Im (S)
Re (S)

7. Laplace Transform
Fourier transform is the projection of Laplace transform on the imaginary axis on the s-plane.
This gives two additional flexibility issues to the Laplace transform:
Analyzing transient behavior of systems
Analyzing unstable systems

8. Region of Convergence (ROC)
Similar to the integral in Fourier transform, the integral in Laplace transform may also not converge for some values of s.
So, Laplace transform of a function is always defined by two entities:
Algebraic expression of X(s).
Range of s values where X(s) is valid, i.e. region of convergence (ROC).
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9. Region of Convergence (ROC)
The ROC consists of those values of
for which the Fourier Transform of
Converges
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10. The Laplace Transform
Transform techniques are an important tool in the analysis of signals and LTI systems.
The Z-transform plays the same role in the analysis of discrete-time signals and LTI systems as the Laplace transform does in the analysis of continuous-time signals and systems.

11. Example (L-Transform)
Compute the Laplace Transform of the following signal:
For what values of a X(s) is valid?
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12. Obtain the Fourier Transform of the signal
(1)

13. Laplace Transform
or with
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14. By comparison with Eqn (1) we recognized Eqn(1) as the Fourier Transform of
And Thus,
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15. Or equivalently, since and
And thus
That is,
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17. Example 2
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18. t

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22. Pole-Zero Plot Given a Laplace transform
Poles of X(s): are the roots of D(s).
- Zeros of X(s): are the roots of N(s).
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23. Example3
Find X(s) for the following x(t).
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24. The set of values of Re{s} for which the Laplace transforms of both terms converges is Re{s} >-1, and thus combining the two terms on the right hand side of the above equation we obtain:
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25. Has poles at p1 =-1 and p2 =-2 and a zero at z =1
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26. Im
ROC
S-plane X X -2 -1 1 Re
pole
zeros

27. Inverse Laplace Transform
Integral of inverse Laplace transform:
However, we will mainly use tables and properties of Laplace transform in order to evaluate x(t) from X(s).
That will frequently require partial fractioning.

28. Laplace Transform and LTI
x(t)=est
y(t)=h(t)* est
In the above system, H(s) is called the transfer function of the system. It is also known as Laplace transform of the impulse response h(t).
LTI
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29. System Characterization by LT
x(t)
Causality
Stability
h(t)
y(t)

30. مع تمنياتي بالتوفيق