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 - 2015 Signals and Systems Lecture 15: Laplace Transform Dr. Iman AbuelMaaly Abdelrahman Course Specifications

Outline Signal Transforms Laplace Transform Region of Convergence Pole-Zero Plot Exercises 2015

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

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) 2015

Laplace Transform In general, for a signal x(t): Is the bilateral Laplace transform, and Is the unilateral Laplace transform.

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)

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

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). 2015

Region of Convergence (ROC) The ROC consists of those values of for which the Fourier Transform of Converges 2015

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.

Example (L-Transform) Compute the Laplace Transform of the following signal: For what values of a X(s) is valid? 2015

Obtain the Fourier Transform of the signal (1)

Laplace Transform or with 2015

By comparison with Eqn (1) we recognized Eqn(1) as the Fourier Transform of And Thus, 2015

Or equivalently, since and And thus That is, 2015

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Example 2 2015

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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). 2015

Example3 Find X(s) for the following x(t). 2015

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: 2015

Has poles at p1 =-1 and p2 =-2 and a zero at z =1 2015

Im ROC S-plane X X -2 -1 1 Re pole zeros

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.

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 2015

System Characterization by LT x(t) Causality Stability h(t) y(t)

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