The Laplace Tranform Prof. Siripong Potiuk Pierre Simon De Laplace 749-827 French Atronomer and Mathematician
Laplace Tranform An extenion of the CT Fourier tranform to allow analyi of broader cla of CT ignal and ytem Handle an important cla of untable ytem whoe ignal are not abolutely integrable, i.e., 2
Condition for CTFT Exitence Applicable for aperiodic ignal of finite and infinite duration which atifie: (a (b Dirichlet' Condition Finite Energy : x( t 2 dt Bilateral Laplace Tranform Satify the condition of abolute integrability by multiplying x(t by a convergence factor e -t for ome value of X ( j { x( t e t } e jt dt CTFT{ x( t e Combine the exponential and let = +j t } 3
Region of Convergence (ROC Convergence i a critical iue in dealing with Laplace tranform X( exit for ome value of, and all An area in the complex or -plane defined by uch that the Fourier integral converge i called the region of convergence (ROC t ROC { j C x( t e dt } Example. Determine the Laplace tranform along with the correponding ROC of the ignal at x( t e u( t, a i real or complex 4
Example 2. Determine the Laplace tranform along with the correponding ROC of the ignal at x( t e u( t, a i real or complex 5
Example 3. Determine the Laplace tranform along with the correponding ROC of the ignal ( a x( t Ce t, j t ( b y( t e 0, t t Rational Laplace Tranform For mot practical ignal, the Laplace tranform can be expreed a a ratio of two polynomial where z, z and N( b0 ( z( z2 ( z X ( D( a ( p ( p ( p of the numerator polynomial p, p 2 2,, z M,, p N are the zeroe of X(,i.e., the root are the pole of X(,i.e., the root of the denominator polynomial. 0 2 M N 6
7 Rational Laplace Tranform N N N M M M N M a a b b b p p p z z z b D N X 0 2 2 0 ( ( ( ( ( ( ( ( ( It i cutomary to normalize the denominator polynomial to make it leading coefficient one, i.e., Alo, X( i a proper rational tranform if pole. # of zeroe # of i.e., M N, The complex -plane + j (Re{} (Im{}
The pole-zero plot of a rational Laplace tranform in the -plane. Laplace Tranform Propertie Parallel many of the CTFT propertie, except for the need to pecify ROC Linearity: Time-hifting 8
Example 4. Determine the Laplace tranform along with the correponding ROC of the ignal x t e u t e u t 2t t ( 3 ( 2 ( Propertie of the ROC 9
0
7 If X( i rational, it ROC i bounded by pole or extend to infinity. Alo, no pole of X( are contained in the ROC. (a x(t right-ided ROC to the right of rightmot pole (b x(t left-ided ROC to the left of leftmot pole (c x(t two-ided ROC a trip between two pole 8 ROC of x(t contain j-axi CTFT of x(t exit 2
Invere Laplace Tranform Tranform back from the -domain to the time domain Generally, computed by For rational Laplace tranform, expand in term of partial fraction and ue table of tranform pair and propertie From the previou example, the time-domain ignal, x(t, reulting from the inverion proce for each ROC i 3
Example 5. Determine the number of all poible ignal that have imilar Laplace tranform below but different ROC. X ( ( 2( 3( 2 4
Convolution Property L Y ( H ( X ( Laplace tranform of output Laplace tranform of input Eigenfunction of LTI Sytem An eigenfunction of a ytem i an input ignal that, when applied to a ytem, reult in the output being the caled verion of itelf. The caling factor i known a the ytem eigenvalue. Complex exponential are eigenfunction of LTI ytem, i.e., the repone of an LTI ytem to a complex exponential input i the ame complex exponential with only a change in amplitude. 5
H( = L{h(t} = Laplace tranform of impule repone H( i called the ytem function or tranfer function Checking Cauality of LTI Sytem A caual LTI ytem ha a caual impule repone (i.e., h(t < 0 for t < 0 ROC of tranfer function for a caual ytem i a RHP, but the convere i not true However, a ytem with rational tranfer function i caual iff it ROC i the plane to the right of the rightmot pole Similar tatement can be aid about an anticaual ytem 6
Checking Stability of LTI Sytem A BIBO table LTI ytem ha an abolutely integrable impule repone (i.e., CTFT of the impule repone converge Conequently, an LTI ytem i BIBO table iff the ROC of it tranfer function include the entire j-axi In term of pole, a caual ytem with rational tranfer function i BIBO table iff all of the pole lie in the LHP (i.e., all pole have negative real part The Unilateral Laplace Tranform - The unilateral Laplace tranform of a CT ignal x(t i defined a X ( x( t e 0 - Equivalent to the bilateral Laplace tranform of x(tu(t - Since x(tu(t i alway a right-ided ignal, ROC of X( alway include the RHP t - Ueful for olving LCCDE with initial condition dt 7