5-59- Control Sytem II FS 28 Lecture 2: The -Tranform From the Laplace Tranform to the tranform The Laplace tranform i an integral tranform which take a function of a real variable t to a function of a complex variable. Intuitively, for control ytem, t repreent time and repreent frequency. Definition. The one-ided Laplace tranform of a ignal x(t) i defined a L (x(t)) X() x() x(t)e t dt. (.) Example. Conider x(t) co(ωt). The Laplace tranform of uch a ignal read L(co(ωt)) From thi equation, one ha e t co(ωt)dt e t co(ωt) ω ω ( e t in(ωt) ω2 2 L(co(ωt)). L(co(ωt)) e t in(ωt)dt + ω ) e t co(ωt)dt (.2) 2 + ω 2. (.3) Some of the known Laplace tranform are lited in Table. Laplace tranform receive a input function, which are defined in continuou-time. In order to analye dicrete-time ytem, one mut derive it dicrete analogue. Dicrete time ignal x(kt ) x[k] are obtained by ampling a continuou-time function x(t). A ample of a function i it ordinate at a pecific time, called the ampling intant, i.e. x[k] x(t k ), t k t + kt, (.4) where T i the ampling period. A ampled function can be expreed through the multiplication of a continuou funtion and a Dirac comb (ee reference), i.e. with D(t) which i a Dirac comb. x[k] x(t) D(t), (.5) Definition 2. A Dirac comb, alo known a ampling function, i a periodic ditribution contructed from Dirac delta function and read D(t) δ(t kt ). (.6)
5-59- Control Sytem II FS 28 Remark. An intuitive explanation of thi, i that thi function i for t kt and for all other cae. Since k i a natural number, i.e. k,...,, applying thi function to a continuou-time ignal conit in conidering information of that ignal paced with the ampling time T. Imagine to have a continuou-time ignal x(t) and to ample it with a ampling period T. The ampled ignal can be decribed with the help of a Dirac comb a x m (t) x(t) δ(t kt ) x(kt ) δ(t kt ) x[k] δ(t kt ), (.7) where we denote x[k] a the k th ample of x(t). Let compute the Laplace tranform of the ampled ignal: where we ued X m () L (x m (t)) ( (b) (c) x m (t)e t dt x[k] ( Thi i an application of Definition. x[k] δ(t kt )e t dt x[k]e kt, δ(t kt )e t dt (.8) (b) The um and the integral can be witched becaue the function f(t) δ(t kt )e t i non-negative. Thi i a direct conequence of the Fubini/Tonelli theorem. If you are intereted in thi, have a look at http://en.wikipedia.org/wiki/fubini% 27_theorem. (c) Thi reult i obtained by applying the Dirac integral property, i.e. δ(t kt )e t dt e kt. (.9) By introducing the variable e T, one can rewrite Equation.8 a X m () x[k] k, (.) which i defined a the tranform of a dicrete time ytem. We have now found the relation between the -tranform and the Laplace tranform and are able to apply the concept to any dicrete-time ignal. 2
5-59- Control Sytem II FS 28 Definition 3. The bilateral tranform of a dicrete-time ignal x[k] i defined a X() Z ((x[k]) x[k] k. (.) Some of the known tranform are lited in Table. x(t) L(x(t))() x[k] X() e at +a e akt e at t 2 kt T ( ) 2 t 2 2 3 (kt ) 2 T 2 (+ ) ( ) 3 in(ωt) co(ωt) ω in(ωkt ) 2 +ω 2 co(ωkt ) 2 +ω 2 in(ωt ) 2 co(ωt )+ 2 co(ωt ) 2 co(ωt )+ 2 Table : Known Laplace and tranform. 2 Propertie In the following we lit ome of the mot important propertie of the tranform. Let X(), Y () be the tranform of the ignal x[k], y[k].. Linearity Z (ax[k] + by[k]) ax() + by (). (2.) Z (ax[k] + by[k]) (ax[k] + by[k]) k ax[k] k + ax() + by (). by[k] k (2.2) 2. Time hifting Z (x[k k ]) k X(). (2.3) Z (x[k k ]) x[k k ] k. (2.4) 3
5-59- Control Sytem II FS 28 Define m k k. It hold k m + k and x[k k ] k x[m] m k k X(). (2.5) 3. Convolution Z (x[k] y[k]) X()Y (). (2.6) Proof. Follow directly from the definition of convolution. 4. Revere time Z (x[ k]) X ( ). (2.7) Z (x[ k]) X r x[ k] k x[r] ( ). ( ) r (2.8) 5. Scaling in domain Z ( a k x[k] ) X (. (2.9) Z ( a k x[k] ) X (. ( k x[k] (2.) 6. Conjugation Z (x [k]) X ( ). (2.) 4
5-59- Control Sytem II FS 28 ( ) X () x[k] k x [k]( ) k. (2.2) Replacing by one get the deired reult. 7. Differentiation in domain Z (kx[k]) X(). (2.3) X() x[k] k linearity of um/derivative x[k] k x[k]( k) k kx[k] k, (2.4) from which the tatement follow. Reference [] Analyi III note, ETH Zurich. [2] http://en.wikipedia.org/wiki/dirac_comb [3] http://ccrma.tanford.edu/~jo/laplace/laplace_4up.pdf 5