Signal Processing and Linear Systems-B.P.Lathi copy

Now t he desired signal f t can be obtained by

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Unformatted text preview: th o f integration (from c ' - jao t o c ' + j ao) in Eq. (6.10) may be different from t hat in Eq. (6.9) t o allow for t he convergence of t he integral in Eq. (6.10). This subtle point will b e discussed later in Sec. 6.8-1. I f y (t) - $=} Y (s), t hen according t o Eq. (6.10) it follows t hat Y (s) = F (s)H(s) (6.11) Here, we have expressed the input f (t) as a sum of exponentials components of the form est. T he system response is t hen o btained by adding the responses t o all these exponential components. T he procedure followed here is identical t o t hat in chapter 2 (where the input is expressed as a sum of impulses) or chapter 4 (where the input is expressed as a sum of exponentials of the form e jwt ). I n conclusion, we have shown t hat for an LTIC s ystem with transfer function H (s), if the input a nd t he o utput a re f (t) a nd y (t), respectively, and if f (t) - $=} F (s), y (t) - $=} Y (s) t hen Y (s) = F (s)H(s) (6.12) We shall derive this result more formally later. Linearity o f the Laplace Transform T he Laplace transform is a linear operator, and t he principle of superposition applies; t hat is, if then f (t) - $=} F(s) (6.13) 6 366 C ontinuous-Time System Analysis Using t he Laplace Transform 6.1 T he Laplace Transform T he p roof is t rivial a nd follows directly from t he definition of t he Laplace transform. This result can be extended t o a ny finite number of terms. T he Region o f Convergence t- Earlier, we discussed t he i ntuitive meaning of t he region of convergence (or region of existence) of t he Laplace transform F (s). Mathematically, t he region of convergence for F (s) is t he s et of values of s ( the region in t he complex plane) for which t he i ntegral in Eq. (6.8b) defining t he direct Laplace transform F (s) converges. T his c oncept will become clear in t he following example . • (a) E xample 6 .1 For the signal f (t) = e -atu(t), find the Laplace transform F (s) and its region of convergence. By definition F (s) Because u (t) = 1: e -atu(t)e-stdt = 0 for t < 0 and u (t) = 1 for t ~ o (- -I _e-Ol u ( -t) 0, (b) F ig. 6 .2 Signals e -atu(t) and _ eatu( - t) have the same Laplace transform but different regions of convergence. F (s)= { OOe-ate-S'dt= (OOe-<s+a)t dt = _ __ e -(s+a)'jOO l io io s+ a (6.14) 0 Role o f the Region of Convergence Note t hat s is complex and as t - + 0 0, the term e -(s+a)' does not necessarily vanish. Here we recall t hat for a complex number z = a + j/3, Now le-j~tl = 1 regardless of the value of /3t. Therefore, as - > 0 0 if a < O. Thus Re z > 0 lim e -zt = t -oo 00 Re z < 0 Clearly t -+ 0 0, e - zt -+ 0 only if a > 0, and e - zt {O lim e-(s+a)t = {O 00 t _oo (6.15) R e(s+a»O R e (s + a) < 0 Use of this result in Eq. (6.14) yields 1 F (s)=s+a + a) > 0 (6.16a) Re s > - a (6.16b) R e(s or e -atu(t) -= _+a 1_ s The region of convergence of F (s) is Re s > - a, as shown in the shaded area in Fig. 6.2a. This fact means that the integral defining F (s) in Eq. (6.14) exists only fo...
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