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

I t should be remembered t hat t he r esults in eqs

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e frequency may have a positive feedback a t some other frequency because of phase shift in t he transmission p ath. I n other words, a feedback system, generally, cannot be described in black a nd w hite terms such as having positive or negative feedback. Let us clarify this s tatement by a n example. Consider t he case G (s)H(s) = l /s(s + 2)(s + 4). T he r oot locus of this system appears in Fig. 6.43. This system shows negative feedback a t lower frequencies. B ut because of p hase shift a t higher frequencies, t he feedback becomes positive. Consider t he loop gain G (s)H(s) a t a frequency w = 2.83 (at s = j2.83). 1 G (jw)H(jw) = j w(jw + 2)(jw + 4) At w = 2.83 1 G (j2.83)H(j2.83) = j2.83(j2.83 + 2)(j2.83 1 _ '1800 = -e 48 J + 4) 1 =-- 48 Recall t hat t he o verall gain (transfer function) T (s) is K G(s) T (s) = 1 + K G(s)H(s) At frequency s = j2.83 (w = 2.83), t he gain is T (j2.83) = KG(J·2.83) K 6 .8 The Bilateral laplace Transform Sit~ations involving noncausal signals a nd/or systems cannot be handled by the (umlateraJ) Laplace transform discussed so far. These cases can be analyzed by t he b ilateral (or t wo-sided) Laplace transform defined by F (s) = I: f (t)e-stdt (6.101a) a nd f (t) c an be obtained from F (s) by t he inverse transformation 1 c jOO 1 f (t)=-. 27rJ + F (s)estds c -joo (6.1Olb) Observe t hat t he u nilateral Laplace transform discussed so far is a special case of t he .bilateral Laplace transform, where t he signals are restricted t o t he c ausal type. Basically, t he two transforms are t he same. For this reason we use t he same notation for the bilateral Laplace transform . . Earl~er we showed t hat t he Laplace transforms of e -atu(t) a nd of - eatu(_t) a re Identical. T he only difference involves their regions of convergence. T he region o f convergence for t he former is Re s > - a; t hat for t he l atter is Re s < - a as illustrated in .Fig. 6.2. Clearly, t he inverse Laplace transform of F (s) is n ot uni~ue unless the regIOn o f convergence is specified. I f we restrict all o ur signals to the causal ~yp:}owever, thi~ ambigui~y does n ot arise. T he inverse transform of l /(s + a) IS e u (t). Thus, I II t he u mlateral Laplace transform, we c an ignore t he region of convergence in determining t he inverse transform of F (s ). We now show t hat any bilateral transform can be expressed in terms of two unilateral transforms. I t is, therefore, possible t o e valuate bilateral transforms from a table of unilateral transforms. Consider t he function f it) a ppearing in Fig. 6.48a. We s eparate f it) i nto two components, h it) a nd h it), r epresenting t he positive time ( causal) c omponent a nd t he negative time ( anticausal) c omponent of f it) respectively (Figs. 6.48b a nd 6.48c): 1 - 48 As long as K r emains below 48, t he system is s table, b ut for K = 48, t he s ystem gain goes t o 0 0, a nd t he system becomes unstable. T he feedback, which was negative below w = 2.83 (because t he phase shift has n ot reached - 180°), becomes positive. I f t here is e nough gain ( K = 48)...
View Full Document

Ask a homework question - tutors are online