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

13 6 366 c ontinuous time system analysis using t he

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Unformatted text preview: is specified. This fact increases t he complexity in using t he L aplace transform. T he complexity is t he result of trying t o h andle causal as well as noncausal signals. I f we r estrict all o ur signals to t he causal type, such an ambiguity does not arise. There is only one inverse transform of F (s) = l /(s + a), namely, e -atu(t). To find f it) from F (s), we need not even specify t he region of convergence. I n summary, if all signals are restricted t o the causal type, then, for a given F (s), t here is only one inverse transform f (t).t T he u nilateral Laplace transform is a special case of the bilateral Laplace transform, where all signals are restricted t o being causal; consequently t he limits of integration for t he integral in Eq. (6.8b) can be taken from 0 t o 0 0. Therefore, the unilateral Laplace transform F (s) of a signal f (t) is defined as F (s) == 1~ f (t)e-stdt (6.18) We choose 0 - (rather t han 0 + used in some texts) as t he lower limit of integration. This convention n ot only ensures inclusion of an impulse function a t t = 0, b ut also allows us t o use initial conditions a t 0 - (rather t han a t 0+) in t he solution of differential equations via t he Laplace transform. In practice, we a re likely to know t he i nitial conditions before the i nput is applied (at 0 -), n ot after t he i nput is applied ( at 0 +). O ther a dvantages of this convention appear on p. 392. T he u nilateral Laplace transform simplifies t he system analysis problem considerably, b ut t he price for this simplification is t hat we c annot analyze noncausal systems or use noncausal inputs. However, in most practical problems this is of little consequence. For this reason, we shall first consider t he u nilateral Laplace transform a nd i ts application t o s ystem analysis. (The bilateral Laplace transform is discussed l ater in Sec. 6.8.) Observe t hat basically there is no difference between t he u nilateral a nd t he bilateral Laplace transform. T he u nilateral transform is t he b ilateral transform t hat deals w ith a subclass of signals s tarting a t t = 0 (causal signals). Therefore, t he expression [(Eq. (6.8a)] for t he inverse Laplace transform remains unchanged. In practice, t he t erm Laplace t ransform means t he unilateral Laplace transform. t Actually, F (s) specifies f(t) within a null function n et) which has the property t hat th~ area under In(t)12 is z ero over any finite interval 0 to t (t > 0) (Lerch's theorem). For example, If two functions are identical everywhere except a t points o f discontinuity, they differ by a null function. roo f(t)e-st dt Jo- roo [f(t)e-<7tj e - jwt dt Jo- Because lejwtl = 1, t he i ntegral on t he r ight-hand side of this equation converges if 1~ If(t)e-O"tl dt < (6.19) 00 Hence t he existence o f t he Laplace transform is g uaranteed if t he integral in (6.19) is finite for some value of (T. Any signal t hat grows no faster t han a n e xponential signal M e<7ot for some M a nd (...
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