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

13 6 366 c ontinuous time system analysis using t he

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: 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 (...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online