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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