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Unformatted text preview: st be a t least 0.25 Hz. Assume the essential
bandwidth ( the folding frequency) of J(t) to be a t least 3 Hz. Do n ot compute the
DFT; just write the appropriate sequence Jk.
2 -3 Choose appropriate values for No a nd T and compute the D FT of the signal e -tu(t).
Use two different criteria for determining the effective bandwidth of e -tu(t). Use the
bandwidth t o b e t hat frequency where the amplitude response drops to 1% of its peak
value (at w = 0). Next, use the 99% energy criterion for determining the bandwidth
(see Example 4.16).
.2-4 Repeat Problem 5.2-3 for t he signal
2 J(t) = t2 + 1 f(t) U
o ( a) 1 1-" " "p
1 2 (b) 1 -.. F ig. P 5.2-5 . 2-5 For the signals J(t) and get) resented in Fig. P5.2-5, write the appropriate sequences
Jk a nd gk necessary for t he computation of the convolution of J(t) and get) using
DFT. Use T = ~. T he F ourier t ransform is a tool which allows us t o r epresent a signal J (t) a s a
continuous s um o f exponentials of t he form e jwt , whose frequencies are r estricted t o
t he i maginary axis in t he c omplex plane (8 = j w). As we saw in C hapters 4 a nd 5,
such a r epresentation is q uite v aluable in t he a nalysis a nd processing of signals. I n
t he a rea o f system analysis, however, t he use of Fourier t ransform leaves much t o
b e desired. F irst, t he F ourier t ransform e xists only for a r estricted class of signals
and, therefore, c annot b e u sed for such i nputs a s growing exponentials. S econd,
t he F ourier t ransform c annot b e u sed easily t o a nalyze u nstable o r even marginally
stable systems. 6.1 The Laplace Transform T he basic reason for b oth t hese difficulties is t hat for some signals, such as
e atu(t) (a > 0 ), t he F ourier t ransform does n ot e xist because o rdinary s inusoids or
exponentials o fthe form e jwt ( on account of t heir c onstant a mplitudes) are incapable
o f s ynthesizing exponentially growing signals. T his p roblem could be resolved i f i t
were possible t o use basis signals of t he form e st ( instead o f e jwt ), where t he c omplex
frequency 8 is n ot r estricted t o j ust t he i maginary axis (as in t he F ourier transform).
This is precisely w hat is done in t he following e xtended t ransform known' as t he
b ilateral L aplace t ransform, w here t he frequency variable 8 = j w is generalized
t o 8 = a + j w. Such g eneralization p ermits u s t o u se exponentially growing sinusoids
t o s ynthesize a signal J (t). Before developing t he m athematical o perations r equired
for s uch a n e xtension, we will find i t i lluminating t o have a n i ntuitive u nderstanding
o f such a generalization. 6.1-1 Intuitive Understanding o f t he Laplace Transform I f a s ignal J (t) shown in Fig. 6.1d is n ot F ourier transformable, we m ay b e
able t o m ake i t F ourier t ransformable b y multiplying i t w ith a d ecaying exponential
such as e-<1t.tFor e xample, a signal e2t u(t) c an b e m ade F ourier t ransformable by
multiplying i t w ith e -<1t w ith a > 2. L et
t This a ssumes t hat If(t)1 < M eat for some real, positive, a nd finite numbers M a nd a. Such
signals a re called exponentia...
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