Unformatted text preview: l-order signals. Signals which d o n ot s atisfy t his c ondition m ay n ot
b e L aplace t ransformable. 361 362 6 C ontinuous-Time S ystem Analysis Using t he Laplace Transform 6.1 T he Laplace Transform 363 4>(t) = f (t)e- ut
as depicted in Fig. 6.1a. T he signal 4>(t) is now Fourier transformable a nd i ts Fourier
components are o f t he form ejwt w ith frequencies w varying from w = - 00 t o 0 0.
T he e xponential components ejwt a nd e - jwt in t he s pectrum a dd t o give a sinusoid
of frequency w. T he s pectrum c ontains a n infinite number of such sinusoids, each
having an infinitesimal amplitude. I t would be very confusing t o d raw all these
components; hence, in Fig. 6.1b, we show j ust two typical components. Addition
of all such components (infinite in number) results in 4>(t), i llustrated in Fig. 6.1a.
T he e xponential spectral components of 4>(t) are of t he form ejwt w ith complex
frequencies j w l ying on t he imaginary axis from w = - 00 t o 0 0, as shown in Fig.
Figure 6.1a shows a signal 4>(t) = f (t)e- ut , Fig. 6.1b shows two of its infinite
spectral components, a nd Fig. 6.1c shows t he location in t he complex plane of the
frequencies of all t he s pectral components of 4>(t). Now, t he desired signal f (t) can
be obtained by multiplying 4>(t) w ith eut . T his fact means t hat we c an synthesize
f (t) by multiplying each spectral component of 4>(t) w ith eO't a nd a dding them. B ut
m ultiplying t he s pectral c omponents of 4>(t) (sinusoids in Fig. 6.1b) with eut results
in t he e xponentially growing sinusoids as shown in Fig. 6.1e. Addition of all such
exponentially growing sinusoids (infinite in number) results in f (t) in Fig. 6.1d.
T he s pectral components of 4>(t) a re of t he form ejwt . Multiplication of these components with eut results in t he s pectral components of the form eutejwt = e(u+jw)t.
Therefore, a component of frequency j w in t he s pectrum of 4>(t) is transformed into
a component of frequency a + j w in t he s pectrum of f (t). T he l ocation in the
complex plane o f t he frequencies a + j w is along a vertical line, depicted in Fig.
I t is clear t hat a signal f (t) c an be synthesized with growing everlasting exponentials lying along t he p ath a + j w, w ith w varying from - 00 t o 0 0. T he value
of a is flexible. For example, if f (t) = e2t u(t), t hen 4>(t) = f (t)e- ut c an be made
Fourier transformable if we choose a > 2. T hus, there are infinite choices for a .
T his means t he s pectrum of f (t) is n ot unique, a nd t here are infinite possible ways
of synthesizing f (t). Nevertheless, a has a certain minimum value ao for a given
f (t) [ao = 2 for f (t) = e2t u(t)]. T he region in t he complex plane where a > ao
(Fig. 6.1g) is c alled t he r egion o f c onvergence (or e xistence) o f t he resulting
transform of f (t ).
All these conclusions reached here by pure heuristic reasoning will now be
derived analytically. T he frequency j w in t he Fourier transform will be generalized cP(t) f (t) = f ( t)e- at t-(a) (d...
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