Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 25

X ( f )

x(t) 0.5 δ ( f )

0.3183 δ ( f +1/4) 0.3183 δ ( f −1/4)

t , sec f , Hz

−1 0 1 2 5

−0.1061 δ ( f −3/4)

Fig. 1.16 A square wave and its Fourier transform (full line) and the envelope of the Fourier
transform (dotted line)

Here T o = 4 sec, f o = 1/4 Hz. The envelope of the X k coefficients in the frequency
domain is obtained by substituting f for kf o = k/4, hence (1/2)sinc(k/2) becomes (1/
2)sinc(2f).

The above rectangular pulse is useful in many applications. Its general form is
given by: P k ¼1 A P T ð t nT o Þ , and its Fourier coefficients are X k = (AT/
T o ){sinc(kf o T), where T is the duration of the ''ON'' state. The envelope of these
coefficients is given by E(f) = (AT/T o )sinc (fT) = X 1p (f)/T o , where X 1p ¼

F P T ð t f g¼ FT of one period.
MATLAB: The rectangular pulse train can be simulated in MATLAB as fol-
lows:

1.2.3.3 The Laplace Transform

The Laplace transform (LT) is a generalization of the Fourier transform in which
one decomposes a signal into the sum of decaying complex exponentials, rather
than simply complex exponentials. The incorporation of amplitude as well as
frequency information into the basis functions of the LT introduces a number of
advantages. Firstly, it enables the LT to deal with a wider class of signals—
including many signals which have no FT (such as the unit ramp t n u(t)). Secondly,
the LT formulation is more naturally suited to analyzing the stability of a system.
Because of these advantages, the LT is the main tool for representing and ana-
lyzing analog (continuous-time) feedback systems, where stability is of extreme
importance. There are two definitions of the LT as detailed below.