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Unformatted text preview: d by t he s um of its orthogonal components
in a complete orthogonal vector space, a signal can also be represented by t he
s um of its orthogonal components in a complete orthogonal signal space. Such a
r epresentation is known as t he g eneralized F ourier series representation. A vector
can be represented by its orthogonal components in many different ways, depending
on t he c oordinate system used. Similarly a signal c an be represented in terms of
different orthogonal signal sets of which t he t rigonometric a nd t he e xponential signal
sets are two examples. We have shown t hat t he t rigonometric a nd e xponential
Fourier series are periodic with period equal t o t hat of t he f undamental in t he
set. In this chapter we have shown how a periodic signal can be represented as a
sum of (everlasting) sinusoids or exponentials. I f t he frequency of a periodic signal
is wo, t hen i t can be expressed as a sum of t he sinusoid of frequency wo a nd its
harmonics (trigonometric Fourier series). We c an r econstruct t he periodic signal
from a knowledge of t he a mplitudes a nd phases of these sinusoidal components
(amplitude a nd phase spectra).
I f a periodic signal has a n even symmetry, t he Fourier series contains only
cosine terms. In contrast, if t he signal has a n o dd symmetry, t he Fourier series
contains only sine terms. I f a periodic signal has a j ump discontinuity, t he signal is
not smooth a nd requires significant high frequency components t o synthesize jumps.
Consequently, its amplitude spectrum decays slowly with frequency as l in. I f t he
signal has no j ump discontinuities, b ut i ts first derivative has a j ump discontinuity,
the signal is s moother, a nd its amplitude s pectrum decays faster as 1 /n 2 • I f n either
the signal nor its first derivative has j ump discontinuities, b ut t he second derivative
has discontinuities, t hen t he signal is even more smooth, a nd its amplitude s pectrum
decays still faster as 1 /n 3 , a nd so on. A sinusoid can be expressed in terms of exponentials. Therefore, t he Fourier series of a periodic signal can also be expressed as a sum of exponentials (the exponential Fourier series). T he e xponential form of t he Fourier series a nd t he expressions
for t he series coefficients are more compact t han t hose of t he t rigonometric Fourier
series. Also, the response of LTIC s ystems t o a n e xponential i nput is simpler t han
that for a sinusoidal input. Moreover, t he e xponential form of representation lends
itself b etter t o m athematical manipulations t han does t he t rigonometric form. For 226 3 Signal R epresentation by Orthogonal Sets t hese reasons, t he e xponential form of t he series is preferred in modern practice in
t he a reas of sigIlals a nd systems.
T he plots o f a mplitudes and angles of various exponential components of t he
F ourier series a s functions of t he frequency are t he exponential Fourier s pectra
( amplitude a nd angle spectra) of t he signal. Because a sinusoid cos wot can be
represented as a. s um of two exponentials, e jwot a nd e  jwot , t he frequencies in t he
e xpon...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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