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Unformatted text preview: y College, Cambridge, who clearly saw t he b ehavior of the
sum of t he F ourier series components in t he periodic sawtooth signal investigated
by Gibbs. l l A pparently, this work was not known t o most people, including Gibbs
a nd Bacher. 3 .5 Exponential Fourier Series I t is shown i n A ppendix 3C t hat t he set of exponentials e jnwot ( n = 0, ± 1, ± 2, . .. )
is o rthogonal over any interval of d uration To = 27r / wo, t hat is,
m fn m =n (3.69) + lin) ~n = [ej(nwot+Bn) (~n eiOn ) = + e -j(nwot+B n )] e inwot + (~n e - jOn) e - jnwot '-,,--' "-v--' Dn. D_ n = D nejnwot + D _ne-jnwot (3.72) T he compact trigonometric Fourier series of a periodic signal j (t) is given by
00 j (t) = Co +L C n cos ( nwot + lin) n =l Use of Eq. (3.72) in t he above equation (and letting Co = D o) yields
00 j (t) = D o + L Dnejnwot + D_ne-jnwot
n =l 00 = L D nejnwot n =-oo which is precisely Eq. (3.70) derived earlier. Observe t he compactness of expressions
(3.70) a nd (3.71) a nd compare t hem t o expressions corresponding to t he trigonometric Fourier series. These two equations clearly demonstrate t he principle virtue
of t he e xponential Fourier series. First, t he form of t he series is more compact.
Second, t he m athematical expression for deriving t he coefficients of t he series is
also compact. T he e xponential series is far more convenient t o h andle t han t he
trigonometric one. In t he system analysis also, t he e xponential form proves more
convenient t han t he t rigonometric form. For these reasons we shall use exponential
(rather t han t rigonometric) representation of signals in t he r est of t he book. 3 Signal Representation by O rthogonal Sets 208 T he c onnection between t he t rigonometric a nd exponential series coefficients
is clear in Eq. (3.72): 3.5 Exponential Fourier Series 209 close connection to t he a mplitudes a nd phases of corresponding components of t he
t rigonometric Fourier series. We therefore plot IDnl vs. w and L Dn vs. w. T his
requires t hat t he coefficients D n b e expressed in polar form as IDnlejLDn.
C omparison of Eqs. (3.51a) a nd (3.71) (for n = 0) shows t hat (3.73)
T he connection between t he t rigonometric a nd t he exponential Series also becomes
clear when we s ubstitute e - jwt = cos w t - j sin w t in Eq. (3.71) t o o btain Do 1 IDnl = I D-nl = 2 Cn 1
1" -( ~ cp(t)e- To = -1 11' j2nt Dn 11' = L D_ n = IDnlej9n a nd D_ n = - On (3.77c) = IDnle-j9n (3.77d) where /Dn/ a re t he m agnitudes a nd L Dn a re t he angles of various exponential
components. From Eqs. (3.77) it follows t hat t he m agnitude s pectrum (/Dnl vs. w)
is a n even function of w and t he angle s pectrum ( LDn vs. w) is a n o dd function of
w when j (t) is a real signal.
For the series in Example 3.6 [Eq. (3.76b)], for instance, dt To e - t/2 e - j2nt d t D o = 0.504 0 e e- ( Dl .2l+j2n)t dt -1 = -1 a nd (3.77b) Thus, where
Dn = n #O = On L Dn E xample 3.6
Find the exponential Fourier series for the signal in Fig. 3.7b (Examp...
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