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Unformatted text preview: le 3.3).
In this case To = 11', WO = 211' ITo = 2, and (3.77a) Equation (3.73) shows t hat, for real j (t), t he twin coefficients D n a nd D _ n are
conjugates, a nd (3.74) • = ao = Co 0.504 = 1 + j4 ' 7596° = 0 .122e-)· = > /Dli = 0.122, L DI = - 75.96° 0 1I'(~+j2n) ~+J2n)t I" D -l 0 0.504
1 + j 4n 0.504
j4 = 1_ ' 7596° = 0 .122e)· => ID_II = 0.122, L D_I = 75.96° and
' 82 87
D2 = 1 + j 8 = 0 .0625e-)·
= > /D21 = 0.0625, LD2 = - 82.87° and
1 + j4n
1 = 0.504 [ 1 + 1+j4 e + _ 1_. 1-J4 j2t + e - j2t + 1+j12 ej6t + ...
_ 1_. e - j4t + _ 1_ e -j6t + ... J
1 + _ 1_ (3.76b) Observe t hat t he coefficients Dn are complex. Moreover, Dn and D - n are conjugates as
expected [see Eq. (3.73)J.
• 3 .5-1 8287
= 0 .0625e)· 0 = > /D-21 = 0.0625, L D_2 = 82.87° (3.76a) cp(t) = 0.504 ' " _ _ _ej2nt
j8 = 1_ Exponential Fourier Spectra I n exponential spectra, we p lot coefficients D n as a function of w . B ut since
D n is complex i n general, we need two plots: t he real and the imaginary p arts of
D n , or t he m agnitude a nd t he angle of D n. We prefer the l atter because of its a nd so on. Note t hat D n a nd D _ n a re conjugates, as expected [see Eqs. (3.77)J.
Figure 3.14 shows t he frequency s pectra ( amplitude a nd angle) of t he exponential Fourier series for t he periodic signal cp(t) in Fig. 3.7b.
We notice some interesting features of these spectra. First, the s pectra exist
for positive as well as negative values of w ( the frequency). Second, t he a mplitude
spectrum is a n even function of w and t he angle s pectrum is a n o dd function of
w. Finally, we see a close connection between these s pectra a nd t he s pectra of t he
corresponding trigonometric Fourier series for <p(t) (Figs. 3.7c a nd d).
W hat is a Negative Frequency? T he existence of t he s pectrum a t negative frequencies is s omewhat disturbing
because, by definition, t he frequency (number of repetitions p er second) is a positive 3 210 0.504 S ignal R epresentation b y O rthogonal S ets 3.5 E xponential F ourier S eries 211 ~ IDn I Cn 16 '.'. en '. '.
0.122 (a) 12 0 -10 -8 --6 -4 10 2 -2 6 3 m~ 12
0 0- (a) - 1t T L Dn ~ It -10 -8 --6 -4 I
- It 2 F ig. 3 .14 ~ (b)
2 -2 16 jDnl 2 4 6 10 m~ L Dn -12
0 11111 3 6 9 12 ( b) Exponential Fourier spectra for t he signal in Fig. 3.7a.
F ig. 3 .15 q uantity. H ow d o we i nterpret a n egative f requency? Using a t rigonometric i dentity,
we c an e xpress a s inusoid o f a n egative f requency - wa a s
cos ( -wat + Ii) = cos (wot - Ii) T his e quation c learly s hows t hat t he f requency of a sinusoid cos (wa t + Ii) is Iwal,
w hich is a p ositive q uantity. T he s ame c onclusion is r eached b y o bserving t hat
e ±jwot = cos wat ± j s in wot T hus, t he f requency o f e xponentials e ±jwot is i ndeed Iwal· H ow d o we t hen i nterpret
t he s pectral p lots for n egative v alues of w? A h ealthier way o f l ooking a t t he s ituation is t o s ay t hat exponentia...
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