Signal Processing and Linear Systems-B.P.Lathi copy

This however is not t he e nd o f t he story b oth

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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 (3.75) 0.504 ' 82 87 D2 = 1 + j 8 = 0 .0625e-)· = > /D21 = 0.0625, LD2 = - 82.87° and D -2 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-J8 1-J12 j4t 1+j8 e 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 1 ~ 0.504 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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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