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

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Unformatted text preview: 3.46) A sinusoid of frequency nwo is called t he n th h armonic of t he sinusoid of frequency wo w hen n is a n i nteger. I n t his s et t he sinusoid of frequency wo, called t he f undamental, serves as a n a nchor of which all t he r emaining t erms a re harmonics. Note t hat t he c onstant t erm 1 is t he Oth h armonic in this s et b ecause cos (0 x wot) = 1. In Appendix 3B we show t hat t his s et is orthogonal over any interval of d uration To = 21f / wo, which is t he p eriod of t he f undamental. Specifically, we have shown t hat r cos nwot cos mwot dt = { 0Ill. }T,o 2 n o;tm m = n o;t 0 (3.47a) 190 3 Signal Representation by Orthogonal Sets 3.4 Trigonometric Fourier Series 191 where [ sin nwot sin mWot dt = }T,o n {OTh #m (3.53a) (3.47b) n =m#O 2 () n =tan - 1 a nd [ sin nwot cos mWot dt = 0 for all n and m (3.47c) }To T he n otation iTo means t he integral over an interval from t = t1 t o tt + To for a ny value of h . T hese equations show t hat t he s et (3.46) is o rthogonal over any contiguous interval of duration To. T his is t he t rigonometric s et, which can be shown to b e a complete set. 6 ,7 Therefore, we can express a signal I (t) by a trigonometric Fourier series over any interval of duration To seconds as ao + L ancos nwot+ bnsin nwot (3.53c) Using t he identity (3.52), the trigonometric Fourier series in Eq. (3.48) can be expressed in t he c ompact f orm of the trigonometric Fourier series as 00 +L C n cos ( nwot + ( )n) (3.54) n =l + b1sin wot + ~ sin 2wot + . . . = (3.53b) Co = ao j (t) = Co t1 :s; t :s; t1 + To (3.48a) 00 I (t) n --;;,-;: ) For consistency, we denote the dc term ao by Co, t hat is I (t) = ao + a1 cos wot + a2 cos 2wot + ... or (-b t1:S; t:s; t1 +To (3.48b) where the coefficients Cn a nd ()n are computed from an a nd bn using Eqs. (3.53). Equation 3.51a shows t hat ao (or Co) is t he average value of j (t) (averaged over one period). This value can often be determined by inspection of j (t). n =l where • E xample 3 .3 271' (3.49) W o=To Using Eq. (3.39), we can determine t he Fourier coefficients ao, an, a nd bn. T hus F ind t he c ompact trigonometric Fourier series for t he e xponential e - t/2 d epicted in Fig. 3.7a over t he s haded i nterval 0 :::; t :::; 11'. B ecause we a re required t o r epresent j (t) b y t he t rigonometric Fourier series over t he i nterval 0 :::; t :::; 11' only, To = 11', a nd t he f undamental frequency is t '+TO an- II I (t) cos nwot dt ::..:!t,'--.,.-:-;..-_ _ _ __ t '+TO Wo = 211' = 2 (3.50) 2 To T herefore cos nwotdt j (t) = ao t, T he integral i n t he d enominator of Eq. (3.50) as seen from Eq. (3.47a) (with m = n) is To/2 when n # O. Moreover, for n = 0, t he denominator is To. Hence 1 ao = r : o a nd 21 21 t '+TO n =l ao '+TO I (t) cos nwotdt n =1,2,3,oO' (3.51b) o 1" 11' bn = e- t2 / dt = 0.504 0 21" =- ~ 11' 1" e - t/2 cos 2 nt d t 0 a nd Arguing t he s ame way, we o btain t 11' t, t, b n=r: =~ (3.51a) I (t) dt L an cos 2nt + bn sin 2nt w here [from Eq. (3.5...
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