Unformatted text preview: 1a)] an
t an = ;:p
.LO I + e t2
/ = 0.504 (   2 2 )
1 + 16n sin 2 ntdt = 0.504 0 (+ 16n
~)
1 T herefore
'+TO I (t)sinnwotdt n = 1 ,2,3, . .. (3.51c) t, j (t) = 0.504 [ 00
2
1 + ~ 1 + 16n 2 (cos 2 nt .
+ 4 nsm 2]
nt) Compact Trigonometric Fourier Series T he t rigonometric Fourier series in Eq. (3.48) contains sine and cosine terms
of t he same frequency. We c an combine t he two terms t o o btain a single sinusoid
o f t he s ame frequency using t he trigonometric identity an cos nwot + bn sin nwot = e n cos (nwot + (}n) (3.52) To find t he c ompact F ourier series, we c ompute i ts coefficients using Eq. (3.53) as Co = ao = 0.504
Cn =
() n J a~ + b~ = 0.504 = t an  1 (b ) =
~ (1+1!n2)2 + (1:;:::2)2 = 0.504 ("h:16n t an  1(  4 n ) =  tan  1 4 n 2) (3.55) 192 3 Signal R epresentation by O rthogonal Sets (a) ~ o  It ~ 193 Periodicity o f t he Trigonometric Fourier Series f {t) 21t 3.4 Trigonometric Fourier Series 21t It I ~ ~112 ~ < p(t) 1 ~ (b) We have shown how a n a rbitrary signal f (t) may b e expressed as a t rigonometric Fourier series over any interval of To seconds. In E xample 3.3, for instance, w~
r epresented e  t / 2 over t he interval from 0 t o 7 r / 2 only. T he Fourier series found in
Eq. (3.56) is e qual t o e  t / 2 over t his i nterval alone. O utside t his interval t he series
is n ot necessarily equal t o e  t / 2 • I t would be interesting t o find o ut w hat h appens
t o t he F ourier series outside this interval. We now show t hat t he t rigonometric
Fourier series is a periodic function o f p eriod To ( the p eriod o f t he f undamental).
Let us denote t he t rigonometric Fourier series o n t he r ighthand side o f Eq. (3.54)
by <.p(t). T herefore
00  2n 0  It 21t It 1_ <.p(t) = Co + a nd r'w I
o <.p(t + To) = Co + L Cn cos [nwo(t + To) + On]
n =l 0.125 T , T 4 2 (c) 6 00 • = Co + = (d) f ~cos(2nt 1 + 16n 2 tan 1 4n) 0 ~ t ~ 7r
o~ t + 0.084 cos (6t  85.24°) + 0.063 cos (8t  86.42°) + . .. Remember t hat t he r ighthand side represents e  t !2 over the interval 0 t o
t his interval, t he two sides need not b e equal. Co +L 0.504
0 I I 2 0.244
0.125
 75.961  82.87 I I 3
0.084 I I I  85.24\ 4 I I 5 0.063
0.0504
8 6.421 87.14 I 6 I 0.042
87.61 I Cn cos (nwot + On) = (3.56a) 7r ~ 7r (3.56b) only. Outside <.p(t) I I 0.036
87.95 ao 7 I I
I • for all t (3.57) This result shows t hat t he t rigonometric Fourier series is a periodic function o f
p eriod To ( the p eriod o f i ts fundamental). For instance, 'P(t), t he Fourier series on
t he r ighthand side o f E q. (3.56), is a periodic function in which t he s egment o f f (t)
in Fig. 3.7a over t he i nterval (0 ~ t ~ 7 r) r epeats periodically every 7r seconds, as
illustrated in Fig. 3 ·7h·t T hus, when we represent a signal f (t) by t he t rigonometric
Fourier series over a certain interval o f d uration To, t he f unction f (t) a nd its Fourier
series <.p(t) need b e e qual only over t hat i nterval o f To seconds. O utsi...
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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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