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Unformatted text preview: ime-inversion of a periodic signal does not affect t he
a mplitude s pectrum, a nd t he p hase s pectrum is also unchanged except for t he change
of sign. 3 .4-5 3 .4-6 ( a) F ind t he t rigonometric Fourier series for t he periodic signal x (t) d epicted i n Fig.
( b) T he s ignal x (t) c an be obtained b y t ime-compressing t he signal cp(t) in Fig. 3.7b
by a factor 2. T hus, x (t) = cp(2t). Hence, t he Fourier series for x (t) c an b e obtained
b y r eplacing t w ith 2t in t he Fourier series [Eq. (3.56)] for cp(t). Verify t hat t he Fourier
series t hus o btained is identical to t hat found in p art ( a).
( c) Show t hat, in general, time-compression of a periodic signal b y a factor a e xpands
t he F ourier s pectra by t he s ame factor a. I n o ther words Co, C n, a nd ()n r emain unchanged, b ut t he f undamental frequency is increased by t he factor a, t hus expanding
t he s pectrum. Similarly time-expansion of a periodic signal by a factor a compresses
its Fourier s pectra by t he factor a.
( a) F ind t he t rigonometric Fourier series for t he periodic signal g(t) in Fig. P3.4-6.
Take a dvantage of t he symmetry. ;1 V-~ ;1
-5 / .~ /'1 ? /l ,
-6 \ . /l ;1
-3 -2 / -2 l'1
1 ./ 2 / 4 (d) / t __ (e) t __ ( f) V2~t . ... Vk ~/4 ;1
6 6 \ 8 F ig. P 3.4-3 F ig. P 3.4-4 ~ ~ ~ ~ l~~t ~ ~ ~
F ig. P 3.4-5 " 230 3 Signal Representation by O rthogonal Sets " '("~ A 6 3 .4-B L F ig. P 3.4-6
( b) O bserve t hat g(t) is i dentical t o f (t) i n Fig. 3.9a left-shifted by 0.5 second.
Therefore, g(t) = f (t + 0.5), a nd t he F ourier series for g(t) c an b e found by replacing
t w ith t + 0 .5 in Eq. (3.63) [the Fourier series for f(t)J. Verify t hat t he F ourier series
t hus o btained is identical t o t hat found in p art ( a).
( c) Show t hat, in general, a time shift o f T seconds of a periodic signal does n ot
affect t he a mplitude s pectrum. However, t he p hase of the n th h armonic is decreased
(increased) by n woT for a delay (advance) of T seconds.
3 .4-1 I f t he two halves o f one period o f a p eriodic signal are of identical s hape e xcept t hat
t he one is t he n egative o f t he o ther, t he p eriodic signal is said t o have a h alf-wave
s ymmetry. I f a p eriodic signal f (t) w ith a period To satisfies t he half-wave s ymmetry
c ondition, t hen
I n this case, show t hat all t he e ven-numbered harmonics vanish, a nd t hat t he o ddnumbered h armonic coefficients a re given by an 41 TO/2 =- To f(t) cos nwot dt a nd 0 bn 41 To 231 O ver a finite interval, a signal c an b e r epresented by more t han one trigonometric
(or exponential) Fourier series. For instance, if we wish t o r epresent f (t) = t over a n
i nterval 0 ::; t ::; 1 b y a Fourier series w ith f undamental frequency Wo = 2, we c an
d raw a pulse f (t) = t over t he i nterval 0 ::; t ::; 1 a nd r epeat t he pulse every 7r seconds
so t hat To = 7r a nd Wo = 2 (Fig. P3.4-8a). I f we w ant t he f undamental frequency wo
t o b e 4, we r epeat t he pulse every rr / 2 seconds. I f we w ant t he series t o c ontain only
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