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

# holt r inehart a nd w inston new york 1989 2 bell e

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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. P3.4-5. ( 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 i' ;1 4 i\ 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 f(t-~)=-f(t) 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 cosine t...
View Full Document

## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online