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

For t he series t o exist t he coefficients ao a n

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t Fig. 3.9a shows t hat t he a verage value (dc) o f f (t) is zero, so t hat ao = O. Also Using t his fact, we c an express t he series in (3.61) as f (t) = iti ::; 2 At { 2 A(1 - t) 1/2 T he d etailed evaluation o f t he a bove integrals shows t hat b oth have a value o f zero. Therefore ~ an = 0 (3.62a) n even n o dd bn = 1 1/2 1 3/2 2 At sin n7l:t d t + - 1/2 On = {O-71: for all n n # 3 ,7,11,15, . .. T he d etailed evaluation o f t hese integrals yields = 3 ,7,11,15,··· tBecause cos ( x ± 71:) = - cos x , we could have chosen the phase 7r or - 7r. In fact, cos ( x ± N7r) - cos x for any odd integral value of N . Therefore, the phase can be chosen as ± N7r where N any convenient odd integer. 2 A(1 - t) sin n7rt d t 1/2 ~ IS b n 8 A. = n 2 7r 2 S In ( ~) 2 n even = {OBA ~ BA -~ n = 1, 5, 9, 13, . .. n = 3, 7, 11, 15, . .. (3.62b) 3 Signal R epresentation by Orthogonal S ets 198 t Trigonometric Fourier Series 199 t erms only a nd t he series for any o dd p eriodic function f (t) consists of sine t erms only. Moreover, because of s ymmetry (even or o dd), t he i nformation of one period of f (t) is i mplicit in only h alf t he p eriod, as seen in Figs. 3.8a a nd 3.9a. In these cases, knowing t he s ignal over a h alf p eriod a nd w hat k ind of s ymmetry (even or odd) is present, we can d etermine t he s ignal waveform over a complete period. For this reason, t he F ourier coefficients in these cases c an b e c omputed by integrating over only h alf t he p eriod r ather t han a c omplete period. To prove t his r esult, recall t hat SA 1t2 C. 3.4 1 ao = - SA 9it To ( b) T 2 jTO/2 bn = To 11 2 o (c) - It 2 F ig_ 3 .9 21 41 To Therefore (3.63) ± sin k t = cos ( kt ' f 90°) Using this identity, Eq. (3.63) can be expressed as :~ [cos (7ft - 90°) + ~ cos (37ft + 90°) + +49 cos (77ft + 90°) + ...J ~ (3.64) The Effect of Symmetry T he F ourier series for t he p eriodic signal in Fig. 3.7b (Example 3.3) consists of sine a nd c osine t erms, b ut t he series for t he s ignal f (t) i n Fig. 3.8a (Example 3.4) consists o f cosine t erms only, a nd t he series for the signal f (t) in Fig. 3.9a ( Example 3.5) consists of sine t erms only. T his o bservation is no accident. We c an show t hat t he F ourier series of any even periodic function f (t) consists of cosine f(t)sinnwotdt (3.65c) TO 2 / f(t)dt (3.66a) TO 2 / f(t) cos nwot dt (3.66b) 0 =0 (3.66c) Similarly, if f (t) is an o dd f unction of t , t hen f (t) cos nwot is a n o dd fUllction of t a nd j (t) s in nwot is a n even function of t. T herefore = an = 0 41 bn = To 90°) In this series all the even harmonics are missing. The phases of odd harmonics alternate from _90° to 90 0 • Figure 3.9 shows amplitude and phase spectra for f (t). • 3.4-1 bn ao ~ cos (57ft - (3.65b) 0 an = - To In order to plot Fourier spectra, the series must be converted into compact trigonometric form as in Eq. (3.54). In this case, sine terms are readily converted into cosine terms with a suitable phase shift. For example, j(t) cos nwot dt - To/2 ao = - + . .. ] (3.65a) Recall also t hat cos nwot is a n even function a nd sin nwot is an ~dd f unction of t. I f f (t) is a n even functi...
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