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

87 n oo r esponse y t t he r esponse

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Unformatted text preview: rom Eqs. (3.88) and (3.89) in the above equation, we obtain L 00 y(t) = 2 " (I _ 4 n2)(j6n + 1) e j 2nt _~ - ,,2 1 ~ (1 _ 4n2)2(36n2 + 1) 00 and the ripple rms value = J Prippl. = 0.05 This shows t hat the rms ripple voltage is 5% of the amplitude of the input sinusoid. • Why Use Exponentials? 00 '" 12 Numerical computation of the right-hand side yields Fig. 3 .20 A full-wave rectifier with a lowpass filter and its output. Therefore + 1) (3.90) n =-oo Kote t hat the out put y( t) is also a periodic signal given by the exponential Fourier series on the right-hand side. The output is numerically computed from the above equation and plotted in Fig. 3.20b. T he e xponential Fourier series is j ust a nother way of representing trigonometric Fourier series (or vice versa). T he two forms c arry i dentical i nformation-no more, no less. T he r easons for preferring t he e xponential form have a lready b een mentioned: T his form is more compact, a nd t he e xpression for deriving t he exponential coefficients is also more compact, as c ompared t o t hose in t he t rigonometric series. Furthermore, t he s ystem response t o e xponential signals is also simpler (more compact) t han t he s ystem r esponse t o sinusoids. I n a ddition, t he m athematical m anipulation a nd h andling of exponential form proves much easier t han t he t rigonometric form in t he a rea of signals as well as systems. For these reasons, in our future discussion we shall use t he e xponential form exclusively. A minor disadvantage of t he e xponential form is t hat i t c annot b e visualized as easily as sinusoids. For intuitive a nd q ualitative u nderstanding, t he sinusoids have t he edge over exponentials. Fortunately, t his difficulty can b e overcome r eadily because of close connection between exponential a nd F ourier s pectra. For t he p urpose of m athematical a nalysis we shall continue t o use exponential signals a nd s pectra, b ut t o u nderstand t he p hysical s ituation i ntuitively or qualitatively we shall speak in t erms o f sinusoids a nd t rigonometric s pectra. T hus, i n f uture discussions, although all m athematical m anipulation will b e i n t erms o f exponential s pectra, we shall speak of exponential a nd sinusoids interchangeably when discussing intuitive and qualitative insights a nd t he u nderstanding o f physical situations. T his is a n i mportant p oint; readers should make a n e xtra effort t o familiarize themselves w ith t he two forms of s pectra, t heir r elationships, a nd t heir convertibility. Dual Personality o f a Signal T he discussion so far shows t hat a p eriodic signal has a d ual p ersonality-the time domain a nd frequency domain. I t c an b e d escribed by its waveform or by its Fourier spectra. T he t ime- and f requency-domain descriptions provide complementary i nsights i nto a signal. For i n-depth p erspective, we need t o u nderstand b oth o f 222 3 Signal Representation by Orthogonal Sets t hese i dentities. I t is i mportant...
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