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

46a t his s ituation d epicted i n fig 210d is valid

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Unformatted text preview: - 0 t_ (0 o o 0 t- t- o t- t- *K L -T 0 t_ o t_ F ig. 2 .13 C onvolution o f f (t) a nd get) i n E xercise E 2.13. W idth of t he Convolved Function T he w idths ( durations) of f (t), g (t), a nd c (t) in Example 2.8 (Fig. 2.10) are 2, 3, a nd 5 respectively. Note t hat t he w idth of c (t) in t his case is t he s um of t he w idths of f (t) a nd g (t). T his observation is n ot a coincidence. Using t he concept of graphical convolution, we can readily see t hat if f (t) a nd g (t) have t he finite widths of T l a nd T2 respectively, t hen t he w idth of c (t) is generally equal to T l + T2. T he reason is t hat t he t ime it takes for a signal of width (duration) T l t o completely pass a nother signal of width (duration) T2 so t hat t hey become nonoverlapping is T l + T2' W hen t he two signals become nonoverlapping, t he convolution goes t o zero. However, t here a re cases where t he two signals f (r) a nd g (t - r) overlap, yet t he a rea u nder t heir p roduct vanishes. Such is t he case of signals in Fig. P 2.4-14 for r > 27r. I n t his case t he w idth property is superficially violated. t E xercise E 2.l0 Rework E xample 2.7 by evaluating get) • f(t) /::, E xercise E 2.11 /::, 'V Use g raphical convolution t o show t hat f(t) * get) = get) /::, E xercise E 2.l2 R epeat P rob. E2.11 for t he functions in Fig. 2.12. 'V /::, E xercise E 2.13 R epeat P rob. E2.11 for t he functions i n Fig. 2.13. 'V * f(t) = crt) in Fig. 2.11. C omputer E xample C 2.4 F ind crt) = f (t) • get) f or t he s ignals i n F ig. 2.9. Some Reflections on the Use o f Impulse Function Fig. 2 .12 C onvolution o f f (t) a nd get) i n E xercise E 2.12. t_ 137 t 1=-1O:.0l:0;t1=t1 ' ; g l=-2*exp(2*tl); t 2=O:.Ol:lO;t2=t2'; g 2=2*exp(-t2); t =[tl;t2J; g =[gl;g2]; f = [ zeros(size(gl)) ; ones(size(g2))]; c =O.Ol * conv(f,g); t =-20:.0l:5;t=t'; p lot(t,c(l:length(t))) (0 Fig. 2 .11 C onvolution o f f (t) a nd get) i n E xercise E 2.1l. k *-=:t- 2.4 S ystem Response t o E xternal I nput: T he Z ero-State Response 'V t Even in s uch cases, t he w idth property may be held valid i f we consider t hat t he region where t he a rea u nder t he p roduct of two nonoverlapping signals vanishes t o become a p art o f crt) (where crt) h appens t o b e zero). Thus, crt) in this case has a n infinite duration, b ut t he value of crt) = 0 for t > 27r. I n t he s tudy of signals a nd s ystems we often come across signals such as impulses, which cannot be generated in practice. O ne wonders why we even consider such signals. T he answer should be clear from our discussion so far in this chapter. Even if t he impulse function has no physical existence, we can compute t he s ystem response h (t) t o t his p hantom i nput according t o t he p rocedure in Sec. 2 .3, a nd knowing h (t), we c an compute t he s ystem response t o any a rbitrary i nput. T he concept of impulse response, therefore, provides a n effective intermediary for computing system response to a n a rbitrary i nput. In addition, t he impulse response h (t) itself provides a...
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