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-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.
/::, 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 ' ;
t =[tl;t2J; g =[gl;g2];
f = [ zeros(size(gl)) ; ones(size(g2))];
c =O.Ol * conv(f,g);
(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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