Question

x(t) ht) 2 2 2.12 Functions x(t) and h(t) have the waveforms shown in Fig. P2.12. Determine and plot y(t) = x(t) *h(t) using the following methods. (a) Integrating the convolution analytically. (b) Integrating the convolution graphically. h 0 0 t(s) t(s) 0 + 1 0 2 Figure P2.12: Waveforms for Problem 2.12.

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Answer 1

a alt) 2 2. 1 a) Analytically ja(t) a (+-2) do Ylt) =alt) * ult) hits in we OR can write nley and mathematical forma alt) = Qult) ault-1 ylt) = a(t) * n(t) > Ý (4) = @Mlt) – Qu(t-1)) * hit gnitat Ihlt) * M (t) above property a Using y Lt) - Qu(t) *nLE) Qu(t) * n(t) ty - 2 J uLtd 2 "(t) dT Y214) ylt) Y, (t) 2 182 (4) Q la Parabolic parabolic - 0 2

2 + 2. a Ramp function integration is parabolic in nature, yet) = a'jae) at af -2 (2-1) dt 13-0) | 14:12 -- (+-)) = f (2) + 434_103448 ylt at² + 2(t-1)² - 4(+-) 22.11 0 +2.2 -2.2t2 2 6) Graphically, - y t) = X(t) * n(t) property a using differentiation = X(t) * n(t) x' (t) 2 ty - t -2 hlt) * 8lt =h(t) hlt) * & lt-to) =hlt-to) - (2)

Using property (1) and (2) - y! (t) = (2814) - 26 (+-1)) * n(t) y! (t) = 2 n(t) - ah(t-1) 1211t) an14-1) 4 4. 1A yllt) = 2 2 ti 3 ty y' (t) 4 O ty 4 YIt) 6 2t? 3 ty