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Unformatted text preview: Assignment #4
ECSE2410 Signals & Systems  Fali 2006 Due Tue 09/ l 2/ 06 This assignment iliustrates many of the basic properties of convolution. 1(20). Carry out the convolution, y(t) m v(t) * w(r) , of the signals shown, and sketch your result. Study
the relationships between these examples to determine the general principles invoived. (a)
V] (I) w: (t)
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0 2 G 2
(b)
V2 (1) W2 (0
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I I
0 2 9 2 4
(C)
1 V3 (1‘). W30)
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t t
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(d) V4 (I)
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G 2 2(25). This example illustrates the distributive property of convolution, namely, that convolution
distributes over addition: x(t)* (h1(t) + hz 0)) = x(t) * h] (t) + x(t) * h2 (t).
For the system shown, let h1(t) = h2(t) = u(t) — u(t —l) . W) Detennine and sketch (:1) [1100+ 112 (I) (b) x0) =i= (h: (r) + hz (0)
Now ﬁnd and sketch (c) x(r) * I110) (d) x(t) * I120) (e) x(r) * h] (I) + x(r) * ha (I) You should be able to show that the sketches in (b) and (e) are the same. Assignment #4  p.2
ECSE24I 0 Signais & Systems  F all 2006 Due Tue 09/12/06 3(20). This example illustrates the associative property of convolution, namely, that the order in which
cascaded terms are convolved is irrelevant: x(t) * (h1(t)* 112 (t)) w: (x0?) * h, (t)) =5= 112(1) . For the system shown, let h} (r) 2 kg (I) = u(t)  u(t m» l) . Determine and sketch (a) WWW) (b) x(r)*(h}(r)*h2{r))
Now ﬁnd and sketch (c) Jeanna) (d) (x(t)*h1(z))*h2(t) You should be able to show that the sketches in (b) and (d) are the same. 4.(20) in video 4.6 (Handling Time Shifts in Convolution), we showed the following: if
y(t) = x(t) =i= h(t) and yé (r) = x(r w to ) * Mt) , then y1 (t) = y(t MIG). In words, ifgne ofthe signals in a
convolution is shifted, then the resultant signal is shifted in the same way. In this problem you are to determine if a simitar type of correspondence holds if the one of the signais
is ﬂipped and shifted. (20(8) Sketch the output y(t) = x(r) * h(t) of a system whose input and impulse response,
respectiveiy, are shown below. 35(1) (b)(8) Sketch the output y1(t) = x(—t + 2) * h(r).
Label all key values on your sketch. (c)(4) Sketch y(—r «1— 2) using your result in (a).
Is it true that yi (I) = y(—t + 2)? 5(15). The input x(r) = e‘ltl is applied to a LTI system whose impuise response is Mt) : u(t) wu(t m I) . (a)(10) Compute analytically (Le. give a formuia for) y(t) = x(r) * h(r) . Use graphicai approach. (b)(5) At what value of I does y(t) reach its maximum value? Justify your answer. (Picture wiil
sufﬁce) ...
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