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Unformatted text preview: EE 224, HW 8, due April 3. 1. It is often useful to represent operations on signals as convolutions. For each of the systemr outputs, ﬁnd the corresponding impulse responses h(t), i.e. ﬁnd h(t) so that
{.W)} = {5605)} * {M0} (a) W) = fioo$(’r)d7
(b) W) = [LIﬂTMT
(C) W) = W) (d) W) = W  T) Hint: Use the deﬁnition of the impulse response. 2. (a) Prove that, for T > 0,
{rect(%)} * {mgn = Tmé» (1) using the graphical (ﬂip—shift—integrate) approach.
(b) Find a different proof of (1) Hint: This problem is solved in handout # 13 for the special case of T = 1 s. Generalize the derivations in handout # 13 to arbitrary T > O.
3. Problem P—9.8 on p. 280 of the textbook.
4. Problem P—9.9 on p. 280 of the textbook. Hint: Use the fact that = tu(t) = r(t) (unit ramp signal). 5. Compute the convolution (ac * h) (t) Where 33(t) and h(t) are 93(t) ': u(t), h(t) = no: — 1) and plot the result. BB 224, HW 8, due April 3. 6. The derivative of a function $05) with respect to t may be written as the convolution:
W) = (20* M) where 6’(t) is the doublet, described in handout # 10. If y(t) = (33*h) (if), use the. above identity
to Show that (93’ * h’W) = Mi). 7. Problem 13—91? on p. 282 of the textbook. (7552;? i , , . . k _\ ...
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 Fall '09
 Digital Signal Processing, Derivative, Impulse response, corresponding impulse responses

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