HW12 - University of Illinois Fall 2003 ECE 210 Homework 12...

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Unformatted text preview: University of Illinois Fall 2003 ECE 210: Homework 12 Due: Wednesday November 19, 2003, 3:00 p.m. Reading: Lecture Notes, Chapter 9 This Homework Set contains six problems and one noncredit exercise 1. Problem 3 2. Problem 6 3. Problem 8, parts (a)-(h). 4. This problem fulfills the promise (threat?) made on page 186 (col. 1) that we will ask you to verify the symmetry and scaling properties of impulse functions in homework problems. We take the definition of the impulse function as a hypothetical function with the property that Z ∞-∞ δ ( t ) g ( t ) dt = g (0) provided that g ( t ) is continuous at t = 0. The basic premise of the verification is that δ ( t ) can be manipulated as an ordinary function wherever it appears in the integrand of an integral. In particular, change-of-variables is permissible. ( a) Show that by a change of variables, we can write Z ∞-∞ δ ( t- a ) g ( t ) dt = Z ∞-∞ δ ( u ) h ( u ) du where h ( u ) = g ( a + u ) and hence Z ∞-∞ δ ( t- a ) g...
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HW12 - University of Illinois Fall 2003 ECE 210 Homework 12...

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