HW3 - conclusions from part(c of problem 2 5 Calculate the...

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EEE 203 – Signals and Systems Homework Assignment #3 Assigned: 4 March 2008 Due: 20 March 2008 Reading: Chapter 4, sections 4.1, 4.3.1 through 4.3.3, 4.3.5 through 4.3.7, and 4.4 through 4.6. Work to hand in: 1. Use linearity and the time-shift theorem to find the Fourier transforms of the following continuous-time signals. Express your answers in terms of the sinc signal. (a) x ( t ) = 2 u ( t - 4) - 2 u ( t - 7) (b) x ( t ) = u ( t + 2) - u ( t + 1) + u ( t - 1) - u ( t - 2) (c) x ( t ) = [ u ( t + 1) - u ( t - 1)] - 2[ u ( t + 2) - u ( t - 2)] 2. Recall that a signal x ( t ) is even if x ( t ) = x ( - t ) for all t ; it is odd if x ( - t ) = - x ( t ) for all t . (a) If x ( t ) is even, show that ˆ x ( ω ) is real for all ω (i.e., its imaginary part is identically zero). (b) If x ( t ) is odd, show that ˆ x ( ω ) is purely imaginary for all ω (i.e., its real part is identically zero). (c) Use these results to determine whether ˆ x ( ω ) is real or purely imaginary for signals (a), (b), and (c) pictured on page 339. 3. Problem 4.17, page 337. 4. Problem 4.23, page 339, parts (a), (b) and (c). Note that your answers should agree with your
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Unformatted text preview: conclusions from part (c) of problem 2. 5. Calculate the energy of the following continuous-time signals. (a) x 1 ( t ) =-3[ u ( t + π )-u ( t-π )] (b) x 2 ( t ) = cos( t ) x 1 ( t ) (c) x 3 ( t ) = [ x 1 * x 2 ]( t ) 6. Calculate the energy of the signal x ( t ) = sinc( at ) with a > 0. 7. Let b > a > 0 and define x 1 ( t ) = sinc( at ) and x 2 ( t ) = sinc( bt ). Calculate [ x 1 * x 2 ]( t ). 8. Define x 1 ( t ) = sinc 2 ( t ) and x 2 ( t ) = cos(Ω t ) x 1 ( t ). Find a value W such that Z ∞-∞ x 1 ( t ) x 2 ( t ) dt = 0 for all | Ω | > W . 9. A signal has Fourier transform ˆ x ( ω ). Obtain an expression, in terms of ˆ x ( ω ) and a sinc signal, for the Fourier transform of the “truncated” signal x ( t )[ u ( t )-u ( t-T )]....
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This note was uploaded on 09/29/2008 for the course EEE 203 taught by Professor Chakrabarti during the Spring '07 term at ASU.

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