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Hwk5f06 - h t = δ t-e-2 t u t(a This system will act as a...

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ECE 3337 Homework 5 Fall 2006 Due: November 1 (Jansen) and 2 (Kalatsky), 2006 1. Obtain the Fourier transform, and sketch the magnitude and phase response of the following func- tions: (a) x 1 ( t ) = 0 for | t | ≥ 1 , x ( t ) = t + 1 for - 1 < t < 0 and x ( t ) = - t + 1 for 0 t < 1. (b) x 2 ( t ) = 5 δ ( t ) exp(2 t 2 ) (c) x 3 ( t ) = exp( - t ) u ( t - 1) (d) x 4 ( t ) = δ ( t - 2) + j exp( - 2 πt ) u ( t ) (e) x 5 ( t ) = x 1 ( t ) · exp( - t ) u ( t ) (f) x 6 ( t ) = 2 x 3 ( t + 1) - 5 x 2 ( t - 1) 2. Obtain the inverse Fourier transform of the following functions: (a) X 1 ( ω ) = 100 100+ (b) X 2 ( ω ) = 10 4 (1+ ) (100+ )(10+ ) (c) X 3 ( ω ) = 1 for | ω | < 10 π and X 3 ( ω ) = 0 elsewhere. (d) X 4 ( ω ) = 1 for 10 < | ω | < 20 π and X 4 ( ω ) = 0 elsewhere. (e) X 1 ( ω - 2 π 10). (f) Which of the above x i ( t ) , i = 1 , . . . , 4 could represent the impulse response of a causal system? 3. The impulse response of a linear, time-invariant system is given by
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Unformatted text preview: h ( t ) = δ ( t )-e-2 t u ( t ) (a) This system will act as a ±lter, i.e., blocking high frequency signal components (low-pass ±lter), passing signals in a certain freqency band only (band-pass), or blocling low frequency compo-nents (high-pass). Determine what type of ±lter this system is by computing and sketching its magnitude response. (b) Determine the output y ( t ) if x ( t ) = 2 cos(10 πt + π/ 4) is applied as input. (c) Determine the Fourier Transform X ( ω ) of input x ( t ) if y ( t ) = exp(-t ) u ( t ) is received at the output. 1...
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