ps4 - x t = e-t 2-2 ≤ t< 2(a Use Matlab to compute the...

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UCSB Fall 2007 ECE 130A: Problem Set 4 Assigned: Thursday, November 1 Due: Tuesday, November 6 Reading: 3.5, 4.1-4.5 Problem 1: Consider the periodic signal x ( t ) = s k = -∞ p ( t - kT ) where p ( t ) = I [0 , 1] ( t ) and T = 2. (a) Find explicit expressions for the complex exponential Fourier series { a k } for x ( t ). (b) Show that we can write x ( t ) = R 0 + s k =1 R k cos ( 0 t + φ k ) where R k 0. Find an explicit expression for the { R k } and { φ k } . (c) De±ne the K th order approximation for x as ˆ x K ( t ) = R 0 + K s k =1 R k cos ( 0 t + φ k ) where K = 1 , 2 , 3 , ... . Use Matlab to plot x ( t ) and ˆ x K ( t ) over 0 t 4, for K = 1 , 10 , 100. Comment on the quality of the approximations for the three values of K . Remark: See Problem 6, Problem Set 1 for hints on how to plot waveforms using Matlab. Problem 2: Let x ( t ) be periodic with period 4, such that
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Unformatted text preview: x ( t ) = e-t 2 ,-2 ≤ t < 2. (a) Use Matlab to compute the complex exponential Fourier series { a k } for-10 ≤ k ≤ 10. (b) Plot x ( t ) and the 10th order approximation ˆ x 10 ( t ) = ∑ 10 k =-10 a k e jkω t for-5 ≤ t ≤ 5. Comment on how good the approximation is. (c) Play around with higher and lower order approximations and report on the results. Problem 3 (a) Fourier transform computation 4.1 (b) Fourier transform application Find an explicit expression for the output of the system in Problem 4.1(b) when the input is sin(10 t )....
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This note was uploaded on 08/06/2008 for the course ECE 130A taught by Professor Madhow during the Fall '07 term at UCSB.

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