Assignment 2, solution

Assignment 2, solution - Assignment 2 ELEC442/ELEC6601...

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Unformatted text preview: Assignment 2 ELEC442/ELEC6601 Concordia University 1 Question 1: Let x(t) be a real continuous time band-limited signal with bandwidth B. Determine the minimum sampling frequency required to avoid aliasing for each of the following signals: a) x(t) b) x(2t) c) ) ( 2 t x d) x(t)*x(t) (Here * means convolution) Solution 1: a) 2B b) 4B c) 4B d) 2B Question 2: Consider the system ) ( ) ( ) ( t x t w t y = where < < < < = T t T T t t w 2 2 1 ) ( is a periodic signal with period of T . Let ) ( = Ω j X for m Ω > Ω . Find the largest value of T such that ) ( t x can be reconstructed from ) ( t y . Determine a system that will perform the reconstruction for the maximum value of T . Solution 2: From Fourier Transform pairs table 4.2 of Signals and Systems book. Please see the tables attached to “Midterm content and Formula sheet for this course), the periodic Square Wave with period “ T ” has following Fourier transform: T k k T k T t T T t k FT π δ 2 ) ( sin 2 2 1 1 1 1 = Ω Ω- Ω Ω → ← ≤ < < ∑ ∞ +-∞ = Taking 4 1 T T = and applying T π 2 = Ω : ∑ ∞ +-∞ =- Ω → ← ≤ < < k FT T k k k T t T T t ) 2 ( 2 sin 2 2 4 4 1 π δ π Assignment 2 ELEC442/ELEC6601 Concordia University 2 Shifting the above signal to the right by 4 T , we get ) ( t w with Fourier transform of ) ( Ω j W as follows. We are using the shifting property of Fourier transform from table 4.1. ∑ ∑ ∞ +-∞ =- ∞ +-∞ = Ω-- Ω =- Ω = Ω → ← k jk k T j FT T k e k k T k k k e j W t w ) 2 ( 2 sin 2 ) 2 ( 2 sin 2 ) ( ) ( 2 4 π δ π π δ π π And we have: ) ( * ) ( 2 1 ) ( ) ( ) ( Ω Ω → ← = j X j W t x t w t y FT π Using property of (.) δ function: ∑ ∞ +-∞ =-- Ω = Ω k jk T k j X e k k j Y )) 2 ( ( 2 sin 2 2 1 ) ( 2 π π π π To reconstruct: max max 2 2 , ), ( ) ( ) ( Ω > Ω < Ω Ω = Ω Ω T j X j Y j H r π To filter the spectrum except the center part of ) ( Ω j X , the filter ) ( Ω j H r should satisfy: ) ( ) ( 2 1 ) ( Ω = Ω Ω j X j X j H r Therefore: Ω- > Ω Ω- < Ω < Ω Ω < Ω = Ω max max max max 2 2 ' 2 ) ( T T care t don j H r π π Question 3: Consider the sinusoidal signal + = 6 4000 sin ) ( π π t t x . This signal is sampled at a sampling rate of π 6000 = Ω s to achieve ) ( t x p . To reconstruct ) ( t x , the sampled signal ) ( t x p is passed through an Assignment 2 ELEC442/ELEC6601 Concordia University 3...
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Assignment 2, solution - Assignment 2 ELEC442/ELEC6601...

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