HW00 - University of Illinois Spring 2011 ECE 361: Problem...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 0: Solutions Linear Systems and Probability Review Note: In this course, we use Fourier transforms with respect to the frequency variable f (measured in Hertz) rather than the radian frequency variable ω = 2 πf (radians per second) that you used in ECE 210. Thus, X ( f ) = Z ∞-∞ x ( t )exp(- j 2 πft ) dt, x ( t ) = Z ∞-∞ X ( f )exp( j 2 πft ) df, and Z ∞-∞ | x ( t ) | 2 dt = Z ∞-∞ | X ( f ) | 2 df. Note also that the unit rectangular pulse rect( · ) and the sinc function sinc( · ) are defined as rect( t ) = ( 1 , | t | < 1 2 , , | t | > 1 2 , and sinc( t ) = sin( πt ) πt , t 6 = 0 , 1 , t = 0 . The sinc function has the useful property that sinc(0) = 1 while sinc( n ) = 0 for nonzero integers n . 1. [Drill exercises on Fourier transforms and linear systems] (a) Z ∞-∞ rect( t/T )exp(- j 2 πft ) dt = Z T/ 2- T/ 2 exp(- j 2 πft ) dt = exp(- j 2 πfT/ 2)- exp( j 2 πfT/ 2)- j 2 πf = sin( πfT ) πf = T · sinc( fT ). Note that narrow pulses in the time domain (small T ) correspond to broad main lobes (from- 1 /T Hz to 1 /T Hz) in the frequency domain. (b) From the duality theorem x ( t ) ↔ X ( f ) ⇔ X ( t ) ↔ x (- f ) and the result of part (a), we have that W · sinc( Wt ) ↔ rect(- f/W ) = rect( f/W ). Setting W = T- 1 , we deduce the transform pair sinc( t/T ) ↔ T · rect( fT ) . Note that the Fourier transform of sinc( t/T ) is nonzero only for | f | < 1 2 T ....
View Full Document

This document was uploaded on 01/24/2012.

Page1 / 3

HW00 - University of Illinois Spring 2011 ECE 361: Problem...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online