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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 0 Linear Systems and Probability Review Due: Thursday January 20 at 9:30 a.m. Reading: Lecture Notes: Lectures 1-3 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) Show that the Fourier transform of rect( t/T ), a pulse of duration T , is T · sinc( fT ). (b) What is the Fourier transform of sinc( t/T )? Note that sinc( t/T ) passes through 0 at (nonzero) integer multiples of T ....
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- Spring '09
- Frequency, LTI system theory, Rectangular function, sinc function