SP_IV_P02_4p - Introduction to Signal Processing 1...

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Unformatted text preview: Introduction to Signal Processing 1 Copyright 2005-2009 Hayder Radha C. Key Properties of the Fourier Transform Linearity If, ( ) ( ) 1 1 x t X and ( ) ( ) 2 2 x t X Then, ( ) ( ) ( ) ( ) 1 2 1 2 ax t bx t aX bX + + . Introduction to Signal Processing 2 Copyright 2005-2009 Hayder Radha Conjugation and Conjugate Symmetry If, ( ) ( ) x t X , Then, ( ) ( ) * * x t X And if ( ) x t is real, ( ) ( ) * X X = . Introduction to Signal Processing 3 Copyright 2005-2009 Hayder Radha Duality If, ( ) ( ) x t X Then, ( ) ( ) 2 X t x . Introduction to Signal Processing 4 Copyright 2005-2009 Hayder Radha Proof Consider comparing the Fourier transform and its inverse, ( ) ( ) j t X x t e d t = ( ) 1 ( ) 2 j t x t X e d = Now consider performing the following operations on the expression for ( ) x t : Introduction to Signal Processing 5 Copyright 2005-2009 Hayder Radha 1. Replacing with t , And, Replacing t with , ( ) 1 ( ) 2 j t x X t e dt = 2. Multiplying by 2 . ( ) ( ) j t x X t e dt 2 = 3. Frequency inversion, i.e. replacing by : ( ) ( ) j t x X t e dt 2 = Introduction to Signal Processing 6 Copyright 2005-2009 Hayder Radha By focusing on: ( ) j t X t e dt It should be clear that this expression is the Fourier transform of the time-domain function ( ) X t . Hence if ( ) x t and ( ) X form an FT pair, then ( ) X t has a Fourier transform ( ) x 2 . Introduction to Signal Processing 7 Copyright 2005-2009 Hayder Radha Example Given the transform pair below we use the duality property to find the Fourier transform of another time function. ( ) ( ) rect sinc 2 x t X t . Introduction to Signal Processing 8 Copyright 2005-2009 Hayder Radha -2-1 1 2 0.5 1 1.5 2 t 1/ x(t)-10-5 5 10-0.4-0.2 0.2 0.4 0.6 0.8 1 / X( ) 1/ Introduction to Signal Processing 9 Copyright 2005-2009 Hayder Radha Applying the duality property, the Fourier transform of ( ) X t is, sinc 2 rect 2 rect 2 t = -10-5 5 10-0.4-0.2 0.2 0.4 0.6 0.8 1 t / x(t) 1/ -2-1 1 2 0.5 1 1.5 2 1/ X( ) 1/ Introduction to Signal Processing 1 Copyright 2005-2009 Hayder Radha Scaling Property If, ( ) ( ) x t X Then, ( ) 1 x at X a a ....
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

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SP_IV_P02_4p - Introduction to Signal Processing 1...

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