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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 ω ⇔ and ( ) ( ) 2 2 x t X ω ⇔ Then, ( ) ( ) ( ) ( ) 1 2 1 2 ax t bx t aX bX ω ω + ⇔ + . Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha Conjugation and Conjugate Symmetry If, ( ) ( ) x t X ω ⇔ , Then, ( ) ( ) * * x t X ω ⇔ − And if ( ) x t is real, ( ) ( ) * X X ω ω − = . Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Duality If, ( ) ( ) x t X ω ⇔ Then, ( ) ( ) 2 X t x π ω ⇔ − . 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 ω ω ∞ − −∞ = ∫ ( ) 1 ( ) 2 j t x t X e d ω ω ω π ∞ −∞ = ∫ Now consider performing the following operations on the expression for ( ) x t : Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha 1. Replacing ω with t , And, Replacing t with ω , ( ) 1 ( ) 2 j t x X t e dt ω ω π ∞ −∞ = ∫ 2. Multiplying by 2 π . ( ) ( ) j t x X t e dt ω π ω ∞ −∞ 2 = ∫ 3. Frequency inversion, i.e. replacing ω by ω − : ( ) ( ) j t x X t e dt ω π ω ∞ − −∞ 2 − = ∫ Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha By focusing on: ( ) j t X t e dt ω ∞ − −∞ ∫ It should be clear that this expression is the Fourier transform of the “time-domain” function ( ) X t . Hence if ( ) x t and ( ) X ω form an FT pair, then ( ) X t has a Fourier transform ( ) x π ω 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 ω ωτ τ τ ⎛ ⎞ ⎛ ⎞ ⇔ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ¡¢£¢¤ ¡¢£¢¤ . Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha -2-1 1 2 0.5 1 1.5 2 t × 1/ τ x(t)-10-5 5 10-0.4-0.2 0.2 0.4 0.6 0.8 1 ω × τ / π X( ω ) × 1/ τ 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 τ ω ω τ π π τ τ − ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⇔ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠-10-5 5 10-0.4-0.2 0.2 0.4 0.6 0.8 1 t × τ / π x(t) × 1/ τ-2-1 1 2 0.5 1 1.5 2 ω × 1/ τ X( ω ) × 1/ π Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha Scaling Property If, ( ) ( ) x t X ω ⇔ Then, ( ) 1 x at X a a ω ⎛ ⎞ ⇔ ⎜ ⎟ ⎝ ⎠ ....
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SP_IV_P02_4p - Introduction to Signal Processing 1...

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