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EE3TP4_15_FTProperties_v3_Lecture 22

# EE3TP4_15_FTProperties_v3_Lecture 22 - Fourier Transform...

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Fourier Transform Properties As we have seen, finding the FT can be difficult. But … there are certain properties that can often make things easier. Also, these properties can sometimes be the key to understanding how the FT can be used in a given application. x t = 1 −∞ X ω e jωt X ω = −∞ x t e jωt dt

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1. Linearity (Supremely Important) If & then x t ↔ X ω y t ↔Y ω [ ax t  by t ] [ aX ω  bY ω ] To see why : F { ax t  by t } = −∞ [ ax t  by t ] e jωt dt = a −∞ x t e jωt dt b −∞ y t e jωt dt By standard Property of Integral of sum of functions Use Defn of FT = X ω = Y ω By Defn of FT F { ax t  by t } = aF { x t } bF { y t } Another way to write this property:
Example Application of “Linearity of FT”: Suppose we need to find the FT of the following signal… x t 1 2 2 2 t Finding this using straight-forward application of the definition of FT is not difficult but it is tedious: F { x t } = 2 1 e jωt dt 2 1 1 e jωt dt 1 2 e jωt dt 1 1 So … we look for short-cuts: One way is to recognize that each of these integrals is basically the same Another way is to break x ( t ) down into a sum of signals on our table!

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X ω = 4 sinc π 2sinc ω π From FT Table we have a known result for the FT of a pulse, so… Break a complicated signal down into simple signals before finding FT: x t p 4 t t p 2 t t Add to get X ω = P 4 ω  P 2 ω x t = p 4 t  p 2 t Mathematically we write: 1 2 2 2 t 1 1 2 2 1 1 1 1
2. Time Shift (Really Important!) If x t ↔ X ω then x t c ↔ X ω e jc ω Shift of Time Signal “Linear” Phase Shift of Frequency Components Used often to understand practical issues that arise in audio , communications , radar , etc. Note : If c > 0 then x ( t c ) is a delay of x ( t ) X ω e jωc ∣=∣ X ω ∣ So… what does this mean ?? First … it does nothing to the magnitude of the FT: That means that a shift doesn’t change “how much” we need of each of the sinusoids we build with { X ω e jcω }= X ω + e jc ω = X ω − This gets added to original phase Line of slope – c Second … it does change the phase of the FT: Phase shift increases linearly as the frequency increases

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x t = 1 −∞ X ω e jωt X ω = −∞ x t e jωt dt
Magnitude of the FT is unaffected! Only phase is shifted.

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Example Application of Time Shift Property: Room Acoustics .
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EE3TP4_15_FTProperties_v3_Lecture 22 - Fourier Transform...

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