102_1_Lecture09 - UCLA Spring 2009-2010 EE102 Systems and...

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UCLA Spring 2009-2010 EE102: Systems and Signals Lecture 09: Fourier Transform Theorems April 27 2010 EE102:Systems and Signals; Spring 09-10, Lee 1
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Linearity Linear combination of two signals f 1 ( t ) and f 2 ( t ) is a signal of the form af 1 ( t ) + bf 2 ( t ) . Linearity Theorem: The Fourier transform is linear; that is, given two signals f 1 ( t ) and f 2 ( t ) and two complex numbers a and b , then af 1 ( t ) + bf 2 ( t ) aF 1 ( j ω ) + bF 2 ( j ω ) . This follows from linearity of integrals: -∞ ( af 1 ( t ) + bf 2 ( t )) e - j ω t dt = a -∞ f 1 ( t ) e - j ω t dt + b -∞ f 2 ( t ) e - j ω t dt = aF 1 ( j ω ) + bF 2 ( j ω ) EE102:Systems and Signals; Spring 09-10, Lee 2
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This easily extends to finite combinations. Given signals f k ( t ) with Fourier transforms F k ( j ω ) and complex constants a k , k = 1 , 2 , . . . K , then K k =1 a k f k ( t ) K k =1 a k F k ( j ω ) . If you consider a system which has a signal f ( t ) as its input and the Fourier transform F ( j ω ) as its output, the system is linear! EE102:Systems and Signals; Spring 09-10, Lee 3
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Linearity Example Find the Fourier transform of the signal f ( t ) = 1 2 1 2 | t | < 1 1 | t | 1 2 This signal can be recognized as f ( t ) = 1 2 rect t 2 + 1 2 rect ( t ) . From linearity and the fact that the transform of rect( t/T ) is T sinc( T ω / (2 π )) , we have F ( ω ) = 1 2 2 sinc(2 ω / (2 π ))+ 1 2 sinc( ω / (2 π )) = sinc( ω / π )+ 1 2 sinc( ω / (2 π )) EE102:Systems and Signals; Spring 09-10, Lee 4
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! 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ! 0.5 0 0.5 1 1.5 2 0 1 2 - 2 - 1 0 2 ! - 2 ! - 4 ! 4 ! ! sinc ( ! / " )+ 1 2 sinc ( ! / ( 2 " )) 1 2 rect ( t / 2 )+ 1 2 rect ( t ) Linearity Example EE102:Systems and Signals; Spring 09-10, Lee 5
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Scaling Theorem Stretch (Scaling) Theorem : Given a transform pair f ( t ) F ( j ω ) , and a real-valued nonzero constant a , f ( at ) 1 | a | F j ω a Proof: Here consider only a > 0 . Negative a left as an exercise. Change variables τ = at -∞ f ( at ) e - j ω t dt = -∞ f ( τ ) e - j ωτ /a d τ a = 1 a F j ω a . If a = - 1 “time reversal theorem:” f ( - t ) F ( - j ω ) EE102:Systems and Signals; Spring 09-10, Lee 6
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Scaling Examples We have already seen that rect( t/T ) T sinc( T ω / 2 π ) by brute force integration. The scaling theorem provides a shortcut proof given the simpler result rect( t ) sinc( ω / 2 π ) . This is a good point to illustrate a property of transform pairs. Consider this Fourier transform pair for a small T and large T , say T = 1 and T = 5 . The resulting transform pairs are shown below to a common horizontal scale: EE102:Systems and Signals; Spring 09-10, Lee 7
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! 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ! 0.2 0 0.2 0.4 0.6 0.8 1 1.2 ! 2 0 2 4 6 ! 2 ! 1 0 1 2 3 4 5 ! ! - 5 ! - 10 ! 0 5 ! 10 ! - 5 ! - 10 ! 0 5 ! 10 ! 0 - 5 5 - 10 10 0 - 5 5 - 10 10 t t sinc ( ! / 2 " ) 5sinc ( 5 ! / 2 " ) rect ( t ) rect ( t / 5 ) EE102:Systems and Signals; Spring 09-10, Lee 8
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The narrow pulse yields a wide transform and a wide pulse yields a narrow spectrum! This example shows that the shorter the pulse (and hence the more pulses we could cram into a transmission channel), the greater the bandwidth required by the transform!!
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