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102_1_Lecture10

# 102_1_Lecture10 - UCLA Win 2009-2010 Systems and Signals...

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UCLA Win 2009-2010 Systems and Signals Lecture 10: Fourier Theorems and Generalized Fourier Transforms April 29 2010 EE102: Systems and Signals; Spr 09-10, Lee 1

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Fourier Transform Notation For convenience, we will write the Fourier transform of a signal f ( t ) as F [ f ( t )] = F ( j ω ) and the inverse Fourier transform of F ( j ω ) as F - 1 [ F ( j ω )] = f ( t ) . Note that F - 1 [ F [ f ( t )]] = f ( t ) at points of continuity of f ( t ) . EE102: Systems and Signals; Spr 09-10, Lee 2
Frequency Domain Convolution There is another version of the convolution theorem that applies when the convolution is in the frequency domain. Frequency Domain Convolution Theorem: If f 1 ( t ) and f 2 ( t ) have Fourier transforms F 1 ( j ω ) and F 2 ( j ω ) , then the product of f 1 ( t ) and f 2 ( t ) has the Fourier transform F [ f 1 ( t ) f 2 ( t )] = 1 2 π -∞ F 1 ( j θ ) F 2 ( j ( ω - θ )) d θ This is the convolution of F 1 ( j ω ) and F 2 ( j ω ) , considered as functions of ω . For convenience, we will write this as F [ f 1 ( t ) f 2 ( t )] = 1 2 π ( F 1 * F 2 )( j ω ) while keeping in mind that the convolution is with respect to ω , not j ω . EE102: Systems and Signals; Spr 09-10, Lee 3

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Multiplication in the time domain corresponds to convolution in the frequency domain. The proof of this theorem is essentially the same as for the time domain convolution theorem. This is a particularly useful for analyzing modulation and demodulation. Example: What is the Fourier transform of sinc 2 ( t ) ? We know the Fourier transform pair sinc ( t ) rect ( ω / 2 π ) . The Fourier transform of sinc 2 ( t ) is then F sinc 2 ( t ) = 1 2 π ( rect ( ω / 2 π ) * rect ( ω / 2 π )) = Δ ( ω / 2 π ) EE102: Systems and Signals; Spr 09-10, Lee 4
We then have the transform pair: sinc 2 ( t ) Δ ( ω / 2 π ) Check that this is consistent with the transform pair Δ ( t ) sinc 2 ( ω / 2 π ) using the duality theorem. EE102: Systems and Signals; Spr 09-10, Lee 5

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Generalized Fourier Transforms: δ Functions A unit impulse δ ( t ) is not a signal in the usual sense (it is a generalized function or distribution). However, if we proceed using the sifting property, we get a result that makes sense: F [
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