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Unformatted text preview: UCLA Win 20092010 Systems and Signals Lecture 10: Fourier Theorems and Generalized Fourier Transforms April 29 2010 EE102: Systems and Signals; Spr 0910, Lee 1 Fourier Transform Notation For convenience, we will write the Fourier transform of a signal f ( t ) as F [ f ( t )] = F ( jω ) and the inverse Fourier transform of F ( jω ) as F 1 [ F ( jω )] = f ( t ) . Note that F 1 [ F [ f ( t )]] = f ( t ) at points of continuity of f ( t ) . EE102: Systems and Signals; Spr 0910, 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 ( jω ) and F 2 ( jω ) , 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 Z ∞∞ F 1 ( jθ ) F 2 ( j ( ω θ )) dθ This is the convolution of F 1 ( jω ) and F 2 ( jω ) , 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 )( jω ) while keeping in mind that the convolution is with respect to ω , not jω . EE102: Systems and Signals; Spr 0910, Lee 3 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 0910, 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....
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