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

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 [
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern