Leo Lam © Signals and Systems EE235 Lecture 25

Leo Lam © x squared equals 9 x squared plus 1 equals y Find value of y

Leo Lam © Today’s menu Fourier Transform

Fourier Transform: Leo Lam © Fourier Formulas: For any arbitrary practical signal And its “coefficients” (Fourier Transform):

Fourier Transform Properties Leo Lam © A Fourier Transform “Pair”: f(t)  F() Re-usable! Scaling Additivity Convolution Time shift time domain Fourier transform

How to do Fourier Transform Leo Lam © Three ways (or use a combination) to do it: –Solve integral –Use FT Properties (“Spiky signals”) –Use Fourier Transform table (for known signals)

FT Properties Example: Leo Lam © Find FT for: We know the pair: So:  G()

More Transform Pairs: Leo Lam © More pairs: time domain Fourier transform

Periodic signals: Transform from Series Leo Lam © Integral does not converge for periodic f n s: We can get it from Fourier Series: How? Find x(t) if Using Inverse Fourier: So

Periodic signals: Transform from Series Leo Lam © We see this pair: More generally, if X(w) has equally spaced impulses: Then: Fourier Series!!!

Periodic signals: Transform from Series Leo Lam © If we know Series, we know Transform Then: Example: We know: We can write:

Leo Lam © Summary Fourier Transform Pairs FT Properties

Duality of Fourier Transform Leo Lam © Duality (very neat): Duality of the Fourier transform: If time domain signal f(t) has Fourier transform F(), then F(t) has Fourier transform 2 f(-) i.e. if: Then: Changed sign

Duality of Fourier Transform (Example) Leo Lam © Using this pair: Find the FT of –Where T=5

Duality of Fourier Transform (Example) Leo Lam © Using this pair: Find the FT of

Convolution/Multiplication Example Leo Lam © Given f(t)=cos(t)e –t u(t) what is F()

More Fourier Transform Properties Leo Lam © Duality Time-scaling Multiplication Differentiation Integration Conjugation time domain Fourier transform Dual of convolution 17

Fourier Transform Pairs (Recap) Leo Lam © Review: