Leo Lam © 2010-2013 Signals and Systems EE235. Fourier Transform: Leo Lam © 2010-2013 2 Fourier Formulas: Inverse Fourier Transform: Fourier Transform:

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Leo Lam © Signals and Systems EE235

Fourier Transform: Leo Lam © Fourier Formulas: Inverse Fourier Transform: Fourier Transform: Time domain to Frequency domain

Fourier Transform (delta function): Leo Lam © Fourier Transform of Standard Fourier Transform pair notation

Fourier Transform (rect function): Leo Lam © Fourier Transform of Plot for T=1? t-T/2 0 T/2 1 Define

Fourier Transform (rect function): Leo Lam © Fourier Transform of Observation: –Wider pulse (in t) taller narrower spectrum –Extreme case: Peak=pulse width (example: width=1) Zero-Crossings:

Fourier Transform - Inverse relationship Leo Lam © Inverse relationship between time/frequency

Fourier Transform - Inverse Leo Lam © Inverse Fourier Transform (Synthesis) Example:

Fourier Transform - Inverse Leo Lam © Inverse Fourier Transform (Synthesis) Example: Single frequency spike in  : exponential time signal with that frequency in t A single spike in frequencyComplex exponential in time

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