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Leo Lam © 2010-2013 Signals and Systems EE235

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Fourier Transform: Leo Lam © 2010-2013 2 Fourier Formulas: Inverse Fourier Transform: Fourier Transform: Time domain to Frequency domain

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Fourier Transform (delta function): Leo Lam © 2010-2013 3 Fourier Transform of Standard Fourier Transform pair notation

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Fourier Transform (rect function): Leo Lam © 2010-2013 4 Fourier Transform of Plot for T=1? t-T/2 0 T/2 1 Define

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Fourier Transform (rect function): Leo Lam © 2010-2013 5 Fourier Transform of Observation: –Wider pulse (in t) taller narrower spectrum –Extreme case: Peak=pulse width (example: width=1) Zero-Crossings:

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Fourier Transform - Inverse relationship Leo Lam © 2010-2013 6 Inverse relationship between time/frequency

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Fourier Transform - Inverse Leo Lam © 2010-2013 7 Inverse Fourier Transform (Synthesis) Example:

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Fourier Transform - Inverse Leo Lam © 2010-2013 8 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

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Fourier Transform Properties Leo Lam © 2010-2013 9 A Fourier Transform “Pair”: f(t) F() Re-usable! Scaling Additivity Convolution Time shift time domain Fourier transform

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How to do Fourier Transform Leo Lam © 2010-2013 10 Three ways (or use a combination) to do it: –Solve integral –Use FT Properties (“Spiky signals”) –Use Fourier Transform table (for known signals)

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FT Properties Example: Leo Lam © 2010-2013 11 Find FT for: We know the pair: So: -8 0 8 G()

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More Transform Pairs: Leo Lam © 2010-2013 12 More pairs: time domain Fourier transform

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Periodic signals: Transform from Series Leo Lam © 2010-2013 13 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

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Periodic signals: Transform from Series Leo Lam © 2010-2013 14 We see this pair: More generally, if X(w) has equally spaced impulses: Then: Fourier Series!!!

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Periodic signals: Transform from Series Leo Lam © 2010-2013 15 If we know Series, we know Transform Then: Example: We know: We can write:

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Leo Lam © 2010-2013 Summary Fourier Transform Pairs FT Properties

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Duality of Fourier Transform Leo Lam © 2010-2013 17 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

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Duality of Fourier Transform (Example) Leo Lam © 2010-2013 18 Using this pair: Find the FT of –Where T=5

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Duality of Fourier Transform (Example) Leo Lam © 2010-2013 19 Using this pair: Find the FT of

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