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Leo Lam © 2010-2012 Signals and Systems EE235

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Leo Lam © 2010-2011 x squared equals 9 x squared plus 1 equals y Find value of y

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Leo Lam © 2010-2011 Today’s menu Fourier Transform Properties (cont’) Duality Loads of examples

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Fourier Transform: Leo Lam © 2010-2011 4 Fourier Transform Inverse Fourier Transform:

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FT Properties Example: Leo Lam © 2010-2012 5 Find FT for: We know the pair: So: -8 0 8 G()

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More Transform Pairs: Leo Lam © 2010-2012 6 More pairs: time domain Fourier transform

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Periodic signals: Transform from Series Leo Lam © 2010-2012 7 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 © 2010-2012 8 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 © 2010-2012 9 If we know Series, we know Transform Then: Example: We know: We can write:

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Leo Lam © 2010-2012 Summary Fourier Transform Pairs FT Properties

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Duality of Fourier Transform Leo Lam © 2010-2011 11 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 © 2010-2011 12 Using this pair: Find the FT of –Where T=5

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Duality of Fourier Transform (Example) Leo Lam © 2010-2011 13 Using this pair: Find the FT of

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Convolution/Multiplication Example Leo Lam © 2010-2011 14 Given f(t)=cos(t)e –t u(t) what is F()

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More Fourier Transform Properties Leo Lam © 2010-2011 15 Duality Time-scaling Multiplication Differentiation Integration Conjugation time domain Fourier transform Dual of convolution 15

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Fourier Transform Pairs (Recap) Leo Lam © 2010-2011 16 1 Review:

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Fourier Transform and LTI System Leo Lam © 2010-2011 17 Back to the Convolution Duality: And remember: And in frequency domain Convolution in time h(t) x(t)*h(t)x(t) Time domain Multiplication in frequency H() X()H() X() Frequency domain input signal’s Fourier transform output signal’s Fourier transform

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Fourier Transform and LTI (Example) Leo Lam © 2010-2011 Delay: LTI h(t) Time domain:Frequency domain (FT): Shift in time Add linear phase in frequency 18

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Fourier Transform and LTI (Example) Leo Lam © 2010-2011 Delay: Exponential response LTI h(t) 19 Delay 3 Using Convolution Properties Using FT Duality

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Fourier Transform and LTI (Example) Leo Lam © 2010-2011 Delay: Exponential response Responding to Fourier Series LTI h(t) 20 Delay 3

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Another LTI (Example) Leo Lam © 2010-2011 Given Exponential response What does this system do? What is h(t)? And y(t) if Echo with amplification 21 LTI

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Another angle of LTI (Example) Leo Lam © 2010-2011 Given graphical H(), find h(t) What does this system do? What is h(t)? Linear phase constant delay 22 magnitude phase 0 0 1 Slope=-5

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Another angle of LTI (Example) Leo Lam © 2010-2011 Given graphical H(), find h(t) What does this system do (qualitatively Low-pass filter. No delay. 23 magnitude phase 0 0 1

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Another angle of LTI (Example) Leo Lam © 2010-2011 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 24 magnitude phase 0 1

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Another angle of LTI (Example) Leo Lam © 2010-2011 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 25 magnitude phase 0 1

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Leo Lam © 2010-2011 Summary Fourier Transforms and examples

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