Leo Lam © Signals and Systems EE235

Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to the Fourier transform of the sinc function? A: "You're such a square!"

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

Fourier Transform: Leo Lam © Fourier Transform Inverse Fourier Transform:

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 9

Fourier Transform Pairs (Recap) Leo Lam © Review:

Fourier Transform and LTI System Leo Lam © 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

Fourier Transform and LTI (Example) Leo Lam © Delay: LTI h(t) Time domain:Frequency domain (FT): Shift in time  Add linear phase in frequency 12

Fourier Transform and LTI (Example) Leo Lam © Delay: Exponential response LTI h(t) 13 Delay 3 Using Convolution Properties Using FT Duality

Fourier Transform and LTI (Example) Leo Lam © Delay: Exponential response Responding to Fourier Series LTI h(t) 14 Delay 3

Another LTI (Example) Leo Lam © Given Exponential response What does this system do? What is h(t)? And y(t) if Echo with amplification 15 LTI

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do? What is h(t)? Linear phase  constant delay 16 magnitude   phase Slope=-5

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do (qualitatively Low-pass filter. No delay. 17 magnitude   phase 0 0 1

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 18 magnitude   phase 0 1

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 19 magnitude   phase 0 1

Leo Lam © Summary Fourier Transforms and examples