1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Leo Lam © 2010-2013 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!"

3. Leo Lam © 2010-2013 Today’s menu Fourier Transform Properties (cont’) Loads of examples

4. Fourier Transform: Leo Lam © 2010-2013 4 Fourier Transform Inverse Fourier Transform:

5. Duality of Fourier Transform Leo Lam © 2010-2013 5 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

6. Duality of Fourier Transform (Example) Leo Lam © 2010-2013 6 Using this pair: Find the FT of –Where T=5

7. Duality of Fourier Transform (Example) Leo Lam © 2010-2013 7 Using this pair: Find the FT of

8. Convolution/Multiplication Example Leo Lam © 2010-2013 8 Given f(t)=cos(t)e –t u(t) what is F()

9. More Fourier Transform Properties Leo Lam © 2010-2013 9 Duality Time-scaling Multiplication Differentiation Integration Conjugation time domain Fourier transform Dual of convolution 9

10. Fourier Transform Pairs (Recap) Leo Lam © 2010-2013 10 1 Review:

11. Fourier Transform and LTI System Leo Lam © 2010-2013 11 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

12. Fourier Transform and LTI (Example) Leo Lam © 2010-2013 Delay: LTI h(t) Time domain:Frequency domain (FT): Shift in time  Add linear phase in frequency 12

13. Fourier Transform and LTI (Example) Leo Lam © 2010-2013 Delay: Exponential response LTI h(t) 13 Delay 3 Using Convolution Properties Using FT Duality

14. Fourier Transform and LTI (Example) Leo Lam © 2010-2013 Delay: Exponential response Responding to Fourier Series LTI h(t) 14 Delay 3

15. Another LTI (Example) Leo Lam © 2010-2013 Given Exponential response What does this system do? What is h(t)? And y(t) if Echo with amplification 15 LTI

16. Another angle of LTI (Example) Leo Lam © 2010-2013 Given graphical H(), find h(t) What does this system do? What is h(t)? Linear phase  constant delay 16 magnitude   phase 0 0 1 Slope=-5

17. Another angle of LTI (Example) Leo Lam © 2010-2013 Given graphical H(), find h(t) What does this system do (qualitatively Low-pass filter. No delay. 17 magnitude   phase 0 0 1

18. Another angle of LTI (Example) Leo Lam © 2010-2013 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 18 magnitude   phase 0 1

19. Another angle of LTI (Example) Leo Lam © 2010-2013 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 19 magnitude   phase 0 1

20. Leo Lam © 2010-2013 Summary Fourier Transforms and examples