1. Leo Lam © 2010-2012 Signals and Systems EE235

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

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

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

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

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

7. 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

8. 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!!!

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

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

11. 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

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

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

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

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

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

17. 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

18. 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

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

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

21. 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

22. 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

23. 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

24. 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

25. 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

26. Leo Lam © 2010-2011 Summary Fourier Transforms and examples