1. Signal and Systems Prof. H. Sameti Chapter 3: Fourier Series Representation of Periodic Signals Complex Exponentials as Eigenfunctions of LTI Systems Fourier Series representation of CT periodic signals How do we calculate the Fourier coefficients? Convergence and Gibbs’ Phenomenon CT Fourier series reprise, properties, and examples DT Fourier series DT Fourier series examples and differences with CTFS Fourier Series and LTI Systems Frequency Response and Filtering Examples and Demos

2. Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations. Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179. Book Chapter 3 : Section 1 Computer Engineering Department, Signal and Systems 2

3. Desirable Characteristics of a Set of “Basic” Signals a. We can represent large and useful classes of signals using these building blocks b.The response of LTI systems to these basic signals is particularly simple, useful, and insightful Previous focus: Unit samples and impulses Focus now: Eigenfunctions of all LTI systems Computer Engineering Department, Signal and Systems 3 Book Chapter 3 : Section 1

4. The eigenfunctions and their properties Eigenfunction in →same function out with a “gain” Computer Engineering Department, Signal and Systems 4 (Focus on CT systems now, but results apply to DT systems as well.) From the superposition property of LTI systems: Now the task of finding response of LTI systems is to determine λ k. Book Chapter 3 : Section 1

5. Complex Exponentials as the Eigenfunctions of any LTI Systems Computer Engineering Department, Signal and Systems 5 eigenvalue eigenfunction Book Chapter 3 : Section 1

6. DT: Book Chapter 3 : Section 1

7. What kinds of signals can we represent as “sums” of complex exponentials? For Now: Focus on restricted sets of complex exponentials CT: DT: s = jω – purely imaginaly, i.e., signals of the form e jωt Magnitude 1 CT & DT Fourier Series and Transforms Periodic Signals Book Chapter 3 : Section 1

8. for all t Computer Engineering Department, Signal and Systems 8 Fourier Series Representation of CT Periodic Signals - smallest such T is the fundamental period - is the fundamental frequency Periodic with period T -periodic with period T -{a k } are the Fourier (series) coefficients -k= 0 DC -k= 1 first harmonic -k= 2 second harmonic Book Chapter 3 : Section 1

9. Computer Engineering Department, Signal and Systems 9 Question #1: How do we find the Fourier coefficients? First, for simple periodic signals consisting of a few sinusoidal terms Book Chapter 3 : Section 1

10.  For real periodic signals, there are two other commonly used forms for CT Fourier series:  Because of the eigenfunction property of e jωt, we will usually use the complex exponential form in 6.003. Book Chapter 3 : Section 1 Computer Engineering Department, Signal and Systems 10 or - A consequence of this is that we need to include terms for both positive and negative frequencies: Remembe r and sin

11. Computer Engineering Department, Signal and Systems 11 Now, the complete answer to Question #1 multiply by Integrate over one period multiply by Integrate over one period denotes integral over any interval of length Here Next, note that Orthogonality Book Chapter 3 : Section 1

12. Computer Engineering Department, Signal and Systems 12 CT Fourier Series Pair (Synthesis equation) (Analysis equation) Book Chapter 3 : Section 1

13. Computer Engineering Department, Signal and Systems 13 Ex: Periodic Square Wave DC component is just the average Book Chapter 3 : Section 1

14.  How can the Fourier series for the square wave possibly make sense?  The key is: What do we mean by  One useful notion for engineers: there is no energy in the difference Book Chapter 3 : Section 1 Computer Engineering Department, Signal and Systems 14 Convergence of CT Fourier Series (just need x(t) to have finite energy per period) ?

15. Computer Engineering Department, Signal and Systems 15 Under a different, but reasonable set of conditions (the Dirichlet conditions) Condition 1. x(t) is absolutely integrable over one period, i. e. And Condition 2. In a finite time interval, x(t) has a finite number of maxima and minima. Ex. An example that violates Condition 2. And Condition 3. In a finite time interval, x(t) has only a finite number of discontinuities. Ex. An example that violates Condition 3. Book Chapter 3 : Section 1

16.  Dirichlet conditions are met for the signals we will encounter in the real world. Then  Still, convergence has some interesting characteristics: Book Chapter 3 : Section 1 Computer Engineering Department, Signal and Systems 16 - The Fourier series = x(t) at points where x(t) is continuous - The Fourier series = “midpoint” at points of discontinuity - As N→ ∞, x N (t) exhibits Gibbs’ phenomenon at points of discontinuity Demo: Fourier Series for CT square wave (Gibbs phenomenon).

17. CT Fourier Series Pairs Book Chapter3: Section2 Computer Engineering Department, Signals and Systems 17

18. Another (important!) example: Periodic Impulse Train Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 18 ─ All components have: (1)the same amplitude, & (2)the same phase.

19. (A few of the) Properties of CT Fourier Series Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 19 Introduces a linear phase shift ∝ t o

20. Example: Shift by half period Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 20 using

21.  Parseval’s Relation  Multiplication Property Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 21 Power in the k th harmonic Average signal power (Both x(t) and y(t) are periodic with the same period T) Energy is the same whether measured in the time-domain or the frequency-domain Proof: X(t) y(t) ckck

22. Periodic Convolution x(t), y(t) periodic with period T Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 22 - Not very meaningful E.g. If both x(t) and y(t) are positive, then

23. Periodic Convolution (continued) Periodic convolution : Integrate over anyone period (e.g. -T/2 to T/2) Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 23 where

24. Periodic Convolution (continued) Facts Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 24 1) z(t) is periodic with period T (why?) From Lecture #2, x(t) = x(t + T) → y(t) = y(t + T) for LTI systems. In the convolution, treat y(t) as the input and x T (t) as h(t) 2) Doesn’t matter which period we choose to integrate over: Periodic 3) Convolution in time Multiplication In frequency!

25. Fourier Series Representation of DT Periodic Signals  x[n] -periodic with fundamental period N, fundamental frequency  Only e jωn which are periodic with period N will appear in the FS  There are only N distinct signals of this form  So we could just use  However, it is often useful to allow the choice of N consecutive values of k to be arbitrary. Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 25 and 2πn2πn

26. DT Fourier Series Representation Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 26 Sum over any N consecutive values of k — This is a finite series - Fourier (series) coefficients Questions: 1) What DT periodic signals have such a representation? 2) How do we find a k ?

27. Answer to Question #1: Any DT periodic signal has a Fourier series representation Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 27 N equations for N unknowns, a 0, a 1, …, a N-1

28. A More Direct Way to Solve for a k Finite geometric series Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 28

29. Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 29 So, from multiply both sides by and then orthogonality

30. DT Fourier Series Pair Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 30 (Synthesis equation) (Analysis equation) Note : It is convenient to think of a k as being defined for all integers k. So: 1) a k+N = a k —Special property of DT Fourier Coefficients. 2) We only use N consecutive values of a k in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)

31. Example #1: Sum of a pair of sinusoids Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 31 - periodic with period N = 16 → ω 0 = π/8

32. Example #2: DT Square Wave Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 32

33. Example #2: DT Square Wave (continued) Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 33

34. Book Chapter3: Section2 Computer Engineering Department, Signal and Systems 34 Convergence Issues for DT Fourier Series: Not an issue, since all series are finite sums. Properties of DT Fourier Series: Lots, just as with CT Fourier Series Example: Frequency shift

35. The Eigenfunction Property of Complex Exponentials  CT  CT System Function:  DT  DT System function: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 35

36. Fourier Series: Periodic Signals and LTI Systems  ,  So, or powers of signals get modified through filter/ system.  , Book Chapter#: Section# Computer Engineering Department, Signal and Systems 36 Includes both amplitude & phase

37. The Frequency Response of an LTI System  CT Frequency Response:  DT Frequency Response: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 37

38. Frequency Shaping and Filtering  By choice of (or ) as a function of , we can shape the frequency composition of the output  Preferential amplification  Selective filtering of some frequencies  Example #1:  For audio signals, the amplitude is much more important than the phase Book Chapter#: Section# Computer Engineering Department, Signal and Systems 38

39. Frequency Shaping and Filtering Book Chapter#: Section# Computer Engineering Department, Signal and Systems 39  Example #2: Frequency Selective Filters  Filter out signals outside of the frequency range of interest  Lowpass Filters: Only show amplitude here. Note for DT:

40. Highpass Filters Book Chapter#: Section# Computer Engineering Department, Signal and Systems 40 Remember:

41. Bandpass Filters  Demo:Filtering effects on audio signals Book Chapter#: Section# Computer Engineering Department, Signal and Systems 41

42. Idealized Filters  CT  DT  Note: |H| = 1 and ∠ H = 0 for the ideal filters in the passbands, no need for the phase plot. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 42 ω c : cutoff frequency

43. Highpass  CT  DT Book Chapter#: Section# Computer Engineering Department, Signal and Systems 43

44. Bandpass  CT  DT Book Chapter#: Section# Computer Engineering Department, Signal and Systems 44 lower cut-off upper cut-off

45. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 45  Example #3: DT Averager/Smoother FIR (Finite Impulse Response) filters

46.  Example #4: Nonrecursive DT (FIR) filters Book Chapter#: Section# Computer Engineering Department, Signal and Systems 46 Rolls off at lower ω as M+N+1 increases

47.  Example #5:Simple DT “Edge” Detector  DT 2-point “differentiator” Book Chapter#: Section# Computer Engineering Department, Signal and Systems 47

48.  Demo: DT filters, LP, HP, and BP applied to DJ Industrial average Book Chapter#: Section# Computer Engineering Department, Signal and Systems 48

49.  Example #6: Edge enhancement using DT differentiator Book Chapter#: Section# Computer Engineering Department, Signal and Systems 49 Courtesy of Jason Oppenheim. Used with permission.

50.  Example #7: A Filter Bank Book Chapter#: Section# Computer Engineering Department, Signal and Systems 50

51.  Demo: Apply different filters to two-dimensional image signals.  Note: To really understand these examples, we need to understand frequency contents of aperiodic signals ⇒ the Fourier Transform Book Chapter#: Section# Computer Engineering Department, Signal and Systems 51 Face of a monkey. Image removed do to copyright considerations

52. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 52