1. 1 Fourier Representations of Signals & Linear Time-Invariant Systems Chapter 3

2. 2 Introduction In the previous chapter, linearity property was exploited to develop the convolution sum and convolution integral. There, the basic idea of convolution is to break up or decompose a signal into sum of elementary function. Then, we find the response of the system to each of those elementary function individually and add the responses to get the overall response. In this chapter, we will express a signal as a sum of real or complex sinusoids instead of sum of impulses.

3. 3 The response of LTI system to sinusoids are also sinusoids of the same frequency but with in general, different amplitude and phase.

4. 4 Complex Sinusoids & Frequency Response of LTI System The response of an LTI system to a sinusoidal input leads to a characterization of system behaviour that is termed the ‘frequency response’ of the system.

5. 5 Fourier Representation for Four Signal Classes There are 4 distinct Fourier representation, each applicable to a different class of signals. These 4 classes are defined by the periodicity properties of a signal and whether it is continuous or discrete.

6. 6 Time propertyPeriodicNonperiodic Continuous-timeFourier Series (CTFS) Fourier Transform (CTFT) Discrete-timeFourier Series (DTFS) Fourier Transform (DTFT) Relationship Between Time Properties of a Signal and the Appropriate Fourier Representations

7. 7 The Continuous-Time Fourier Series (CTFS)

8. 8 Objectives To develop methods of expressing periodic signals as linear combination of sinusoids, real or complex. To explore the general properties of these ways of expressing signals. To apply these methods to find the responses of systems to arbitrary periodic signals.

9. 9 Representing a Signal The Fourier series represents a signal as a linear combination of complex sinusoids The responses of LTI system to sinusoids are also sinusoids of the same frequency but with, in general, different amplitude and phase. Expressing signals in this way leads to frequency domain concept, thinking of signals as function of frequency instead of time.

10. 10 Periodic Excitation and Response

11. 11 Aperiodic Excitation and Response

12. 12 Basic Concept & Development of the Fourier Series

13. 13 Linearity and Superposition If an excitation can be expressed as a sum of complex sinusoids the response can be expressed as the sum of responses to complex sinusoids (same frequency but different multiplying constant).

14. 14 Continuous- Time Fourier Series Concept

15. 15 Conceptual Overview The Fourier series represents a signal as a sum of sinusoids. Consider original signal x(t), which we would like to present as a linear combination of sinusoids as illustrated by the dash line.

16. 16 Conceptual Overview (cont…) The best approximation to the dashed-line signal using a constant + one sinusoid of the same fundamental frequency as the dashed-line signal is the solid line. + =

17. 17 Conceptual Overview (cont…) The best approximation to the dashed-line signal using a constant + one sinusoid of the same fundamental frequency as the dashed-line signal + another sinusoid of twice the fundamental frequency of the dashed-line signal is the solid line.

18. 18 Conceptual Overview (cont…) The best approximation to the dashed-line signal using a constant + three sinusoids is the solid line. In this case, the third sinusoid has zero amplitude, indicating that sinusoid at that frequency does not help the approximation.

19. 19 Conceptual Overview (cont…) The best approximation to the dashed-line signal using a constant + four sinusoids is the solid line (the forth fundamental frequency is three times fundamental frequency of the dashed-line signal). This is a good approximation which gets better with the addition of more sinusoids at higher integer multiples of the fundamental frequency.

20. 20 Trigonometric Form of CTFS In the example above, each of the sinusoids used in the approximation above is of the form cos(2Пkf F t+θ) multiplied by a constant to set its amplitude. So we can use trigonometry identity: cos(a+b) = cos(a)cos(b) - sin(a)sin(b) sin(a+b) = sin(a)cos(b) + cos(a)sin(b) Therefore, we can reformulate this functional form into: cos(2Пkf F t+θ)= cos(θ) cos(2Пkf F t) - sin(θ)sin(2Пkf F t)

21. 21 Trigonometric Form of CTFS (cont…) The summation of all those sinusoids expressed as cosines and sines are called the continuous-time Fourier Series (CTFS). In the CTFS, the higher frequency sines and cosines have frequencies that are integers multiples of fundamental frequencies. The multiple is called the harmonic number, k.

22. 22 If we have function cos(2Пkf F t) or sin(2Пkf F t) i) k is harmonic number ii) kf F is highest frequency. If the signal to be represented is x(t), the amplitude of the k th harmonic sine will be designed X s [k] and the amplitude of the k th harmonic cosine will be designed X c [k]. X s [k] and X c [k] are called sine and cosine harmonic function respectively. Component of CTFS

23. 23 Complex Sinusoids form of CTFS Every sine and cosine can be replaced by a linear combination of complex sinusoids cos(2Пkf F t) = (e j2Пkf F t + e -j2Пkf F t )/2 sin(2Пkf F t) = (e j2Пkf F t - e -j2Пkf F t )/j2

24. 24 Component of CTFS (cont…)

25. 25 CT Fourier Series Definition

26. 26 The Trigonometric CTFS The fact that, for a real-valued function x(t) also leads to the definition of an alternate form of the CTFS, the so-called trigonometric form. where

27. 27 The Trigonometric CTFS Since both the complex and trigonometric forms of the CTFS represent a signal, there must be relationships between the harmonic functions. Those relationships are

28. 28 Periodicity of the CTFS

29. 29 The dash line are periodic continuations of the CTFS representation The illustrations show how various kinds of signals are represented by CTFS over a finite time.

30. 30 The dash line are periodic continuations of the CTFS representation

31. 31 Linearity of the CTFS These relations hold only if the harmonic functions X of all the component functions x are based on the same representation time.

32. 32 Magnitude and Phase of X[k] A graph of the magnitude and phase of the harmonic function as a function of harmonic number is a good way of illustrating it.

33. 33 CTFS of Even and Odd Functions

34. 34 Numerical Computation of the CTFS How could we find the CTFS of this signal which has no known functional description? Numerically. Unknown

35. 35 Numerical Computation of the CTFS We don’t know the function x(t), but if we set of N F samples over one period starting at t=0, the time between the samples is T s T F /N F, and we can approximate the integral by the sum of several integrals, each covering a time of lenght T s.

36. 36 Numerical Computation of the CTFS (cont…) Samples from x(t)

37. 37 Numerical Computation of the CTFS (cont…) where

38. 38 Convergence of the CTFS To examine how the CTFS summation approaches the signal it represents as the number of terms used in the sum approaches infinity. We do this by examining the partial sum.

39. 39 Convergence of the CTFS (cont…) For continuous signals, convergence is exact at every point. A Continuous Signal Partial CTFS Sums

40. 40 Convergence of the CTFS (cont…) For discontinuous signals, convergence is exact at every point of continuity. Discontinuous Signal Partial CTFS Sums

41. 41 Convergence of the CTFS (cont…) At points of discontinuity the Fourier series representation converges to the mid-point of the discontinuity.

42. 42 CTFS Properties Linearity

43. 43 CTFS Properties Time Shifting

44. 44 CTFS Properties (cont…) Frequency Shifting (Harmonic Number Shifting) A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential. Time Reversal

45. 45 CTFS Properties (cont…) Time Scaling If a is an integer,

46. 46 CTFS Properties (cont…) Time Scaling (continued)

47. 47 CTFS Properties (cont…) Change of Representation Time (m is any positive integer)

48. 48 CTFS Properties (cont…) Change of Representation Time (cont..)

49. 49 CTFS Properties (cont…) Time Differentiation

50. 50 Time Integration is not periodic CTFS Properties (cont…)

51. 51 CTFS Properties (cont…) Multiplication-Convolution Duality

52. 52 CTFS Properties (cont…) Conjugation Parseval’s Theorem The average power of a periodic signal is the sum of the average powers in its harmonic components.