Lecture 11: Introduction to Fourier Series Sections 2.2.3, 2.3

1. Lecture 11:Introduction to Fourier SeriesSections 2.2.3, 2.3

2. Trigonometric Fourier Series • Outline • Introduction • Visualization • Theoretical Concepts • Qualitative Analysis • Example • Class Exercise

3. Introduction • What is Fourier Series? • Representation of a periodic function with a weighted, infinite sum of sinusoids. • Why Fourier Series? • Any arbitrary periodic signal, can be approximated by using some of the computed weights • These weights are generally easier to manipulate and analyze than the original signal

4. Periodic Function • What is a periodic Function? • A function which remains unchanged when time-shifted by one period • f(t) = f(t + To) for all values of t • What is To To To

5. Properties of a periodic function 1 • A periodic function must be everlasting • From –∞ to ∞ • Why? • Periodic or Aperiodic?

6. Properties of a periodic function • You only need one period of the signal to generate the entire signal • Why? • A periodic signal cam be expressed as a sum of sinusoids of frequency F0 = 1/T0 and all its harmonics

7. Visualization Can you represent this simple function using sinusoids? Single sinusoid representation

8. amplitude New amplitude amplitude Fundamental frequency 2nd Harmonic 4th Harmonic Visualization To obtain the exact signal, an infinite number of sinusoids are required

9. Period Cosine terms Sine terms Theoretical Concepts

10. Theoretical Concepts

11. A -2 -1 0 1 2 A -A -1 0 1 2 -2 DC Offset Average Value = 0 What is the difference between these two functions? Average Value ?

12. DC Offset If the function has a DC value:

13. Qualitative Analysis • Is it possible to have an idea of what your solution should be before actually computing it? For Sure

14. -A -1 1 2 A A -1 0 1 2 -2 Properties – DC Value • If the function has no DC value, then a0 = ? DC? DC?

15. f(-t) = f(t) A 0 π π/2 3π/2 -A f(-t) = -f(t) A 0 π π/2 3π/2 A Properties – Symmetry • Even function • Odd function

16. Even Even Odd Even = = Odd Even Properties – Symmetry If f(t) is even: • Note that the integral over a period of an odd function is? X X

17. Odd Odd Even Odd = = Odd Even Properties – Symmetry If f(t) is odd: • Note that the integral over a period of an odd function is zero. X X

18. Properties – Symmetry • If the function has: • even symmetry: only the cosine and associated coefficientsexist • odd symmetry: only the sine and associated coefficientsexist • even and odd: both terms exist

19. -A -1 1 2 A Properties – Symmetry • If the function is half-wave symmetric, then only odd harmonics exist Half wave symmetry: f(t-T0/2) = -f(t)

20. -A -1 1 2 A A -1 0 1 2 -2 Properties – Discontinuities • If the function has • Discontinuities: the coefficients will be proportional to 1/n • No discontinuities: the coefficients will be proportional to 1/n2 • Rationale: Which function has discontinuities? Which is closer to a sinusoid?

21. -A -1 1 2 A Example • Without any calculations, predict the general form of the Fourier series of: DC? Half wave symmetry? Yes, only odd harmonics No, a0 = 0; Discontinuities? Symmetry? No, falls of as 1/n2 Even, bn = 0; Prediction an 1/n2 for n = 1, 3, 5, …;

22. zero for n even Example • Now perform the calculation

23. Example • Now compare your calculated answer with your predicted form DC? Half wave symmetry? Yes, only odd harmonics No, a0 = 0; Discontinuities? Symmetry? No, falls of as 1/n2 Even, bn = 0;

24. A -1 0 1 2 -2 Class exercise • Discuss the general form of the solution of the function below and write it down • Compute the Fourier series representation of the function • With your partners, compare your calculations with your predictions and comment on your solution

25. Spectral Lines

26. Spectral Lines • Gives the frequency composition of the function • Amplitude, phase of sinusoidal components • Could provide information not found in time signal • E.g. Pitch, noise components • May help distinguish between signals • E.g speech/speaker recognition

27. Spectral Lines Example QUESTIONS -- DC Yes____ ao = ? No_____ ao = 0 Symmetry Even____ an = ? bn = 0 Odd____ an = 0 bn = ? Nether even nor odd ____ an = ? bn = ? Halfwave symmetry Yes_____ only odd harmonics No______ all harmonics Discontinuities Yes_____ proportional to1/n No______ proportional to1/n2 Note ? means find that variable. Comment on the general form of the Fourier Series coefficients [an and/or bn.] X X X X

28. Spectral Lines Example

29. Spectral Lines Example

30. Spectral Lines Example