1. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Trigonometric Fourier Series  Outline –Introduction –Visualization –Theoretical Concepts –Qualitative Analysis –Example –Class Exercise

2. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesIntroduction  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

3. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Periodic Function  What is a periodic Function? –A function which remains unchanged when time-shifted by one period f(t) = f(t + T o ) for all values of t  What is T o ToToToTo ToToToTo

4. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties of a periodic function 1  A periodic function must be everlasting –From –∞ to ∞  Why?  Periodic or Aperiodic?

5. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 F 0 = 1/T 0 and all its harmonics

6. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesVisualization Can you represent this simple function using sinusoids? Single sinusoid representation

7. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesVisualization To obtain the exact signal, an infinite number of sinusoids are requiredamplitude Fundamental frequency New amplitude 2 nd Harmonic amplitude 4 th Harmonic

8. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Theoretical Concepts (6) (6) Period Cosine terms Sine terms

9. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Theoretical Concepts (6) (6)

10. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series DC Offset What is the difference between these two functions?A012 -2 -A A012 -2 Average Value = 0 Average Value ?

11. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series DC Offset If the function has a DC value:

12. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Qualitative Analysis  Is it possible to have an idea of what your solution should be before actually computing it? For Sure

13. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – DC Value  If the function has no DC value, then a 0 = ?12 -A A DC?A012 -2 DC?

14. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Symmetry A A 0 π/2 π 3π/2 f(-t) = -f(t)  Even function  Odd function 0 -A A π/2 π 3π/2 f(-t) = f(t)

15. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Symmetry  Note that the integral over a period of an odd function is? If f(t) is even: Even OddX=Odd Even EvenX=Even

16. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Symmetry  Note that the integral over a period of an odd function is zero. If f(t) is odd: Odd EvenX=Odd Odd OddX=Even

17. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Symmetry  If the function has: –even symmetry: only the cosine and associated coefficients exist –odd symmetry: only the sine and associated coefficients exist –even and odd: both terms exist

18. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Symmetry  If the function is half-wave symmetric, then only odd harmonics exist Half wave symmetry: f(t-T 0 /2) = -f(t) 12 -A A

19. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties – Discontinuities  If the function has –Discontinuities: the coefficients will be proportional to 1/n –No discontinuities: the coefficients will be proportional to 1/n 2  Rationale:12 -A A A012 -2 Which is closer to a sinusoid? Which function has discontinuities?

20. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesExample  Without any calculations, predict the general form of the Fourier series of:12 -A A DC? No, a 0 = 0; Symmetry? Even, b n = 0; Half wave symmetry? Yes, only odd harmonics Discontinuities? No, falls of as 1/n 2 Prediction a n  1/n 2 for n = 1, 3, 5, …;

21. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesExample  Now perform the calculation zero for n even

22. ES 240: Scientific and Engineering Computation. Introduction to Fourier SeriesExample  Now compare your calculated answer with your predicted form DC? No, a 0 = 0; Symmetry? Even, b n = 0; Half wave symmetry? Yes, only odd harmonics Discontinuities? No, falls of as 1/n 2

23. ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 solutionA012 -2