Fourier Series : Dr. Mohammed Saheb Khesbak Page 34
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1 Fourier Series : Dr. Mohammed Saheb Khesbak Page 34
2 Dr. Mohammed Saheb Khesbak Page 35
3 Example 1: Dr. Mohammed Saheb Khesbak Page 36
4 Dr. Mohammed Saheb Khesbak Page 37
5 Dr. Mohammed Saheb Khesbak Page 38
6 Derivation of the Euler Formulas: Theorem 1: Proof: Dr. Mohammed Saheb Khesbak Page 39
7 Example 2: Dr. Mohammed Saheb Khesbak Page 40
8 Example 3: Dr. Mohammed Saheb Khesbak Page 41
9 Example 3: Dr. Mohammed Saheb Khesbak Page 42
10 Exercises of Fourier Series Showing the details of your work, find the Fourier series of the given f(x), which is assumed to have period 2π. Dr. Mohammed Saheb Khesbak Page 43
11 Even and Odd Functions : Half-Range Expansions : Theorem 1 Dr. Mohammed Saheb Khesbak Page 44
12 Dr. Mohammed Saheb Khesbak Page 45
13 Example 1 Example 2 Dr. Mohammed Saheb Khesbak Page 46
14 Dr. Mohammed Saheb Khesbak Page 47
15 Half-Range Expansions: Half range expansions are Fourier series. We want to represent f(x) in Fig 267a by a Fourier series. Dr. Mohammed Saheb Khesbak Page 48
16 Dr. Mohammed Saheb Khesbak Page 49
17 Dr. Mohammed Saheb Khesbak Page 50
18 Dr. Mohammed Saheb Khesbak Page 51
19 Fourier Series Summary and Notes: Dr. Mohammed Saheb Khesbak Page 52
20 Review of integration by parts: Dr. Mohammed Saheb Khesbak Page 53
21 EXAMPLE 2 Find Dr. Mohammed Saheb Khesbak Page 54
22 EXAMPLE 3: Thus; EXAMPLE 4: Dr. Mohammed Saheb Khesbak Page 55
23 Then; Dr. Mohammed Saheb Khesbak Page 56
24 Exercises and Examples Find the Fourier series expansion. Dr. Mohammed Saheb Khesbak Page 57
25 eq (a) Now; we use the Integration by parts method eq (b) Using Integration By Parts again Dr. Mohammed Saheb Khesbak Page 58
26 Substituting in equation (b0 and (a) to get ; Dr. Mohammed Saheb Khesbak Page 59
27 eq (c) Substituting in to equation (c), get; Dr. Mohammed Saheb Khesbak Page 60
28 Home Work1: Show that the Fourier Series expansion of x 3 is: Home Work2: ANSWER:,, Example 3 : Find the F.S. expansion for ; Solution: X is an even function, so Dr. Mohammed Saheb Khesbak Page 61
29 Since the function is even and the sine is odd, then b n =0. It will be shown in details below; Dr. Mohammed Saheb Khesbak Page 62
30 Home Work3: Find the Fourier Series Expansion of the function: Answer: Example 4: Find the F.S. expansion for ; Dr. Mohammed Saheb Khesbak Page 63
31 Home Work4: Find the Fourier Series Expansion of the function: Answer Dr. Mohammed Saheb Khesbak Page 64
32 Exercises: 1- Find the Fourier series expansion for the periodic function f(t) in Fig Find the half range Fourier series expansion for the non-periodic function m(t) in Fig.2. Fig ρ 2- f(t) 4 t Fig.2 4 m(t) 2 t 3- Find the Fourier series expansion for the periodic function f(t) in Fig Find the half range Fourier series expansion for the non-periodic function m(t) in Fig.4. 1 f(t) 4 m(t) - 4 Fig.3 ρ 2 4 t Fig.4 2 t 5- Find the Fourier series expansion for the periodic function f(t) in Fig Find the half range Fourier series expansion for the non-periodic function m(t) in Fig.6. Fig ρ 1- f(t) 2 t -4 2 m(t) t Fig.6 Dr. Mohammed Saheb Khesbak Page 65
33 Fourier Transform Let f(t) be the time domain non-periodic function. Then the frequency domain Fourier transform FT (exponential form) corresponding function is ( ) ( ) While the Inverse Fourier Transform (IFT) is defined as; Example: ( ) ( ) Find the FT of f(t)=1 Solution: ( ) [ ] ( ) ( ) Example: Find the FT of ( ) { where a is a constant Solution: ( ) ( ) Dr. Mohammed Saheb Khesbak Page 66
34 ( ) ( ) [ ( ) ] ( ) Exercises: 9- ( ) 10- ( ) Dr. Mohammed Saheb Khesbak Page 67
35 Some Useful Identities Dr. Mohammed Saheb Khesbak Page 68
36 Some Useful Integrations Dr. Mohammed Saheb Khesbak Page 69
37 Dr. Mohammed Saheb Khesbak Page 70
3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
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