3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series

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1 Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2 <... < x n = b such that f is continuous. Piecewise continuous function Definition 2 A function f is said to be periodic, with a period p if f(x + p) = f(x) for all x. Examples: 1 Periodic function f (x) is periodic if it is defined for all real (x) and if there is some positive numbers (T) such as f (x, T) = f (x), for all (x) If T > 0, then the least value of T is called the period of f (x). 1

2 Example 1: The period of sin x and cos x is (2 ), where; sin (x +2 ) = sin (x +4 ) = sin (x +6 ) =... = sin (x) cos (x +2 ) = cos (x +4 ) = cos (x +6 ) =... = cos (x) but the least period of both functions is 2. Generally, the period of sin (nx) or cos (nx) is. The period of the function tan (x) is. Where; tan (x + ) = tan (x + 2 ) = tan (x + n ) =... = tan (x) Now, is the smallest period. 2 Fourier series Let f be a piecewise continuous function defined on [, ]. Then one can express f as a linear combination of sine and cosine functions as follows : ( ) ( ) ( ) 2

3 where a o, a n and b n are constants to be determined. In order to find a o ; ( ) To find a n ; ( ) ( ) While b n ; ( ) ( ) For n = 1, 2, 3, Example 2: let us consider the function Solution: = 0 3

4 = 0. ( [ ( ) ( )] [ ( ) ( )] ) n B

5 Thus the Fourier series of f is given as As a result, the general form of the Fourier series is or; ( ) (1) Where a o, a n and b n are constants, and they are called the Fourier coefficients of the series. Now, if f (x) is a periodic function having a period of (2 ), it can be represented by Eq. (1). Thus, Eq. (1) is called Fourier Series. Example 3: Let us consider the function f defined as follows 5

6 Exercise 1: Let us consider the function 1. ( ) { 2. ( ) { 6

7 2.1 Periodicity To be certain we all know what we re talking about, a function f (t) is periodic of period T if there is a number T > 0 such that Consider the following function which has the graph as shown down Then, 7

8 2.2 Fourier Sine series: An odd function is a function with the property f ( x) = f (x). For instance: 1. f (x) = x 3. let x = -1, then (-1) 3 = - (1) 3 2. f (x) = sin (x). let x = - /2, then sin (- /2) = - sin ( /2). Note: The integral of an odd function over a symmetric interval is zero. Let us calculate the Fourier coefficients of an odd function: Since a n = 0, all the cosine functions will not appear in the Fourier series of an odd function. The Fourier series of an odd function is an infinite series of odd functions (sine): 8

9 However, we are only interested in what happens [0, ]. In this interval f is identical to its odd extension: Exercise 2: f (x) = sin (x), - < x < 2.3 Fourier Cosine Series: An even function is a function with the property f ( x) = f (x). The sine coefficients of a Fourier series will be zero for an even function, f (x) = x 2, let x = -1, then (-1) 2 = (1) 2 f (x) = cos (x), let x = -, then cos (- ) = cos ( ). The Fourier series of an even function is an infinite series of even functions (cosines): The coefficients of the cosines may be evaluated using information about f (x) only on [0, ], since 9

10 If the function f is only given for 0 x and not necessarily even, then f can be extended as an even function, which called the even extension of f. Moreover, the Fourier series of the even extension of f only involves cosines: In the interval [0, ], the function f is identical to its even extension : Example 4: Let us consider the function f (x) = 1 on [0, ]. The Fourier cosine series has coefficients 10

11 Example 5: find the Fourier complex transform ( ) { Solution: ( ) ( ) ( ) [ ( ) ( ) ] Since, [cos (x) cos (nx)] = [cos (n 1)x + cos (n + 1)x] [ ( ) ( ) ] [ ( ) ( ) ( ) ( ) ] [ ( ) ( ) ( ) ( ) ] Example 6: Obtain the Fourier series of periodicity (10) for the following function ( ) { Solution: the period of the function f(x) = 10 = 2L. Thus, L = 5. Thus, the integration is from c = 5, to c + 2L. ( ) ( ) [ ( ) ( ) ] 11

12 [ ] ( ) [ ( ) ] [ ] ( ) [ ( ) ] [ ] ( ) ( ) (( ) ( ) ) Exercise 3: 1. Find the Fourier complex transform ( ) { 2. Find the Fourier complex transform ( ) { Brief solution: ( ) ( ) ( ) 12

13 [ ( ) ] 3 Complex form of Fourier series (FT) = ( ( ) ( ) ) Where, c k = c * k from the last equation. 4 Euler formula If f (x) is a periodic function having a period of (2 ), and f (x) having the interval of (c, c+2 ) where c is real constant. Then the Fourier coefficients become: ( ) (2) 13

14 ( ) ( ) (3) (4) For n = 1, 2, 3,. the last three equations are called Euler Fourier formula. 5 Fourier Transform It transforms from a complex form to Fourier series and Fourier integral. We are about to make the transition from Fourier series to the Fourier transform. Transition means we pass from periodic to non-periodic functions. A nonperiodic function is a limiting case of a periodic function as the period becomes longer and longer. Actually, this process doesn t immediately produce the desired result. Recall that for a general function f (t) of period T the Fourier series has the form The n th Fourier coefficient is given by 14

15 Example 7: First, we need to find the fundamental period, T. Since f (t) repeats itself 2 times from t = 0 to t = 1, the period is T = 0.5. We'll find the complex form of the Fourier Series, which is more useful in general than the entirely real Fourier Series representation. the cosine function can be rewritten (via Euler's identity) as: 15

16 Let's look at the first integral in last equation, when n does not equal 1, note that the integral becomes zero: A simple way to evaluate the integral for n = 1 is to plug in n = 1 before the integration is done: Using the same type of analysis, you can quickly figure out that Hence, all the coefficients are zero except for the n = 1 and n = 1 terms. Finally, let's evaluate the infinite complex Fourier Sum with the calculated coefficients and see that it gives f (t): 16

17 And we see that the Fourier Representation g(t) yields exactly what we were trying to reproduce, f (t). This might seem stupid, but it will work for all reasonable periodic functions, which makes Fourier Series a very useful tool. Example 8: The exponential function and its Fourier transform. 17

18 Solution: ( ) ( ) ( ) Fourier transform of the rectangular function Where sinc (x) = sin (x) / x Rectangular wave (t) and its Fourier transform 18