Introduction to Fourier Series PDF

www.covenantuniversity.edu.ng Raising a new Generation of Leaders GEC 310 ENGINEERING MATHEMATICS III FOURIER SERIES Lecturer: Mrs Oni 2  Outline 1 Introduction: History and applications o...

www.covenantuniversity.edu.ng Raising a new Generation of Leaders GEC 310 ENGINEERING MATHEMATICS III FOURIER SERIES Lecturer: Mrs Oni 2  Outline 1 Introduction: History and applications of Fourier series Sinusoidal and Non-sinusoidal periodic functions Analytic description of periodic functions Integrals of periodic functions Fourier Series 3  Introduction Fourier Series was invented by a French mathematician Jean Baptiste Joseph Fourier. He was also a physicist and an historian. Applications of Fourier Series: Signal processing, Image processing, Heat distribution mapping, Wave simplification, Radiation measurements etc. 4  Introduction Signal and Image processing can be used in: Security systems Medicine Communications Pattern Recognition Forensics Video Analytics Object tracking Machine/Robot Vision Military Industry 5  Introduction Most Periodic Functions can be expressed in infinite series. Fourier series enables us to represent a Periodic Function as an infinite Trigonometric Series in sine and cosine terms. Examples of Periodic Functions are: heart beat signals, wave signals, sound signals, speech signals, electronic signals e.t.c. 6 Introduction 7  Periodic Functions A function f(x) is said to be periodic if its function values repeat at regular intervals of the independent variable. The regular interval between repetitions is the period of oscillations. A periodic signal is a signal that repeats its pattern at a certain period. 8  Sinusoidal Periodic Functions These are Periodic functions in sine wave forms. For example, The Graph of y=A sin nx. The graph below shows that y=sin x x increases from 0∘ to 360∘ or 2π radians. The period is the distance between the beginning of the first waveform and the beginning of next waveform. 9  Sinusoidal Periodic Functions Generally, when y=A sin nx The amplitude is A. The period is 360∘/n or 2π/n radians. 10  Sinusoidal Periodic Functions Given the function y = 3sin 4x, determine the period and the amplitude. The period of the function is 2π/n Therefore, the period is 2π/4 = π/2 The amplitude of the function is | a |. Therefore, the amplitude is 3. 11  Sinusoidal Periodic Functions y=5sin 2x The amplitude is 5. The period is 180∘ or π radians 12  Graphing Periodic Function Graph y = sin x Sketch the Graph y = cos x 13 Writing the Equation of Periodic Function Amplitude Period =2π/n π= π =2π/n n=2 The equation as a function of sine is y = 2sin 2x 14 Writing the Equation of Periodic Function 15 Sinusoidal Periodic Functions 16  Non-sinusoidal Periodic Functions These are functions that do not appear sinusoidal but are actually periodic in nature. Examples are shown below: 17 Non-sinusoidal Periodic Functions 18 Analytic description of Periodic Functions 19 Analytic description of Periodic Functions 20  Note the following: cos0 = 1 cosπ = cos1800 = -1 cos2π = cos3600 = 1 cos3π = cos5400 = -1 cos4π = cos7200 = 1 cos5π = cos9000 =- 1 cos6π = cos10800 = 1 cos7π = cos12600 = -1 21 Integrals of Periodic Functions 22 Integrals of Periodic Functions 23 Integrals of Periodic Functions 24 Integrals of Periodic Functions 25 Integrals of Periodic Functions 26  Integrals of Periodic Functions When m = n 27  Fourier Series Fourier Series is an infinite series representation of periodic function in terms of the trigonometric sine and cosine functions. Fourier series is to be expressed in terms of periodic functions- sines and cosines. Fourier series is a very powerful method to solve ordinary and partial differential equations. 28  Dirichlet Conditions of Fourier Series x(t) should be periodic x(t) must be integrable over any given interval x(t) must have finite number of minimum and maximum in any given interval X(t) must have a finite number of discontinuities in any given interval 29  Fourier Series A function f(x) is said to be periodic function with period T >0 if for all x , f(x) = f(x+ T ) and T is the least of such values. If f(x) = f(x+ T ) = f(x+2 T ) =……. Then T is called the period of the function f(x) Examples sin x = sin(x+ 2π) = sin(x+ 4π )=…. sin x is a periodic function of f(x) sin x, cos x are periodic functions with period 2π. 30  Fourier Series According to Fourier Theorem, any practical periodic function can be represented as an infinite sum of sine or cosine functions that are integral multiples of the fundamental frequency. Therefore f(x) can be expressed as 31  Determination of Fourier Coefficient Fourier Series expansion of f(x) 32  Harmonic Series A function f(x) is sometimes expressed as a series of number of different sine components. The component with the largest period is the first harmonic or fundamental of f(x) Harmonic are integer multiples of the fundamental frequency. i.e. w, 2w, 3w, 4w … 33 Harmonic Series In General 34  Fourier Series Where Fourier Coefficients 35 Useful Integrals 36  Examples 1. Find the Fourier representation of To find 37  Examples To find 38  Examples To find 39  Examples 2. Find the Fourier series representation of To find 40  Examples To find 41  Examples To find 42  Examples Fourier Series expression f(x) 43  FUNCTION DEFINED IN TWO OR MORE SUB RANGES 3. Find the Fourier series of the function 44  FUNCTION DEFINED IN TWO OR MORE SUB RANGES Find 45 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 46 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 47 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 48 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 4. Obtain the Fourier expansion for function f(x) of period 2π 49 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 50 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 51 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 52 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 53 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 54 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 55 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 56  FUNCTION DEFINED IN TWO OR MORE SUB RANGES 5. Find the Fourier series of the function Find the Fourier series of the function 57  FUNCTION DEFINED IN TWO OR MORE SUB RANGES To Calculate 58 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 59 FUNCTION DEFINED IN TWO OR MORE SUB RANGES 60  FUNCTION DEFINED IN TWO OR MORE SUB RANGES Fourier Series expansion 61  Class Exercises Find the Fourier series of the functions defined as 1) 2) 3) 4) 62  EVEN AND ODD FUNCTIONS A function f(x) is said to be even function if f(-x) = f(x) The graph of such a function is symmetrical with respect to y axis ( f(x)axis ). e.g. x2 ,cos x, x4 + cos2x+2 The area under such curve from –π to π is double the area from 0 to π 63 Determine if the given function is Even, Odd or Neither. f(x) = 1- x2 f(-x) = 1- (-x)2 = 1-x2 The function is Even 64  Note The graph of f(x) is symmetrical about y axis f(x) contains only even powers of x and may contain only cos x The sum of two even functions is even The product of two even function is even 65  ODD FUNCTION A function f(x) is called odd function if f(-x) = -f(x) The graph of an odd function is symmetrical about origin. The area under the curve from –π to π is zero 66 Determine if the given function is Even, Odd or Neither. f(x) = x5 + x f(-x) = (-x)5 + (-x) = -x5 – x = -(x5 + x) -f(x) = -(x5 + x) = - x5 – x =-(x5 + x) The function is Odd 67  Note The graph of f(x) is symmetric about the origin (lies in the opposite quadrants 1st and 3rd ) f(x) contains only odd powers of x and may contain only sinx The sum of two odd function is odd The product of two odd function is even The product of even and odd function is odd 68 The waveform is neither even nor odd because is not symmetrical either about y axis or x axis. 69 Determine the given function is Even, Odd or Neither. f(x) = 2x -x2 if f(-x) = 2(-x) –(-x)2 =-2x -x2 =-(2x +x2) -f(x) = -1(2x -x2 ) =-2x+ x2 The function is Neither even nor odd 70  PRODUCT & SUM OF EVEN AND ODD FUNCTIONS Products of Functions Even × Even = Even Odd × Odd = Even Odd × Even = Odd Sums of Functions Even + Even = Even Odd + Odd = Odd Odd + Even = Neither 71  Class Exercises State whether each of the following functions is odd, even or neither 1) f(x) =1 - x4 2) f(x) =x5 + x 3) f(x) =x4/3 – 4 4) X3cosx 5) (2x+3)sin4x 6) X3ex 72  HALF RANGE : Even and ODD functions The Fourier Expansions for Even and Odd functions gives rise to cosine and sine half range Fourier Expansions. If we are only given values of a function f(x) over half of the range [0; π], we can define two different extensions of f to the full range [-π; π], which yield distinct Fourier Expansions. The even extension gives rise to a half range cosine series, while the odd extension gives rise to a half range sine series. 73  Expansion of Even Function As f(x) and cos nx are both even functions, the product of f(x). cos nx is also an even function As sin nx is an odd function so f(x).sin nx is also an odd function. We need not to calculate bn. The series of even function may contain a constant and cosine term only. 74  Expansion of Odd Function If f(x). sin nx is even function The series of the odd function will only contain sine term 75  Procedure 1. Identify whether the given function f(x) is even or odd function in the given interval. 2. If f(x) is even, calculate only ao and an (no need to calculate bn) 3. If f(x) is odd, calculate only bn. 76  Examples 6. Find the Fourier series expansion of the periodic functions of period 2π 77  Examples Contd. The function is an even function thus bn =0 Fourier series expansion 78  Examples Contd. 7. Obtain the Fourier expression for 79  Examples Contd. The function f(x) = x3 is an odd function Therefore 80 8. Describe the waveform below and write out the Fourier series expression 81  The waveform is symmetrical about y-axis, therefore the function is even and there will be no sine terms in the series, bn = 0. 82 83  Examples Contd. 9. Determine the Fourier series for the periodic function defined by 84  Examples Contd. The square wave is a an even function. Calculate 85  Examples Cont. Calculate 86 Examples Contd. 87 10. Find the sine series for the function f(x) = eax for 0 < x < π where a is a constant 88 when n =1 n =2 Sine Fourier series expansion 89  ASSIGNMENT 10. Obtain the Fourier series for the square wave 90  Fourier series of function with arbitrary period p=2L Instead of a period of 2, many functions have an arbitrary period, say a period of 2L. In order to convert the Fourier series defined earlier for these functions, a change of variable is needed. Replace the variable x by (/L)x. when x=L the new variable equals to ; when x= -L, it equals to - . Therefore, the previous formulas can be used by simply making the change Let f(x) be defined in the interval (-L,L) and outside the interval by f(x +2L) =f(x) i.e. f(x) is 2L periodic. The Fourier series expansion for f(x) is given by 91 92 11. Determine the half range Fourier sine series for the function defined by : 93  Sine Series 94 95 When n is even bn =0 96 Fourier Series Expansion 97  Fourier Series of functions with period T For functions with Period T, each cycle is completed in T seconds and the frequency f Hertz (i.e. oscillations per second) of the periodic function is given by: The Angular Velocity is The angle, x radaians at any time is therefore x=wt 98  Fourier Series of functions with period T The Fourier series that is used to represent functions with Period T is expressed as: 99  Fourier Series of functions with period T The Fourier constant and coefficients are: 100  Fourier Series of functions with period T Determine the Fourier series for a periodic function defined by 101 Fourier Series of functions with period T 102 Fourier Series of functions with period T 103 Fourier Series of functions with period T 104  Fourier Series of functions with period T Remember that cos0 = 1 cos1π = cos1800 = -1 cos2π = cos3600 = 1 cos3π = cos5400 = -1 cos4π = cos7200 = 1 105  Fourier Series of functions with period T Simplify the expression 106 Fourier Series of functions with period T 107