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EE2006_1_FourierSeries EE2006 Engineering Mathematics I Assoc. Prof. Wang Han Room : S2 –B2b-49 Phone: 6790-4506 Email: hw@ ntu.edu.sg Slides Adopted from Prof. N. Sundararajan Topics : Fourier Analysis – Fourier Series ; Fourier Transform Textbook : 1. Kreyzig, E...

EE2006_1_FourierSeries
EE2006 Engineering Mathematics I Assoc. Prof. Wang Han Room : S2 –B2b-49 Phone: 6790-4506 Email: hw@ ntu.edu.sg Slides Adopted from Prof. N. Sundararajan Topics : Fourier Analysis – Fourier Series ; Fourier Transform Textbook : 1. Kreyzig, E., Advanced Engineering Mathematics, 8th edition, John Wiley. Reference: 2. Glyn James, Modern Engineering Mathematics, Fourth Edition, 2008, Prentice Hall. 3. Ravish R Singh, Mukul Bhatt, Engineering Mathematics, A Tutorial Approach, 2010, McGraw Hill. 1. Fourier Analysis 1.1 Periodic Functions 1.2 Fourier Series & Euler Formulae 1.3 Fourier Series for Functions with Period p = 2L 1.4 Even and Odd Functions 1.5 Fourier Series for 'Even' and 'Odd' Functions 1.6 Half-range Expansions 1.7 Exponential Form of Fourier Series Fourier Transform : 1.8 Fourier Transform & Properties 1.9 Convergence of Fourier Transform 1.10 Fourier Transform of Elementary Functions 1.11 Some Observations about Fourier Transform 1.12 Properties of Fourier Transform 1.13 Discrete Fourier Transform 1.1 Periodic Functions A function f(t) is said to be periodic if it is defined for all real t, and if there is some positive number T such that; f(t + T) = f(t), for all t T f ( t ) t Periodic function T 'PERIOD' of f ( t ) Periodic Functions Example: sin t, cos t, f = c, const. Non-periodic Functions Ex: f(t) = t, t2, t3, ln t. Note: i.e. for any integer n, T "fundamental period" 2T, 3T, 4T, .... nT are also periods of f(t) Note: If f(t) and g(t) have periods T, then h(t) = a f(t) + b g(t); (a, b, are constants) also has the period T. Periodic Functions with Period 2 (T = 2) Ex: 1, cos x, sin x, cos 2x, sin 2x ..... cos nx, sin nx, ..... Trigonometric Series a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + .... where (a0, a1, a2, ...., b0, b1, b2 .... are real constants) where an, bn are coefficients. Examples 2 cos x 0 1  1 2 cos 2 x 0 1  1 2 sin x 0 1  1 2 sin 2 x 0 1  1 2 cos x 0 1  1 1.2 Fourier Series & Euler Formulae Aim: To represent a given period function f(x) with period 2 by the trigonometric series. Problem : given f(x), find a0, an and bn for n = 1, 2, ... ? Fourier Series f(x) – periodic function with period 2. Fourier Coefficients a0, an, bn for n = 1, 2, ... Euler Formulae in the interval [-,+]: (p=2) Example 1: Given a periodic function f(x + 2) = f(x) Find its Fourier series representation and Fourier coefficients. 2 k k 0 x f ( x )Solution  Note: a0 is the average value of f(x) in [, ] an= 0 ; b2 = 0 ; ; b4 = 0 Fourier series of f(x) is f(x) = b1sinx + b2 sin2x + b3 sin3x + … 2        Period Frequency=1/period (2  /5)       (2  /3)       Try this at home: This property of functions is called “orthogonal”. Example 2: Find the Fourier series of in the interval of [-,]. Solution: Hence, f(x) = a0 + a1cosx + a2cos2x + a3cos3x + … + b1sinx + b2 sin2x + b3 sin3x + … 1.3 Fourier Series for Functions with Period p = 2L Periodic functions in applications have (rarely) the period 2, but some other period (L - length of a vibrating string - rod in heat conduction) i.e. p = 2L 2 Fourier Series for such function f(x), L < x < L and f(x + 2L) = f(x) is given by where Fourier Coefficients a0, an, bn are: Examples 1. Function with period p = 2L. Find the Fourier Series of the function  3  2  1 1 2 3 k f ( x ) x The period p = 2L = 4 L = 2 f(x ) ; n = 1, , n = 2, a2 = 0 n = 3, , n = 4, an = 0 an = 0, for n is even 4/5    4/3    4    Period Frequency 2. Find the Fourier series of Solution: The Fourier series of f(x) with period p=2l=2 is given by, Hence, 1.4 Even and Odd Functions · A function f(t) is even if - graph of such a function will be symmetric about y axis.  t f ( t ) Examples: f(t) = cos t since cos(t) = cos t · A function g(t) is odd if - graph of such a function is symmetric about x axis.  t g ( t ) Examples: g(t) = sin t as sin(t) = sin t · If f(t) is even and has period T, then · If g(t) is odd and has period T, then · The product of even function f(t) and an odd function g(t) is odd. Let g(t) = f(t)g(t) q(t) = f(t)g(t) = f(t) – g(t) = f(t)g(t) = q(t) q(t) is odd. · Note: odd . odd = even even . even = even odd . even = odd Examples: (1) x2 is even in (, ) sinx is odd x2 sinx is odd (2) sinx . sinx = sin2x = even · Note: Most functions are neither "even" or "odd". Example: ex, x2 – x ... 1.5 Fourier Series for 'Even' and 'Odd' Functions Case 1: If f(x) is an even function with period p=2L (no sine terms), then The product of two even functions is even, and Case 2: If f(x) is an odd function with period p=2L, then a0=0, an=0, …(1) Example 2: Find the Fourier series of f(x)=x|x| in the interval (-1,1). Solution: f(x) = x|x| f(-x)= -x|x| = -f(x) Hence f(x) is an odd function. So, a0=0, an=0 Theorem: * The Fourier coefficients of a sum f1 + f2 are the sums of the corresponding coefficients of f1 and f2. * The Fourier coefficients of Cf(t) are C x Fourier coefficients of f(t). Example: Find the Fourier Series of the function f(x) = x + , < x < and f(x + 2) = f(x) Solution: We can write f = f1 + f2 where f1 = x and f2 = . Now f2 = form f1 = x odd : 1.6 Half-range Expansions Question: What is a half-range expansion (Fourier Series) of the given function f(x)? Answer: A function is given in the interval f(x) , 0 x L L 0 f ( x ) To find a Fourier Series, the function has to be 'periodic'. Hence, the given function f(x), 0 x L has to be made periodic. Before doing it, we have to define the 'period'. Normally, the period is defined as p = 2L, i.e. the function is defined in L x L. But the function is given only in f(x), 0 x L Hence, in order to get the period of 2L, we have to extend the function in L x 0. This extension can be done in two ways (1) Odd extension in L x 0 Or (2) Even extension in L x 0. What is 'odd' extension? L 0 f ( x )  LDefine f(x) = f(x) in L x 0 L 0 f ( x )  L x'Even' Extension: f(x) = f(x), L x 0 Once the function f(x) is defined in L x L either by odd or even extension, these can be extended with a period 2L as shown below. Fig. (a) Function f(x) given on an interval 0 x L, (b) its even extension to the full "range" (interval) L x L (heavy curve) and the periodic extension of period 2L to the x-axis, (c) its odd extension to L x L (heavy curve) and the periodic extension of period 2L to the x-axis. For these periodic functions, Fourier Series can be found in the normal way. They are referred as "Half-range" Expansions and are called "odd" or "even" depending on the extension in L x 0. "Even" (cosine) Half-range Expansion where "Odd" (sine) Half-range Expansion where Example 1: The function f(x) is given below in 0 x L. Find the two possible half-range Fourier expansions for f(x) k L x f ( x ) 'Triangle' Answer: 'Even' Extension k x f ( x ) 'Odd' Extension L k x f ( x ) Solution: (a) Even periodic extension: Now i.e. , , an = 0 if n 2, 6, 10… half-range expansion (even) (b) Odd periodic extension: Example 2: Find half-range cosine series of f(x)=x(-x) in the interval (0,) and hence deduce that And Solution: The half-range cosine series of f(x) with period 2 is given by (L= 1.7 Complex Form of Fourier Series Euler formulae: ej = cos + jsin; e-j = cos - jsin So Given that Replace sin, cos with its complex form, we have where, Thus the alternative complex form of Fourier series is Case 1: when c=0, Case 2: when c=-L, Example 1: Find the complex form of Fourier series f(x) = 2x in the interval ) . Solution: L= , hence, Example 2: Find the complex form of Fourier series of Solution: 2L=2, hence L=1 For n=0, Hence, f ( t ) k t -L LExample 3: f(t + 2L) = f(t), hence period p=2L ; Solution: When n=0, c 0 c 1 c 2 c 3 c 4 c 5 nw 0 0.2 c n w 0 2 w 0 3 w 0 4 w 0 5 w 0 0 Suppose that a = 5 and k = 1 The Fourier series is Summary · Non sinusoidal, but periodic, waveforms are frequently encountered in the operation of electrical systems. · Most periodic waveforms can be represented as Fourier Series. · Any symmetry that exists in the waveform of a periodic function greatly simplifies the task of finding the Fourier coefficients. · Since the Fourier series translates the problem from time domain to the frequency domain, the frequency response of a system can be readily used to predict the waveform of the output. 1-36
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