APPLIED MATHEMATICS Part 4: Fourier Analysis

Transcription

1 APPLIED MATHEMATICS Part 4: Fourier Analysis

2 Contents 1 Fourier Series, Integrals and Transforms Periodic Functions. Trigonometric Series Fourier Series Functions of Any Period p =2L Even and O dd Functions Complex Fourier Series Half-Range Expansion Forced Oscillations. Optional Approximation by Trigonometric Polynomials Fourier Integrals Fourier Cosine and Sine Transforms Fourier Transform Table of Fourier Transforms

3 Chapter 1 Fourier Series, Integrals and Transforms 2

4 1.1 Periodic Functions. Trigonometric Series f(x) isperiodiciff(x) isdefinedforx R and if p> such that f(x + p) =f(x) x, p is called period of f(x). f(x + np) =f(x) for all integers n (np) is also period of f(x). If f(x) andg(x) haveperiodp h(x) =af(x)+bg(x) also has period p, wherea, b are constants. Trigonometric series with period p =2π: f(x) = a + a 1 cos x + b 1 sin x + a 2 cos 2x + b 2 sin 2x +... = a + (a n cos nx + b n sin nx) where a n and b n are real constants, called coefficients of the series. If f(x) converges the sum of the series will be a function with period 2π. 3

5 1.2 Fourier Series Trigonometric series to represent periodic function f(x), where the coefficients are determined from f(x) a : Euler formulas for Fourier coefficients a n and b n f(x) =a + (a n cos nx + b n sin nx) f(x)dx = π a + (a n cos nx + b n sin nx) dx = a dx + ( an cos nxdx + b n a = 1 2π = 2πa + f(x)dx sin nxdx) 4

6 a n (n ) :(n =1, 2, 3...) = = a = f(x)cosmxdx (m >) a + cos mπdx }{{} [ an( 1 2 (1 +b n 2 (a n cos nx + b n sin nx) cos mxdx + [ an cos nx cos mxdx + b n sin nx cos mxdx] }{{} cos(n + m)xdx sin(n + m)xdx }{{} cos(n m)xdx) sin(n m)xdx }{{} )] 1 π 2 cos(n m)xdx = f(x)cosmxdx = a mπ π if n = m if n m a m = 1 π f(x)cosmxdx 5

7 b n (n ) :(n =1, 2, 3...) = f(x)sinmxdx a + = a = = b m π sin mxdx }{{} [ an( 1 2 (a n cos nx + b n sin nx) sin mxdx + ( an cos nx sin mxdx + b n sin nx sin mxdx) }{{} sin(n + m)xdx 1 2 (1 +b n 2 cos(n m)xdx 1 2 sin(n m)xdx }{{} cos(n + m)xdx }{{} ) )] = b m = 1 π f(x)sinmxdx 6

8 Ex : Square wave f(x) = k k <x< <x<π and f(x +2π) =f(x) a = 1 2π f(x)dx = a n = 1 π = 1 π = 1 π f(x)cosnxdx [ ( k)cosnxdx + k cos nxdx ] k sin nx n + k sin nx n π = b n = 1 π = 1 π f(x)sinnxdx [ = 1 k π = k nπ = 2k nπ 4k = nπ ( k)sinnxdx + k sin nxdx ] cos nx n k cos nx n π [1 cos( nπ) cos nπ +1] (1 cos nπ) odd n (n =1, 3, 5...) even n (n =2, 4, 6...) f(x) = 4k π (sin x sin 3x sin 5x +...) 7

9 Orthogonality of Trigonometric System: Functions of f(x) andg(x), f(x) g(x) are orthogonal on some interval a x b, if b a f(x) g(x)dx = For examples, 1, cos x, sin x, cos 2x, sin 2x, cos 3x, sin 3x,...are orthogonal on the interval x π, i.e., cos mx cos nxdx = m sin mx sin nxdx = m cos mx sin nxdx = n n Note that we have used these properties in deriving Euler s formulas. Convergence and Sum of Fourier Series: Iff(x) =a + (a n cos nx + b n cos nx) = means the series converges and the sum represents f(x). 8

10 Theorem 1 : (1) periodic function f(x), f(x) =f(x =2π) and (2) f(x) is piecewise continuous onx [, π],and (3) f(x) has left-hand and right-hand derivatives at every x [, π] Fourier series of f(x) converges, and the sum of the series is f(x), except at x where f(x) is discontinuous, at such point the sum is the average of the left-and right-hand limits of f(x) atx. Note : If f(x) is discontinuous at x = x left-hand limit right-hand limit f(x ) = lim f(x h) h f(x + ) = lim f(x + h) h left-hand derivative right-hand derivative df dx x x + df dx = lim h f(x h) f(x ) h = lim h f(x + h) f(x +) h 9

11 Ex : f(x) = x 2 x<1 x/2 x>1 f(1 ) = 1 f(1 + ) = 1 2 df =2 dx 1 df = 1 dx

12 Ex : Square wave f(x) = 4k π (sin x sin 3x +...) At x = π 2 f(x) = 4k π ( ) } {{ } π/4 = k At x = 3π 2 f(x) = 4k π ( ) } {{ } /4 = k At x = f(x) = 4k π (+++...)= 11

13 Proof : We consider only the convergence of the theorem for a continuous function f(x) having continuous 1st and 2nd derivatives. f(x) =a + (a n cos nx + b n sin nx) a = 1 2π f(x)dx a n = 1 π b n = 1 π f(x)cosnxdx (n =1, 2,...) f(x)sinnxdx (n =1, 2,...) By using integration by part udv = uv vdu a n = 1 π f(x)cosnxdx = 1 nπ f(x)sinnx π 1 }{{} nπ = 1 n 2 π f (x)cosnx π }{{} 1 n 2 π f (x)sinnxdx f (x)cosnxdx 12

14 Since f(x) is continuous for x [, π] f (x) <M a n = 1 n 2 π 1 n 2 π Similarly < 1 n 2 π = 2M n 2 f (x)cosnxdx f (x) }{{} <M Mdx cos nx dx }{{} 1 b n < 2M n 2 f(x) f(x) = a + (a n cos nx + b n sin nx) a + ( a n + b n ) = a +2 = finite value 2M n 2 the series converges 13

15 1.3 Functions of Any Period p =2L So far we only consider periodic functions with period p =2π. For periodic function f(x) with period p =2L, the Fourier series for f(x) is: f(x) =a + where the coefficients are: a = 1 L 2L f(x)dx L ( a n cos nπ L x + b n sin nπ L x ) a n = 1 L b n = 1 L L L f(x)cosnπx L dx L L f(x)sinnπx L dx 14

16 Proof : Rescaling the variable by v πx L x = Lv π x = ±L v = π(±l) L f(x) =g(v) =g = ±π ( ) πx L = a + (a n cos nv + b n sin nv) a = 1 2π = 1 2L g(v)dv = 1 2π l L f(x)dx ( ) L g πx ( π L L L dx) a n = 1 π = 1 L g(v)cosnvdv = 1 π L L f(x)cosnπx L dx ( ) L g πx cos n πx ( ) π L L L L dx b n = 1 π g(v)... Note : Shift of the period +l +l L L L+l L+l eg : 2π eg : 2L 15

17 Ex : Half-Wave rectifier clips negative portion of the waves L = π ω P = 2π ω u(t) = L <t< E sin ωt <t<l a = 1 2L u(t)dt = ω 2L 2π /ω E sin ωtdt = E π a n = 1 L 2L u(t)cos nπt L dt = ω π /ω E sin ωt cos nωtdt = n =1, 3, 5,... 2E n =2, 4, 6,... (n 1)(n +1)π b n = 1 L = 2L u(t)sin nπt L dt = wπ /w E sin wt sin nwtdt E 2 n =1 otherwise 16

18 1.4 Even and Odd Functions even function: g( x) =g(x) L g(x)dx L L =2 g(x)dx odd function: h( x) = h(x) L h(x)dx = L Products of odd and even functions: g 1 (x) g 2 (x) =g 1 ( x) g 2 ( x) g(x) h(x) =g( x) ( h( x)) = g( x) h( x) h 1 (x) h 2 (x) =( h 1 ( x)) ( h 2 (x)) = h 1 ( x) h 2 ( x) Fourier coefficients of odd and even functions: f(x) =a = (a n cos nx + b n sin nx) a = 1 2π 2π f(x)dx = foroddf(x) a n = 1 π 2π f(x)cosnxdx = foroddf(x) b n = 1 π 2π f(x)sinnxdx = forevenf(x) 17

19 Theorem : Fourier Cosine Series (for even periodic function f(x) withp =2L) f(x) =a + a n cos nπ L x a = 1 L a n = 2 L L f(x)dx L f(x)cosnπx L dx Fourier Sine Series (for odd periodic function f(x) with period p =2L) f(x) = b n sin nπ L x b n = l L L f(x)sinnπx L dx 18

20 Theorem : The Fourier coefficients of (f 1 + f 2 ) are the sums of the corresponding Fourier coefficients of f 1 and f 2. The Fourier coefficients of cf, wherec is a real constant, are c times the corresponding Fourier coefficients of f. Ex : Saw tooth wave f(x) = }{{} x + }{{} π for π<x<π, f(x +2π) =f(x) f 1 f 2 f 1 = x = b n sin nπ L x (odd function) b n = 2 π f 1(x)sinnxdx = 2 π x sin nxdx = 2 cos nπ n f 2 = π = a + a = π, a n = a n cos nπ L x (even function) f(x) = π + 2 n cos nπ sin nπ L x = π +2 sin x 1 2 sin 2x sin 3x... 19

21 1.5 Complex Fourier Series Euler formula: e ix =cosx + i sin x e ix =cosx isin x e inx =cosnx + i sin nx e inx =cosnx i sin nx cos nx = 1 2 (einx + e inx ) sin nx = 1 2i (einx e inx ) For Fourier series: f(x) =a + (a n cos nx + b n sin nx) (a n,b n R) a n cos nx + b n sin nx = 1 2 a n(e inx + e inx )+ 1 2i b n(e inx e inx ) = 1 2 (a n ib n ) e inx + 1 }{{} 2 (a n + ib n ) }{{} c n d n f(x) =c + (c n e inx + d n e inx ) e inx 2

22 where c = a R c n = 1 2 (a n ib n ) = 1 2π = 1 2π f(x)(cos nx i sin nx)dx f(x)e inx dx d n = 1 2 (a n + ib n ) = 1 2π = 1 2π c n,d n C f(x)(cos nx + i sin nx)dx f(x)einx dx Note : c n = d n f(x) =c + (c n e inx + c n e inx ) = c n e inx n= where c n = 1 2π f(x)e inx dx 21

23 Complex form of Fourier series of real function: For real function f(x) with period 2L, f(x) = n= c n e inπx L where c n = 1 L 2L L f(x)e inπx L dx 22

24 1.6 Half-Range Expansion In practical applications, f(x) are given on some interval only, eg. f(x) isextendedasaneven periodic function f 1 (x) with period 2L. f 1 (x) can be represented by a Fourier cosine series with period 2L. f(x) isextendedasanodd periodic function f 2 (x) with period 2L. f 2 (x) can be represented by a Fourier sine series with period 2L. f(x) is extended as a periodic function f 3 (x) with period L. f 3 (x) can be represented by a Fourier Series with period L. Note : f(x) =f 1 (x) =f 2 (x) =f 3 (x) ONLYwithin<x<L 23

25 Ex : f(x) = (1) Even extension: 2k L x <x<l 2 2k L (L x) L 2 <x<l f 1 (x) =a + a n cos nπ L x a = 1 L = 1 L L f(x)dx L 2 2k L x dx + L L 2 2k L (L x)dx ( x L x) = 1 L 2k L L 2 xdx + 2k L L 2 x( d x) = k 2 a n = 2 L L f(x)cosnπx L dx =...= 4k n 2 π 2 ( 2cos nπ 2 cos nπ 1 ) 24

26 (2) Odd extension: f 2 (x) = b n sin nπ L x b n = 2 L L f(x)sinnπx L = 8k n 2 π 2 sin nπ 2 (3) Periodic extension with period p = L: f 3 (x) =a + (a n cos 2nπ L x + b n sin 2nπ L )x a = 1 L a n = 2 L b n = 2 L L f(x)dx L f(x)cos2nπ L dx L f(x)sin2nπ L xdx 25

27 1.7 Forced Oscillations. Optional 26

28 1.8 Approximation by Trigonometric Polynomials Application of Fourier series: (1) solution of partial differential equation, (2) approximation. Given a periodic function f(x) withp =2π f(x) = a + N (a n cos nx + b n sin nx) F (x) (1) i.e., f(x) is approximated by a trigonometric polynomial of order N. Question : Is (1) the best approximation to f(x)? Def : Best approximation means minimal error E Def : Error E e.g. max f(x) F (x) We choose total square error of f for the approximated function F : Note : E = (f F )2 dx Max f F measures the goodness according to the maximal difference at the particular point. (f F )2 dx measures the goodness of agreement on the whole interval x π. 27

29 Proof : F (x) is the best approximation of f(x) f(x) =a + (a n cos nx + b n sin nx) F (x) =α + N (α n cos nx + β n sin nx) E = (f F )2 dx = f 2 dx 2 ffdx+ F 2 dx ffdx = π(2α a + α 1 a α N a N + β 1 b β N b N ) F 2 dx = π(2α 2 + α α2 N + β β 2 N ) π cos nx sin nx dx = π cos mx cos nx sin mx cos nx dx = m n π cos mx sin nxdx = cos2 nx sin 2 nx dx = π 28

30 E = f 2 dx 2π [ 2αa + N If we take α n = a n and β n = b n E = f 2 dx π [ 2a 2 + N E E = π { 2(α a ) 2 + N }{{} E E, (α n a n + β n b n ) ] + π [ 2α 2 + N (αn 2 + β2 n )] (a 2 n + b 2 n) ] [ (αn a n ) 2 } {{ } +(β n β n ) 2 ]} }{{} i.e. the error is minimum (E = E ) if and only if α = a,..., β n = b n. Theorem : Minimum square error The total square error of trigonometric polynomial relative to f on x π is minimum if and only if the coefficients of the trigonometric polynomial are the Fourier coefficients of f. Notes : (1) Convergence of the trigonometric polynomial: E = π f 2 dx π 2a 2 + N (a 2 n + b 2 n) N E } {{ } the more terms in partial sum of the Fourier series the better the approximation. 29

31 (2) Bessel inequality: 1 π f(x)2 dx 2a 2 + N (a 2 n + b 2 n) (3) Parseval s theorem (identity): 1 π f(x)2 dx =2a 2 + (a 2 n + b2 n ) 3

32 1.9 Fourier Integrals Fourier series periodic function Fourier Integral non-periodic function (p = 2L ) f L (x) =a + (a n cos w n x + b n sin w n x) w n = nπ L a = 1 L 2L f L L( x)d x a n = 1 L b n = 1 L L L f L( x)cosw n xd x n =1, 2,... L L f L( x)sinw n xd x n =1, 2,... f L (x) = 1 L 2L f L L( x)d x + 1 L [ cos w n x L f L L( x)cosw n xd x +sinw n x ] L f L L( x)sinw n xd x w w n+1 w n = (n +1)π L nπ L = π L 1 L = w π f L (x) = 1 L 2L f L L( x)d x + 1 π [ (cos w n x) w L L f L( x)cosw n xd x +(sinw n x) w ] L f L L( x)sinw n xd x 31

33 f(x) = lim L f L(x) lim w nπ n = lim L L L w lim w = lim L L lim L f L(x) = lim L + lim L π L dw 1 L 2L f L L( x)d x 1 π w [ (cos w n x) L L f L( x)cosw n xd x +(sinw n x) ] L f L L( x)sinw n xd x f(x) = 1 π dw [ cos wx f( x)cosw xd x }{{} πa(w) +sinwx f( x)sinw xd x ] }{{} πb(w) where f(x) = [A(w)coswx + B(w)sinwx]dw A(w) = 1 π B(w) = 1 π f( x)cosw xd x f( x)sinw xd x This is a representation of f(x) by a Fourier integral, where f(x) is assumed to be absolutely integrable, i.e., f(x) dx exists. 32

34 Theorems : Fourier Integral If f(x) is (1) piecewise continuous in every finite interval, (2) has a right-hand and a left-hand derivative at every point, (3) absolutely integrable, then f(x) can be represented by a Fourier integral. At the points where f(x) is discontinuous, the value of the Fourier integral equals to the average of the left-hand and right-hand limits of f(x). 33

35 Ex : f(x) = 1 if x < 1 if x > 1 A(w) = 1 π f( x)cosw xd x = 1 π 2 cos w xd x = 2sinw wπ B(w) = 1 π f( x)sinw xd x = f(x) = [A(w)coswx + B(w)sinwx]dw = 2 cos wx sin w dw π w 34

36 f(x) = [A(w)coswx + B(w)sinwx]dw A(w) = 1 π B(w) = 1 π f( x)cosw xd x f( x)sinw xd x Fourier cosine integral: If f(x) is an even function, then f(x) = A(w)coswxdw A(w) = 2 π f( x)cosw xd x B(w) = Fourier sine integral: If f(x) is an odd function, then f(x) = A(w) = B(w)sinwxdw B(w) = 2 π f( x)sinw xd x 35

37 1.1 Fourier Cosine and Sine Transforms Sometimes, it is easier to solve differential and integral equations in the transformed space rather than in the original physical space. Laplace transform, Fourier transform, Hankel transform, Hilbert transform,... The Fourier cosine integral of a non-periodic even function f(x): f(x) = A(w)coswxdw A(w) = 2 π f( x)cosw xd x let A(w) 2 π ˆf c (w) f(x) = 2 π ˆf c (w)coswxdw ˆfc (w) = 2 π f(x)coswxdx ˆf c (w) is called Fourier cosine transform of f(x) f(x) is called inverse Fourier cosine transform of ˆf c (w) Sometimes we write: F c (f) = ˆf c and Fc 1 ( ˆf c )=f 36

38 Similarly, the Fourier sine integral of a non-periodic odd function f(x): f(x) = B(w)sinwxdw B(w) = 2 π f( x)sinw xd x let B(w) 2 π ˆf s (w) f(x) = 2 π ˆf s (w)sinwxdw ˆfs (w) = 2 π f(x)sinwxdx ˆf s (w) is called Fourier sine transform of f(x) f(x) is called inverse Fourier sine transform of ˆf s (w) Sometimes we write: F s (f) = ˆf s and F 1 s ( ˆf s )=f 37

39 Linearity F c (af + bg) = 2 π [af(x)+bg(x)] cos wxdx = a 2 π f(x)coswxdx + b 2 π g(x)coswxdx = af c (f)+bf c (g) Similarly, F s (af + bg) =af s (f)+bf s (g) 38

40 Transform of Derivative F c (f (x)) = = 2 π 2 π f (x)coswxdx [ f(x)coswx x= + w f(x)sinwxdx ] = 2 π f() + wf s(f(x)) (Assume f(x), as x ) F s (f (x)) = = 2 π f (x)sinwxdx 2 π [f(x)sinwx] = wf c (f(x)) w f(x)coswxdx F c (f (x)) = 2 π f () + wf s (f (x)) = 2 π f () w 2 F c (f(x)) F s (f (x)) = wf c (f (x)) = 2 π wf() w2 F s (f(x)) 39

41 4

42 1.11 Fourier Transform where f(x) = [A(w)coswx + B(w)sinwx]dw A(w) = 1 π B(w) = 1 π f( x)cosw xd x f( x)sinw xd x f(x) = 1 π dw f( x)[cos w x cos wx +sinw x sin wx]d x = 1 π = 1 2π dw d xf( x)cos(wx w x) }{{} dw even function of w = 1 2π dw d xf( x)eiw(x x) }{{} complex Fourier integral (w [, ] [, ]) d xf( x)[cos(wx w x)+isin(wx w x) ] }{{} odd function 41

43 f(x) = 1 2π 1 2π Fourier transform of f(x): f( x)e iw x d x } {{ } ˆf(w) ˆf(w) = 1 2π f(x)e iwx dx Inverse Fourier transform of ˆf(w): f(x) = 1 2π ˆf(w)e iwx dw Again f(x) mustbe (1) piecewise continuous on every finite interval (2) f(x) is absolutely integrable e iwx dw We also write the Fourier transform pair as: F(f(x)) = ˆf(w) F 1 ( ˆf(w)) = f(x) 42

44 Spectrum Consider the oscillation a spring-mass system: my + ky = ( ), where y(t) is the displacement, m is the mass and k is the spring constant. ( )y dt = 1 2 my ky2 }{{} total energy = E = constant The solution y(t) of the ordinary differential equation (*) is: y(t) =a 1 cos k m t + b 1 sin k m t = c 1 e iw t + c 1 e iw t where ω = k m, c 1 = 1 2 (a 1 ib 1 ), c 1 = 1 2 (a 1 + ib 1 )=c 1 The total energy can be represented by: E = 1 2 m(c 1iω e iωt c 1 iω e iωt ) k(c 1e iωt + c 1 e iωt ) 2 =2kc 1 c 1 =2k c 1 2 i.e. E c

45 If y = f(x) = n= c n e iw nx c n 2 energy component of frequency w n We have a series of c n 2 for different w n. We call this discrete spectrum. Similarly, if y = f(x) = 1 2π ˆf(w)e iwx dw ˆf(w) 2 dw energy within [w, w + dw] ˆf(w) 2 total energy We have a function of ˆf(w) 2. We called this continuous spectrum. 44

46 Properties of Fourier Transform Linearity: F(af + bg) = af(f)+ bf(g) a and b are constants. Fourier Transform of derivative of f(x): Proof : F(f (x)) = iwf (f(x)) (f as x ) F(f (x)) = 1 2π f (x)e iwx dx = 1 2π [ f(x)e iwx ( iw) f(x)e iwx dx ] =+iwf(f(x)) 45

47 Convolution For f(x) andg(x), the convolution of f and g is defined as: h(x) =(f g) = f(p)g(x p)dp = f(x p)g(p)dp Theorem : Convolution Theorem If f and g are piecewise continuous, bounded and absolutely integrable, then Proof : F(f g) = 2πF(f)F(g) F(f g) = 1 [ f(p)g(x 2π p)dp] e iwx dx = 1 [ f(p)g(x 2π p)e iwx dx ] dp x p q x = p + q F(f g) = 1 2π f(p)g(q)e iw(p+q) dqdp = 1 2π f(p)e iwp dp g(q)e iwq dq = 2πF(f)F(g) 46

48 F(f g) = 2π F(f) F(g) }{{}}{{} ˆf ĝ f g = ˆf(w)ĝ(w)e iwx dx 47

49 1.12 Table of Fourier Transforms 48