ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil equtions with boundry conditions (s will be reviewed in Chpter 8). Fourier trnsforms re often used to extrct frequency informtion from time series dt. For lck of time in this course, only brief introduction is provided here. Sections in this Chpter: 7.0 Orthogonl Functions 7.0 Definitions of Fourier Series 7.03 Hlf-Rnge Fourier Series 7.04 Frequency Spectrum Sections for reference only, not exminble in this course: 7.05 Complex Fourier Series 7.06 Fourier Integrls 7.07 Complex Fourier Integrls 7.08 Some Fourier Trnsforms 7.09 Summry of Fourier Trnsforms
ENGI 940 7.0 - Orthogonl Functions Pge 7.0 7.0 Orthogonl Functions The inner product (or sclr product or dot product) of two vectors u nd v is defined in Crtesin coordintes in 3 by 3 uv u v u v u v u v k The inner product possesses the four properties: Commuttive: u v v u Sclr multipliction: u v u v, Positive definite: k k 3 3 k k k 0 if uu 0 if u0 u 0 Associtive: u v w u v u w Vectors uv, re orthogonl if nd only iff uv 0. A pir of non-zero orthogonl vectors intersects t right ngles. The inner product of two rel-vlued functions f nd f on n intervl [, b] my be defined in wy tht lso possesses these four properties: b, f f f x f x dx Two functions f nd f re sid to be orthogonl on n intervl [, b] if their inner product is zero: b f, f f x f x dx 0 A set of rel-vlued functions x x x x 0,,,, n is orthogonl on the intervl [, b] if the inner product of ny two of them is zero: b, x xdx 0 m n m n m n If, in ddition, the inner product of ny function in the set with itself is unity, then the set is orthonorml: b 0 m n m, n m xn xdx mn m n where mn is the Kronecker delt symbol.
ENGI 940 7.0 - Orthogonl Functions Pge 7.03 3 Just s ny vector in my be represented by liner combintion of the three Crtesin bsis vectors, (which form the orthonorml set {i, j, k}), so rel vlued function f (x) defined on [, b] my be written s liner combintion of the elements of x, x, x, on [, b]: n infinite orthonorml set of functions 0 f x c x c x c x 0 0 To find the coefficients c n, multiply f (x) by n(x) nd integrte over [, b]: b b b f x x dx c x x dx c x x dx n 0 0 n n m 0 b m m n c x x dx But the { n(x) } re n orthonorml set. Therefore ll but one of the terms in the infinite series re zero. The exception is the term for which m = n, where the integrl is unity. Therefore nd b cn f xn xdx b n n n 0 f x f x x dx x If the set is orthogonl but not orthonorml, then the form for f (x) chnges to b f xn xdx b n f x x n 0 n x dx The orthogonl set { n(x)} is complete if the only function tht is orthogonl to ll members of the set is the zero function f (x) 0. An expnsion of every function f (x) in terms of n orthogonl or orthonorml set { n(x)} is not possible if { n(x)} is not complete. Also note tht generlised form of n inner product cn be defined using weighting function w(x), so tht, in terms of complete orthogonl set { n(x)}, b w x f x x dx f x x n b n n 0 wxn x dx We shll usully be concerned with the cse w(x) only.
ENGI 940 7.0 - Orthogonl Functions Pge 7.04 Exmple 7.0. Show tht the set { sin nx } n is orthogonl but not orthonorml nd not complete on [, +].
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.05 7.0 Definitions of Fourier Series Exmple 7.0. nx nx orthonorml on [, ]. Show tht the set, cos, sin, n is orthogonl but not
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.06 Exmple 7.0. (continued)
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.07 Using the results from Exmple 7.0., we cn express most rel-vlued functions f (x) defined on (, ), in terms of n infinite series of trigonometric functions: The Fourier series of f (x) on the intervl (, ) is where nd 0 nx nx f x ncos bnsin n n x n f x dx n cos, 0,,, 3, n x bn f x dx n sin,,, 3, The { n, b n } re the Fourier coefficients of f (x). Note tht the cosine functions (nd the function ) re even, while the sine functions re odd. If f (x) is even (f ( x) = + f (x) for ll x), then b n = 0 for ll n, leving Fourier cosine series (nd perhps constnt term) only for f (x). If f (x) is odd (f ( x) = f (x) for ll x), then n = 0 for ll n, leving Fourier sine series only for f (x).
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.08 Exmple 7.0. Expnd f x 0 x 0 x 0 x in Fourier series.
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.09 Exmple 7.0. (Additionl Notes lso see "www.engr.mun.c/~ggeorge/940/demos/") The first few prtil sums in the Fourier series n f x cos nx sin nx x 4 n n n re S0 4 S cos sin 4 x x S cos sin sin 4 x x x S 3 cos sin sin cos3 sin 3 4 x x x 9 x 3 x nd so on. The grphs of successive prtil sums pproch f (x) more closely, except in the vicinity of ny discontinuities, (where systemtic overshoot occurs, the Gibbs phenomenon).
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7.0 Exmple 7.0.3 Find the Fourier series expnsion for the stndrd squre wve, x 0 f x 0 x The grphs of the third nd ninth prtil sums (contining two nd five non-zero terms respectively) re displyed here, together with the exct form for f (x), with periodic extension beyond the intervl (, +) tht is pproprite for the squre wve. y S3 x
ENGI 940 7.0 - Full Rnge Fourier Series Pge 7. Exmple 7.0.3 (continued) y S9 x Convergence At ll points x x in (, ) where f (x) is continuous nd is either differentible or the limits lim x xo f o x nd lim f x x xo At finite discontinuities, (where the limits both exist, the Fourier series converges to f (x). lim x xo f x nd lim f x x xo both exist), the f xo f xo Fourier series converges to, f x lim f x nd f x lim f x ). (using the bbrevitions o o x xo x xo f (x) not continuous continuous but continuous nd t x = x o not differentible differentible f xo f xo In ll cses, the Fourier series t x xo converges to (the red dot).
ENGI 940 7.03 - Hlf Rnge Fourier Series Pge 7. 7.03 Hlf-Rnge Fourier Series A Fourier series for f (x), vlid on [0, ], my be constructed by extension of the domin to [, ]. An odd extension leds to Fourier sine series: where n x f x bn sin n n x bn f x dx n 0 sin,,, 3, An even extension leds to Fourier cosine series: 0 n x f x n cos n where n x n f xcos dx, n 0,,, 3, 0 nd there is utomtic continuity of the Fourier cosine series t x = 0 nd t x =.
ENGI 940 7.03 - Hlf Rnge Fourier Series Pge 7.3 Exmple 7.03. Find the Fourier sine series nd the Fourier cosine series for f (x) = x on [0, ]. Fifth order prtil sum of the Fourier sine series for f (x) = x on [0, ]
ENGI 940 7.03 - Hlf Rnge Fourier Series Pge 7.4 Exmple 7.03. (continued)
ENGI 940 7.03 - Hlf Rnge Fourier Series Pge 7.5 Exmple 7.03. (continued) Third order prtil sum of the Fourier cosine series for f (x) = x on [0, ] Note how rpid the convergence is for the cosine series compred to the sine series. for cosine series nd y S x y S x 3 for sine series for f (x) = x on [0, ] 5
ENGI 940 7.04 - Frequency Spectrum Pge 7.6 7.04 Frequency Spectrum The Fourier series my be combined into single cosine series. et p be the fundmentl period. If the function f (x) is not periodic t ll on [, ], then the fundmentl period of the extension of f (x) to the entire rel line is p. Define the fundmentl frequency. p The Fourier series for f (x) on [, ] is, from pge 7.07, nx nx f x b n x b n x 0 0 n cos n sin n cos n sin n n where n x n f x dx f x n x dx n nd n x bn f x sin dx f x sin n x dx, n,, 3, cos cos, 0,,, 3, bn et the phse ngle n be such tht tnn, n bn n so tht sinn nd cosn c c where the mplitude is n c b. n n n Also, in the trigonometric identity cos Acos B sin Asin B cos A B replce A by n x nd B by n. Then n, cos n x b sin n x c cos cos n x c sin sin n x n n n n n n bn cn cos n x n, where, cn n bn nd tnn p Therefore the phse ngle or hrmonic form of the Fourier series is f x cn n x 0 cos n n n
ENGI 940 7.04 - Frequency Spectrum Pge 7.7 Exmple 7.04. Plot the frequency spectrum for the stndrd squre wve, x 0 f x 0 x From Exmple 7.0.3, the Fourier series for the stndrd squre wve is n n 4 f x sin n x sin k x k n k
ENGI 940 7.04 - Frequency Spectrum Pge 7.8 Exmple 7.04. Plot the frequency spectrum for the periodic extension of f x x x,
ENGI 940 7.04 - Frequency Spectrum Pge 7.9 Exmple 7.04. (continued) (which converges very rpidly, s this third prtil sum demonstrtes) The hrmonic mplitudes re n 0 n 0 cn n 0 n even, n n 4 n n odd n The frequencies therefore diminish rpidly:
ENGI 940 7.05 - Complex Fourier Series Pge 7.0 7.05 Complex Fourier Series [for reference only, not exminble] j Note tht the Euler identity e cos jsin leds to jn x jn x jn x jn x e e e e cos nx nd sin nx j The Fourier series for the periodic extension of f (x) from the originl intervl [, +p) is 0 f x ncos nx bnsin nx, n where nd p p p n f xcos nx dx, bn f xsin nx dx p p The Fourier series becomes jn x jn x jn x jn x 0 e e e e f x n bn j n jb jnx jb jnx e 0 n n n n e n n jnx * d0 dne dn e jnx n jbn where dn for ll non-negtive integers n. p p n jbn dn f xcos nx dx j f xsin nx dx p p p cos sin p jn x f x nx j nx dx f x e dx p p * p jn x p j n x dn f x e dx f x e dx dn p p Therefore the entire Fourier series my be re-written more concisely s p jnx jnx f x dne, where dn f xe dx p n The numbers {..., d, d, d 0, d, d,... } re the complex Fourier coefficients of f. The hrmonic mplitudes of f re just the mgnitudes { d n } for n = 0,,,....
ENGI 940 7.05 - Complex Fourier Series Pge 7. Exmple 7.05. Find the complex Fourier series expnsion for the stndrd squre wve, x 0 f x 0 x p 0 d0 f x dx dx dx 0 0 For n 0, 0 jn x jn x jn x dn f xe dx e dx e dx 0 0 jnx jn x e e j jn jn e e jn jn n 0 jn jn j e e j j n n n n n or 0 n even dn j n odd n In its most compct form, the complex Fourier series for the squre wve is k f x e j k The mplitude spectrum is the set n, dn n, j k x, for odd n only. n
ENGI 940 7.06 - Fourier Integrls Pge 7. 7.06 Fourier Integrls [for reference only, not exminble] The Fourier series my be extended from (, ) to the entire rel line. n et n n n The Fourier series for f (x) on (, ) is f x f t dt n nt nx f tcos dt cos nt nx f tsin dt sin f x f t dt n n cos cos f t t dt nx f tsin nt dt sin nx Now tke the limit s 0: The first integrl converges to some finite number, so the first term vnishes in the limit. The summtion becomes n integrl over ll frequencies in the limit: f x 0 0 f tcost dt cosx d f tsin t dt sin x d Therefore the Fourier integrl of f (x) is where the Fourier integrl coefficients re cos sin f x A x B x d 0 A f tcos t dt nd B f tsin t dt provided f x dx converges.
ENGI 940 7.06 - Fourier Integrls Pge 7.3 Exmple 7.06. Find the Fourier integrl of f x x 0 otherwise From the functionl form nd from the grph of f (x), it is obvious tht f (x) is piecewise smooth nd tht f x dx converges to the vlue sin t sin A f tcos t dt cos t dt The function f (x) is even B = 0 for ll. Therefore the Fourier integrl of f (x) is sin f x cos x d 0 It lso follows tht 0 x sin cos x d x 0 otherwise Fourier series nd Fourier integrls cn be used to evlute summtions nd definite integrls tht would otherwise be difficult or impossible to evlute. For exmple, setting x = 0 in Exmple 7.06., we find tht 0 sin t dt t
ENGI 940 7.07 - Complex Fourier Integrls Pge 7.4 7.07 Complex Fourier Integrls [for reference only, not exminble] cos sin f x A x B x d 0 jx jx jx jx e e e e A B d 0 j A jb jx A jb jx e e d 0 * j x j C e C e x d, where C A jb 0 But nd jx 0 t j t * cos sin C f t dt t j sin t cos f t dt C 0 jx C e d C e d By convention, the fctor of is extrcted from the coefficients. Therefore the complex Fourier integrl of f (t) is jt f t C e dt where the complex Fourier integrl coefficients re j t C f te dt (which is lso the Fourier trnsform of f, f F [f (t)]() ). is the frequency of the signl f (t).
ENGI 940 7.08 - Fourier Trnsforms Pge 7.5 7.08 Some Fourier Trnsforms [for reference only, not exminble] If ˆf is the Fourier trnsform of f (t), then nd the inverse Fourier trnsform is ˆ jt f f te dt ˆ jt f t f e d Exmple 7.08. Find the Fourier trnsform of the pulse function k t f t k H t H t 0 otherwise From the functionl form nd from the grph of f (t), it is obvious tht f (t) is piecewise smooth nd tht f t dt converges to the vlue k. jt ˆ jt jt k e k f f te dt k e dt j Therefore ˆ sin f k e j e j j
ENGI 940 7.08 - Fourier Trnsforms Pge 7.6 Exmple 7.08. (continued) The trnsform is rel. Therefore the frequency spectrum follows quickly:
ENGI 940 7.08 - Fourier Trnsforms Pge 7.7 Exmple 7.08. Find the Fourier trnsform of the tringle function k t t f t 0 otherwise From the functionl form nd from the grph of f (t), it is obvious tht f (t) is piecewise smooth nd tht f t dt converges to the vlue k. 0 jt k jt k jt 0 fˆ f t e dt t e dt t e dt 0 k t jt k t j t e e j j j j k j j j j 0 e 0 e k j j k e e cos 0 Therefore An equivlent form is fˆ fˆ k cos sin k
ENGI 940 7.08 - Fourier Trnsforms Pge 7.8 Exmple 7.08. (continued) The Fourier trnsform of the tringle function hppens to be rel nd non-negtive, so f ˆ fˆ. tht it is its own frequency spectrum
ENGI 940 7.08 - Fourier Trnsforms Pge 7.9 Exmple 7.08.3 Time Shift Property: et ˆ jt f F f t f te dt. Then j uto F f t t f t t e dt f u e du jt o o o o o ˆ jt jt jt ju jt e f u e du e f t e dt e f Therefore the time shift property of Fourier trnsforms is jto f t to e f t F F There re mny other properties of Fourier trnsforms to explore, (such s smpling, windowing, filtering, Fourier [Co]sine Trnsforms, discrete Fourier trnsforms nd Fst Fourier trnsforms) nd their pplictions to signl nlysis. However, there is insufficient time in this course to proceed beyond this introduction.
ENGI 940 7.09 - Fourier Trnsforms Pge 7.30 7.09 Summry of Fourier Trnsforms [for reference only, not exminble] In this tble, > 0. e f (t) ˆf F f t t e t 4j te t te t H t j 4 t e e Pulse (or gte) k H t H t Tringle k t t 0 otherwise Time shift k sin k cos j t o ˆ e f f (t t o ) Scling f (t) ˆf
ENGI 940 7.09 - Fourier Trnsforms Pge 7.3 Tble of Fourier Trnsforms (continued) f (t) ˆf F f t Time Differentition f (n) (t) Frequency Differentition t n f (t) Time Integrtion t f x dx Time Convolution f * g Frequency Convolution f. g Dirc delt t t n ˆ j f [provided f continuous] j n n d n d ˆf fˆ j [provided f ˆ 0 0] fˆ gˆ fˆ gˆ e e j t t j e t j sgn() Shnnon Smpling Theorem The entire signl f (t) my be reconstructed from the discrete smple t times t,,, 0,,, : n sin t n f t f n t n END OF CHAPTER 7
ENGI 940 7 - Fourier Series & Trnsforms Pge 7.3 [Spce for dditionl notes]