Fourier series. (sine and cosine)( ) ... : w h ere 2 (1 1)
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1 Fourier series ( (sie ad osie( ( (... u A si t A si t u A o s t A o s t w h ere (.
2 (.(frequey ( (omplex( i t e os t i si t ( A z A e i t ( ( ( 3 z (Argad.(Argad diagram (
3 z x iy A (os t i si t ( 4 A=r A x y r or i t i t zz z Ae Ae A r ( 5 u u A e A e i t i t u u u A e A e i t u e ( A A e i t i A A e ( A A os i ( A si M e i where M ( A A os ( A si u M e e M e i t i i ( t i t u M (os( t i si ( t x iy x M os( t y M si( t ( 6 i 3
4 . ( x A si t B si t ( 7.(3 (3 (7 ( x f t f t ( 8. si 4wt si 3wt f(x 4
5 (7 si wt... f ( t a a os t a os t... a os t b si t b si t... b si t ( 9 ( ( (harmois w,3w,. w (4 5
6 (5 ( (6 6
7 (7 ( ( (quality. (Fourier (8. trasformatio x A s i t ( t t t t ( ( t t. ( kx wt x, t f ( x v t f ( x t ( x, t f ( x t ( 7
8 ( ( ( (8. (- or or x L x L t ad so o 8 ( si x,,3... os x,,,3, 4... ( (- from trigoometry si A si B os( A B os( A B os A os B os( A B os( A B si A os B si( A B si( A B
9 si os ( k x ( k x si si dx ( x ( x si si dx ( x si dx ( 3 x whe k kx dx x si si where k if kx dx ( 4 x dx d dx d si si k d k=,,,3.. where k os( k os( k d si( k si( k ( k ( k whe ( k ad ( k ad whe k ;, k,,3,...( 5 x 3 os os kx dx where k ;, k,,,3...( 6 3 si( k si( k ( k ( k ( -4 k= ( 7 (- 6 ( -4 ( - 6 9
10 4 4 4 si si x os x 4 x si si whe k x,,3,... where x kx si dx x dx x dx os dx x dx x os os ( 8 5,,3, 5 =k= x kx 5 os os dx whe k d x. k...( 9 ( -6 ( (- 9 f ( t a a os t a os t... a os t b si t b si t... b si t ( 9. 3 f ( t a ( a os t b si t (3 b a t (3- t
11 os k t dt (3- f ( t os k t dt a os k t dt ( a os t os k t dt b si t os k t dt f ( t os k t dt a os k t dt a os t os k t dt b si t os k t dt (3 a ( 3- =k f ( t os k t dt a os t os k t dt whe k f ( t os t dt a a f ( t os t dt,,,3... (3 3 si k t dt (3- f ( t si k t dt a si k t dt ( a os t si k t dt b si t si k t dt f ( t os k t dt a si k t dt a os t si k t dt b si t si k t dt (3 4 f ( t si k t dt b si t si k t dt whe k f ( t si t dt b b f ( t si t dt,,3... (3 3 b (3-4 =k a f ( t os t dt f ( t dt a f ( t dt (3 4 a a = (3-3 a
12 (( f(t f ( t a a e a e a e... a e i t i t i 3 t i t 3 a e a e a e... a e i t i t i 3 t i t 3 i t f ( t a e (3 5 where (3-5 ( i i t e e dt i t whe e dt e dt i t f ( t dt a e dt i t i t ( i e dt e e i i i whe e i f ( t dt a a f ( t dt f ( t (3 6 f(t (3-6 e i t (3-5 i t i t i t ( f t e dt a e e dt a i t a f ( t e dt (3 7 (3-7 i t i t i t ( f t e dt a e e dt a i t a f ( t e dt (3 8 a - a a (3-5 a i t e a - a - a (3-5
13 a a * or a a * A os t B si t C os( t C os t os C si t si os si ta where A B C C C A B B A A f t C t or ( os( A f t C t ( si( (3 9 i t i t i t f ( t a e a e a a e i t f ( t a a e a e ( os si (3-5 f ( t a ( a a os t i ( a a si t A a a B i a a i t A a A f t A t B t (3 3
14 i t i t A a a f ( t e dt f ( t e dt i t i t A f ( t ( e e dt A f ( t (os t dt A f ( t (os t dt (3 i t i t B i a a i f ( t e dt f ( t e dt i t i t B f ( t i ( e e dt B f ( t (si t dt B f ( t (si t dt (3 ( f(t (-L,L t t f(t.. t (-L,L f ( t ( t f ( t t f ( t. 4 (effetive (root-mea-square value i t f ( t a e (4 f eff f ( t dt (4 4
15 . (4- r r i t ir t i t ir t f ( t ae are aare e r r r i r t f t aare r ( (4 3 (4-3 (ofusio r i r t f t a are dt r, r ( (4 5 i r t f ( t a a e dt (4 6 r eff r r r=- i t i (6-4 im t im t e dt e if m im whe m r e dt e dt i r t f t a a e dt r eff ( r r eff (4 7 f ( t a a a a (4 8 where a a * eff f ( t a a a a (4 9 +r = m (4-7 f ( t a e ir t f ( t b e (4 r r r i ( r t p f ( t f ( t dt ab r e dt r r i t p a b. p a b (4 r r m r if r r r. 5
16 (Parseval's theorem. 5 (3 - f ( t a ( a os t b si t (5 f(t f ( t d t a f ( t dt ( a f ( t os t dt b f ( t si t dt f ( t d t a a b (5 w here f ( t os t dt a f ( t si t dt b f ( t dt a (4- r i ( r t p f ( t f ( t dt ab r e dt r p a b p a b whe r r m r. if r r r f ( t f * ( t b A a a a A ib a A ib a * p f t f t dt a a a B i a a ib a a (4- * ( ( (5 3 (5- a A ib A ib A ib A B 4 (5-3 (3 - (
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