Hilbert Inner Product Space (2B) Young Won Lim 2/23/12

Transcription

1 Hilber nner Produc Space (2B)

2 Copyrigh (c) Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version 1.2 or any laer version published by he Free Sofware Foundaion; wih no nvarian Secions, no Fron-Cover exs, and no Back-Cover exs. A copy of he license is included in he secion eniled "GNU Free Documenaion License". Please send correcions (or suggesions) o youngwlim@homail.com. his documen was produced by using OpenOffice and Ocave.

3 rigonomeric deniies cosθ cosϕ = 1 (cos(θ ϕ) + cos(θ + ϕ)) 2 sin θ sin ϕ = 1 (cos(θ ϕ) cos(θ + ϕ)) 2 sin θ cos ϕ = 1 (sin(θ + ϕ) + sin(θ ϕ)) 2 cosθ sin ϕ = 1 (sin(θ + ϕ) sin(θ ϕ)) (1 + cos(θ + ϕ)) when θ=ϕ (1 cos(θ + ϕ)) when θ=ϕ (sin(θ + ϕ)) when θ=ϕ (sin(θ + ϕ)) when θ=ϕ +π π cos n x cos m x dx = 0 (n m) +π π cos n x cos m x dx = π (n = m) +π π sin n x sin m x dx = 0 (n m) +π π sin n x sin m x dx = π (n = m) +π π +π π sin n x cos m x dx = 0 cos n x sin m x dx = 0 n, m : ineger 3

4 rigonomeric Orhogonaliy f ( x) = a 0 + (a k cos kx + b k sin kx) k=1 a 0 = 1 2π +π π f ( x) dx a k = 1 π π +π f ( x) cos kx dx +π π +π π +π π +π π cos n x cos m x dx = 0 (n m) sin n x sin m x dx = 0 (n m) sin n x cos m x dx = 0 cos n x sin m x dx = 0 b k = 1 π π +π f ( x) sin kx dx +π π cos n x cos m x dx = π (n = m) k = 1, 2, 3,... +π π sin n x sin m x dx = π (n = m) n, m : ineger a k f ( x) cos k x = a 0 cos k x + (a m cos m x cos k x + b m sin m x cos k x) m=1 b k f ( x) sin k x = a 0 sin k x + (a m cos m x sin n x + b m sin m x sin k x) m=1 4

5 nner Produc Space Hilber Space real / complex inner produc space f, g = a b f () g ()d complex conjugae y, x = x, y linear a x 1 + b x 2, y = a x 1, y + b x 2, y Norm x = x, x Cauchy-Schwarz nequaliy x, y x y posiive semidefinie x, x 0 5

6 Orhogonal Funcions (1) e j nω 0 x() = + C k e + j kω 0 k= C k = 1 0 x( ) e j k ω 0 d k =..., 2, 1, 0, +1, +2,... fundamenal frequency f 0 = 1 ω 0 = 2 π f 0 = 2 π n-h harmonic frequency f n = n f 0 ω n = 2π f n = 2π n e j m ω 0, e j n ω 0 = 0 e + j(m n)ω 0 d = 0 (m n) (m = n) m, n : ineger 6

7 Orhogonal Funcions (2) e j nω 0 e j m ω 0, e j n ω 0 = 0 e + j(m n)ω 0 d = 0 (m n) (m = n) m, n : ineger e + j m ω 0 e j n ω 0 = (cos mω 0 + j sin m ω 0 ) (cos nω 0 j sin nω 0 ) = {cos m ω 0 cos nω 0 + sin m ω 0 sin nω 0 } + j {sin mω 0 cos nω 0 cos mω 0 sin nω 0 } = cos{m ω 0 nω 0 } + j sin {mω 0 sin nω 0 } = cos {(m n)ω 0 } + j sin {(m n)ω 0 } 1 0 (m = n) 7

8 nner Produc Examples (1) f 0 = 1/ ω 0 = 2π/ e j 1ω 0 e + j(1 1)ω 0 = e + j 0 ω 0 e j 1ω 0 e + j(1+1)ω 0 = e + j 2ω 0 e j( 1)ω 0 e + j(1 2)ω 0 = e + j( 1)ω 0 e j 2 ω 0 e + j(1+2)ω 0 = e + j 3 ω 0 e j( 2)ω 0 8

9 nner Produc Examples (2) f 0 = 1/ ω 0 = 2π/ e j 1ω 0 e j 1ω 0, e j1 ω 0 = 0 e + j(1 1)ω 0 d = e j 1ω 0 e j 1ω 0, e j 1 ω 0 = 0 e + j(1+1)ω 0 d = 0 e j 1 ω 0 e j 1ω 0, e j 2ω 0 = 0 e + j(1 2)ω 0 d = 0 e j 2 ω 0 e j 1ω 0, e j 2 ω 0 = 0 e + j(1+2)ω 0 d = 0 e j 2 ω 0 9

10 Orhogonal Funcions cos n ω 0 cos m ω 0, cos nω 0 = 0 cos m ω0 cos nω 0 d = m, n : ineger 0 (m n) /2 (m = n) cos m ω 0 cos n ω 0 = 1 2 { cos(m n)ω 0 + cos(m+n)ω 0 } 1 (m = ±n) 10

11 nner Produc Examples (1) cos n ω 0 f 0 = 1/ ω 0 = 2π/ cos 1ω {1 + cos2 ω 0 } cos1ω {1 + cos2 ω 0 } cos( 1)ω {cos1 ω 0 + cos 3ω 0 } cos 2 ω {cos1 ω 0 + cos 3ω 0 } cos( 2)ω 0 11

12 nner Produc Examples (2) cos n ω 0 f 0 = 1/ ω 0 = 2π/ cos 1ω 0 cos1ω 0, cos1 ω 0 = {cos(1 1)ω 0 + cos(1+1)ω 0 } d = {1 + cos 2ω 0 } d = 2 cos1ω 0, cos( 1)ω 0 = {cos(1+1)ω 0 + cos(1 1)ω 0 } d = {1 + cos 2ω 0 } d = 2 cos1ω 0, cos 2ω 0 = {cos(1 2)ω 0 + cos(1+2)ω 0 } d = {cos1ω 0 + cos 3ω 0 } d = 0 cos1ω 0, cos( 2)ω 0 = {cos(1+2)ω 0 + cos(1 2)ω 0 } d = {cos1ω 0 + cos 3ω 0 } d = 0 cos1ω 0 cos( 1)ω 0 cos 2 ω 0 cos( 2)ω 0 12

13 Orhogonal Funcions sin n ω 0 sin m ω 0, sin nω 0 = 0 sin m ω0 sin nω 0 d = m, n : ineger 0 (m n) /2 (m = n) sin m ω 0 sin nω 0 = 1 2 { cos(m n)ω 0 cos(m+n)ω 0 } 1 (m = ±n) 13

14 nner Produc Examples (1) sin n ω 0 f 0 = 1/ ω 0 = 2π/ sin 1ω {1 cos2 ω 0 } sin 1ω {cos 2ω 0 1} sin( 1)ω {cos1 ω 0 cos 3ω 0 } sin 2 ω {cos 3 ω 0 cos 1ω 0 } sin( 2)ω 0 14

15 nner Produc Examples (2) sin n ω 0 f 0 = 1/ ω 0 = 2π/ sin 1ω 0 sin1 ω 0, sin 1 ω 0 = {cos(1 1)ω 0 cos(1+1)ω 0 } d = {1 cos 2ω 0 } d = 2 sin1 ω 0, sin( 1)ω 0 = {cos(1+1)ω 0 cos(1 1)ω 0 } d = {cos2 ω 0 1} d = 2 sin1 ω 0, sin 2ω 0 = {cos(1 2)ω 0 cos(1+2)ω 0 } d = {cos1ω 0 cos 3ω 0 } d = 0 sin1 ω 0, sin( 2)ω 0 = {cos(1+2)ω 0 cos(1 2)ω 0 } d = {cos 3ω 0 cos1ω 0 } d = 0 sin 1ω 0 sin( 1)ω 0 sin 2 ω 0 sin( 2)ω 0 15

16 nfinie Se of Orhogonal Funcions {1, cos ω 0, cos 2 ω 0,, cos nω 0, } Linear Combinaion: + g ( ) = a n cosnω n=0 even funcion only + h() = { sin ω 0, sin 2ω 0,, sin n ω 0, } b n sin n ω n=1 Linear Combinaion: odd funcion only cos m ω 0, cos nω 0 = 0 cos m ω0 cos nω 0 d = m > 0, n > 0 : ineger 0 (m n) /2 (m = n) sin m ω 0, sin nω 0 = 0 sin m ω0 sin nω 0 d = m > 0, n > 0 : ineger 0 (m n) /2 (m = n) 16

17 Compleeness (1) {1, cosω 0, cos2 ω 0, cos 3ω 0,, sin ω 0, sin 2 ω 0, sin 3 ω 0, } Linear Combinaion: + f () = n=0 + a n cos n ω + b n sin nω n=1 Can be even / odd funcion When he collecion of orhogonal funcions are complee, every funcion can be expanded via a linear combinaion of hese infinie se of funcions cos m ω 0, cos nω 0 = 0 cos m ω0 cos nω 0 d = m > 0, n > 0 : ineger 0 (m n) /2 (m = n) sin m ω 0, sin nω 0 = 0 sin m ω0 sin nω 0 d = m > 0, n > 0 : ineger 0 (m n) /2 (m = n) 17

18 Compleeness (2) {1, cosω 0, cos2 ω 0, cos 3ω 0,, sin ω 0, sin 2 ω 0, sin 3 ω 0, } Linear Combinaion: + f () = n=0 + a n cos n ω + b n sin nω n=1 even / odd funcion periodic funcion periodic coninuous ime funcion f() = 2 π ω 0 a n b n weigh of cos nω 0 weigh of sin n ω 0 ime ω 0 2 ω 0 n ω 0 infinie bu discree se of angular frequency ang freq ω 18

19 Compleeness (3) + f () = n=0 + a n cos n ω + b n sin nω a 1 = 0.5, a 2 = 1.0, a 3 = 1.5 n=1 a 4 = a 5 = = 0 b 1 = b 2 = = cos 1 ω cos 2ω cos 3 ω cos1 ω cos 2 ω cos 3ω 0.5cos1 ω, cos1 ω = cos 2ω, cos1ω = 0 1.5cos3ω, cos1ω = 0 0.5cos1 ω, cos 2ω = 0 1.0cos 2ω, cos2 ω = cos3ω, cos 2ω = 0 0.5cos1 ω, cos 3ω = 0 1.0cos 2ω, cos3ω = 0 1.5cos3ω, cos3 ω = f () cos1 ω cos 2 ω cos 3ω 0.5cos1 ω, cos1 ω = cos 2ω, cos2 ω = cos3ω, cos3 ω =

20 Hilber Space Complee Orhogonal Sysem {, e j nω 0,, e j ω 0, 1, e + j ω 0,, e + j nω 0, } e j m ω 0, e j n ω 0 = 0 e + j(m n)ω 0 Complee BiOrhogonal Sysem d = 0 (m n) (m = n) m, n : ineger {1, cosω 0, cos2ω 0, cos 3ω 0,, sin ω 0, sin 2 ω 0, sin 3 ω 0, } cos m ω 0, cos nω 0 = 0 cos m ω0 cos nω 0 d = m > 0, n > 0 : ineger sin m ω 0, sin nω 0 = 0 sin m ω0 sin nω 0 d = m > 0, n > 0 : ineger 0 (m n) /2 (m = n) 0 (m n) /2 (m = n) 20

21 Cauchy-Schwarz nequaliy For all vecors x and y of an inner produc space x, y 2 x, x y, y x, y x y he equaliy holds if and only if x and y are linearly dependen maximum a b x() y()d a b x( ) x()d a b y() y()d nner produc is maximum when y = k x 21

22 Orhogonaliy r 0 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 0 r 0 r 0 r 1 r 0 r 2 r 0 r 3 r 0 r 4 r 0 r 5 r 0 r 6 r 0 r 7 r 1 r 0 r 1 r 1 r 1 r 2 r 1 r 3 r 1 r 4 r 1 r 5 r 1 r 6 r 1 r 7 r 2 r 0 r 2 r 1 r 2 r 2 r 2 r 3 r 2 r 4 r 2 r 5 r 2 r 6 r 2 r 7 = r 3 r 0 r 4 r 0 r 3 r 1 r 4 r 1 r 3 r 2 r 4 r 2 r 3 r 3 r 4 r 3 r 3 r 4 r 4 r 4 r 3 r 5 r 4 r 5 r 3 r 6 r 4 r 6 r 3 r 7 r 4 r 7 r 5 r 0 r 5 r 1 r 5 r 2 r 5 r 3 r 5 r 4 r 5 r 5 r 5 r 6 r 5 r 7 r 6 r 0 r 6 r 1 r 6 r 2 r 6 r 3 r 6 r 4 r 6 r 5 r 6 r 6 r 6 r 7 r 7 r 0 r 7 r 1 r 7 r 2 r 7 r 3 r 7 r 4 r 7 r 5 r 7 r 6 r 7 r 7 r i H, r i H = r i r i = N r i H, r j H = r i r j = 0 (i j) 22

23 Complex Vecor nner Produc Hermiian inner produc x, y = x H y = x i y i Norm of Hermiian inner producs x H : conjugae ranspose x = x, x = x H x = x i x i he lengh of a vecor x = (a 1 + j b 1 a 2 + j b 2 n) a n + j b x, x = x H x = x i x i (a 1 j b 1 a 2 j b 2 a n j b n ) (a1+ j b1 a 2 + j b 2 n) a n + j b = n i=1 a i 2 + b i 2 23

24 he 1s ow of he DF Marix 0 cycle W 8 k n = e j 2 8 k n k = 0, n = 0, 1,..., 7 samples of samples of cos = cos sin = sin measure = 2 f f s X[0] measures how much of he +0 ω componen is presen in x. 24

25 he 3rd ow of he DF Marix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 = cos 2 sin 2 = sin 2 measure = 2 f f s X[2] measures how much of he +2 ω componen is presen in x. 25

26 he 4h ow of he DF Marix 3 cycles W 8 k n = e j 2 8 k n k = 3, n = 0, 1,..., 7 samples of samples of cos 3 = cos 3 sin 3 = sin 3 measure = 2 f f s X[3] measures how much of he +3 ω componen is presen in x. 26

27 he 5h ow of he DF Marix 4 cycles W 8 k n = e j 2 8 k n k = 4, n = 0, 1,..., 7 samples of samples of cos 4 = cos 4 sin 4 = sin 4 measure = 2 f f s X[4] measures how much of he +4 ω componen is presen in x. 27

28 he 6h ow of he DF Marix 3 cycles W 8 k n = e j 2 8 k n k = 5, n = 0, 1,..., 7 samples of samples of cos 3 = cos 3 sin 3 = sin 3 measure = 2 f f s X[5] measures how much of he 3 ω componen is presen in x. 28

29 he 7h ow of he DF Marix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 = cos 2 sin 2 = sin 2 measure = 2 f f s X[6] measures how much of he 2 ω componen is presen in x. 29

30 he 8h ow of he DF Marix 1 cycle W 8 k n = e j 2 8 k n k = 7, n = 0, 1,..., 7 samples of samples of cos = cos sin = sin measure = 2 f f s X[7] measures how much of he 1 ω componen is presen in x. 30

31 Any Period p = 2L g(v) = a 0 + (a k cos kv + b k sin kv) k=1 f ( x) = a 0 + ( a k π k cos k=1 L x + b k sin k π L x ) a 0 = 1 2π +π π g(v) dv a k b k = 1 π π = 1 π π +π g(v) cos kv dv +π g(v) sin kv dv k = 1, 2,... a 0 = 1 2 L +L L f ( x) dx a k b k = 1 +L k π x L L f ( x) cos L = 1 +L k π x L L f ( x) sin L k = 1, 2, 3,... dx dx v : [, ] x : [ L, L] v = L x dv = L dx 31

32 ime and Frequency f ( x) = a 0 + ( a k π k cos k=1 L x + b k sin k π L x ) x() = a 0 + ( a 2 π k k cos k=1 + b k sin 2π k ) a 0 = 1 2 L +L L f ( x) dx a k b k = 1 +L k π x L L f ( x) cos L = 1 +L k π x L L f ( x) sin L k = 1, 2, 3,... dx dx a 0 = 1 0 x() d a k = 2 2 π k 0 x() cos b k = 2 2 π k 0 x( ) sin k = 1, 2,... d d x : [ L, L] : [0, ] 2 L = Coninuous ime Periodic Signal x 32

33 Harmonic Frquency x() = a 0 + ( a 2 π k k cos k=1 + b k sin 2π k ) x() = a 0 + ( a k cos(2π k f 0 ) + b k sin(2π k f 0 )) k=1 a 0 = 1 0 x() d a k = 2 2 π k 0 x() cos b k = 2 2 π k 0 x( ) sin k = 1, 2,... d d a 0 = 1 0 x() d a k = 2 0 x() cos (2π k f 0 ) d k = 1, 2,... b k = 2 0 x( ) sin (2π k f 0 ) d k = 1, 2,... : [0, ] : [0, ] resoluion frequency f 0 = 1 n-h harmonic frequency f n = n f 0 = n 1 33

34 adial Frequency x() = a 0 + (a k cos(k 2π f 0 ) + b k sin (k 2π f 0 )) n=1 x() = a 0 + ( a k cos(k ω 0 ) + b k sin( k ω 0 )) n=1 a 0 = 1 0 x() d a k = 2 0 x() cos (k 2π f 0 ) d k = 1, 2,... b k = 2 0 x( ) sin (k 2π f 0 ) d k = 1, 2,... a 0 = 1 0 x() d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d k = 1, 2,... : [0, ] : [0, ] linear frequency angular (radial) frequency f ω = 2 π f 34

35 Complex Fourier Series Coefficiens x() = a 0 + (a k cos(k ω 0 ) + b k sin(k ω 0 )) k=1 x() = A 0 + ( A k e j k ω 0 k=1 + B k e j k ω 0 ) a 0 = 1 0 x() d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d k = 1, 2,... A 0 = 1 0 x d A k = 1 j k ω 0 x () e 0 B k = 1 + j k ω 0 x () e 0 d d : [0, ] : [0, ] eal coefficiens a 0, a k, b k, k = 1, 2,... one-sided specrum only posiive frequencies Complex coefficiens A 0, A k, B k, k = 1, 2,... wo-sided specrum Boh pos and neg frequencies 35

36 Euler Equaion (1) x() = a 0 a 0 = (a k cos(k ω 0 ) + b k sin(k ω 0 )) k=1 x() d a k cos(k ω 0 ) + b k sin(k ω 0 ) = a k 1 2 (e j k ω 0 + e j k ω 0 ) + b k 1 2 j (e j k ω 0 e j k ω 0 ) a k = 2 0 x() cos (k ω0 ) d x( ) sin (k ω0 ) d = (a k j b k ) 2 e j k ω 0 + (a k + j b k ) 2 e j k ω 0 b k = 2 0 k = 1, 2,... = A k e j k ω 0 + B k e j k ω 0 e j e j = cos j sin = cos j sin cos = e j e j 2 sin = e j e j 2 j x() = A 0 zero freq pos freq neg freq + ( A k e j k ω 0 k=1 A 0 = a 0 + B k e j k ω 0 ) A k = 1 2 (a k j b k ) B k = 1 2 (a k + j b k ) only posiive frequencies 36

37 Euler Equaion (2) x() = a 0 a 0 = (a k cos(k ω 0 ) + b k sin(k ω 0 )) k=1 x() d A k = 1 0 x() (cos (k ω0 ) j sin (k ω 0 )) d B k = 1 0 x() (cos (k ω0 ) + j sin (k ω 0 )) d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d k = 1, 2,... n jn A k = 1 j k ω 0 x () e 0 n jn B k = 1 + j k ω 0 x () e 0 d d x() = A 0 zero freq pos freq neg freq + ( A k e j k ω 0 k=1 A 0 = a 0 + B k e j k ω 0 ) A k = 1 2 (a k j b k ) B k = 1 2 (a k + j b k ) only posiive frequencies x() = A 0 + ( A k e + j k ω 0 k=1 x() = ( A k e + j k ω 0 k=0 + B k e j k ω 0 ) + B k e j k ω 0 ) 37

38 Complex Fourier Series x() = a 0 + (a k cos(k ω 0 ) + b k sin(k ω 0 )) k=1 x() = ( A k e + j k ω 0 k=0 + B k e j k ω 0 ) a 0 = 1 0 x() d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d k = 1, 2,... A k = 1 j k ω 0 x() e 0 k = 0, 1, 2,... B k = 1 + j k ω 0 x() e 0 k = 1, 2,... d d x() = A 0 + ( A k e j k ω 0 k=1 + B k e j k ω 0 ) x() = + C k e + j k ω 0 k= A 0 = a 0 C k = 1 0 x() e j k ω 0 d A k = 1 2 (a k j b k ) k = 2, 1, 0, +1, +2,... B k = 1 2 (a k + j b k ) k = 1, 2,... C k = A 0 (k = 0) A k (k > 0) B k (k < 0) 38

39 Single-Sided Specrum x() = a 0 + k=1 (a k cos(k ω 0 ) + b k sin(k ω 0 )) x() = g 0 + g k cos(k ω 0 + ϕ k ) k=1 a 0 = 1 0 x() d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d g 0 = a 0 g k = a k 2 + b k 2 ϕ k = an 1 ( b k a k ) k = +1, +2,... k = +1, +2,... cos(α+β) = cos(α) cos(β) sin(α) sin(β) g k cos(k ω 0 + ϕ k ) = g k cos(ϕ k ) cos(k ω 0 ) g k sin(ϕ k ) sin(k ω 0 ) a k cos( k ω 0 ) + b k sin(k ω 0 ) a k = g k cos(ϕ k ) b k = g k sin(ϕ k ) a k 2 + b k 2 = g k 2 b k a k = an (ϕ k ) 39

40 Phasor epresenaion (1) x() = a 0 + (a k cos(k ω 0 ) + b k sin(k ω 0 )) k=1 x() = g 0 + g k cos(k ω 0 + ϕ k ) k=1 a 0 = 1 0 x() d a k = 2 0 x() cos (k ω0 ) d b k = 2 0 x( ) sin (k ω0 ) d k = 1, 2,... g 0 = a 0 g k = a k 2 + b k 2 ϕ k = an 1 ( b k a k ) k = 1, 2,... x() = g 0 x() = g 0 x() = g 0 + g k cos(k ω 0 + ϕ k ) k=1 + g k {e + j(k ω 0 + ϕ ) k } k=1 + k=1 {g k e + j ϕ k e + jk ω 0 } x() = X 0 X 0 = g 0 X k + k=1 = g k e + j ϕ k k = 1, 2,... {X k e + j k ω 0 } 40

41 Phasor epresenaion (2) x() = g 0 x() = g 0 x() = g 0 + k=1 + k=1 + k=1 g k 2 (e+ j( k ω 0 + ϕ k) + e j( k ω 0 + ϕ k) ) ( g k 2 e+ j ϕ k e + jk ω 0 + g k 2 e j ϕ k e j k ω 0) ( g k e + j ϕk 2 e + jk ω 0 + g k e j ϕ k 2 e j k ω 0) x() = g 0 g 0 = a 0 g k = a k 2 + b k 2 ϕ k + g k cos(k ω 0 + ϕ k ) k=1 = an 1 ( b k a k ) k = 1, 2,... x() = + C k e + j k ω 0 k= x() = X 0 + k=1 {X k e + j k ω 0 } C k C k = g k e + j ϕ k 2 (k > 0) = g k e j ϕ k (k < 0) 2 k = 2, 1, 0, +1, +2,... X 0 = g 0 X k = g k e + j ϕ k k = 1, 2,... 41

42 wo-sided Specrum x() = + C k e + j k ω 0 k= x() = + C k e + j k ω 0 k= C k = 1 0 x( ) e j k ω 0 d C k = 1 0 x( ) e j k ω 0 d k =..., 2, 1, 0, +1, +2,... k =..., 2, 1, 0, +1, +2,... C k = a 0 (k = 0) 1 2 k jb k ) (k > 0) 1 2 k + jb k ) (k < 0) C k = a 0 (k = 0) 1 2 k e + j ϕ k (k > 0) 1 2 k e j ϕ k (k < 0) C k = a 0 (k = 0) 1 2 a 2 2 k + b k (k 0) Arg (C k ) = an 1 ( b k /a k ) (k > 0) an 1 (+b k /a k ) (k < 0) C k = a 0 (k = 0) 1 2 g k ( k 0) Arg (C k ) = +ϕ k (k > 0) ϕ k (k < 0) Power Specrum wo-sided Periodogram C k 2 + C k 2 = 1 2 g k 2 = 1 (a 2 2 k + b 2 k ) 2 C k = g k = a 2 2 k + b k One-Sided 42

43 CFS of mpulse rain (1) p() = n= δ( n s ) Fourier Series Expansion of mpulse rain p() = + k = a k e + j k ω s Fourier Series Coefficiens 3 s 2 s s ω s = 2π s s 2 s 3 s a k = 1 + s /2 s s /2 = 1 + s /2 s s /2 δ( ) e j k ω s d δ() e j k ω s 0 d s s s s s s s = 1 + s /2 s s /2 δ() d = 1 s 3ω s 2 ω s ω s ω s 2 ω s 3 ω s 43

44 CFS of mpulse rain (2) p() = 1 + s k= a k e + j k ω s = 1 + s k = (cos k ω s j sin k ω s ) cos 2π 1 sin 2π cos 2π k k =1 40 sin 2 π k k =1 44

45 CFS of mpulse rain (3) p() = 1 + s k= a k e + j k ω s = 1 + s k = (cos k ω s j sin k ω s ) cos 2π 1 sin 2π e-13 5e e-14-1e cos 2π k k = sin 2π k k = 40 45

46 nner Produc Space Hilber Space real / complex inner produc space f, g = a b f () g ()d complex conjugae y, x = x, y linear a x 1 + b x 2, y = a x 1, y + b x 2, y Norm x = x, x Cauchy-Schwarz nequaliy x, y x y posiive semidefinie x, x 0 46

47 Orhogonaliy x() = + C k e + j kω 0 k= C k = 1 0 x( ) e j k ω 0 d k =..., 2, 1, 0, +1, +2,... fundamenal frequency f 0 = 1 ω 0 = 2 π f 0 = 2 π n-h harmonic frequency f n = n f 0 ω n = 2π f n = 2π n e j m ω 0, e j n ω 0 = 0 e + j(m n)ω 0 d = 0 (m n) (m = n) m, n : ineger 47

48 nner Produc Examples f 0 = 1/ ω 0 = 2π/ e j 1ω 0 e j 1ω 0, e j1 ω 0 = 0 e + j(1 1)ω 0 d = e j 1ω 0 e j 1ω 0, e j 1 ω 0 = 0 e + j(1+1)ω 0 d = 0 e j 1 ω 0 e j 1ω 0, e j 2ω 0 = 0 e + j(1 2)ω 0 d = 0 e j 2 ω 0 e j 1ω 0, e j 2 ω 0 = 0 e + j(1+2)ω 0 d = 0 e j 2 ω 0 48

49 Cauchy-Schwarz nequaliy For all vecors x and y of an inner produc space x, y 2 x, x y, y x, y x y he equaliy holds if and only if x and y are linearly dependen maximum a b x() y()d a b x( ) x()d a b y() y()d nner produc is maximum when y = k x 49

50 Orhogonaliy r 0 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 0 r 0 r 0 r 1 r 0 r 2 r 0 r 3 r 0 r 4 r 0 r 5 r 0 r 6 r 0 r 7 r 1 r 0 r 1 r 1 r 1 r 2 r 1 r 3 r 1 r 4 r 1 r 5 r 1 r 6 r 1 r 7 r 2 r 0 r 2 r 1 r 2 r 2 r 2 r 3 r 2 r 4 r 2 r 5 r 2 r 6 r 2 r 7 = r 3 r 0 r 4 r 0 r 3 r 1 r 4 r 1 r 3 r 2 r 4 r 2 r 3 r 3 r 4 r 3 r 3 r 4 r 4 r 4 r 3 r 5 r 4 r 5 r 3 r 6 r 4 r 6 r 3 r 7 r 4 r 7 r 5 r 0 r 5 r 1 r 5 r 2 r 5 r 3 r 5 r 4 r 5 r 5 r 5 r 6 r 5 r 7 r 6 r 0 r 6 r 1 r 6 r 2 r 6 r 3 r 6 r 4 r 6 r 5 r 6 r 6 r 6 r 7 r 7 r 0 r 7 r 1 r 7 r 2 r 7 r 3 r 7 r 4 r 7 r 5 r 7 r 6 r 7 r 7 r i H, r i H = r i r i = N r i H, r j H = r i r j = 0 (i j) 50

51 Complex Vecor nner Produc Hermiian inner produc x, y = x H y = x i y i Norm of Hermiian inner producs x H : conjugae ranspose x = x, x = x H x = x i x i he lengh of a vecor x = (a 1 + j b 1 a 2 + j b 2 n) a n + j b x, x = x H x = x i x i (a 1 j b 1 a 2 j b 2 a n j b n ) (a1+ j b1 a 2 + j b 2 n) a n + j b = n i=1 a i 2 + b i 2 51

52 he 1s ow of he DF Marix 0 cycle W 8 k n = e j 2 8 k n k = 0, n = 0, 1,..., 7 samples of samples of cos = cos sin = sin measure = 2 f f s X[0] measures how much of he +0 ω componen is presen in x. 52

53 he 3rd ow of he DF Marix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 = cos 2 sin 2 = sin 2 measure = 2 f f s X[2] measures how much of he +2 ω componen is presen in x. 53

54 he 4h ow of he DF Marix 3 cycles W 8 k n = e j 2 8 k n k = 3, n = 0, 1,..., 7 samples of samples of cos 3 = cos 3 sin 3 = sin 3 measure = 2 f f s X[3] measures how much of he +3 ω componen is presen in x. 54

55 he 5h ow of he DF Marix 4 cycles W 8 k n = e j 2 8 k n k = 4, n = 0, 1,..., 7 samples of samples of cos 4 = cos 4 sin 4 = sin 4 measure = 2 f f s X[4] measures how much of he +4 ω componen is presen in x. 55

56 he 6h ow of he DF Marix 3 cycles W 8 k n = e j 2 8 k n k = 5, n = 0, 1,..., 7 samples of samples of cos 3 = cos 3 sin 3 = sin 3 measure = 2 f f s X[5] measures how much of he 3 ω componen is presen in x. 56

57 he 7h ow of he DF Marix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 = cos 2 sin 2 = sin 2 measure = 2 f f s X[6] measures how much of he 2 ω componen is presen in x. 57

58 he 8h ow of he DF Marix 1 cycle W 8 k n = e j 2 8 k n k = 7, n = 0, 1,..., 7 samples of samples of cos = cos sin = sin measure = 2 f f s X[7] measures how much of he 1 ω componen is presen in x. 58

59 eferences [1] hp://en.wikipedia.org/ [2] J.H. McClellan, e al., Signal Processing Firs, Pearson Prenice Hall, 2003 [3] G. Beale, hp://eal.gmu.edu/~gbeale/ece_220/fourier_series_02.hml [4] C. Langon, hp://