Hilbert Inner Product Space (2B) Young Won Lim 2/7/12

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1 Hilbert nner Product Space (2B)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no nvariant Sections, no Front-Cover exts, and no Back-Cover exts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. his document was produced by using OpenOffice and Octave.

3 rigonometric dentities 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 : integer 3

4 rigonometric Orthogonality f ( x) = a 0 + (a k cos kx + b k sin kx) 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 : integer 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 Product Space Hilbert Space real / complex inner product space f, g = a b f (t) g (t)d t complex conjugate 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-Schwartz nequality x, y x y positive semidefinite x, x 0 5

6 Orthogonality x(t) = + C k e + j kω 0 t k= C k = 1 0 x(t ) e j k ω 0 t dt k =..., 2, 1, 0, +1, +2,... fundamental frequency f 0 = 1 ω 0 = 2 π f 0 = 2 π n-th harmonic frequency f n = n f 0 ω n = 2π f n = 2π n e j m ω 0 t, e j n ω 0 t = 0 e + j(m n)ω 0 t d t = 0 (m n) (m = n) m, n : integer 6

7 Orthogonality (2) e j m ω 0 t, e j n ω 0 t = 0 e + j(m n)ω 0 t d t = 0 (m n) (m = n) m, n : integer e + j m ω 0 t e j n ω 0 t = (cos mω 0 t + j sin m ω 0 t) (cos nω 0 t j sin nω 0 t) = {cos m ω 0 t cos nω 0 t + sin m ω 0 t sin nω 0 t } + j {sin mω 0 t cos nω 0 t cos mω 0 t sin nω 0 t } = cos{m ω 0 t nω 0 t} + j sin {mω 0 t sin nω 0 } = cos {(m n)ω 0 t} + j sin {(m n)ω 0 t } 1 0 (m = n) 7

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

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

10 Cauchy-Schwartz nequality For all vectors x and y of an inner product space x, y 2 x, x y, y x, y x y he equality holds if and only if x and y are linearly dependent maximum a b x(t) y(t)d t a b x(t ) x(t)d t a b y(t) y(t)d t nner product is maximum when y = k x 10

11 Orthogonality 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) 11

12 Complex Vector nner Product Hermitian inner product x, y = x H y = x i y i Norm of Hermitian inner products x H : conjugate transpose x = x, x = x H x = x i x i the length of a vector 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 12

13 he 1st ow of the DF Matrix 0 cycle W 8 k n = e j 2 8 k n k = 0, n = 0, 1,..., 7 samples of samples of cos t = cos t sin t = sin t measure t = 2 f t f s t X[0] measures how much of the +0 ω component is present in x. 13

14 he 3rd ow of the DF Matrix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 t = cos 2 t sin 2 t = sin 2 t measure t = 2 f t f s t X[2] measures how much of the +2 ω component is present in x. 14

15 he 4th ow of the DF Matrix 3 cycles W 8 k n = e j 2 8 k n k = 3, n = 0, 1,..., 7 samples of samples of cos 3 t = cos 3 t sin 3 t = sin 3 t measure t = 2 f t f s t X[3] measures how much of the +3 ω component is present in x. 15

16 he 5th ow of the DF Matrix 4 cycles W 8 k n = e j 2 8 k n k = 4, n = 0, 1,..., 7 samples of samples of cos 4 t = cos 4 t sin 4 t = sin 4 t measure t = 2 f t f s t X[4] measures how much of the +4 ω component is present in x. 16

17 he 6th ow of the DF Matrix 3 cycles W 8 k n = e j 2 8 k n k = 5, n = 0, 1,..., 7 samples of samples of cos 3 t = cos 3 t sin 3 t = sin 3 t measure t = 2 f t f s t X[5] measures how much of the 3 ω component is present in x. 17

18 he 7th ow of the DF Matrix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 t = cos 2 t sin 2 t = sin 2 t measure t = 2 f t f s t X[6] measures how much of the 2 ω component is present in x. 18

19 he 8th ow of the DF Matrix 1 cycle W 8 k n = e j 2 8 k n k = 7, n = 0, 1,..., 7 samples of samples of cos t = cos t sin t = sin t measure t = 2 f t f s t X[7] measures how much of the 1 ω component is present in x. 19

20 Any Period p = 2L g(v) = a 0 + (a k cos kv + b k sin kv) f ( x) = a 0 + ( a k π k cos 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 20

21 ime and Frequency f ( x) = a 0 + ( a k π k cos L x + b k sin k π L x ) x(t) = a 0 + ( a 2 π k k cos t + b k sin 2π k t ) 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(t) dt a k = 2 2 π k t 0 x(t) cos b k = 2 2 π k t 0 x(t ) sin k = 1, 2,... dt dt x : [ L, L] t : [0, ] 2 L = Continuous ime Periodic Signal x t 21

22 Harmonic Frquency x(t) = a 0 + ( a 2 π k k cos t + b k sin 2π k t ) x(t) = a 0 + ( a k cos(2π k f 0 t) + b k sin(2π k f 0 t)) a 0 = 1 0 x(t) dt a k = 2 2 π k t 0 x(t) cos b k = 2 2 π k t 0 x(t ) sin k = 1, 2,... dt dt a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (2π k f 0 t) dt k = 1, 2,... b k = 2 0 x(t ) sin (2π k f 0 t) dt k = 1, 2,... t : [0, ] t : [0, ] resolution frequency f 0 = 1 n-th harmonic frequency f n = n f 0 = n 1 22

23 adial Frequency x(t) = a 0 + (a k cos(k 2π f 0 t) + b k sin (k 2π f 0 t )) n=1 x(t) = a 0 + ( a k cos(k ω 0 t ) + b k sin( k ω 0 t)) n=1 a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k 2π f 0 t) dt k = 1, 2,... b k = 2 0 x(t ) sin (k 2π f 0 t) dt k = 1, 2,... a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt k = 1, 2,... t : [0, ] t : [0, ] linear frequency angular (radial) frequency f ω = 2 π f 23

24 Complex Fourier Series Coefficients x(t) = a 0 + (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = A 0 + ( A k e j k ω 0t + B k e j k ω 0 t) a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt k = 1, 2,... A 0 = 1 0 x t dt A k = 1 j k ω 0 x (t) e 0 t B k = 1 + j k ω 0 x (t) e 0 t dt dt t : [0, ] t : [0, ] eal coefficients a 0, a k, b k, k = 1, 2,... one-sided spectrum only positive frequencies Complex coefficients A 0, A k, B k, k = 1, 2,... two-sided spectrum Both pos and neg frequencies 24

25 Euler Equation (1) x(t) = a 0 a 0 = (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) dt a k cos(k ω 0 t ) + b k sin(k ω 0 t) = a k 1 2 (e j k ω 0t + e j k ω 0 t ) + b k 1 2 j (e j k ω 0t e j k ω 0t ) a k = 2 0 x(t) cos (k ω0 t) dt x(t ) sin (k ω0 t) dt = (a k j b k ) 2 e j k ω 0t + (a k + j b k ) 2 e j k ω 0 t b k = 2 0 k = 1, 2,... = A k e j k ω 0t + B k e j k ω 0t e j t e j t = cos t j sin t = cos t j sin t cos t = e j t e j t 2 sin t = e j t e j t 2 j x(t) = A 0 zero freq pos freq neg freq + ( A k e j k ω 0t A 0 = a 0 + B k e j k ω 0 t) A k = 1 2 (a k j b k ) B k = 1 2 (a k + j b k ) only positive frequencies 25

26 Euler Equation (2) x(t) = a 0 a 0 = (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) dt A k = 1 0 x(t) (cos (k ω0 t) j sin (k ω 0 t)) dt B k = 1 0 x(t) (cos (k ω0 t) + j sin (k ω 0 t)) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt k = 1, 2,... n jn t A k = 1 j k ω 0 x (t) e 0 t n jn t B k = 1 + j k ω 0 x (t) e 0 t dt dt x(t) = A 0 zero freq pos freq neg freq + ( A k e j k ω 0t A 0 = a 0 + B k e j k ω 0 t) A k = 1 2 (a k j b k ) B k = 1 2 (a k + j b k ) only positive frequencies x(t) = A 0 + ( A k e + j k ω 0t x(t) = ( A k e + j k ω 0 t k=0 + B k e j k ω 0t ) + B k e j k ω 0 t) 26

27 Complex Fourier Series x(t) = a 0 + (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = ( A k e + j k ω 0 t k=0 + B k e j k ω 0t ) a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt k = 1, 2,... A k = 1 j k ω 0 x(t) e 0 t k = 0, 1, 2,... B k = 1 + j k ω 0 x(t) e 0 t k = 1, 2,... dt dt x(t) = A 0 + ( A k e j k ω 0t + B k e j k ω 0 t) x(t) = + C k e + j k ω 0 t k= A 0 = a 0 C k = 1 0 x(t) e j k ω 0 t dt 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) 27

28 Single-Sided Spectrum x(t) = a 0 + (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = g 0 + g k cos(k ω 0 t + ϕ k ) a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt g 0 = a 0 g k = a k 2 + b k 2 ϕ k = tan 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 t + ϕ k ) = g k cos(ϕ k ) cos(k ω 0 t) g k sin(ϕ k ) sin(k ω 0 t) a k cos( k ω 0 t ) + b k sin(k ω 0 t) 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 = tan (ϕ k ) 28

29 Phasor epresentation (1) x(t) = a 0 + (a k cos(k ω 0 t) + b k sin(k ω 0 t)) x(t) = g 0 + g k cos(k ω 0 t + ϕ k ) a 0 = 1 0 x(t) dt a k = 2 0 x(t) cos (k ω0 t) dt b k = 2 0 x(t ) sin (k ω0 t) dt k = 1, 2,... g 0 = a 0 g k = a k 2 + b k 2 ϕ k = tan 1 ( b k a k ) k = 1, 2,... x(t) = g 0 x(t) = g 0 x(t) = g 0 + g k cos(k ω 0 t + ϕ k ) + g k {e + j(k ω 0 t + ϕ ) k } + {g k e + j ϕ k e + jk ω 0 t } x(t) = X 0 X 0 = g 0 X k + = g k e + j ϕ k k = 1, 2,... {X k e + j k ω 0t } 29

30 Phasor epresentation (2) x(t) = g 0 x(t) = g 0 x(t) = g g k 2 (e+ j( k ω 0t + ϕ k) + e j( k ω 0t + ϕ k) ) ( g k 2 e+ j ϕ k e + jk ω 0t + g k 2 e j ϕ k e j k ω 0t) ( g k e + j ϕk 2 e + jk ω 0t + g k e j ϕ k 2 e j k ω 0t) x(t) = g 0 g 0 = a 0 g k = a k 2 + b k 2 ϕ k + g k cos(k ω 0 t + ϕ k ) = tan 1 ( b k a k ) k = 1, 2,... x(t) = + C k e + j k ω 0 t k= x(t) = X 0 + {X k e + j k ω 0t } 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,... 30

31 wo-sided Spectrum x(t) = + C k e + j k ω 0 t k= x(t) = + C k e + j k ω 0 t k= C k = 1 0 x(t ) e j k ω 0 t dt C k = 1 0 x(t ) e j k ω 0 t dt 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 ) = tan 1 ( b k /a k ) (k > 0) tan 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 Spectrum 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 31

32 CFS of mpulse rain (1) p(t) = n= δ(t n s ) Fourier Series Expansion of mpulse rain p(t) = + k = a k e + j k ω st Fourier Series Coefficients 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(t ) e j k ω st dt δ(t) e j k ω s 0 dt s s s s s s s = 1 + s /2 s s /2 δ(t) dt = 1 s 3ω s 2 ω s ω s ω s 2 ω s 3 ω s 32

33 CFS of mpulse rain (2) p(t) = 1 + s k= a k e + j k ω s t = 1 + s k = (cos k ω s t j sin k ω s t) cos 2π 1 t sin 2π 1 t cos 2π k t k =1 40 sin 2 π k t k =1 33

34 CFS of mpulse rain (3) p(t) = 1 + s k= a k e + j k ω s t = 1 + s k = (cos k ω s t j sin k ω s t) cos 2π 1 t sin 2π 1 t e-13 5e e-14-1e cos 2π k t k = sin 2π k t k = 40 34

35 nner Product Space Hilbert Space real / complex inner product space f, g = a b f (t) g (t)d t complex conjugate 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-Schwartz nequality x, y x y positive semidefinite x, x 0 35

36 Orthogonality x(t) = + C k e + j kω 0 t k= C k = 1 0 x(t ) e j k ω 0 t dt k =..., 2, 1, 0, +1, +2,... fundamental frequency f 0 = 1 ω 0 = 2 π f 0 = 2 π n-th harmonic frequency f n = n f 0 ω n = 2π f n = 2π n e j m ω 0 t, e j n ω 0 t = 0 e + j(m n)ω 0 t d t = 0 (m n) (m = n) m, n : integer 36

37 nner Product Examples f 0 = 1/ ω 0 = 2π/ e j 1ω 0 t e j 1ω 0 t, e j1 ω 0t = 0 e + j(1 1)ω 0 t d t = e j 1ω 0 t e j 1ω 0 t, e j 1 ω 0t = 0 e + j(1+1)ω 0 t d t = 0 e j 1 ω 0 t e j 1ω 0 t, e j 2ω 0t = 0 e + j(1 2)ω 0 t d t = 0 e j 2 ω 0 t e j 1ω 0 t, e j 2 ω 0t = 0 e + j(1+2)ω 0 t d t = 0 e j 2 ω 0 t 37

38 Cauchy-Schwartz nequality For all vectors x and y of an inner product space x, y 2 x, x y, y x, y x y he equality holds if and only if x and y are linearly dependent maximum a b x(t) y(t)d t a b x(t ) x(t)d t a b y(t) y(t)d t nner product is maximum when y = k x 38

39 Orthogonality 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) 39

40 Complex Vector nner Product Hermitian inner product x, y = x H y = x i y i Norm of Hermitian inner products x H : conjugate transpose x = x, x = x H x = x i x i the length of a vector 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 40

41 he 1st ow of the DF Matrix 0 cycle W 8 k n = e j 2 8 k n k = 0, n = 0, 1,..., 7 samples of samples of cos t = cos t sin t = sin t measure t = 2 f t f s t X[0] measures how much of the +0 ω component is present in x. 41

42 he 3rd ow of the DF Matrix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 t = cos 2 t sin 2 t = sin 2 t measure t = 2 f t f s t X[2] measures how much of the +2 ω component is present in x. 42

43 he 4th ow of the DF Matrix 3 cycles W 8 k n = e j 2 8 k n k = 3, n = 0, 1,..., 7 samples of samples of cos 3 t = cos 3 t sin 3 t = sin 3 t measure t = 2 f t f s t X[3] measures how much of the +3 ω component is present in x. 43

44 he 5th ow of the DF Matrix 4 cycles W 8 k n = e j 2 8 k n k = 4, n = 0, 1,..., 7 samples of samples of cos 4 t = cos 4 t sin 4 t = sin 4 t measure t = 2 f t f s t X[4] measures how much of the +4 ω component is present in x. 44

45 he 6th ow of the DF Matrix 3 cycles W 8 k n = e j 2 8 k n k = 5, n = 0, 1,..., 7 samples of samples of cos 3 t = cos 3 t sin 3 t = sin 3 t measure t = 2 f t f s t X[5] measures how much of the 3 ω component is present in x. 45

46 he 7th ow of the DF Matrix 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 samples of samples of cos 2 t = cos 2 t sin 2 t = sin 2 t measure t = 2 f t f s t X[6] measures how much of the 2 ω component is present in x. 46

47 he 8th ow of the DF Matrix 1 cycle W 8 k n = e j 2 8 k n k = 7, n = 0, 1,..., 7 samples of samples of cos t = cos t sin t = sin t measure t = 2 f t f s t X[7] measures how much of the 1 ω component is present in x. 47

48 eferences [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] G. Beale, [4] C. Langton,