[1] 1.1 x(t) t x(t + n ) = x(t) (n = 1,, 3, ) { x(t) : : 1 [ /, /] 1 x(t) = a + a 1 cos πt + a cos 4πt + + a n cos nπt + + b 1 sin πt + b sin 4πt = a

13/7/1 II ( / A: ) (1) 1 [] (, ) ( ) ( ) ( ) etc. etc. 1. 1

[1] 1.1 x(t) t x(t + n ) = x(t) (n = 1,, 3, ) { x(t) : : 1 [ /, /] 1 x(t) = a + a 1 cos πt + a cos 4πt + + a n cos nπt + + b 1 sin πt + b sin 4πt = a + ( a n cos nπt + b n sin nπt ) n=1 + + b n sin nπt + x(t) (Fourier series) a n, b n (1) a n = x(t) cos nπt dt (n =, 1,, ) () b n = x(t) sin nπt dt (n = 1,, ) (3) a n, b n

) π x(t) = { x(t + π) = x(t) 1 ( π < t < ) 1 ( < t < π) ) = π a = 1 π a n = 1 π b n = 1 π x(t)dt = 1 { ( 1)dx + π π π π { π = 1 π = 1 π π x(t) cos ntdt = 1 π π π ( 1) cos ntdt + π } dx = 1 ( π + π) = π π [ 1 ] n sin nt + 1 [ ] 1 π sin nt = (n ) π π n x(t) sin ntdt = 1 { ( 1) sin ntdt + π π π π [ ] 1 cos nt 1 [ ] 1 π cos nt = (1 cos nπ) n π π n nπ = nπ {1 ( 1)n } (n = 1,, ) } cos ntdt } sin ntdt (sin t + 13 sin 3t + 15 sin 5t + 17 sin 7t + ) x(t) = 4 π x(t). g( t) = g(t) g(t) h( t) = h(t) h(t) cos sin = = = g(t)dt = g(t)dt, h(t)dt = 3

(a) 4 π sin t, (b) 4 π sin t + 4 3π sin 3t, (c) 4 π sin t + 4 4 3π sin 3t + 5π sin 5t, (d) n 1. (1) x(t) = n= e ±iθ = cos θ ± i sin θ cos θ = 1 sin θ = 1 i (e iθ + e iθ) (e iθ e iθ) { A n e inπt/ + B n e inπt/ } (4) A n = a n ib n = 1 x(t)e inπt/ dt (5) B n = a n + ib n = 1 x(t)e inπt/ dt (6) n x(t) = C n e inπt/ (7) n= C n = 1 x(t)e inπt/ dt (8) (complex Fourier series) 4

[ /, /] /n (n = 1,, ) harmonic wave e inπt/ (= cos(nπt/ ) + i sin(nπt/ )) C n 1.3 f n = n/, f = 1/ (7) (8) { } 1 x (t) = x(t)e inπt/ dt e inπt/ n= = n= { x(t)e iπf nt dt } e iπf nt f x(t) = lim x (t) f x(t) = X(f) = X(f)e iπft df (9) x(t)e iπft dt (1) (Fourier integral) (Fourier transform) (9) (1) f ω = πf x(t) = X(ω)e iωt dω (11) X(ω) = 1 x(t)e iωt dt (1) π ( X(ω)/π X(ω) ) 5

t x, ω k f(x) = F (k)e ikx dk (13) F (k) = 1 f(x)e ikx dx (14) π 1/π ( ) ) x(t) ( ) ( a > ) { e at (t > ) x(t) = (t < ) ) X(f) = = [ x(t)e ift dt = e (a+if)t a + if ] = 1 a + if e at e ift dt 1.4 ( ) x(t) = X(f)e iπft df X(f) f e iπft X(f) (Energy Spectrum Density; ESD) f X(f) Φ(f) = X(f) (15) (Power Spectrum Density; PSD) ( /, /) X(f) 6

X(f) P (f) = lim X (f) X(f) X(f) X (f) = lim (16) x 1 = lim x (t)dt = P (f)df (17) f f + df x P (f)df (15) (16) P (f) P P(f) df df f 3 [].1 x(t) x(t) = x(t ± n ) (n =, 1,, ) n x(t) 7

x = x(t) y = x(t + τ) x(t) τ C(t, τ) = E[x(t)x(t + τ)] (18) (auto-correlation function) τ (lag) ( ) 1 E[f k (t)] = lim N N N f k (t) ( ) k=1 1 E[f k (t)] = lim f k (t)dt 1 C(τ) = x(t)x(t + τ) = lim x(t)x(t + τ)dt (19) C(τ) t τ. (1) (18) τ = τ 1 C( τ) = lim x(t)x(t τ)dt 1 τ = lim x(t 1 )x(t 1 + τ)dt 1 τ t 1 = t τ () τ = τ {x(t) ± x(t + τ)} C(τ) = C( τ) () 1 lim {x(t) ± x(t + τ)} dt 8

1 = lim x 1 (t)dt + lim x 1 (t + τ)dt ± lim x(t)x(t + τ)dt = C() ± C(τ) > C() > C(τ) (τ ) (1) (3) τ C(τ) (τ ) ().3 x(t) / < t < / x(t) = (t > / ) x(t) = X(ω) = 1 π X(ω)e iωt dω x(t)e iωt dt 1 C(τ) = lim x(t)x(t + τ)dt { 1 } = lim x(t) X(ω)e iω(t+τ) dω dt { 1 } = lim X(ω)e iωτ x(t)e iωt dt dω = { lim πx(ω)x } (ω) e iωτ dω (3) x(t)e iωt dt = x(t)e iωt dt = πx (ω) τ = C() = x { x πx(ω)x } (ω) = lim dω S(ω)dω (16) f ω, P S 9

S πx(ω)x (ω) π X(ω) S(ω) = lim = lim (4) x(t) X(ω) S(ω) () (3) C(τ) S(ω) C(τ) = S(ω)e iωτ dω (5) S(ω) = 1 C(τ)e iωτ dτ (6) π (Wiener-Khintchine) 3 (Wiener-Khintchine) ω > x = S(ω) G(ω) ( G(ω)dω = G(ω) = S(ω) ) S(ω)dω S(ω) : (two-sided spectrum) G(ω) : (one-sided spectrum) 1

[3] 3.1 (1) x(t) [-, ] x(t) (window function) W (t) { 1 ( t ) W (t) = ( t > ) W (boxcar fnction ) x(t) W (t)x(t) x(t) P (f) = (7) W (t)x(t)e iπft dt (8) W (t) x(t) F (x(t)) F (W (t)x(t)) = F (W (t)) F (x(t)) W (t) Q (f) = W (t)e iπft dt = [ e e iπft iπft dt = iπf ] = sin πf πf (9) ( (b)) Q(f) Q(f) = W (t) = W (t)e iπft dt (3) Q(f)e iπtf df (31) f = ( ) 11

W 1 1 Q 1 / Q / W 1 1-1 1 t/ (a) f (b) 4 (a) W, W 1, (b) Q, Q 1. W 1 = 1 t (3) W = 1 ( 1 + cos πt ) ( Hunning) (33) W 3 =.54 +.46 cos πt ( Humming) (34) ( ) sin πf Q 1 (f) = (35) πf Q (f) = 1 Q (f) + 1 4 ( {Q f + 1 ) ( + Q f 1 )} ( Q 3 (f) =.54Q (f) +.3 {Q f + 1 ) ( + Q f 1 )} (36) (37) W 3 W 1-1 1 t/ (a) 1 Q / Q /1.8 1 f (b). -.4 5 (a) W, W 3, (b) Q, Q 3. 1

() ( f S ) ( ) 6 t f N = 1 t = 1 f S (38) (Nyquist frequency) (= t = 1/f N ) ( ) t (aliasing) 7 13

x(t) t C(τ) τ = n t (n: ) P (f) P (f) = τ n= e iπfn τ C(n τ) (39) C(n τ) = 1 τ 1 τ e iπfn τ P (f)df (4) τ = t, f N = 1/ τ P (f) C(τ) = ( + = C(n τ) = 1 τ 1 τ = 1 τ 3 τ 1 τ 1 τ [ 1 + τ 1 τ + k= e iπfn τ P (f)df 3 τ 1 τ + ) e iπfn τ P (f)df e iπ( ( ) ] k k τ +f)n τ P τ + f df e iπfn τ [ k= e iπkn = 1 (41) (4) P (f) = k= P P ( ) ] k τ + f df (41) ( ) k τ + f = P (f) + P (f N f) + P (f N + f) + P (4f N f) + P (4f N + f) + (4) 14

C(τ) τ P (f) P (f) P (f) f N = 1/ t f > f N f < f N f f N P 8 f N (folding) (3) x(t) X(f) P (f) = lim X(f) x(t) X(f) = X R(f) + X I (f) ( X R X I X(f) ) x(t) X R (f) X I (f) ˆP (f) Gauss k = χ - (C.V.) C.V. = k = 1 1% ( ) ( ) ( ) 15

3. 1. Blackman-urkey 195 Wiener-Khinchine 1965 Cooley ukey FF( ). ( ) Cooley and ukey (1965) (FF=Fast Fourier ransform) (FF ) Wiener-Khinchine Blackman-urkey FF 3. MEM( ) Burg (1967) MEM 16