Junji OHTSUBO 2012
FFT FFT
SN
sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c], b = Im[c] (1.3)
r = a 2 + b 2 (1.4a) θ = tan 1 b a (1.4b) 1.1 cosθ=a/r sinθ=b/r r c c = r θ arg(c)=θ (Euler) c = rexp(iθ) sinθ = cosθ = exp(iθ) exp( iθ) 2i exp(iθ) + exp( iθ) 2 = Im c r = Re c r (1.5a) (1.5b) (1.5c) (1.5a) (1.5a) c c* c* = rexp( iθ) = r{cos( θ) + isin( θ)} = a ib (1.6) (1.2) A 0 ψ = Re[A 0 exp{i(kx ωt + φ)}] (1.7) A= A 0 exp{i(kx-ωt+φ)} (1.2) cos
I =<ψ 2 >= ω π /ω A 2 2π 0 cos 2 (kx ωt + φ)dt = A 0 (1.8) π /ω 2 2 (1.11) (1.2) A I = AA* 2 = A 2 0 2 (1.9) (1.5a) A 0 φ ψ 1 = A 01 cos(kx ωt + φ 1 ) = Re[A 01 exp{i(kx ωt) + iφ 1 }] (1.10a) ψ 2 = A 02 cos(kx ωt + φ 2 ) = Re[A 02 exp{i(kx ωt) + iφ 2 }] (1.10b) ψ =ψ 1 +ψ 2 = Re[A 1 + A 2 ] (1.11) = A 01 exp(iφ 1 ) + A 02 exp(iφ 2 ) cos{kx ωt + arg(a 1 ) arg(a 2 )} ψ 1 = 4cos(kx ωt) ψ 2 = 3cos(kx ωt + π /2) cos ψ =ψ 1 +ψ 2 = Re[4 exp{i(kx ωt)} + 3exp{i(kx ωt) + i π 2 }] = Re[ 4 + i3 exp{i(kx ωt + tan 1 3 4 )}] = 5cos(kx ωt + tan 1 3 4 ) (1.12) 1.2
sin cos 1.2 f(x) x 0 f(x+x 0 )=f(x) f(x) f (x) = a 0 x 0 + 2 x 0 {a n cos(n2πνx) + b n sin(n2πνx)} (1.13) n =1 ν=1/x 0 cos sin a n b n a n = b n = x 0 / 2 f (x)cos(n2πνx)dx (n=0,1,2, ) (1.14a) x 0 / 2 x 0 / 2 f (x)sin(n2πνx)dx (n=1,2,3, ) (1.14b) x 0 / 2 cos sin (1.13) ν
(1.13) cos sin cos sin (1.13) f(x) f (x) = a 0 x 0 + 1 x 0 (a n ib n )exp(i 2πnx ) + 1 (a x n + ib n )exp( i 2πnx )(1.15) 0 n =1 A 0 = a 0 A n = a n ib n A n = a n + ib n f (x) = 1 x 0 x 0 n =1 x 0 (1.16a) (1.16b) (1.16c) A n exp(i 2πnx ) (1.17) n = x 0 n A n f(x) A n A n = x f (x)exp i 2nπx 0 / 2 dx (1.18) x 0 / 2 x 0 (1.17) A -n =A n * f(x) (1.13) (1.17) (1.17) x 0 1/2
f(x) 1 0 x 0 /2 x 0 x 1 1.3 1.3 1 nx 0 x < (n + 1 f (x) = 2 )x 0 1 (n + 1 (1.19) 2 )x 0 x < (n +1)x 0 (1.17) (1.18) f (x) = 2 1 iπ 2n 1exp{i2π(2n 1) x } (1.20) n = x 0 (1.19) (1.20) 1.4(a) (1.20) (1.13) f (x) = 4 1 π 2n 1 sin{2π(2n 1) x } (1.21) x 0 n =1 n 1 4 1.4(b) n 1 5 n=5 1.3 1.4 (a) (b)
(1.13) (1.17) 1/x 0 (1.17 ) a n b n (1.17) A n f(x) 1.5 (1.20) 1.5 1.5 cos sin f(x) {ψ n } f (x) = F n ψ n (x) (1.22) n = F n ψ n (x)ψ * n (x)dx = δ nm (1.23) δ nm (Kronecker) n=m δ nm =1 δ nm =0 f(x) F n
F n = f (ξ)ψ * n (ξ)dξ (1.24) sin cos sin cos (1.14) (1.23) exp (1.17) (1.13) sin cos sin cos sin cos (1.13) sin cos f(x) 1 1.3 (1.19) (1.21) (1.20)
δ(x) = 0 (x 0) δ(x)dx =1 (2.1) x=0 x=0 2.1 du(x) dx = 0 (x 0) (2.2a) du(x) dx dx = [ u(x) ] =1 (2.2b) (2.1) ε(ε 0) 1 x ε 2ε δ(x) = 0 x > ε (2.3) (2.1) 2.1
f(x) (2.3) f (x)δ(x)dx = 1 ε f (x)dx = f (θ) f (0) (ε 0) (2.4) 2ε ε ε [ ε,ε] f(x) f(θ)={f(ε)+f( ε)}/2 (2.4) x=0 δ(x) a f(x) (2.4) f (x)δ(x a)dx = f (a) (2.5) (2.5) f(x) 2.1 δ f (x)δ( x)dx = f ( x)δ(x)dx = f (0) (2.6) δ(x) = exp( i2πνx)dν (2.7) f(x)
{ψ n } f (x) = F n ψ n (x) (2.8) n = F n ψ n (x)ψ * n (x)dx = δ nm (2.9) δ nm (Kronecker) n=m δ nm =1 n m δ nm =0 F n f(x) F n = f (ξ)ψ * n (ξ)dξ (2.10) f(x) f (x) = f (ξ) ψ * n (ξ)ψ n (x)dξ (2.11) n = (2.5) δ(ξ x) = ψ * n (ξ)ψ n (x) (2.12) n = ψ n (x) = 1 2N exp(i nπ N x) (2.13) δ(ξ x) = 1 2N exp{ i nπ (ξ x)} (2.14) N n = N (2.14) δ(ξ x) = exp{ i2πν n (ξ x)}δν n exp{ i2πν(ξ x)}dν (2.15) n = (2.7) Δν n = ν n +1 ν n
(2.7) t=0 0 DC 0 1 δ 2 δ δ(x) = N exp( N 2 πx 2 ) (N ) (2.16) 1 x 1 δ(x) = Nrect(Nx) = 2N 0 x > 1 (N ) (2.17) 2N δ(x) = Nsinc(Nx) = N sinπnx πnx (N ) (2.18) rect(x) rectangular sinc(x) sinc sinc(x) = sin(πx) /(πx) 2.2 (2.5)
2.2 δ (a) (b) (c)sinc (1.13) x 0 (1.14) (1.18) n/x 0 (1.18) A n ν=n/x 0 (ω=2πν) F(ν) (1.17) f (x) = F(ν)exp(i2πνx)dν (2.19) f(x) f(x) exp( i2πνx) [, ] f (x)exp( i2πνx)dx = dx = F(ν')δ(ν' ν)dν' dν'f(ν')exp{i2π(ν' ν)x} (2.20) = F(ν) (2.19) (2.20) f(x) F(ν) (2.20) f(x) (2.19) F(ν) 2.3(a) [ a/2,a/2] 1 x a /2 f (x) = 0 x > a/2 (2.21)
(2.20) F(ν) = a / 2 exp( i2πνx)dx = asinc(aν) (2.22) a / 2 F(ν) 2.3(b) (2.21) ν f(x) F(ν) F(ν) F(ν) = F(ν) exp{iφ(ν)} (2.23) φ(ν) F(ν) 2 f(x) Φ(ν) = F(ν) 2 (2.24) f(x) ν 2.3 (a) (b)
FT FT -1 FT[αg(x) + βf (x)] = αg(ν) + βf(ν) (2.25) FT[g(ax)] = 1 a G(ν a ) (2.26) α β a (2.26) a 1/a a FT[g(x a)] = exp( i2πaν)g(ν) (2.27) a a (shift invariant) FT 1 [FT[g(x)]] = g(x) (2.28) FT[FT[g(x)]] = g( x) (2.29) 2 FT[g(x, y)] = FT[g x (x)g y (y)] = FT[g x (x)]ft[g y (y)] (2.30) df (x) exp( i2πνx)dx = i2πνf(ν) (2.31) dx f(x) f (±) 0 FT[ d n dx n f (x)] = (i2πν)n F(ν) (2.32)
(FFT: Fast Fourier Transform) FFT (2.20) dx f(x) F(ν) 2π cos sin cos sin 1 FFT 2 (2.20) x ν Δx Δν F(nΔν) = f (lδx)exp( i2πnδν lδx)δx (2.33) l = n Δx Δν 4 n l N Δx=1 (2.33)
N 1 nl F n = f l W N (2.34) l =0 Δν = 1 NΔx = 1 N W N nl (2.35) = exp i2π nl = cos 2π nl isin 2π nl (2.36) N N N (2.34) N F 0 = F N F N 1 f l (N-1)/2 N/2 FFT (2.19) (2.34) N 1 N 1 N 1 N 1 nl F l W N = f k W kl nl (k n)l N W N = f k W N l =0 l =0 k =0 k =0 N 1 l =0 N 1 l =0 nl F l W N (2.37) N 1 = f k Nδ kn = Nf n k =0 f n = 1 N 1 nl F N l W N (2.38) l =0 1/N (2.35) 1/N N (2.34) (DFT: Discrete Fourier Transform)
(2.34) n 0 N 1 f W N 2 (2.36) cos sin N 2 N=8 F 0 f 4 W 0 8 F 2 f 4 W 8 8 = f 4 W 0 8 F 4 f 4 W 16 8 = f 4 W 0 8 F 6 f 4 W 24 8 = f 4 W 0 8 (2.34) N 2 N N DFT cos sin 1 cos sin cos sin cos sin 1/8 cos sin FFT FFT 2.4(a) (Cooley-Tukey) FFT 2.4(b) N=8 2 2 N=2 3 =8 3 2 2.4(a) A B W s W t N=8 2.4(b) W m m=8 W m m W F 3 F 3 = f 0 + f 1 W 3 + f 2 W 6 + f 3 W 6+3 + f 4 W 4 + f 5 W 4 +3 + f 6 W 4 +6 + f 7 W 4 +6+3 = f 0 W 0 + f 1 W 3 + f 2 W 6 + f 3 W 9 + f 4 W 12 + f 5 W 15 + f 6 W 18 + f 7 W 21 (2.39)
F 3 (2.34) FFT 2 N 2 n n 2.4 (b) f F 2.4 FFT (a) (b)
FFT 2 FFT FFT 2 (2.26) (2.27) (2.28) (2.29) (2.31)
3.1 f(x) g(x) f (z) = f (ξ)δ(z ξ)dξ (3.1) (3.1) ( ) (convolution) ξ (4.1) f ξ x (3.1) δ z ξ S (3.1) S g(x) g(x) = S[ f (z)] = f (ξ)s[δ(z ξ)]dξ (3.2) -
z z x S z (3.2) S S[δ(z ξ)] = h(x,ξ) (3.3) h(x) h(x) 3.1 (3.3) f(ξ) g(x) = f (ξ)h(x,ξ)dξ (3.4) h(x,ξ) = h(x ξ) (3.5)
g(x) = f (ξ)h(x ξ)dξ = f (x) h(x) (3.6) f(x) h(x) (3.6) (3.6) (4.1) f(ξ) ξ x g(ξ-x) f(ξ) ξ x h(x-ξ) (3.6) h h(ξ-x) h(x-ξ) h(ξ-x) 3.2 n=0 n=2 (3.6) g 3.2 f h f(0) n=2
3.3 (a) (b) (c) 2 g 0 (2) = h(m) f (0) (3.7) m =0 f(1) f(2) 2 g 1 (2) = h(m 1) f (1) (3.8a) m =0 2 g 2 (2) = h(m 2) f (2) (3.8b) m =0 h(m) m h(m)=0 n=2 g(2) 2 g(2) = h(n m) f (m) (3.9) m =0 3.3 3.3(b)
3.3(c) 3.3(a) (3.6) f(x) F(ν) 3.1 h(x) g(x) H(ν) G(ν) (3.6) (3.6) h(x) g(x) f(x) f(x) g(x) G(ν) = { f (ξ)h(x ξ)dξ}exp( i2πνx)dx = f (ξ)h(x ξ)exp{ i2πν(x ξ) i2πνξ}dξdx = [h(x ξ)exp{ i2πν(x ξ)}dx] f (ξ)exp( i2πνξ)dξ (3.10) = H(ν)F(ν)
H(ν) G(ν) F(ν)=G(ν)/H(ν) F(ν) f(x) H(ν) ν F(ν) T(ν) =1/ H(ν) 5.2 H(ν) (transfer function) Φ F (ν) = F(ν) 2 Φ G (ν) = G(ν) 2 (3.10) Φ G (ν) = H(ν) 2 Φ F (ν) (3.11) 3 H(ν)F(ν) h(x) f (x)
f(x) x f(x) x' f(x') f(x) g(x) 1 R(x) = lim T T T / 2 f (ξ) f * (ξ x)dξ (4.1) T / 2 T ξ (4.1) 6 < > R(x) =< f (ξ) f * (ξ x) > (4.2) (4.1) x 6 f(x) f(x) R(x) = f (ξ) f * (ξ x)dξ = f (ξ + x) f * (ξ)dξ = R( x) (4.3) (4.2) (4.1) x=0 R(0) = f (ξ) 2 dξ (4.3) R(0)
4.1(a) R(x) = rect(ξ)rect(ξ x)dξ = 1/ 2 1/ 2 x 1/ 2+x 1/ 2 dξ =1 x x 0 dξ =1+ x x < 0 (4.4) 4.1(b) 4.2(a) (4.1) 4.2(b) 4.1 (a) (b) 4.2 (a) (b) f(x) g(x) R(x) = f (ξ)g * (ξ x)dξ (4.5)
x (2.24) Φ(ν)exp(i2πνx)dν = 2 f (x)exp( i2πνx)dx exp(i2πνx)dν = { f (x 1 )exp( i2πνx 1 ) f * (x 2 )exp(i2πνx 2 ) dx 1 dx 2 }exp(i2πνx)dν = f (x 1 ) f * (x 2 )exp{i2πν(x 2 x 1 + x) }dνdx 1 dx 2 = f (x 1 ) f * (x 2 )δ{x 2 (x 1 x)}dx 1 dx 2 = f (x 1 ) f * (x 1 x)dx 1 = R(x) (4.6) (Wiener-Khintchine theorem) x=0 Φ(ν)dν = F(ν) 2 dν = f (x) 2 dx (4.7) (Parseval theorem) f(x) g(x) f (t)g * (t)dt = F(t)G * (t)dt (4.8) f(x) g(x) f(x)=g(x) (4.7)
4.3 f(x) 4.3 x X f s (x) f s (x) f s (x) = comb( x ) f (x) (4.9) X comb(x) comb(x) = δ(x n) (4.10) n= AD
(4.9) f s (x) F s (ν) = FT[comb( x )] F(ν) (4.11) X comb comb (4.10) comb(x)exp( i2πνx)dx = exp( i2πnν) (4.12) n= comb comb(x) = a n exp( i2πnx) (4.13) n= a n 1/ 2 a n = comb(x)exp( i2πnx)dx =1 (4.14) 1/ 2 (4.10) (4.13) (4.11) F s (ν) = δ(ν n X ) F(ν) = F(ν n ) (4.15) X n= n= (4.11) comb FT[comb( x )] = Xcomb(Xν) = Xδ(Xν n) (4.16) X (4.15) δ(xν n)g(ν)dν = 1 X δ(ν n )g(ν)dν (4.17) X
δ 4.4 (4.15) 4.4 F(ν) ν=1/x B 1/2X ν=0 X 1 2B (4.18) X=1/2B 4.4 H(ν) = rect( ν 2B ) (4.19) X=1/2B (4.19) (4.15) F s (ν)h(ν) = F(ν) (4.20)
(4.20) f(x) (4.20) f (x) = [comb( x ) f (x)] h(x) = { X δ( n= = f ( n 2B )sinc{2b(x n 2B )} n= x X n) f (x)} {2Bsinc(2Bx)} (4.21) (2.64) 4.5 X=1/2B sinc (Shannon) 4.5 4.5 B X << 1/2B (4.19) B X >>1/2B (2.62)
X =1/2B X =1/2B X =1/2B 4 (4.8)
f(x) h(x) g(x) = h(x) f (x) = h(ξ x) f (ξ)dξ (5.1) 5.1 (a) (b) h(x) g(x) f(x) (5.1) (deconvolution) 5.1 f(x) h(x)
5.1(a) h(x) 5.1(b) g(x) h(x) f(x) (5.1) N g(x) f(x) i g i f i h(x) {h ij } (5.1) N g i = h(i j)x( j) = h ij x j (5.2) j =1 j =1 N (5.2) h h(1-n) h(n-1) g x g x h H (5.2) g = H f (5.3) 5.2 5.2 H 5.1 f(x) H H 0 f H H ij f i = N j =1 g j H ji H (5.4)
(5.4) H 0 (Jaccobi) H D 0 D -1 k+1 f (k +1) = f (k ) + D 1 (g H f (1) ) (5.5) (5.5) f (k +1) = f (k ) g = H f (k +1) = H f f (0) g f (0) = g D 5.2 h 11 =h 0 (5.5) f (k +1) = f (k ) + 1 (g H f (1) ) (5.6) h 0 (5.6) 1/h 0 d d/h d 0 h 0 <d<h 0 (5.3) (5.6) f (k+1) i f (k+1) j j=1 i 1 (Gauss-Seidel ) f (k+1) j j=i 1 (5.6) N (k f +1) (k 1 = f ) 1 + 1 (k (g h 1 h 1 j f ) j ) (5.7a) 0 j =1
i 1 N (k f +1) (k i = f ) i + 1 (k (g h i h 1 j f +1) (k j h 1 j f ) j ) (i = 2,3,,N) (5.7b) 0 j =1 j =i (5.1) (5.6) H(ν) 0 H(ν) 0 H(ν)=0 3.1 H(ν) G(ν) F(ν) F(ν)= G(ν)/H(ν) g(x) G(ν) T i (ν)=1/h(ν) f(x) T i (ν) H(ν) H
5.3 g(x) n(x) u(x) g(x) 5.3
n(x) g(x) n(x) u(x) = g(x) + n(x) (5.8) u(x) g(x) u(x) t(x) g (x) g (x) = t(x) u(x) (5.9) 2 e = g (x) g(x) 2 dx (5.10) (5.10) 2 2 2 E =< G (ν) G(ν) 2 >=< T(ν){G(ν) + N(ν)} G(ν) 2 > (5.11) < > N(ν) < N >=< N * >= 0 E = G 2 + T 2 ( G 2 + < N 2 >) T G 2 T * G 2 (5.12) ν T E = Γ T G 2+ 2 G 2 G 4 Γ Γ (5.11) Γ = G 2 + < N 2 > (5.12) T(ν) = G(ν) 2 G(ν) 2 + < N(ν) 2 > (5.13)
T(ν)=1 g(x) G(ν) G(ν) u(x) h(x) u(x) f(x) (5.1) u(x) = g(x) + n(x) = h(x) f (x) + n(x) (5.14) u(x) f(x) t(x) n(x,y) t(x)
f (x) = t(x) u(x) (5.15) 2 e = f (x) f (x) 2 dx (5.16) 2 (5.14) U = HF + N (5.17) (5.16) 2 E =< F F 2 >=< T(HF + N) F 2 > (5.18) 2 E = F 2 + T 2 ( H 2 F 2 + < N 2 >) T * F 2 H * T F 2 H (5.19) E = Γ T H* F 2+ 2 F 2 H 2 F 4 (5.20) Γ Γ Γ = H 2 F 2 + < N 2 > (5.21) T T(ν) = F(ν) 2 H * (ν) F(ν) 2 H(ν) 2 + < N(ν) 2 > (5.22) < N(ν) 2 >= 0 T T(ν) = T i (ν) =1/H(ν) N(ν) F(ν) H(ν) T i (ν)=1/h(ν)
H(ν) (5.22) T(ν) = H * (ν) H(ν) 2 + < N(ν) 2 > F(ν) 2 (5.23) α =< N(ν) 2 > / F(ν) 2 5.4 5.4(a) sinc H 0 T i(ν)=1/h(ν) 5.4(b) 0 5.4(c) (5.23) H 0 H 0 5.4 (a) (b) (c) (5.23) 0 5 (5.3) H f (5.4)
X x (sample) X X <x P X (x) P X (x) (probability distribution) x - + P X ()=1 x x 0 P X (x)=p(x) P (x) x (probability density distribution) p(x) = dp(x) dx (6.1) x x+δx p(x)δx 6.1 6.1(a) x x+δx (
6.1 (a) b ) 6.1(b) p(x)δx x x+δx p(x) x P (x) p(x) =1 (6.2) x 6.1 X x x x(t) x x x(t) x(t)
x(t) [0,T] 1 x = lim T T T x(t)dt (6.3) 0 6.2 [0,T] N {x k (t)}(k=1,2,,n) t=t 0 ensemble average 1 x 0 = lim N N N x k (t 0 ) (6.4) k =1 x = x 0 < x >= x x x 6.2
x 1 σ 2 = lim N N N (x k x ) 2 (6.5) k =1 σ 2 =< (x k x ) 2 > (6.6) x Δx x k p(x k ) x k x k p(x k )Δx x x = xp(x)dx (6.7) x σ 2 = (x x ) 2 p(x)dx (6.8) σ x σ 2 2 n x n m n =< x n >= x n p(x)dx (6.9) 1 2 m 1 = x (6.10a) m 2 = σ 2 2 + m 1 (6.10b)
2 2 2 2 1 X<x Y<y 2 P X,Y (x,y) P X,Y (x,y) x y p X,Y (x,y) x y p X,Y (x,y) = p X (x)p Y (y) (6.11) 2 y x p X Y (x y) = p X,Y (x, y) p Y (y) y x 2 x (6.12) x = xp X,Y (x,y)dxdy = xp X (x)dx (6.13) x σ 2 x = (x x ) 2 p X,Y (x,y)dxdy = (x x ) 2 p X (x)dx (6.14) y x y R xy C xy (covariance) R xy =< xy >= xyp X,Y (x, y)dxdy (6.15)
C xy =< (x x )(y y ) >= (x x )(y y ) p X,Y (x, y)dxdy (6.16) 6.3 (a) (b) (Normal distribution) (Gaussian distribution) x p(x) = 1 2πσ exp (x x ) 2 2 2σ 2 (6.17) σ 2 ±σ 68.5% ±3σ 99.7 6.3 6.3(a)
4.2 6.3(b) 6.3(a) 2 2 2 1 p(x, y) = 2πσ 1 σ 2 1 r exp 1 (x x ) 2 2 2(1 r 2 2 ) σ 1 2r(x x )(y y ) (y y )2 + 2 σ 1 σ 2 σ 2 (6.18) r x y r =< (x x )(y y ) > r=0 2 p(x, y) = p(x) p(y) x n n < x n >= 1 x n k = x n (6.19) k =1 < x n > x (n ) <x n > n x n σ 2 n =< (x n x ) 2 >=< 1 (x n k x ) 2 > (6.20) k =1 x < (x i x )(x j x ) >= 0 (6.19) (6.20)
σ 2 n =< (x n x ) 2 >=< 1 n = 1 n < 1 n n k =1 n k =1 (x k x ) 2 + 1 n x k x n i j 2 >=< 1 n (x n k x ) k =1 (x i x )(x j x ) >= 1 n < 1 n 2 > n k =1 (x k x ) 2 >= σ2 n (6.21) n 1 1/n (central limit theorem) (large number theorem) x = x 1 + x 2 + + x n (6.22) x i n x x SN
R Δν 2 < v 2 >= 4k B TRΔν (6.23) k B 2 < i 2 i 2 >= 2eI Δν (6.24) e I 1/f Signal-to-Noise ratio: SN SNR SN P S P N SNR =10log 10 P S P N (6.25) [db] SNR 0dB 0dB
1 6 p(x) = σ 2π exp x 2 2σ 2 x 2 p(x)dx σ 1 p(x) = σ 2π exp x 2 2σ 2 x σ 0 e ax 2 dx e ax 2 dx = π 2 2 a 0 1 (1) a 0 x 2 e ax 2 dx (2) a = 1 2σ 2 0 x 2 e x 2 2σ 2 dx x 2 p(x)dx σ
SN SN 7.1 RC 7.1(a) 7.1(b) H(ω) 7.1 RC (a)rc (b)
H(ω) = R R 0 1 1+ iωrc (7.1) h(t) = 1 R 0 C exp( t RC ) (7.2) V in RC 3dB RC 0 i s i i N N S = s i = Ns (7.3) i=1 N s s 1 s N i N i=1
n i N 0 σ n 2 = 1 n N σ 2 N =< n i i=1 2 k =1 (x k x ) 2 >= Nσ 2 (7.4) σ 2 N N SN = S σ N = N s σ N SN N (7.5) N 1 7.1 RC RC h(x) t x f(x) g(x) g(x) = h(x) f (x) (7.6) 7.6
7.6 f(x) AD f(i) i w(i) i g(i) = 1 W m w(i + j) f ( j) 7.7 j = m m W = w( j) 7.8 j = m j (7.7) g N g(i) f(j) N+2m 7.2 w(j) w( j) =1 j [ m, +m] M=2m+1 i g(i) = 1 M m f (i + j) 7.9 j = m 1/N SN 1/ N M
7.2 2 7.3 i 2m+1 2 g( j) = a( j i) 2 + b( j i) + c j = m + i,,m + i (7.10) a b c 2 m +1 e = {g(x) f (x)} 2 (7.11) j = m +1 7.3 a b c w(j) (7.7)
7 (m=3) w(j) w(±3) = 2 w(±2) = 3 w(±1) = 6 w(0) = 7 (7.8) W W=21 1 f(i) s(i) n(i) f (i) = s(i) + n(i) (7.12) 0 σ n 2 f (i) 2 σ f 2 (i) =< { f (i) f (i)} 2 > (7.13) s (i) s (i) = σ 2 2 f σ n 2 { f (i) f (i)} + f (i) (7.14) σ f f (i) =< f (i) > (7.15) (7.14) (7.14) s (i) = f (i) σ f 2 ~ σ n 2 s (i) = f (i) (7.6) f(x) h(x) g(x)
G(ν) = H(ν)F(ν) (7.16) H(ν) (7.7) H(ν) = sinc(x) = sinπx πx (7.17) (7.17) 0 FFT (7.10) i 2m+1 g'( j) = 2a( j i) + b (7.18) i
g'(i) = b m=3 2 w(±3) = 3 w(±2) = 2 w(±1) = 1 w(0) = 0 1 1 0 (7.12) s(x) s(x) = Acos(ω 0 t +θ) (7.19) R n (x) R(x) = A2 2 cosω x + R 0 n (x) (7.20) 7.4 7.4(a) n(x) 7.4(b) s(x) R(x) 7.4(c) x=0 0
R n (x) R(x) x R(x) 7.4 (a) (b) (c) ν 0 =ω 0 /2π (7.20) Φ(ν) = A2 4 δ(ν ν 0 ) + C (7.21) (7.21) C
(7.6) (7.6) H(ν) G(ν) = H(ν)F(ν) f(x) F(ν) G(ν) H(ν) G(ν) f(x) h(x) g(x) g(ξ) = f (ξ')h(ξ ξ')dξ'= f (ξ ξ')h(ξ')dξ' (7.21) R fg (x) = f (ξ)g(ξ + x)dξ = f (ξ x)g(ξ)dξ (7.22) R fg (7.21) ξ
R fg (x) = f (ξ x) f (ξ ξ')h(ξ')dξdξ' = f (ξ) f (ξ + x ξ')h(ξ')dξdξ' = R ff (x ξ')g(ξ')dξ' (7.23) R ff Φ fg (ν) = H(ν)Φ ff (ν) (7.24) Φ ff Φ fg (ν) = H(ν) H(ν) s(x) (matched filter) f(x) n(x) f (x) = s(x x 0 ) + n(x) (7.25) x 0 s(x) R fs (x) = f (ξ)s( ξ x)dξ = R ss (x x 0 ) (7.26) x=x 0 s(x)
7 7.1 RC (7.1) (7.2)