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3 15 5 22

1 2 1.1... 2 1.2... 4 1.3... 5 1.4... 8 1.5 (Matlab )... 11 2 15 2.1... 15 2.2... 16 2.3... 17 3 19 3.1... 19 3.2... 2 3.3... 21 3.4... 22 3.5... 23 3.6... 24 3.7... 25 3.8 Daubechies... 26 4 29 4.1... 29 4.2... 34 4.3 ( )... 35 4.4... 39 4.5... 41 1

1 1.1 x(t) (non-stationary process) (time variant process) x(t) t t t [t T/2, t + T/2] T T t ±ω x(t) =sin(ω t) t = x(t) [ T/2, T/2] x T/2 (t) x(t) 1, T P T/2 (t) = 2 t T 2, others x T/2 (t) =x(t)p T/2 (t) F x T/2 (t) =x(t)p T/2 (t) (1.1) X T/2 (ω) =F[x T/2 (t)] = F[x(t)] F[P T/2 (t)] P T/2 (t) F[P T/2 (t)] = T sinc(tω/2) = (2/ω)sin(Tω/2) x(t) F[x(t)] ±ω (δ(ω ω )+δ(ω + ω )) X T/2 (ω) X T/2 (ω) = F[x(t)] F[P T/2 (t)] = (δ(ω ω )+δ(ω + ω )) T sinc(tω/2) = T sinc(t (ω ω )/2) + T sinc(t (ω + ω )/2) (1.2) sinc sinc T T 2

5 T=2 T=4 T=1 Power spectrum 4 3 2 1 Power spectrum 8 6 4 2 Power spectrum 2 15 1 5.8.9.1.11.12 Frequency [Hz].8.9.1.11.12 Frequency [Hz].8.9.1.11.12 Frequency [Hz] 1.1:.1[Hz] T 1.1[Hz] T = 2, 4, 1[sec] X T/2 (ω) 2 1[Hz] T 1.1.1[Hz].1[Hz] T (Matlab ex1.m) x(t) T sinc X(ω) =F[x(t)] T ω ω 1 ω ω 1 T T T 1a.1[Hz].12[Hz] T = 2, 4, 1[sec] X T/2 (ω) 2 2 1[Hz] T 1.2 T 2 2 T = 4, 1 (Matlab ex1a.m) 3

Power spectrum 1 8 6 4 2 T=2 Power spectrum 8 6 4 2 T=4 Power spectrum 25 2 15 1 5 T=1.8.9.1.11.12 Frequency [Hz].8.9.1.11.12 Frequency [Hz].8.9.1.11.12 Frequency [Hz] 1.2:.1[Hz].12[Hz] T 1.2 P T/2 (t) t = h(t) x(t) X(t, ω) = x(t )h(t t)e iωt dt (1.3) (Short term Fourier transform) X(t, ω) (Short term Fourier spectrum) (Spectrogram) t ω X(t, ω) t t (Time-Frequency Analysis) 2 P T/2 (t) h(t) t ω h(t) h(t) H(ω) =F[h(t)] h(t) t 2 H(ω) ω 2 t 2 = ω 2 = t 2 h(t) 2 dt h(t) 2 dt ω 2 H(ω) 2 dω H(ω) 2 dω (1.4) (1.5) 4

t ω 1 2 (1.6) 1/2 h(t) h(t) =ae bt2 (1.7) H(ω) π ω2 H(ω) =a e 4b (1.8) b h(t) G(t, ω) = x(t )ae b(t t) 2 e iωt dt (1.9) (Gabor transform) g ω,t (t) e b(t t ) 2 ω 2 1.1[Hz].4[Hz].3[Hz].1[Hz] 1[Hz] h(t) T T =32, 64, 128[sec] 1.3 T =32, 64, 128[sec] (Matlab ex2.m) 1.3 x(t) x(t) R(t, τ) =E[x(t + τ/2)x (t τ/2)] (1.1) τ t x(t) E[ ] x(t) 5

Rectangle T=32[sec] Rectangle T=64[sec] Rectangle T=128[sec] Frequency [Hz].1.2.3.4 Frequency [Hz].1.2.3.4 Frequency [Hz].1.2.3.4.5 5 1 Time [sec] Blackman T=32[sec].5 5 1 Time [sec] Blackman T=64[sec].5 5 1 Time [sec] Blackman T=128[sec] Frequency [Hz].1.2.3.4 Frequency [Hz].1.2.3.4 Frequency [Hz].1.2.3.4.5 5 1 Time [sec].5 5 1 Time [sec].5 5 1 Time [sec] 1.3:.1[Hz].4[Hz].3[Hz].1[Hz] T =32, 64, 128[sec] x(t) t 1 T/2 R(τ) = lim x(t + τ/2)x (t τ/2) dt (1.11) T T T/2 t 1 (1.1) x(t + τ/2)x (t τ/2) (1.12) τ t ω W (t, ω) = x(t + τ/2)x (t τ/2)e iωτ dτ (1.13) W (t, ω) (Wigner distribution) 1 6

1 T/2 1 lim W (t, ω) dt = lim T T T/2 T T { = = T/2 lim T T/2 T/2 1 T T/2 R(τ) e iωτ dτ x(t + τ/2)x (t τ/2)e iωτ dτ dt } x(t + τ/2)x (t τ/2)dt e iωτ dτ = X(ω) 2 (1.14) x(t) x(t) ω W (t, ω) dω = = = x(t + τ/2)x (t τ/2)e iωτ dτ dω { } e iωτ dω x(t + τ/2)x (t τ/2)dτ δ(τ)x(t + τ/2)x (t τ/2)dτ = x(t) 2 (1.15) W (t, ω) x(t) ω 1,ω 2 2 ω 1,ω 2 t 1,t 2 t 1,t 2 3.1[Hz].4[Hz] 1 1[Hz].1[Hz].4[Hz].1[Hz].4[Hz] 1 W (t, ω) 1.4(a).3[Hz].1[Hz] W (t, ω) (b) (Matlab ex3.m) 7

.5 (a).5 (b).4.4.3.3.2.2 Frequency [Hz].1.1 Frequency [Hz].1.1.2.2.3.3.4.4.5 2 4 6 8 1 Time [sec].5 2 4 6 8 1 Time [sec] 1.4: (a).1[hz].4[hz] (b).1[hz].4[hz].3[hz].1[hz] 1.4 x(t) W (t, ω) 2 Φ(t, ω) 2 C(t, ω) =W (t, ω) Φ(t, ω) = W (t,ω )Φ(t t,ω ω ) dt dω (1.16) (Cohen s class) 2 Φ(t, ω) W (t, ω) Φ(t, ω) =δ(t)δ(ω) (1.16) C(t, ω), W(t, ω), Φ(t, ω) 2 M(θ, τ) = 1 2π A(θ, τ) = 1 2π φ(θ, τ) = 1 2π (1.16) C(t, ω)e iθt e iωτ dt dω (1.17) W (t, ω)e iθt e iωτ dt dω (1.18) Φ(t, ω)e iθt e iωτ dt dω (1.19) M(θ, τ) =A(θ, τ)φ(θ, τ) (1.2) A(θ, τ) x(t) W (t, ω) 2 (1.13) A(θ, τ) = x(t + τ/2)x (t τ/2)e iθt dt (1.21) 8

A(θ, τ) x(t) A(θ, τ) 2 φ(θ, τ) 2 C(t, ω) = A(θ, τ)φ(θ, τ)e iθt e iωτ dθ dτ (1.22) h(t) x(t )h(t t) R(t, τ) = x(t + τ/2)h(t t + τ/2)x (t τ/2)h (t t τ/2)dt (1.23) x(t) x(t )h(t t) (1.23) x(t )h(t t) X(t, ω) 2 X(t, ω) 2 (1.23) R(t, τ) X(t, ω) 2 = R(t, τ)e iωτ dτ (1.24) h(t) Φ( t, ω) C(t, ω) C(t, ω) 2 Φ(t, ω) φ(θ, τ) - (Chui - Williams distribution) φ(θ, τ) ( (2π) 2 θ 2 τ 2 ) φ(θ, τ) =exp, σ > (1.25) σ t 1 t 2 ω 1 ω 2 x(t) A(θ, τ) (θ, τ) (ω 1 ω 2,t 1 t 2 ) (ω 2 ω 1,t 2 t 1 ) C(t, ω) A(θ, τ) φ(θ, τ) 2 φ(θ, τ) =1 - (1.25) θ τ φ(θ, τ) 4 T = 124/8[msec] 9

(a) Rectangle window (b) Blackman window 5 5 1 1 Frequency [Hz] 15 2 25 Frequency [Hz] 15 2 25 3 3 35 35 4 2 4 6 8 Time [sec] 4 2 4 6 8 Time [sec] 1.5: 1[sec] ( 8[kHz]) 1.4 (a) (b) 1[Hz] 2[Hz] 15[Hz] 3[Hz] 1

1.5 (Matlab ) 1 Matlab 1.1 1.5 function ex1 % 1 set(, DefaultAxesFontName, TimesNewRoman ); set(, DefaultAxesFontSize,8); set(, DefaultTextFontSize,8); LL=[2 4 1]; for k=1:3 subplot(1,3,k) sub1(ll(k)) title([ T= num2str(ll(k))]) end set(gcf, PaperUnits, centimeters ) set(gcf, PaperPosition,[ 6 16 5]) print -depsc ex1.eps function sub1(l) N=2^14; x=zeros(1,n); y=sin((1:l)*(2*pi*.1)); x(1:length(y))=y; xx=fftshift(fft(x)); px=abs(xx).^2/l; f=[-n/2:n/2-1]/n; plot(f,px, Linewidth,.2) axesresize([1.8]) xlim([.8.12]); ylim([ max(px)]); xlabel( Frequency [Hz] ); ylabel( Power spectrum ); --------------------------------------------------------- function ex1a % 1a set(, DefaultAxesFontName, TimesNewRoman ); set(, DefaultAxesFontSize,8); set(, DefaultTextFontSize,8); LL=[2 4 1]; for k=1:3 subplot(1,3,k) sub1(ll(k)) title([ T= num2str(ll(k))]) end set(gcf, PaperUnits, centimeters ) set(gcf, PaperPosition,[ 6 16 5]) print -depsc ex1a.eps function sub1(l) N=2^14; x=zeros(1,n); y=sin((1:l)*(2*pi*.1))+sin((1:l)*(2*pi*.12)); x(1:length(y))=y; xx=fftshift(fft(x)); 11

px=abs(xx).^2/l; f=[-n/2:n/2-1]/n; plot(f,px, Linewidth,.2) axesresize([1.8]) xlim([.8.12]); ylim([ max(px)]); xlabel( Frequency [Hz] ); ylabel( Power spectrum ); --------------------------------------------------------- function ex2 % 2 set(, DefaultAxesFontName, TimesNewRoman ); set(, DefaultAxesFontSize,8); set(, DefaultTextFontSize,8); t=:999; x=chirp(t,.1,t(end),.4)+chirp(t,.3,t(end),.1); subplot(2,3,1) nfft=32; window=rectwin(nfft); stft(x,nfft,window); title([ Rectangle T= int2str(nfft) [sec] ]); subplot(2,3,2) nfft=64; window=rectwin(nfft); stft(x,nfft,window); title([ Rectangle T= int2str(nfft) [sec] ]); subplot(2,3,3) nfft=128; window=rectwin(nfft); stft(x,nfft,window); title([ Rectangle T= int2str(nfft) [sec] ]); subplot(2,3,4) nfft=32; window=blackman(nfft); stft(x,nfft,window); title([ Blackman T= int2str(nfft) [sec] ]); subplot(2,3,5) nfft=64; window=blackman(nfft); stft(x,nfft,window); title([ Blackman T= int2str(nfft) [sec] ]); subplot(2,3,6) nfft=128; window=blackman(nfft); stft(x,nfft,window); title([ Blackman T= int2str(nfft) [sec] ]); set(gcf, PaperUnits, centimeters ) set(gcf, PaperPosition,[ 6 16 1]) print -depsc ex2.eps function stft(x,nfft,window) x=[zeros(1,nfft/2) x zeros(1,nfft/2)]; [y,f,t]=specgram(x,nfft,1,window,nfft-1); zz=abs(y); zzmax=max(zz(:)); zzmin=min(zz(:)); pz=fix((zz-zzmin)/(zzmax-zzmin)*256); pz=uint8(255-pz); image(t,f,pz) 12

colormap(gray(256)); ylabel( Frequency [Hz] ); xlabel( Time [sec] ); --------------------------------------------------------- function ex3 % 3 set(, DefaultAxesFontName, TimesNewRoman ); set(, DefaultAxesFontSize,8); set(, DefaultTextFontSize,8); t=:999; subplot(1,2,1) x=chirp(t,.1,t(end),.4); wigner(x); title( (a) ) subplot(1,2,2) x=chirp(t,.1,t(end),.4) + chirp(t,.3,t(end),.1); wigner(x); title( (b) ); set(gcf, PaperUnits, centimeters ) set(gcf, PaperPosition,[ 6 16 7]) print -depsc2 ex3.eps function wigner(x) n=length(x); maxtau=n/1; % less than n x=[zeros(n,1) ; x(:) ; zeros(n,1)]; z=zeros(n,2*maxtau+1); for t=1:n for tau=-maxtau:maxtau; z(t,tau+maxtau+1) = x(t+n+ceil(tau/2))*x(t+n-ceil(tau/2)); end end zz=abs(fft(z,[],2)); zz=fftshift(zz,2) ; zzmax=max(zz(:)); zzmin=min(zz(:)); pz=fix((zz-zzmin)/(zzmax-zzmin)*256); pz=uint8(pz); f=:size(pz,2)-1; image([ 1],[-.5.5],pz) colormap(hsv(256)); ylabel( Frequency [Hz] ); xlabel( Time [sec] ); function y=shift(x,n) x=x(:) ; y=[x(n:end) x(1:n-1)]; --------------------------------------------------------- function ex4 % 4 set(, DefaultAxesFontName, TimesNewRoman ); set(, DefaultAxesFontSize,8); set(, DefaultTextFontSize,8); [x,fs,nbits]=wavread( onkai.ume.wav ); subplot(1,2,1) stft(x,124,boxcar(124),fs); title( (a) Rectangle window ) 13

subplot(1,2,2) stft(x,124,blackman(124),fs); title( (b) Blackman window ); set(gcf, PaperUnits, centimeters ) set(gcf, PaperPosition,[ 6 16 7]) print -depsc2 ex4.eps function pz=stft(x,nfft,window,fs) x=x(:); x=[zeros(nfft/2,1) ; x ; zeros(nfft/2,1)]; [y,f,t]=specgram(x,nfft,fs,window,nfft/4*3); zz=abs(y); zzmax=max(zz(:)); zzmin=min(zz(:)); pz=fix(((zz-zzmin)/(zzmax-zzmin)).^.35*255); pz=uint8(255-pz); image(t,f,pz) colormap(gray(256)); xlabel( Time [sec] ); ylabel( Frequency [Hz] ); 14

2 2.1 ψ(t) 2 a b a b { ψ a,b (t) a( ),b R }, ψ a,b (t) 1 ( ) t b ψ (2.1) a a R ψ(t) a 1/ a ψ a,b (t) 2 ψ a,b (t) x(t) W (a, b) = ψ a,b (t)x(t) dt (2.2) a, b x(t) ψ(t) (wavelet) (2.2) x(t) W (a, b) (wavelet transform) h(t) W (a, b) x(t) x(t) = W (a, b) ψ a,b (t) da db (2.3) ψ a,b (t) ψ a,b (t) 1 ψ(t) ψ(t) ψ a,b (t) ψ a,b (t) 1 a ψ ( t b a ), a( ),b R (2.4) (inverse wavelet transform) ψ(t) ψ(t) (dual wavelet) ψ(t) 15

2 ψ(t) Ψ(f) Ψ(f) 2 C ψ df < (2.5) f (admissible condition) ψ(t) Ψ() = ψ(t)dt = ψ(t) ψ(t) ψ(t) ψ (t) ψ(t) (2.3) x(t) = 1 1 C ψ a 2 W (a, b)ψa,b(t) da db (2.6) 2.2(a) 2.2 a, b a, b 2 L 2 (R) Z {ψ k (t) L 2 (R) k Z} x(t) L 2 (R) 2 <A,B< L 2 (R) (frame) ( ) 2 ψ k (t)x(t)dt k Z A B (2.7) x(t) 2 dt A, B (frame bound) A = B (tight) ψ k (t) (complete) A > x(t) x(t) x(t) L 2 (R) x(t) = ( ) ψ k (t)x(t)dt ψ k (t) (2.8) k Z { ψ k (t) k Z} { ψ k (t) k Z} {ψ k (t) k Z} (dual frame) {ψ k (t) k Z} (Riesz basis) 2.2(b) ψ(t) a, b {ψ i,j (t) i, j Z }, ψ i,j (t) 2 i/2 ψ(2 i t j) (2.9) 16

x(t) W i,j = ψ i,j (t)x(t) dt, i, j Z (2.1) (discrete wavelet transform) ψ i,j (t) L 2 (R) (2.8) x(t) = W i,j ψi,j (t) (2.11) i,j Z ψ i,j (t) ψ(t) {ψ i,j (t) i, j Z} L 2 (R) { ψ i,j (t) i, j Z} { ψ i,j (t) ψ 1, i = k and j = l k,l (t) dt = (2.12), others ψ i,j (t) ψ(t) ψ i,j (t) =2 i/2 ψ(2 i t j), i,j Z (2.13) ψi,j (t) = ψ i,j (t) (2.12) { ψ i,j (t)ψk,l(t) 1, i = k and j = l dt =, others (2.14) ψ(t) (orthogonal wavelet) x(t) W i,j = ψ i,j (t)x(t) dt =2 i/2 ψ(2 i t j)x(t) dt, i, j Z (2.15) x(t) = W i,j ψi,j(t) =2 i/2 W i,j ψ (2 i t j) (2.16) i,j Z i,j Z (orthogonal wavelet transform) (inverse orthogonal wavelet transform) 2.2(c) 2.3 (Haar function) 1, t<1/2 ψ(t) = 1, 1/2 t<1, others (2.17) 17

i,j i,j i,j i,j i,j 2.1: ψ(t) C m t t k ψ(t) dt =, k m (2.18) Meyer Mayer Daubechies Daubechies 18

3 3.1 x(t) ψ(t) x(t) t x(t) 2 L 2 (R) 2 x(t) L 2 (R) 2 j, j Z A 2 j A 2 j [x(t)] 2 j 2 j A 2 j 1. A 2 j V 2 j L 2 (R) A 2 j [x(t)] 2 j A 2 j [A 2 j [x(t)]] A 2 j [x(t)] A 2 j A 2 j = A 2 j V 2 j L 2 (R) 2 j 2. A 2 j V 2 j A 2 j [x(t)] 2 j 2 x(t) y(t) x(t) 2 dt A 2 j [x(t)] x(t) 2 dt, y(t) V 2 j (3.1) 3. x(t) 2 j+1 x(t) 2 j V 2 j V 2 j+1, j Z (3.2) 4. 2 j V 2 j x(t) x(2 k t) 2 j+k V 2 j+k x(t) V 2 j x(2 k t) V 2 j+k, j, k Z (3.3) 19

5. A 2 j [x(t)] 2 j V 2 j I 2 (Z) I 6. x(t) L 2 (R) 2 j k 2 j k 2 j A 2 j [x(t 2 j k)] x(t) 2 j 2 j k A 2 j [x(t)] t=t 2 j k 5. A 2 j [x(t 2 j k)] = A 2 j [x(t)] t=t 2 j k, j, k Z (3.4) I[A 2 j [x(t)]] = α i I[A 2 j [x(t 2 j k)]] = α i k, i, j, k Z (3.5) 7. x(t) 2 j x(t) + x(t) + lim V 2 j + j = V 2 j is dense in L 2 (R) j= + (3.6) lim V 2 j j = V 2 j = { } j= 1. 7. A 2 j L 2 (R) 2 j 3. 7. {V 2 j j Z} L 2 (R) (multi-resolution approximation) 3.2 {V 2 j j Z} L 2 (R) { 2 j/2 φ(2 j t n) n Z} (3.7) V 2 j φ(t) V 1 L 2 (R) φ(t) (scaling function) x(t) L 2 (R) 2 j A 2 j [x(t)] A 2 j [x(t)] = 2 j/2 A 2 j x n φ(2 j t n) (3.8) n Z A 2 j x n 2 j/2 φ (2 j t m) t {2 j/2 φ(2 j t n) n Z} A 2 j x n =2 j/2 φ (2 j t n)a 2 j [x(t)] dt, n Z (3.9) 2

{2 j/2 φ(2 j t n) n Z} V 2 j A 2 j V 2 j (3.9) A 2 j [x(t)] x(t) (3.9) A 2 j x n =2 j/2 φ (2 j t n)x(t) dt, n Z (3.1) x(t) 2 j (3.1) (3.1) 2 j/2 φ (2 j t n) x(t) 2 j (LPF) 2 j/2 φ(2 j t n) 2 j LPF (3.1) x(t) 2 j/2 φ(2 j t) sinc 2 j/2 φ(2 j t)=sin(2 j πt)/(2 j πt) LPF x(t) 2 j A 2 j x n x(t) x(t) x(t) 2 j A 2 j [x(t)] LPF A 2 j x n 2 j 2 =1 A 1 x n 3.3 2 =1 x(t) A 1 x n x(t) A 2 j x n,j< x(t) A 2 j x n,j< 3. 5. A 1 x n V 2 V 2 1 V 2 V 2 1 V 2 φ(t) V 2 1 { 2 1/2 φ(2t n) n Z} φ(t) = l ( n)2 1/2 φ(2t n) (3.11) n Z l ( n) n l ( n) { 2 1/2 φ(2t n) n Z} l ( n) = 2 1/2 φ (2t n)φ(t) dt (3.12) (3.11) φ(t) 2 (two scale) φ(t) φ(t) 1 1 21

3.1: A 2 j x n A 2 j+1x n ( ) A 2 x n A 2 j x n,j < (3.11) t 2 j t k x(t) (3.1) A 2 j x n = l ( k)a 2 j+1x 2n+k k Z 2n + k = m A 2 j x n = m Z l (2n m)a 2 j+1x m (3.13) (3.13) l (n) FIR L A 2 j x n A 2 j+1x n A 2 j x n A 2 j+1x n FIR L 1 3.1 A 1 x n L A 2 j x n,j< 3.1 3.4 2 j+1 V 2 j+1 2 j V 2 j V 2 j V 2 j+1 V 2 j+1 V 2 j O 2 j V 2 j+1 = V 2 j O 2 j (3.14) L 2 (R) O 2 j D 2 j x(t) L 2 (R) O 2 j D 2 j [x(t)] { 2 j/2 ψ(2 j t n) n Z} (3.15) O 2 j ψ(t) O 1 L 2 (R) (3.15) (2.9) ψ(t) 22

D 2 j [x(t)] O 2 j { 2 j/2 ψ(2 j t n) n Z} D 2 j [x(t)] = 2 j/2 n Z D 2 j x n ψ(2 j t n) (3.16) D 2 j x n 2 j/2 ψ(2 j t m) { 2 j/2 ψ(2 j t n) n Z} D 2 j x n =2 j/2 ψ (2 j t n)x(t) dt (3.17) A 2 j+1x n A 2 j x n x(t) 1 A 1 x n J A 2 J x n { D 2 j x n J j 1} V 1 V 1 = O 2 1 O 2 2 O 2 J V 2 J (3.18) V 1 (orthogonal wavelet decomposition) 3.5 ψ(t) O 1 V 2 { 2 1/2 φ(2t n) n Z} ψ(t) = h ( n)2 1/2 φ(2t n) (3.19) n Z h ( n) n h ( n) { 2 1/2 φ(2t n) n Z} h ( n) = 2 1/2 φ (2t n)ψ(t) dt (3.2) (3.19) t 2 j t k x(t) (3.1),(3.17) D 2 j x n = h (2n k)a 2 j+1x k (3.21) k Z (3.21) h (n) FIR H D 2 j x n A 2 j+1x n D 2 j x n A 2 j+1x n FIR H 1 3.2 A 1 x n L 2:1 ( j 1) H 2:1 D 2 j x n, j < 3.2 23

3.2: D 2 j x n A 2 j+1x n ( ) A 2 x n D 2 j x n,j < 3.6 V 2 = V 1 O 1 φ(2t) V 2 V 1 { φ(t n) n Z} O 1 { ψ(t n) n Z} φ(2t k) =2 1/2 l 1 (k 2n)φ(t n)+2 1/2 h 1 (k 2n)ψ(t n) (3.22) n Z n Z l 1 (k 2n),h 1 (k 2n) l 1 (n),h 1 (n) l 1 (n),h 1 (n) n l 1 (k 2n),h 1 (k 2n) (3.22) 2 1/2 l 1 (k 2n),h 1 (k 2n) {φ(t n),ψ(t n) n Z} l 1 (k 2n) =2 1/2 φ (t n)φ(2t k) dt (3.23) h 1 (k 2n) =2 1/2 ψ (t n)φ(2t k) dt (3.24) (3.22) t 2 j t x(t) (3.1),(3.17) A 2 j+1x n = l 1 (n 2k)A 2 j x k + h 1 (n 2k)D 2 j x k (3.25) k Z k Z x(t) (orthogonal wavelet reconstruction) (3.25) l 1 (n) h 1 (n) FIR L 1,H 1 A 2 j+1x n A 2 j x n D 2 j x n A 2 j+1x n A 2 j x n D 2 j x n 24

3.3: A 2 j x n D 2 j x n A 2 j+1x n FIR L 1,H 1 3.3 (3.13) (3.21) (3.25) (subband filter) L H L 1 H 1 3 3.4 2 =1 x(t) φ(t) ψ(t) l (t), l 1 (t), h (t), h 1 (t) 3.7 (3.12) (3.2) (3.23) (3.24) FIR L,H,L 1,H 1 l (n), h (n), l 1 (n), h 1 (n) (3.23) (3.24) n = (3.12) (3.2) l (n) l 1 (n) h (n) h 1 (n) l 1 (n) =l( n) (3.26) h 1 (n) =h ( n) (3.27) V 1 O 1 { φ(t n) n Z} { ψ(t n) n Z} φ (t m)ψ(t)dt =, m Z (3.28) 25

3.4: 3 (3.11),(3.19) (3.28) l (n)h (k) n Z k Z φ (t 2m + n)φ(t + k) dt =, m Z (3.29) (3.29) k n 2m k = n 2m l (n)h (n 2m) =, m Z (3.3) n Z (3.3) h (n) =( 1) 1 n l (1 n) (3.31) l 1 (n) h 1 (n) h 1 (n) =( 1) 1 n l 1 (1 n) (3.32) L H L 1 H 1 3.8 Daubechies x(t) 26

Meyer φ(t) ψ(t) L,L 1,H,H 1 Daubechies Daubechies l (t),l 1 (t),h (t),h 1 (t) Daubechies N {l (n) n Z} N Daubechies l (n) =, n <, n>2n 2N 1 l (n) =2 1/2 k= 2N 1 l(n) 2 (3.33) =1 k= 2N 1 k= l (n)l (n 2m) =, m (3.33) l (n) Daubechies N =1,...,1 3.1 Daubechies (3.33) l (n),n 2 (3.11) φ(x)dx =1 φ(t) φ(x) ψ(x) N 1 Daubechies 5 N Daubechies 27

3.1: Daubechies L l (n) N n l (n) N n l (n) 1 2 1/2 7 13.3537138 1 1 2 1/2 8.54415842243 2.482962913145 8 1.31287159914 2 1.83651633738 8 2.67563736297 2 2.22414386842 8 3.585354683654 2 3.12949522551 8 4.1582915256 3.3326755295 8 5.28415542962 3 1.8689159311 8 6.472484574 3 2.45987752118 8 7.12874742662 3 3.1351121 8 8.17369312 3 4.85441273882 8 9.4488253931 3 5.35226291882 8 1.1398127917 4.2337781339 8 11.87469447 4 1.71484657553 8 12.487352993 4 2.638876793 8 13.39174373 4 3.27983769417 8 14.67544946 4 4.18734811719 8 15.117476784 4 5.3841381836 9.3877947364 4 6.3288311667 9 1.243834674613 4 7.159741785 9 2.6482312369 5.1612397974 9 3.6572887851 5 1.63829269797 9 4.133197385825 5 2.72438528438 9 5.293273783279 5 3.13842814591 9 6.9684783223 5 4.24229488766 9 7.14854749338 5 5.32244869585 9 8.3725681479 5 6.7757149384 9 9.6763282961 5 7.624149213 9 1.25947115 5 8.1258751999 9 11.22361662124 5 9.3335725285 9 12.472324758 6.1115474335 9 13.428153682 6 1.49462389398 9 14.1847646883 6 2.7511339821 9 15.23385764 6 3.3152535179 9 16.251963189 6 4.226264693965 9 17.3934732 6 5.129766867567 1.26675791 6 6.975165587 1 1.188176878 6 7.2752286553 1 2.52721188932 6 8.3158239318 1 3.68845939454 6 9.55384221 1 4.281172343661 6 1.4777257511 1 5.249846424327 6 11.1773185 1 6.195946274377 7.778525485 1 7.12736934336 7 1.396539319482 1 8.935736464 7 2.7291329846 1 9.71394147166 7 3.46978228745 1 1.29457536822 7 4.143963929 1 11.3321267459 7 5.22436184994 1 12.366553567 7 6.7139219267 1 13.1733175483 7 7.861269151 1 14.1395351747 7 8.3829936935 1 15.199245295 7 9.16574541631 1 16.685856695 7 1.1255998556 1 17.116466855 7 11.429577973 1 18.9358867 7 12.1816474 1 19.1326423 28

4 4.1 x(t) N x(t) = x n (t) (4.1) n=1 x n (t) x(t) x(t) x(t) =x 1 (t) x 2 (t) x N (t) N X(ω) = X n (ω) (4.2) n=1 X(ω) =F[x(t)], X n (ω) = F[x n (t)] (4.2) (cepstrum analysis) (spectrum) spec X(ω) x n (t) X(ω) (Complex Cepstrum) x(t) X(ω) 2 x n (t) X n (ω) 2 X(ω) 2 (Power Cepstrum) (4.2) 2 N X(ω) 2 = X n (ω) 2 (4.3) n=1 2 N log X(ω) = log X n (ω) (4.4) n=1 x pc (t) F 1 [log X(ω) ] = 29 N F 1 [log X n (ω) ] (4.5) n=1

t (frequency) fre que (quefrency) x pc (t) X n (ω) 2 (filter) fil (lifter) (high) (low) (long) (short) x(t) y(t) s ay(t) x(t) =y(t)+ay(t s), a < 1 (4.6) y(t) h(τ) =δ(τ)+aδ(τ s) x(t) =h(t) y(t) y(t) x(t) y(t) s x(t) y(t) y(t) y(t s) y(t) x(t), h(t), y(t) X(ω), H(ω), Y(ω) (4.6) ( X(ω) =Y (ω)h(ω) =Y (ω) 1+ae iωs) (4.7) X(ω) 2 = Y (ω) 2 1+ae iωs 2 = Y (ω) 2 (1 + ae iωs )(1 + ae iωs ) = Y (ω) 2( ) 1+a 2 +2acos(ωs) (4.8) ( log X(ω) 2 =log Y (ω) 2 +log(1+a 2 )+log 1+ 2a ) 1+a 2 cos(ωs) (4.9) 2a a < 1 < 1 log(1+x) Taylor 1 <x 1 1+a 2 x x2 2 + x3 3 x4 4 + log X(ω) 2 =log Y (ω) 2 +log(1+a 2 )+ ( 1) n+1 ( ) n 2a n 1+a 2 cos n (ωs) (4.1) 2 t = 3 cos(ωs), cos(2ωs), cos(3ωs),... s x pc (t) x pc (t) =y pc (t)+a δ(t)+ n=1 A n (δ(t ns)+δ(t + ns)) (4.11) n=1 3

A n, n Z y(t) Y (ω) 2 X(ω) 2 s x(t) x(t) x(t) x(t) x(t) x(t) x(t) (4.11) 3 (long) a 1 2a/(1 + a 2 ) 1 3 x(t) 6 y(t) =cos (2π 2 ) 21 t e t 8, t =, 1,...,23 1, t = h(t) =.7, t =1, others x(t) =y(t) h(t) x(t) y(t) h(t) x(t) N =64, 128, 256 y(t) 4.1(a) y(t) (b) (c) x(t) (c) (b) x(t) N = 128 (d) (d) (e) 1 1 2 (e) (f) y(t) (b) x(t) N =64, 128, 256 4.2 31

1 (a) original signal y(t) 2 (b) power spectrum of y(t).5 15 1.5 5 1 5 1 15 2 time 5 5 frequency 1 (c) observed signal x(t) 6 (d) power stpectrum of x(t).5 5 4 3.5 2 1 1 1 2 3 time 5 5 frequency 1 (e) power cepstrum of x(t) 2 (f) estimated power spectrum of y(t).5 15 1 5.5 5 5 quefrency 5 5 frequency 4.1: 6 32

1 (a) power cepstrum N=64 2 (b) estimated power spectrum N=64.5 15 1 5.5 2 2 quefrency 2 2 frequency 1 (c) power cepstrum N=128 2 (d) estimated power spectrum N=128.5 15 1 5.5 5 5 quefrency 5 5 frequency 1 (e) power cepstrum N=256 2 (f) estimated power spectrum N=256.5 15 1 5.5 1 5 5 1 quefrency 1 5 5 1 frequency 4.2: 33

4.2 x(t) x x(t) log(x) log x + i arg(x) (4.12) x cc (t) F 1 [log F[x(t)]] = F 1 [log X(ω) + i arg(x(ω))] (4.13) arg(x(ω)) ( π, π) arg(x) x pc (t) x cc (t) x(t) x pc (t) x pc (t)=f 1 [log X(ω) 2 ] = F 1 [log(x(ω)x (ω))] = F 1 [log X(ω)+logX (ω)] (4.14) X (ω) =X( ω) x(t) x pc (t) x pc (t) =x cc (t)+x cc ( t) (4.15) 2 x(t) =y(t)+ay(t s), a < 1 x(t) 2 Taylor ( log X(ω) =logy (ω)+log 1+ae iωs) (4.16) log X(ω) =logy (ω)+ a n ( 1) n+1 e inωs (4.17) n! n=1 34

x(t) x cc (t) x cc (t)=f 1 [log X(ω)] a n ( 1) n+1 = y cc (t)+ δ(t ns) (4.18) n! n=1 s s y(t) Y (ω) X(ω) Y (ω) y(t) 7 y(t) =cos (2π 2 ) 21 t e t 8, t =, 1,...,23 1, t = h(t) =.7, t =1, others x(t) =y(t) h(t) x(t) y(t) h(t) x(t) N =64 4.3(a) y(t) (b) (c) x(t) (c) (a) x(t) N =64 (d) (d) (e) 1 1 2 (e) (f) (g) (a) 4.3 ( ) (4.9) (4.16) a > 1 (4.16) ( ) log X(ω) =logy (ω)+log ae iωs (1 + a 1 e iωs ) (4.19) a 1 < 1 Taylor Taylor a n ( 1) n+1 log X(ω) =logy (ω)+loga iωs + e inωs (4.2) n! x cc (t) =y cc (t)+δ(t)loga + F 1 a n ( 1) n+1 [ iωs]+ δ(t + ns) (4.21) n! n=1 n=1 35

1 (a) original signal y(t) 5 (b) power spectrum of y(t).5.5 1 5 1 15 2 time 5 2 2 frequency 1 (c) observed signal x(t) 1 (d) power spectrum of x(t).5 5.5 5 1 1 2 3 time 1 2 2 frequency 1 (e) complex cepstrum of x(t) 5 (f) estimated power spectrum of y(t).5.5 1 2 4 6 quefrency 5 2 2 frequency 1 (g) estimated y(t).5.5 1 5 1 15 2 time 4.3: 7 36

(4.18) a > 1 2,3 3 x(t) x(t) z X(z) X(z) = n a n b Az r (1 a i z 1 ) (1 b i z) i=1 n c i=1 n d (1 c i z 1 ) (1 d i z) i=1 i=1 (4.22) a i, b i, c i, d i < 1 i a i,c i z b i,d i z A z r r < r> (4.22) n a n b log X(z)=logA + r log z + log(1 a i z 1 )+ log(1 b i z) i=1 i=1 n d n c log(1 c i z 1 ) log(1 d i z) (4.23) i=1 i=1 log(1 + x) Taylor log(1 + x) = ( 1) n=1 n+1 xn n for x < 1 (4.24) n a log X(z)=logA + r log z n c + i=1 n=1 ( )= n=1 n a log X(z)=logA + r log z n c + i=1 n= c n i n z n + n= i=1 n=1 n d where, u(n) = i=1 n= c n i n z n u(n 1) + i=1 n=1 a n i n z n n b i=1 n=1 b n i n zn d n i n zn (4.25) ( )u(n 1) (4.26) { 1, n, n < a n i n z n u(n 1) n d i=1 n= n b i=1 n= b n i n zn u(n 1) d n i n zn u(n 1) (4.27) 37

4,6 n n log X(z)=logA + r log z + n c n= i=1 n a n= i=1 c n i n u(n 1)z n a n i n u(n 1)z n + n d n= i=1 n b n= i=1 b n u( n 1)z n in d n in u( n 1)z n (4.28) z z (4.22) x(n) x cc (n)=δ(n)loga + r( 1)n n n c + i=1 c n nd i n u(n 1) n a {u(n 1) + u( n 1)} i=1 i=1 a n nb i n u(n 1) + i=1 b n u( n 1) in d n u( n 1) (4.29) in n r( 1) n n b b n nd i + n n d n i for n< n i=1 i=1 x cc (n) = log A for n = r( 1) n n a a n nc i n n + c n i for n> n i=1 i=1 (4.3) (4.3) a i,b i,c i,d i r( 1)n n ω 2 (4.3) x cc (n) x(n) n < n > x pc (n), for n> x cc (n) =.5x pc (n), for n (4.31), for n<, for n> x cc (n) =.5x pc (n), for n (4.32) x pc (n), for n< (exponential window) w(t) = { α n, <α<1 for t N 1, others (4.33) 38

x(t) w(t) z z X(α 1 z) a s aα s α.96.99 8 y(t) =cos (2π 2 ) 21 t e t 8, t =, 1,...,23 1, t = h(t) = 2, t =1, others x(t) =y(t) h(t) x(t) y(t) h(t) x(t) N =64 h(t) x(t) 4.4(a) y(t) (b) (c) x(t) (c) (a) x(t) N =64 (d) (d) (e) 1 1 2 (e) (f) (g) (a) h(t) 1 7 h(t) 4.3(e), 1, 2, 3,... 4.4(e), 1, 2, 3,... (e) 4.4 x(t) X(ω) ˆX(ω) = isgn(ω)x(ω) (4.34) X(ω) (Hilbert transform) 1, x > sgn(x) =, x = 1, x < x(t) ˆx(t) ˆX(ω) isgn(ω) 1/πt ˆx(t)=F 1 [ ˆX(ω)] = F 1 [ isgn(ω)x(ω)] 39

1 (a) original signal y(t) 5 (b) power spectrum of y(t).5.5 1 5 1 15 2 time 5 2 2 frequency 5 (c) observed signal x(t) 2 (d) power spectrum of x(t) 1 1 5 1 2 3 time 2 2 2 frequency 1 (e) complex cepstrum of x(t) 1 (f) estimated power spectrum of y(t).5 5.5 5 1 6 4 2 quefrency 1 2 2 frequency 2 (g) estimated y(t) 1 1 2 5 1 15 2 time 4.4: 8 4

4.1: x(t) ˆx(t) ŷ(t) y(t) y(t)e iωt iy(t)e iωt y(t)cos(ωt) y(t)sin(ωt) y(t)sin(ωt) y(t)cos(ωt) = 1 π = F 1 [ isgn(ω)] F 1 [X(ω)] = 1 πt x(t) x(τ) dτ (4.35) t τ 4.1 X(ω) ˆX(ω) x(t) ˆx(t) x(t) ˆx(t) x(t) ˆx (t) dt = = 1 2π = = = F 1 [X(ω) ˆX ( ω)] dt = i X(ω ω ) ˆX ( ω )dω e iωt dω dt x(t)e iω t dt ˆX ( ω )dω X( ω ) ˆX ( ω )dω ( i)sgn(ω )X( ω )X ( ω )dω sgn(ω) X(ω) 2 dω = (4.36) 4.5 x(t) ˆx(t) x(t) ˆx(t) x + (t)=x(t)+iˆx(t) (4.37) x (t)=x(t) iˆx(t) (4.38) x + (t) x (t) x + (t) x + (t), x (t) X + (ω),x (ω) X + (ω)=x(ω)+i( isgn(ω))x(ω) =X(ω)+sgn(ω)X(ω) (4.39) X (ω)=x(ω) i( isgn(ω))x(ω) =X(ω) sgn(ω)x(ω) (4.4) 41

X + (ω) = X (ω) = 2X(ω), ω > X(ω), ω =, ω <, ω > X(ω), ω = 2X(ω), ω < (4.41) (4.42) x + (t) x (t) e ±iωt e ±iωt =cos(ωt) ± i sin(ωt) (4.43) x + (t) X + (ω) X + (t) 2πx + ( ω) X + (t) F[X + (t)] = 2πx + ( ω) = 2π(x( ω)+iˆx( ω)) = 2π(x(ω) iˆx(ω)) (4.44) X + (t) (4.41) t < X + (t) h(t) t< x(t) x + (t) x(t) X(ω) (4.41) X + (ω) x(t) N 2X(ω), ω {1, 2,...,N/2 1} X + (ω) = X(ω), ω {,N/2}, ω {N/2+1,...,N 1} x(t) N 2X(ω), ω {1, 2,...,(N 1)/2} X + (ω) = X(ω), ω =, ω {(N 1)/2+1,...,N 1} cos(ωt) m(t) (Amplitude Modulation) x(t) =m(t)cos(ωt) m(t) x(t) x(t) m(t) x(t) 4.1 ˆx(t) =m(t)sin(ωt) 42

x(t) x ± (t) = m(t)cos(ωt) ± im(t)sin(ωt) = m(t)e ±iωt (4.45) x ± (t) = m(t) m(t), t x(t) m(t), t x(t) x + (t) =x(t)+iˆx(t) 1 ˆx(t) φ m (t) tan x(t) (4.46) ω m (t) d dt φ m(t) (4.47) 9 sin(ω c t), ω c =2π.2 m(t) =.3sin(ω m t)+.5, ω m =2π.4 t t =, 1,...,999 4.5(a) m(t) (b) x(t) (c) x(t) m(t) (d) 43

1 (a) carrier wave 1 (b) signal m(t).5.8.6.4.5.2 1 2 4 6 8 1 time 2 4 6 8 1 time (c) Amplitude modulated signal x(t) and the Hilbert transform 1.5.5 1.8.6.4.2 (d) Demodulated signal 1 2 4 6 8 1 time 2 4 6 8 1 time 4.5: 9 44

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