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1 8 2 3 4 ( ) 2 / 72

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sample.m Fs = 1000; T = 1/Fs; L = 1000; t = (0:L-1)*T; % Sampling frequency % Sample time % Length of signal % Time vector y=1+0.7*sin(2*pi*50*t)+sin(2*pi*120*t)+2*randn(size(t)); plot(fs*t(1:1000),y(1:1000)) xlabel( time (milliseconds) ) ( ) 6 / 72

sample.m fft NFFT = 2^nextpow2(L); Y = fft(y,nfft)/l; ( ) 7 / 72

sample.m f = Fs/2*linspace(0,1,NFFT/2+1); % Plot single-sided amplitude spectrum. plot(f,2*abs(y(1:nfft/2+1))) xlabel( Frequency (Hz) ) ylabel( Amplitude Spectrum ) ( ) 8 / 72

(discrete Fourier transform; DFT) g 0, g 1,, g N 1 G 0, G 1,, G N 1 N 1 G k = g n w kn n=0 i 2π/N w = e ( ) 9 / 72

8 8 Im i 2π/8 w = e i -1 -i w 1 Re ( ) 10 / 72

8 8 Im i 2π/8 w = e Im -1 i -i w 1 Re w 4 w 5 w 3 w 2 w 6 w 7 Re w 0 w 8 w w 9 ( ) 10 / 72

8 8 Im i 2π/8 w = e Im -1 i -i w 1 Re w -4 w -3 w -5 w -6 w -2 w -1 Re w 0 w -8 w w -7 ( ) 10 / 72

8 8 G 0 = g 0 w 0 0 + g 1 w 0 1 + g 2 w 0 2 + + g 6 w 0 6 + g 7 w 0 7 G 1 = g 0 w 1 0 + g 1 w 1 1 + g 2 w 1 2 + + g 6 w 1 6 + g 7 w 1 7 G 2 = g 0 w 2 0 + g 1 w 2 1 + g 2 w 2 2 + + g 6 w 2 6 + g 7 w 2 7. G 6 = g 0 w 6 0 + g 1 w 6 1 + g 2 w 6 2 + + g 6 w 6 6 + g 7 w 6 7 G 7 = g 0 w 7 0 + g 1 w 7 1 + g 2 w 7 2 + + g 6 w 7 6 + g 7 w 7 7 ( ) 11 / 72

8 8 G 0 G 1 G 2. G 6 G 7 = w 0 0 w 0 1 w 0 2 w 0 6 w 0 7 w 1 0 w 1 1 w 1 2 w 1 6 w 1 7 w 2 0 w 2 1 w 2 2 w 2 6 w 2 7........ w 6 0 w 6 1 w 6 2 w 6 6 w 6 7 w 7 0 w 7 1 w 7 2 w 7 6 w 7 7 g 0 g 1 g 2. g 6 g 7 ( ) 12 / 72

8 8 G 0 G 1 G 2. G 6 G 7 = w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 6 w 7 w 0 w 2 w 4 w 12 w 14........ w 0 w 6 w 12 w 36 w 42 w 0 w 7 w 14 w 42 w 49 g 0 g 1 g 2. g 6 g 7 ( ) 13 / 72

8 8 F 8 = w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 6 w 7 w 0 w 2 w 4 w 12 w 14........ w 0 w 6 w 12 w 36 w 42 w 0 w 7 w 14 w 42 w 49 ( ) 14 / 72

8 8 G 0 G 1 G 2. G 6 G 7 = F 8 g 0 g 1 g 2. g 6 g 7 ( ) 15 / 72

g 0 g 1 g 2. g 6 g 7 = F 1 8 G 0 G 1 G 2. G 6 G 7 ( ) 16 / 72

F 8 = w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 6 w 7 w 0 w 2 w 4 w 12 w 14........ w 0 w 6 w 12 w 36 w 42 w 0 w 7 w 14 w 42 w 49 ( ) 17 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 ( ) 18 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 F 8 F 8 (0, 0) = w 0 + w 0 + w 0 + + w 0 + w 0 = 8 ( ) 18 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 F 8 F 8 (1, 0) = w 0 + w 1 + w 2 + + w 6 + w 7 = 0 ( ) 18 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 F 8 F 8 (2, 0) = w 0 + w 2 + w 4 + + w 12 + w 14 = 0 ( ) 18 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 F 8 F 8 (0, 1) = w 0 + w 1 + w 2 + + w 6 + w 7 = 0 ( ) 18 / 72

F 8 F 8 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 7 w 0 w 2 w 4 w 14....... w 0 w 6 w 12 w 42 w 0 w 7 w 14 w 49 F 8 F 8 (1, 1) = w 0 + w 0 + w 0 + + w 0 + w 0 = 8 ( ) 18 / 72

{ 8 (i, j) = (0, 0), (1, 1),, (7, 7) F 8 F 8 (i, j) = 0 F 8 F 8 = 8 I 8 8 F 1 8 = 1 8 F 8 ( ) 19 / 72

g 0 g 1 g 2. g 6 g 7 = 1 8 F 8 G 0 G 1 G 2. G 6 G 7 ( ) 20 / 72

g 0 g 1 g 2. g 6 g 7 = 1 8 w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 6 w 7 w 0 w 2 w 4 w 12 w 14........ w 0 w 6 w 12 w 36 w 42 w 0 w 7 w 14 w 42 w 49 G 0 G 1 G 2. G 6 G 7 ( ) 21 / 72

(fast Fourier transform; FFT) DFT Cooley and Tukey, 1965 N DFT 2 (N/2) DFT FFT DFT N log 2 N FFT DFT N/ log 2 N 4 2 8 16 2.00 16 4 64 256 4.00 64 6 384 4096 10.67 256 8 2048 65536 32.00 1024 10 10240 1048576 102.40 ( ) 22 / 72

8 G 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 G 1 w 0 w 1 w 2 w 3 w 4 w 5 w 6 w 7 G 2 w 0 w 2 w 4 w 6 w 8 w 10 w 12 w 14 G 3 G 4 = w 0 w 3 w 6 w 9 w 12 w 15 w 18 w 21 w 0 w 4 w 8 w 12 w 16 w 20 w 24 w 28 G 5 w 0 w 5 w 10 w 15 w 20 w 25 w 30 w 35 G 6 w 0 w 6 w 12 w 18 w 24 w 30 w 36 w 42 G 7 w 0 w 7 w 14 w 21 w 28 w 35 w 42 w 49 g 0 g 1 g 2 g 3 g 4 g 5 g 6 g 7 ( ) 23 / 72

G 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 g 0 G 1 w 0 w 2 w 4 w 6 w 1 w 3 w 5 w 7 g 2 G 2 w 0 w 4 w 8 w 12 w 2 w 6 w 10 w 14 g 4 G 3 G 4 = w 0 w 6 w 12 w 18 w 3 w 9 w 15 w 21 g 6 w 0 w 8 w 16 w 24 w 4 w 12 w 20 w 28 g 1 G 5 w 0 w 10 w 20 w 30 w 5 w 15 w 25 w 35 g 3 G 6 w 0 w 12 w 24 w 36 w 6 w 18 w 30 w 42 g 5 G 7 w 0 w 14 w 28 w 42 w 7 w 21 w 35 w 49 g 7 ( ) 24 / 72

w 2 = e i 2π/4 4 DFT = = w 0 w 0 w 0 w 0 w 0 w 2 w 4 w 6 w 0 w 4 w 8 w 12 w 0 w 6 w 12 w 18 (w 2 ) 0 (w 2 ) 0 (w 2 ) 0 (w 2 ) 0 (w 2 ) 0 (w 2 ) 1 (w 2 ) 2 (w 2 ) 3 (w 2 ) 0 (w 2 ) 2 (w 2 ) 4 (w 2 ) 6 (w 2 ) 0 (w 2 ) 3 (w 2 ) 6 (w 2 ) 9 = F 4 (4 DFT ) ( ) 25 / 72

w 8 = 1 = = = F 4 w 0 w 8 w 16 w 24 w 0 w 10 w 20 w 30 w 0 w 12 w 24 w 36 w 0 w 14 w 28 w 42 w 0 w 0 w 0 w 0 w 0 w 2 w 4 w 6 w 0 w 4 w 8 w 12 w 0 w 6 w 12 w 18 ( ) 26 / 72

= = = w 0 w 0 w 0 w 0 w 1 w 3 w 5 w 7 w 2 w 6 w 10 w 14 w 3 w 9 w 15 w 21 w 0 w 1 w 0 w 1 w 2 w 3 w 2 w 3 F 4 w 0 w 0 w 0 w 0 w 0 w 2 w 4 w 6 w 0 w 4 w 8 w 12 w 0 w 6 w 12 w 18 ( ) 27 / 72

= = = w 4 w 4 w 4 w 4 w 5 w 7 w 9 w 11 w 6 w 10 w 14 w 18 w 7 w 13 w 19 w 25 w 4 w 5 w 4 w 5 w 6 w 7 w 6 w 7 F 4 w 0 w 0 w 0 w 0 w 0 w 2 w 4 w 6 w 0 w 4 w 8 w 12 w 0 w 6 w 12 w 18 ( ) 28 / 72

8 G 0 g 0 G 1 G 2 = F g 2 4 g 4 + G 3 g 6 G 4 G 5 G 6 G 7 = F 4 g 0 g 2 g 4 g 6 + w 0 w 1 w 4 w 5 w 2 w 3 w 6 w 7 F 4 F 4 g 1 g 3 g 5 g 7 g 1 g 3 g 5 g 7 ( ) 29 / 72

8 G 0 g 0 G 1 G 2 = F g 2 4 g 4 + G 3 g 6 G 4 G 5 G 6 G 7 = F 4 g 0 g 2 g 4 g 6 + w 0 w 1 w 4 w 5 w 2 w 3 w 6 w 7 F 4 F 4 g 1 g 3 g 5 g 7 g 1 g 3 g 5 g 7 ( ) 29 / 72

8 G 0 g 0 G 1 G 2 = F g 2 4 g 4 + G 3 g 6 G 4 G 5 G 6 G 7 = F 4 g 0 g 2 g 4 g 6 + w 0 w 1 w 4 w 5 w 2 w 3 w 6 w 7 F 4 F 4 g 1 g 3 g 5 g 7 g 1 g 3 g 5 g 7 ( ) 29 / 72

8 G 0 g 0 G 1 G 2 = F g 2 4 g 4 + G 3 g 6 G 4 G 5 G 6 G 7 = F 4 g 0 g 2 g 4 g 6 + w 0 w 1 w 4 w 5 w 2 w 3 w 6 w 7 F 4 F 4 g 1 g 3 g 5 g 7 g 1 g 3 g 5 g 7 ( ) 29 / 72

C 8 D u 8 = w 0 w 1 4 4 w 2 w 3 D8 u D8 d + +, Dd 8 = 8 w 4 w 5 w 6 w 7 ( ) 30 / 72

4 Q 0 Q 2 Q 4 Q 6 = F 4 g 0 g 2 g 4 g 6 [ Q0 Q 2 [ Q4 Q 6 ] ] = F 2 [ g0 g 4 = F 2 [ g0 g 4 ] [ w 0 + ] [ w 4 + w 2 w 6 ] ] F 2 [ g2 g 6 F 2 [ g2 g 6 ] ] ( ) 31 / 72

4 Q 0 Q 2 Q 4 Q 6 = F 4 g 0 g 2 g 4 g 6 [ Q0 Q 2 [ Q4 Q 6 ] ] = F 2 [ g0 g 4 = F 2 [ g0 g 4 ] [ w 0 + ] [ w 4 + w 2 w 6 ] ] F 2 [ g2 g 6 F 2 [ g2 g 6 ] ] ( ) 31 / 72

4 Q 1 Q 3 Q 5 Q 7 = F 4 g 1 g 3 g 5 g 7 [ Q1 Q 3 [ Q5 Q 7 ] ] = F 2 [ g1 g 5 = F 2 [ g1 g 5 ] [ w 0 + ] [ w 4 + w 2 w 6 ] ] F 2 [ g3 g 7 F 2 [ g3 g 7 ] ] ( ) 32 / 72

4 Q 1 Q 3 Q 5 Q 7 = F 4 g 1 g 3 g 5 g 7 [ Q1 Q 3 [ Q5 Q 7 ] ] = F 2 [ g1 g 5 = F 2 [ g1 g 5 ] [ w 0 + ] [ w 4 + w 2 w 6 ] ] F 2 [ g3 g 7 F 2 [ g3 g 7 ] ] ( ) 32 / 72

C 4 2 D4 u + 2 D4 d + 4 [ w D4 u 0 = w 2 ] [ 1 = i ] [ w, D4 d 4 = w 6 ] [ 1 = i ] ( ) 33 / 72

2 [ P0 P 4 [ P2 P 6 [ P1 P 5 [ P3 P 7 ] ] ] ] = = = = [ 1 w 0 1 w 4 [ 1 w 0 1 w 4 [ 1 w 0 1 w 4 [ 1 w 0 1 w 4 ] [ g0 g 4 ] [ g2 g 6 ] [ g1 g 5 ] [ g3 g 7 ] ] ] ] ( ) 34 / 72

C 2 1 D2 u + 1 D2 d + 2 D u 2 = w 0 = 1, D d 2 = w 4 = 1 ( ) 35 / 72

g000 g100 g010 g110 g001 g101 g011 g111 C2 C2 C2 C2 2 2 2 2 C4 C4 4 4 C8 8 G000 G001 G010 G011 G100 G101 G110 G111 ( ) 36 / 72

g 0 g 4 g 2 g 6 g 1 = g 5 g 3 g 7 g 000 g 100 g 010 g 110 g 001 g 101 g 011 g 111 G 0 G 1 G 2 G 3 G 4 G 5 G 6 G 7 = G 000 G 001 G 010 G 011 G 100 G 101 G 110 G 111 ( ) 37 / 72

8 4 4 4 4 (8 FFT ) = 2 (4 FFT ) + 8 4 2 2 2 2 (4 FFT ) = 2 (2 FFT ) + 4 2 FFT 2 [ ] [ ] [ ] [ ] P0 1 +1 g0 g0 + (+1) g = = 4 P 4 1 1 g 4 g 0 + ( 1) g 4 ( ) 38 / 72

(2 FFT ) = 2 = 2 log 2 2 (4 FFT ) = 2 2 + 4 = 8 = 4 log 2 4 (8 FFT ) = 2 8 + 8 = 24 = 8 log 2 8 (16 FFT ) = 2 24 + 16 = 64 = 16 log 2 16. (N FFT ) = N log 2 N 256 256 = 2 8+8 DFT: 2 8+8 2 8+8 FFT: 2 8+8 (8 + 8) 256 256/(8 + 8) = 4096 ( ) 39 / 72

6 i 2π/6 w = e G 0 G 1 G 2 G 3 G 4 G 5 = w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 1 w 2 w 3 w 4 w 5 w 0 w 2 w 4 w 6 w 8 w 10 w 0 w 3 w 6 w 9 w 12 w 15 w 0 w 4 w 8 w 12 w 16 w 20 w 0 w 5 w 10 w 15 w 20 w 25 g 0 g 1 g 2 g 3 g 4 g 5 ( ) 40 / 72

6 i 2π/6 w = e G 0 G 1 G 2 G 3 G 4 G 5 = w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 2 w 4 w 1 w 3 w 5 w 0 w 4 w 8 w 2 w 6 w 10 w 0 w 6 w 12 w 3 w 9 w 15 w 0 w 8 w 16 w 4 w 12 w 20 w 0 w 10 w 20 w 5 w 15 w 25 g 0 g 2 g 4 g 1 g 3 g 5 ( ) 41 / 72

6 w 0 w 0 w 0 w 0 w 2 w 4 w 0 w 4 w 8 w 0 w 0 w 0 w 1 w 3 w 5 w 2 w 6 w 10 w 3 w 9 w 15 w 4 w 12 w 20 w 5 w 15 w 25 = w 0 w 6 w 12 w 0 w 8 w 16 w 0 w 10 w 20 w 0 = w 1 w 3 = w 4 w 2 w 5 = F 3 F 3 F 3 ( ) 42 / 72

9 i 2π/9 w = e G 0 G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 = w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 3 w 6 w 1 w 4 w 7 w 2 w 5 w 8 w 0 w 6 w 12 w 2 w 8 w 14 w 4 w 10 w 16 w 0 w 9 w 18 w 3 w 12 w 21 w 6 w 15 w 24 w 0 w 12 w 24 w 4 w 16 w 28 w 8 w 20 w 32 w 0 w 15 w 30 w 5 w 20 w 35 w 10 w 25 w 40 w 0 w 18 w 36 w 6 w 24 w 42 w 12 w 30 w 48 w 0 w 21 w 42 w 7 w 28 w 49 w 14 w 35 w 56 w 0 w 24 w 48 w 8 w 30 w 56 w 16 w 40 w 64 g 0 g 3 g 6 g 1 g 4 g 7 g 2 g 5 g 8 ( ) 43 / 72

9 = = = = = = F 3 w 0 w 1 F 3, = w 3 w 5 w 6 w 7 w 2 w 5 w 8 F 3, = F 3, = w 0 w 2 w 6 w 8 w 12 w 14 w 4 w 10 F 3 w 16 F 3 F 3 ( ) 44 / 72

( ) 45 / 72

( ) 46 / 72

40x40 4x4 ( ) 47 / 72

(pixel) picture element 4 4 8 0 ( ) 255 ( ) ( ) 48 / 72

640 420 8 640 420 8 = 640 420 = 268800 = 268800/1024 = 262.5 ( ) 49 / 72

FFT FFT ( ) 50 / 72

g m,n (m, n = 0, 1, 2,, N 1) i 2π/N w = e G j,k = m = m g m,n w jm w kn n n i 2π(jm+kn)/N g m,n e ( ) G(ξ, η) = g(x, y) e i(ξx+ηy) dx dy ( ) 51 / 72

(matched filter) Input G inp Reference DDFT G G inp ref G ref DIDFT ( ) 52 / 72

(matched filter) (x, y) g(x, y) g(x, y) g(x, y) G(ξ, η) = F[g(x, y)] = g(x, y) e i(ξx+ηy) dx dy g(x, y) = F 1 [G(ξ, η)] = G(ξ, η) e i(ξx+ηy) dξ dη ( ) 53 / 72

(matched filter) g ref x x 0 y y 0 g inp g ref (x x 0, y y 0 ) = g inp (x, y) G ref (ξ, η) G inp (ξ, η) G ref (ξ, η) e i(x 0ξ+y 0 η) = G inp (ξ, η), ξ, η [ ] F 1 Ginp (ξ, η) = δ(x x 0, y y 0 ). G ref (ξ, η) ( ) 54 / 72

(matched filter) ( ) 55 / 72

(phase only correlation; POC) Input G inp Reference DDFT G G inp ref = G G inp inp G G ref ref G ref DIDFT ( ) 56 / 72

(phase only correlation; POC) G(ξ, η) G (ξ, η) G(ξ, η) G (ξ, η) = G(ξ, η), G(ξ, η) = G(ξ, η) G (ξ, η) G ref (ξ, η) G inp (ξ, η) G ref (ξ, η) e i(x 0ξ+y 0 η) = G inp (ξ, η), ξ, η G ref (ξ, η) e i(x 0ξ+y 0 η) = G inp (ξ, η), ξ, η ( ) 57 / 72

(phase only correlation; POC) G ref G ref G inp / G ref G ref = G ref, G ref 2 = G ref G ref G inp = G inp/ G inp G ref G ref / G ref = G inp G ref G ref G inp = G inp G ref G ref G ref G inp G ref = G inpg ref G ref G ref G inp G ref = G inpg ref G inp G ref [ ] F 1 Ginp G inp G ref = δ(x x 0, y y 0 ) G ref ( ) 58 / 72

(phase only correlation; POC) ( ) 59 / 72

F N F 1 N = (1/N) F N N log 2 N ( ) 60 / 72

x(t) = c 0 1 + a 1 cos t + b 1 sin t +a 2 cos 2t + b 2 sin 2t +a 3 cos 3t + b 3 sin 3t + a 1 cos t + b 1 sin t (a 1 ib 1 )(cos t + i sin t) a 2 cos 2t + b 2 sin 2t (a 2 ib 2 )(cos 2t + i sin 2t). x(t) = c 0 1 + c 1 e it + c 2 e 2it + c 3 e 3it + ( ) 61 / 72

x(t) = c 0 1 + c 1 e it + c 2 e 2it + c 3 e 3it + ( ) 62 / 72

c 1 = a 1 ib 1 r 1 α 1 a 1 = r 1 cos α 1, b 1 = r 1 sin α 1 c 1 e it = a 1 cos t + b 1 sin t = r 1 (cos t cos α 1 sin t sin α 1 ) = r 1 cos(t α 1 ) c 1 c 1 c 1 ( ) 63 / 72

c 0 2π 0 1 dt = 2π, 2π 0 2π 0 [ e e it it dt = i [ e e 2it 2it dt = 2i. ] t=2π t=0 ] t=2π t=0 = 0, = 0, 2π 0 x(t) dt = c 0 2π c 0 = 1 2π x(t) dt 2π 0 ( ) 64 / 72

c 1 e it x(t) e it = c 0 e it + c 1 1 + c 2 e it + c 3 e 2it + 2π 0 x(t) e it dt = c 1 2π c 1 = 1 2π x(t) e it dt 2π 0 ( ) 65 / 72

c 2 e 2it x(t) e 2it = c 0 e 2it + c 1 e it + c 2 1 + c 3 e it + c 4 e 2it + 2π 0 x(t) e 2it dt = c 2 2π c 2 = 1 2π x(t) e 2it dt 2π 0 ( ) 66 / 72

< f (t), g(t) >= 2π 0 f (t) g(t) dt < x(t), 1 > = < x(t), e it > = < x(t), e 2it > = 2π 0 2π 0 2π 0 x(t) dt x(t) e it dt x(t) e 2it dt ( ) 67 / 72

x(t) = c 0 1 + c 1 e it + c 2 e 2it + c 3 e 3it + c 0 = 1 2π < x(t), 1 > c 1 = 1 2π < x(t), eit > c 2 = 1 2π < x(t), e2it > c 3 = 1 2π < x(t), e3it >. ( ) 68 / 72

< 1, 1 >=< e it, e it >=< e 2it, e 2it >= = 2π 0 1 dt = 2π < 1, e it >=< e it, e 2it >=< e 2it, e 3it >= = < 1, e 2it >=< e it, e 3it >=< e 2it, e 4it >= = 2π 0 2π 0. e it dt = 0, e 2it dt = 0, ( ) 69 / 72

g 0 = 1, g 1 = e it, g 2 = e 2it, g 3 = e 3it, < g i, g j >= { 2π i = j 0 i j g i g j ( ) 70 / 72

(8 ) h = 2π/8 f (0) = f 0, f (h) = f 1, f (2h) = f 2,, f (7h) = f 7 2π 0 f (t) dt = f 0 h + f 1 h + f 2 h + + f 7 h w = e i (2π/8) 2π 0 x(t) e it dt = (x 0 1)h + (x 1 w)h + (x 2 w 2 )h + + (x 7 w 7 )h c 1 = h { x0 + x 1 w + x 2 w 2 + + x 7 w 7} 2π ( ) 71 / 72

c 1 x 0 + x 1 w + x 2 w 2 + + x 7 w 7 c 0 x 0 + x 1 + x 2 + + x 7 c 1 x 0 + x 1 w + x 2 w 2 + + x 7 w 7 c 2 x 0 + x 1 w 2 + x 2 w 4 + + x 7 w 14 c 3 x 0 + x 1 w 3 + x 2 w 6 + + x 7 w 21. c 7 x 0 + x 1 w 7 + x 2 w 14 + + x 7 w 49 ( ) 72 / 72

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