Введение в цифровую обработку сигналов

  • Published on
    28-Jul-2015

  • View
    307

  • Download
    7

Embed Size (px)

Transcript

<p> ( )</p> <p> , 2002</p> <p> , </p> <p>1</p> <p> .......................................................................................................................................3 ..........................................................................................................................................................3 .......................................................................................................................................................4 ...............................................................................................................................................................4 ................................................................................................................5 ......................................................................................................................................5 () ......................................................................................................................6 .......................................................................................................................................................7 ...............................................................................................................................................................8 .............................................................................................................................8 .....................................................................................................................................................11 .............................................................................................................................................................12 .................................................................................................................................................................12 .....................................................................................................................................................13 .............................................................................................................................................................13 ...........................................................................................................................................................14 ..................................................................................................15 ................................................................................................................................................15 ............................................................................................................................................18 ................................................................................................................................................18 .....................................................................................................................................................19 .............................................................................................................................................................20 ................................................................................................................................................20 ......................................................................................................................................20 . .......................................................................................23 ......................................................................................................................................................24 ...................................................................................................................................................29 .....................................................................................................................................................30 ...........................................................................................................................................................30 ............................................................................32 ............................................................................................................................................32 - ...........................................................................................................................34 ..................................................................................................................35 .................................................................................................39 ............................................................................................................................................41 ..............................................................................41 ..............................................................................................................41 ..........................................................................................41 ...............................................................................................................................41 ........................................................................................................................42 .............................................................................................................................43</p> <p>2</p> <p> (). , . . - . , , x(t). . . x(t) y(t). : x(t ) y (t ) . , .. x(t ) y (t ) , x(t + T ) y (t + T ) . , , . . . , , , , . , : x1 (t ) y1 (t ) x 2 (t ) y 2 (t ) , x1 (t ) + x 2 (t ) y1 (t ) + y 2 (t ) . t. . , ( ), 2 , ( ) , .</p> <p>3</p> <p> : 1. () . 2. . ( ). , .. (-). ? , , .. t = t = + . ( - , , - 0), . . .</p> <p>x(t ) y (t ) , 1. , y(t)=cos(t)+sin(3t). ? x(t)=sin(t)+sin(2t), </p> <p>2. x(t)=2sin(t)-cos(3t). ? 3. , ( ) .</p> <p>1. , .. sin(3t), . . , sin(2t) , .. . 2. , A sin(t + ) + B sin(t + ) , A, B, - . 3. x1 (t ) 1 y1 (t ) 2 z1 (t ) x 2 (t ) 1 y 2 (t ) 2 z 2 (t ) . 1 x1 (t ) + x 2 (t ) y1 (t ) + y 2 (t ) , , y1 (t ) + y 2 (t ) 2 z1 (t ) + z 2 (t ) . , 1 2 x1 (t ) + x 2 (t ) z1 (t ) + z 2 (t ) , . </p> <p>4</p> <p> (, ) ( ). . . , . ( ) . . , , , . () . , - , (, analogue-to-digital converter, ADC). , , . : ? . , . , . , . () . ( ). , .. . , , . , . , F ( F), F . --: , F, 2F </p> <p>5</p> <p> . . . , , . , , 20 . , 40 , - . , . , , - (, digital-to-analogue converter, DAC). , , - , -. - 44100 . , CD 44100 .</p> <p> () , ( )? . , , , , , . (aliasing). , , 20 , - (, ) 39 , . 44.1 . , , 44.1 = 22.05 ( 2 ). .. 22.05 , , , 5 . , 5 . , . , , . , , , . ? , 6</p> <p> . . , , . , (-, low-pass filters) , , , . (cutoff frequency) . . -, .. . - . , , - . 44.1 , 22 . , , 22 (, ), , 22 ( ). , 22 . , , ( ). - . , . . , 11 5.5 . ( , ). - , .</p> <p>1. , 8 . ? ? 2. 44 - . 24 . ? ? ? 3. , - ?</p> <p>7</p> <p>1. , 8 0 4 . 4 , 4 . 2. 22 , . .. 24 , , , 22 24 . ( 22 ) 20 22 . , 20 22 . , , .. 20 . 3. , .. . .</p> <p> , , , . x[n] ( ), y[n] (. 1). , .</p> <p>3</p> <p>2</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>. 1. : , . , , .. . , . - ( - ).</p> <p>8</p> <p>1, n = 0 - () [n] = , .. 0, n 0 (. 2).1</p> <p>3</p> <p>2</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>. 2. -. , , (. 3). , x[n] x[n] =i = </p> <p> x[i] [n i] . - </p> <p>+</p> <p> , x[i] . n, x[n]=x[n]*1, .. 0, - .</p> <p>. 3. -. -. - . [n] h[n] (. 4). , h[n] ( -), . , , h[n]. (. 5).</p> <p>9</p> <p> [n]3 2 1</p> <p>1</p> <p>h[n]0 1 2 3 3 2 1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>. 4. -.</p> <p>x[n]3 2 1 0 1 2 3</p> <p>x[1] [n + 1]3 2 1 0 1 2 3</p> <p>x[1] h[n + 1]3 2 1 0 1 2 3</p> <p>x[0] [n]3 2 1 0 1 2 3</p> <p>x[0] h[n]3 2 1 0 1 2 3</p> <p>x[1] [n 1]3 2 1 0 1 2 3</p> <p>x[1] h[n 1]3 2 1 0 1 2 3</p> <p>y[n]3 2 1 0 1 2 3</p> <p>. 5. h[n].</p> <p>10</p> <p> y[n] x[n] : y[n] =k = </p> <p> x[n k ] h[k ] . h[n] </p> <p>+</p> <p> (impulse response) , .. (-). . . x[i,j], (i,j) . - (0,0) . , - h[i,j], h[i,j]=const (0,0) 3 . , h[i,j] 1 ( const). , ( ). , [i m, j n] (-, (m,n)) h[i-m, j-n] . , 3 . . PSF point spread function, .. . , -. , , . , ( ) , , . , , . , . , ( ), . , , , . .</p> <p>1. x[n] x[n] ( ). 2. : x[n] 2 x[n] . 3. h[n] = [n 1] ?</p> <p>11</p> <p>1 4. h[n] = [n] + [n 2] ? 2</p> <p>1. -: h[n] = [n] . . 2. h[n] = 2 [n] . 3. . : y[n]=x[n-1]. 4. ( ). 2 , 2 .</p> <p>, . . h ( ), . , . , 0. , , , ( 3 ). : y[n] =</p> <p>k = </p> <p> x[n k ] h[k ]</p> <p>+</p> <p> (convolution). , . : y[n] = x[n] h[n] . h[n] (kernel) . , . , , (.. ). , , . x[n] </p> <p>12</p> <p> 0 N-1 ( N). h[n] m1 m2 , M ( M = m1 + m2 + 1 ). , y[n], m1 N 1 + m2 . , N+M-1, .. . , M-1 , M . : 1. x[n] y[n] = y[n] x[n] (.. ). 2. ( x[n] y[n]) z[n] = x[n] ( y[n] z[n]) (.. , , ( y[n] z[n]) , y[n] z[n]). 3. x[n] y[n] + x[n] z[n] = x[n] ( y[n] + z[n])</p> <p>1. x[n], [A, B], h[n], [C, D]. , . 2. , N M.</p> <p>1. x[n] A [A+C, A+D]. x[n] B [B+C, B+D]. , [A+C, B+D]. (B-A)+(D-A)+1, N+M-1. 2. N M . , M (M ). , M*N .</p> <p>13</p> <p> . , , . , . , , . (correlation). x[n], g[n] . x[n] g[nk] k. ( ) . , y[k], , x[n] k g[n]. : y[k ] =i = </p> <p> g[i] x[i k ] .</p> <p>+</p> <p> , .. i, g[i] . , , ( ). y[n] , n , n . , . , , . : y[n] =</p> <p>k = </p> <p> x[n + k ] g[k ]</p> <p>+</p> <p> , . , , y[n] =k = </p> <p> x[n k ] g[k ] .</p> <p>+</p> <p>, h[k]=g[-k]. , , , . - (cross-correlation). (autocorrelation) , . , .</p> <p>14</p> <p> , (). . , . . , (.. , , ). , . (Fourier transform) ( , .. ). . 1. . 2. . 3. . 4. . , , . . x[n] N . (.. ) : x[n] = Ak cosk =0 N 2</p> <p>2kn N 2 2kn N 2 2k (n + k ) + Bk sin = Ck cos N N N k =0 k =0</p> <p> . , (DC offset) . , . , . Ak Bk (spectrum). , . .</p> <p>15</p> <p>1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>1</p> <p>1</p> <p>1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>1</p> <p>1</p> <p>1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>1</p> <p>1</p> <p>1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>1</p> <p>1</p> <p>1</p> <p>1</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>0</p> <p>1</p> <p>2</p> <p>3</p> <p>4</p> <p>5</p> <p>6</p> <p>7</p> <p>8</p> <p>1</p> <p>1</p> <p>. 6. 8- . , . .</p> <p>16</p> <p> . 6 , 8 . 8 , . . , Ak Bk . , , . (.. ) . (inverse Fourier transform). ( ; ). , .. Ak Bk . N 2kn 2kn sin , cos n , k = 0,..., N N 2 N. , () , . Ak Bk . , Ak Bk ( , ): Ak = 2 N</p> <p> x[i] cosi =0</p> <p>N 1</p> <p>N 2ki , k = 1,..., 1 N 2 N 2ki , k = 0, N 2</p> <p>Ak = Bk =</p> <p>1 N</p> <p> x[i] cosi =0</p> <p>N 1</p> <p>2 N</p> <p> x[i] sini =0</p> <p>N 1</p> <p>N 2ki , k = 0,..., N 2</p> <p> : N , N+2 ? : B0 BN 2 (.. ), . , , , . , </p> <p>17</p> <p> . .</p> <p> ( N 2 ) . : N log 2 N . (, FFT, fast Fourier transform). , () ( , , ). , . N . , . , .. , . : , N . , . N (FFT size).</p> <p> . . x[n], n=0,,N-1 , N . X[k], k=0,N-1 , N . ( j = 1 ): X [k ] = x[n] e jnk ( 2n=0 N 1 N)</p> <p>x[n] =</p> <p>1 N</p> <p> X [k ] ek =0</p> <p>N 1</p> <p>jnk ( 2 N )</p> <p> , N/2+1 ( ) , , . , .</p> <p> , , . </p> <p>18</p> <p> 2k1n1 2k2 n2 cos . N1 N2</p> <p>hksink 2 (n1 , n2 ) = sin 1,</p> <p>2k1 n1 2k 2 n 2 sin N1 N2</p> <p>hkcosk 2 (n1 , n2 ) = cos 1,</p> <p> N1xN2 , . k1 k2 ( , ). , k1 = 0,,N1-1; k2 = 0,,N2-1. n1 n2 - . , n1 = 0,,N1-1; n2 = 0,,N2-1. ( x[n1,n2] , X[k1,k2] ): X [k1 , k2 ] =N 1 1N 2 1 n1 = 0 n 2 = 0</p> <p> x[n , n ] e1 2 N 1 1N 2 1 k1 = 0 k 2 = 0 1</p> <p> jn1 k1 ( 2 N 1 ) jn 2 k 2 ( 2 N 2 )</p> <p>e</p> <p>x[n1 , n2 ] =</p> <p>1 N1 N 2</p> <p> X [k , k ] e2</p> <p>jn1 k1 ( 2 N 1 )</p> <p>e jn 2 k 2 ( 2</p> <p>N2 )</p> <p> . , , .. . , . . , ( ). . , , .</p> <p>1. 44100 . 4096. ? ( ) ? 2. ? , 4 ?</p> <p>19</p> <p>1. 4096 0.0929 . (0 ) , . 4096 , 0.0929 . , 10.77 . 21.53 ....</p>