Cwtlianxuxiaobo.rar_小波连续变换_连续小波变换

/* Fast Fourier/Cosine/Sine Transform dimension :one data length :power of 2 decimation :frequency radix :4, 2 data :inplace table :use functions cdft: Complex Discrete Fourier Transform rdft: Real Discrete Fourier Transform ddct: Discrete Cosine Transform ddst: Discrete Sine Transform dfct: Cosine Transform of RDFT (Real Symmetric DFT) dfst: Sine Transform of RDFT (Real Anti-symmetric DFT) function prototypes void cdft(int, int, double *, int *, double *); void rdft(int, int, double *, int *, double *); void ddct(int, int, double *, int *, double *); void ddst(int, int, double *, int *, double *); void dfct(int, double *, double *, int *, double *); void dfst(int, double *, double *, int *, double *); -------- Complex DFT (Discrete Fourier Transform) -------- [definition] <case1> X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n <case2> X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n (notes: sum_j=0^n-1 is a summation from j=0 to n-1) [usage] <case1> ip[0] = 0; // first time only cdft(2*n, 1, a, ip, w); <case2> ip[0] = 0; // first time only cdft(2*n, -1, a, ip, w); [parameters] 2*n :data length (int) n >= 1, n = power of 2 a[0...2*n-1] :input/output data (double *) input data a[2*j] = Re(x[j]), a[2*j+1] = Im(x[j]), 0<=j<n output data a[2*k] = Re(X[k]), a[2*k+1] = Im(X[k]), 0<=k<n ip[0...*] :work area for bit reversal (int *) length of ip >= 2+sqrt(n) strictly, length of ip >= 2+(1<<(int)(log(n+0.5)/log(2))/2). ip[0],ip[1] are pointers of the cos/sin table. w[0...n/2-1] :cos/sin table (double *) w[],ip[] are initialized if ip[0] == 0. [remark] Inverse of cdft(2*n, -1, a, ip, w); is cdft(2*n, 1, a, ip, w); for (j = 0; j <= 2 * n - 1; j++) { a[j] *= 1.0 / n; } . -------- Real DFT / Inverse of Real DFT -------- [definition] <case1> RDFT R[k] = sum_j=0^n-1 a[j]*cos(2*pi*j*k/n), 0<=k<=n/2 I[k] = sum_j=0^n-1 a[j]*sin(2*pi*j*k/n), 0<k<n/2 <case2> IRDFT (excluding scale) a[k] = (R[0] + R[n/2]*cos(pi*k))/2 + sum_j=1^n/2-1 R[j]*cos(2*pi*j*k/n) + sum_j=1^n/2-1 I[j]*sin(2*pi*j*k/n), 0<=k<n [usage] <case1> ip[0] = 0; // first time only rdft(n, 1, a, ip, w); <case2> ip[0] = 0; // first time only rdft(n, -1, a, ip, w); [parameters] n :data length (int) n >= 2, n = power of 2 a[0...n-1] :input/output data (double *) <case1> output data a[2*k] = R[k], 0<=k<n/2 a[2*k+1] = I[k], 0<k<n/2 a[1] = R[n/2] <case2> input data a[2*j] = R[j], 0<=j<n/2 a[2*j+1] = I[j], 0<j<n/2 a[1] = R[n/2] ip[0...*] :work area for bit reversal (int *) length of ip >= 2+sqrt(n/2) strictly, length of ip >= 2+(1<<(int)(log(n/2+0.5)/log(2))/2). ip[0],ip[1] are pointers of the cos/sin table. w[0...n/2-1] :cos/sin table (double *) w[],ip[] are initialized if ip[0] == 0. [remark] Inverse of rdft(n, 1, a, ip, w); is rdft(n, -1, a, ip, w); for (j = 0; j <= n - 1; j++) { a[j] *= 2.0 / n; } . -------- DCT (Discrete Cosine Transform) / Inverse of DCT -------- [definition] <case1> IDCT (excluding scale) C[k] = sum_j=0^n-1 a[j]*cos(pi*j*(k+1/2)/n), 0<=k<n <case2> DCT C[k] = sum_j=0^n-1 a[j]*cos(pi*(j+1/2)*k/n), 0<=k<n [usage] <case1> ip[0] = 0; // first time only ddct(n, 1, a, ip, w); <case2> ip[0] = 0; // first time only ddct(n, -1, a, ip, w); [parameters] n :data length (int) n >= 2, n = power of 2 a[0...n-1] :input/output data (double *) output data a[k] = C[k], 0<=k<n ip[0...*] :work area for bit reversal (int *) length of ip >= 2+sqrt(n/2) strictly, length of ip >= 2+(1<<(int)(log(n/2+0.5)/log(2))/2). ip[0],ip[1] are pointers of the cos/sin table. w[0...n*5/4-1] :cos/sin table (double *) w[],ip[] are initialized if ip[0] == 0. [remark] Inverse of ddct(n, -1, a, ip, w); is a[0] *= 0.5; ddct(n, 1, a, ip, w); for (j = 0; j <= n - 1; j++) { a[j] *= 2.0 / n; } . -------- DST (Discrete Sine Transform) / Inverse of DST -------- [definition] <case1> IDST (excluding scale) S[k] = sum_j=1^n A[j]*sin(pi*j*(k+1/2)/n), 0<=k<n <case2> DST S[k] = sum_j=0^n-1 a[j]*sin(pi*(j+1/2)*k/n), 0<k<=n [usage] <case1> ip[0] = 0; // first time only ddst(n, 1, a, ip, w); <case2> ip[0] = 0; // first time only ddst(n, -1, a, ip, w); [parameters] n :data length (int) n >= 2, n = power of 2 a[0...n-1] :input/output data (double *) <case1> input data a[j] = A[j], 0<j<n a[0] = A[n] output data a[k] = S[k], 0<=k<n <case2> output data a[k] = S[k], 0<k<n a[0] = S[n] ip[0...*] :work area for bit reversal (int *) length of ip >= 2+sqrt(n/2) strictly, length of ip >= 2+(1<<(int)(log(n/2+0.5)/log(2))/2). ip[0],ip[1] are pointers of the cos/sin table. w[0...n*5/4-1] :cos/sin table (double *) w[],ip[] are initialized if ip[0] == 0. [remark] Inverse of ddst(n, -1, a, ip, w); is a[0] *= 0.5; ddst(n, 1, a, ip, w); for (j = 0; j <= n - 1; j++) { a[j] *= 2.0 / n; } . -------- Cosine Transform of RDFT (Real Symmetric DFT) -------- [definition] C[k] = sum_j=0^n a[j]*cos(pi*j*k/n), 0<=k<=n [usage] ip[0] = 0; // first time only dfct(n, a, t, ip, w); [parameters] n :data length - 1 (int) n >= 2, n = power of 2 a[0...n] :input/output data (double *) output data a[k] = C[k], 0<=k<=n t[0...n/2] :work area (double *) ip[0...*] :work area for bit reversal (int *) length of ip