/* * Free FFT and convolution (C) * * Copyright (c) 2017 Project Nayuki. (MIT License) * https://www.nayuki.io/page/free-small-fft-in-multiple-languages * * Permission is hereby granted, free of charge, to any person obtaining a copy of * this software and associated documentation files (the "Software"), to deal in * the Software without restriction, including without limitation the rights to * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of * the Software, and to permit persons to whom the Software is furnished to do so, * subject to the following conditions: * - The above copyright notice and this permission notice shall be included in * all copies or substantial portions of the Software. * - The Software is provided "as is", without warranty of any kind, express or * implied, including but not limited to the warranties of merchantability, * fitness for a particular purpose and noninfringement. In no event shall the * authors or copyright holders be liable for any claim, damages or other * liability, whether in an action of contract, tort or otherwise, arising from, * out of or in connection with the Software or the use or other dealings in the * Software. */ #include #include #include #include #include #include "fft.h" // Private function prototypes static size_t reverse_bits(size_t x, int n); static void *memdup(const void *src, size_t n); bool Fft_transform(double real[], double imag[], size_t n) { if (n == 0) return true; else if ((n & (n - 1)) == 0) // Is power of 2 return Fft_transformRadix2(real, imag, n); else // More complicated algorithm for arbitrary sizes return Fft_transformBluestein(real, imag, n); } bool Fft_inverseTransform(double real[], double imag[], size_t n) { return Fft_transform(imag, real, n); } bool Fft_transformRadix2(double real[], double imag[], size_t n) { // Length variables bool status = false; int levels = 0; // Compute levels = floor(log2(n)) for (size_t temp = n; temp > 1U; temp >>= 1) levels++; if ((size_t)1U << levels != n) return false; // n is not a power of 2 // Trignometric tables if (SIZE_MAX / sizeof(double) < n / 2) return false; size_t size = (n / 2) * sizeof(double); double *cos_table = malloc(size); double *sin_table = malloc(size); if (cos_table == NULL || sin_table == NULL) goto cleanup; for (size_t i = 0; i < n / 2; i++) { cos_table[i] = cos(2 * M_PI * i / n); sin_table[i] = sin(2 * M_PI * i / n); } // Bit-reversed addressing permutation for (size_t i = 0; i < n; i++) { size_t j = reverse_bits(i, levels); if (j > i) { double temp = real[i]; real[i] = real[j]; real[j] = temp; temp = imag[i]; imag[i] = imag[j]; imag[j] = temp; } } // Cooley-Tukey decimation-in-time radix-2 FFT for (size_t size = 2; size <= n; size *= 2) { size_t halfsize = size / 2; size_t tablestep = n / size; for (size_t i = 0; i < n; i += size) { for (size_t j = i, k = 0; j < i + halfsize; j++, k += tablestep) { size_t l = j + halfsize; double tpre = real[l] * cos_table[k] + imag[l] * sin_table[k]; double tpim = -real[l] * sin_table[k] + imag[l] * cos_table[k]; real[l] = real[j] - tpre; imag[l] = imag[j] - tpim; real[j] += tpre; imag[j] += tpim; } } if (size == n) // Prevent overflow in 'size *= 2' break; } status = true; cleanup: free(cos_table); free(sin_table); return status; } bool Fft_transformBluestein(double real[], double imag[], size_t n) { bool status = false; // Find a power-of-2 convolution length m such that m >= n * 2 + 1 size_t m = 1; while (m / 2 <= n) { if (m > SIZE_MAX / 2) return false; m *= 2; } // Allocate memory if (SIZE_MAX / sizeof(double) < n || SIZE_MAX / sizeof(double) < m) return false; size_t size_n = n * sizeof(double); size_t size_m = m * sizeof(double); double *cos_table = malloc(size_n); double *sin_table = malloc(size_n); double *areal = calloc(m, sizeof(double)); double *aimag = calloc(m, sizeof(double)); double *breal = calloc(m, sizeof(double)); double *bimag = calloc(m, sizeof(double)); double *creal = malloc(size_m); double *cimag = malloc(size_m); if (cos_table == NULL || sin_table == NULL || areal == NULL || aimag == NULL || breal == NULL || bimag == NULL || creal == NULL || cimag == NULL) goto cleanup; // Trignometric tables for (size_t i = 0; i < n; i++) { unsigned long long temp = (unsigned long long)i * i; temp %= (unsigned long long)n * 2; double angle = M_PI * temp / n; // Less accurate version if long long is unavailable: double angle = M_PI * i * i / n; cos_table[i] = cos(angle); sin_table[i] = sin(angle); } // Temporary vectors and preprocessing for (size_t i = 0; i < n; i++) { areal[i] = real[i] * cos_table[i] + imag[i] * sin_table[i]; aimag[i] = -real[i] * sin_table[i] + imag[i] * cos_table[i]; } breal[0] = cos_table[0]; bimag[0] = sin_table[0]; for (size_t i = 1; i < n; i++) { breal[i] = breal[m - i] = cos_table[i]; bimag[i] = bimag[m - i] = sin_table[i]; } // Convolution if (!Fft_convolveComplex(areal, aimag, breal, bimag, creal, cimag, m)) goto cleanup; // Postprocessing for (size_t i = 0; i < n; i++) { real[i] = creal[i] * cos_table[i] + cimag[i] * sin_table[i]; imag[i] = -creal[i] * sin_table[i] + cimag[i] * cos_table[i]; } status = true; // Deallocation cleanup: free(cimag); free(creal); free(bimag); free(breal); free(aimag); free(areal); free(sin_table); free(cos_table); return status; } bool Fft_convolveReal(const double x[], const double y[], double out[], size_t n) { bool status = false; double *ximag = calloc(n, sizeof(double)); double *yimag = calloc(n, sizeof(double)); double *zimag = calloc(n, sizeof(double)); if (ximag == NULL || yimag == NULL || zimag == NULL) goto cleanup; status = Fft_convolveComplex(x, ximag, y, yimag, out, zimag, n); cleanup: free(zimag); free(yimag); free(ximag); return status; } bool Fft_convolveComplex( const double xreal[], const double ximag[], const double yreal[], const double yimag[], double outreal[], double outimag[], size_t n) { bool status = false; if (SIZE_MAX / sizeof(double) < n) return false; size_t size = n * sizeof(double); double *xr = memdup(xreal, size); double *xi = memdup(ximag, size); double *yr = memdup(yreal, size); double *yi = memdup(yimag, size); if (xr == NULL || xi == NULL || yr == NULL || yi == NULL) goto cleanup; if (!Fft_transform(xr, xi, n)) goto cleanup; if (!Fft_transform(yr, yi, n)) goto cleanup; for (size_t i = 0; i < n; i++) { double temp = xr[i] * yr[i] - xi[i] * yi[i]; xi[i] = xi[i] * yr[i] + xr[i] * yi[i]; xr[i] = temp; } if (!Fft_inverseTransform(xr, xi, n)) goto cleanup; for (size_t i = 0; i < n; i++) { // Scaling (because this FFT implementation omits it) outreal[i] = xr[i] / n; outimag[i] = xi[i] / n; } status = true; cleanup: free(yi); free(yr); free(xi); free(xr); return status; } static size_t reverse_bits(size_t x, int n) { size_t result = 0; for (int i = 0; i < n; i++, x >>= 1) result = (result << 1) | (x & 1U); return result; } static void *memdup(const void *src, size_t n) { void *dest = malloc(n); if (n > 0 && dest != NULL) memcpy(dest, src, n); return dest; }