// create FFT plan for regular DFT
// _nfft : FFT size
// _x : input array [size: _nfft x 1]
// _y : output array [size: _nfft x 1]
// _dir : fft direction: {LIQUID_FFT_FORWARD, LIQUID_FFT_BACKWARD}
// _method : fft method
FFT(plan) FFT(_create_plan_rader)(unsigned int _nfft,
TC * _x,
TC * _y,
int _dir,
int _flags)
{
// allocate plan and initialize all internal arrays to NULL
FFT(plan) q = (FFT(plan)) malloc(sizeof(struct FFT(plan_s)));
q->nfft = _nfft;
q->x = _x;
q->y = _y;
q->flags = _flags;
q->type = (_dir == LIQUID_FFT_FORWARD) ? LIQUID_FFT_FORWARD : LIQUID_FFT_BACKWARD;
q->direction = (_dir == LIQUID_FFT_FORWARD) ? LIQUID_FFT_FORWARD : LIQUID_FFT_BACKWARD;
q->method = LIQUID_FFT_METHOD_RADER;
q->execute = FFT(_execute_rader);
// allocate memory for sub-transforms
q->data.rader.x_prime = (TC*)malloc((q->nfft-1)*sizeof(TC));
q->data.rader.X_prime = (TC*)malloc((q->nfft-1)*sizeof(TC));
// create sub-FFT of size nfft-1
q->data.rader.fft = FFT(_create_plan)(q->nfft-1,
q->data.rader.x_prime,
q->data.rader.X_prime,
LIQUID_FFT_FORWARD,
q->flags);
// create sub-IFFT of size nfft-1
q->data.rader.ifft = FFT(_create_plan)(q->nfft-1,
q->data.rader.X_prime,
q->data.rader.x_prime,
LIQUID_FFT_BACKWARD,
q->flags);
// compute primitive root of nfft
unsigned int g = liquid_primitive_root_prime(q->nfft);
// create and initialize sequence
q->data.rader.seq = (unsigned int *)malloc((q->nfft-1)*sizeof(unsigned int));
unsigned int i;
for (i=0; i<q->nfft-1; i++)
q->data.rader.seq[i] = liquid_modpow(g, i+1, q->nfft);
// compute DFT of sequence { exp(-j*2*pi*g^i/nfft }, size: nfft-1
// NOTE: R[0] = -1, |R[k]| = sqrt(nfft) for k != 0
// (use newly-created FFT plan of length nfft-1)
T d = (q->direction == LIQUID_FFT_FORWARD) ? -1.0 : 1.0;
for (i=0; i<q->nfft-1; i++)
q->data.rader.x_prime[i] = cexpf(_Complex_I*d*2*M_PI*q->data.rader.seq[i]/(T)(q->nfft));
FFT(_execute)(q->data.rader.fft);
// copy result to R
q->data.rader.R = (TC*)malloc((q->nfft-1)*sizeof(TC));
memmove(q->data.rader.R, q->data.rader.X_prime, (q->nfft-1)*sizeof(TC));
// return main object
return q;
}
// create FFT plan
// _nfft : FFT size
// _x : input array [size: _nfft x 1]
// _y : output array [size: _nfft x 1]
// _dir : fft direction: {LIQUID_FFT_FORWARD, LIQUID_FFT_BACKWARD}
// _method : fft method
FFT(plan) FFT(_create_plan_rader2)(unsigned int _nfft,
TC * _x,
TC * _y,
int _dir,
int _flags)
{
// allocate plan and initialize all internal arrays to NULL
FFT(plan) q = (FFT(plan)) malloc(sizeof(struct FFT(plan_s)));
q->nfft = _nfft;
q->x = _x;
q->y = _y;
q->flags = _flags;
q->type = (_dir == LIQUID_FFT_FORWARD) ? LIQUID_FFT_FORWARD : LIQUID_FFT_BACKWARD;
q->direction = (_dir == LIQUID_FFT_FORWARD) ? LIQUID_FFT_FORWARD : LIQUID_FFT_BACKWARD;
q->method = LIQUID_FFT_METHOD_RADER2;
q->execute = FFT(_execute_rader2);
unsigned int i;
// compute primitive root of nfft
unsigned int g = liquid_primitive_root_prime(q->nfft);
// create and initialize sequence
q->data.rader2.seq = (unsigned int *)malloc((q->nfft-1)*sizeof(unsigned int));
for (i=0; i<q->nfft-1; i++)
q->data.rader2.seq[i] = liquid_modpow(g, i+1, q->nfft);
#if 0
// compute larger FFT length greater than 2*nfft-4
// NOTE: while any length greater than 2*nfft-4 will work, use
// nfft_prime as smallest 'simple' FFT (mostly small factors)
//
// TODO: devise better score (fewer factors is better)
// score(n) = n / sum(factors(n).^2)
float gamma_max = 0.0f; // score
unsigned int nfft_prime_opt = 0;
unsigned int num_steps = 10;// + q->nfft;
for (i=1; i<=num_steps; i++) {
unsigned int n_hat = 2*q->nfft - 4 + i;
// compute factors
unsigned int k;
unsigned int num_factors = 0;
unsigned int m = n_hat;
float gamma = 0.0f;
do {
for (k=2; k<=m; k++) {
if ( (m % k) == 0) {
m /= k;
num_factors++;
gamma += k*k;
break;
}
}
} while (m > 1);
// compute score:
//float gamma = (float)n_hat / (float)num_factors;
//float gamma = 1e3f * (float)num_factors / (float)n_hat;
gamma = (float)n_hat / gamma;
if (gamma > gamma_max) {
gamma_max = gamma;
nfft_prime_opt = n_hat;
}
}
q->data.rader2.nfft_prime = nfft_prime_opt;
#else
// compute larger FFT length greater than 2*nfft-4
// NOTE: while any length greater than 2*nfft-4 will work, use
// nfft_prime = 2 ^ nextpow2( 2*nfft - 4 ) to enable
// radix-2 transform
unsigned int m=0;
q->data.rader2.nfft_prime = (2*q->nfft-4)-1;
while (q->data.rader2.nfft_prime > 0) {
q->data.rader2.nfft_prime >>= 1;
m++;
}
q->data.rader2.nfft_prime = 1 << m;
#endif
//printf("nfft_prime = %u\n", q->data.rader2.nfft_prime);
// assert(nfft_prime > 2*nfft-4)
// allocate memory for sub-transforms
q->data.rader2.x_prime = (TC*)malloc((q->data.rader2.nfft_prime)*sizeof(TC));
q->data.rader2.X_prime = (TC*)malloc((q->data.rader2.nfft_prime)*sizeof(TC));
// create sub-FFT of size nfft-1
q->data.rader2.fft = FFT(_create_plan)(q->data.rader2.nfft_prime,
q->data.rader2.x_prime,
q->data.rader2.X_prime,
LIQUID_FFT_FORWARD,
q->flags);
// create sub-IFFT of size nfft-1
q->data.rader2.ifft = FFT(_create_plan)(q->data.rader2.nfft_prime,
q->data.rader2.X_prime,
q->data.rader2.x_prime,
LIQUID_FFT_BACKWARD,
q->flags);
// compute DFT of sequence { exp(-j*2*pi*g^i/nfft }, size: nfft_prime
// NOTE: R[0] = -1, |R[k]| = sqrt(nfft) for k != 0
// (use newly-created FFT plan of length nfft_prime)
T d = (q->direction == LIQUID_FFT_FORWARD) ? -1.0 : 1.0;
for (i=0; i<q->data.rader2.nfft_prime; i++)
q->data.rader2.x_prime[i] = cexpf(_Complex_I*d*2*M_PI*q->data.rader2.seq[i%(q->nfft-1)]/(T)(q->nfft));
FFT(_execute)(q->data.rader2.fft);
// copy result to R
q->data.rader2.R = (TC*)malloc(q->data.rader2.nfft_prime*sizeof(TC));
memmove(q->data.rader2.R, q->data.rader2.X_prime, q->data.rader2.nfft_prime*sizeof(TC));
// return main object
return q;
}
