475 lines
16 KiB
Plaintext
475 lines
16 KiB
Plaintext
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// qracodes.c
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// Q-ary RA codes encoding/decoding functions
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//
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// (c) 2016 - Nico Palermo, IV3NWV - Microtelecom Srl, Italy
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// ------------------------------------------------------------------------------
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// This file is part of the qracodes project, a Forward Error Control
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// encoding/decoding package based on Q-ary RA (Repeat and Accumulate) LDPC codes.
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//
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// qracodes is free software: you can redistribute it and/or modify
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// it under the terms of the GNU General Public License as published by
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// the Free Software Foundation, either version 3 of the License, or
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// (at your option) any later version.
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// qracodes is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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// You should have received a copy of the GNU General Public License
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// along with qracodes source distribution.
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// If not, see <http://www.gnu.org/licenses/>.
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#include <stdio.h>
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#include <math.h>
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#include "npfwht.h"
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#include "pdmath.h"
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#include "qracodes.h"
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int qra_encode(const qracode *pcode, int *y, const int *x)
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{
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int k,j,kk,jj;
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int t, chk = 0;
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const int K = pcode->K;
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const int M = pcode->M;
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const int NC= pcode->NC;
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const int a = pcode->a;
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const int *acc_input_idx = pcode->acc_input_idx;
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const int *acc_input_wlog = pcode->acc_input_wlog;
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const int *gflog = pcode->gflog;
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const int *gfexp = pcode->gfexp;
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// copy the systematic symbols to destination
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memcpy(y,x,K*sizeof(int));
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y = y+K; // point to check symbols
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// compute the code check symbols as a weighted accumulation of a permutated
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// sequence of the (repeated) systematic input symbols:
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// chk(k+1) = x(idx(k))*alfa^(logw(k)) + chk(k)
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// (all operations performed over GF(M))
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if (a==1) { // grouping factor = 1
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for (k=0;k<NC;k++) {
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t = x[acc_input_idx[k]];
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if (t) {
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// multiply input by weight[k] and xor it with previous check
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t = (gflog[t] + acc_input_wlog[k])%(M-1);
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t = gfexp[t];
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chk ^=t;
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}
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y[k] = chk;
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}
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#ifdef QRA_DEBUG
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// verify that the encoder accumulator is terminated to 0
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// (we designed the code this way so that Iex = 1 when Ia = 1)
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t = x[acc_input_idx[k]];
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if (t) {
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t = (gflog[t] + acc_input_wlog[k])%(M-1);
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t = gfexp[t];
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// accumulation
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chk ^=t;
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}
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return (chk==0);
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#else
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return 1;
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#endif // QRA_DEBUG
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}
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else { // grouping factor > 1
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for (k=0;k<NC;k++) {
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kk = a*k;
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for (j=0;j<a;j++) {
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jj = kk+j;
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// irregular grouping support
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if (acc_input_idx[jj]<0)
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continue;
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t = x[acc_input_idx[jj]];
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if (t) {
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// multiply input by weight[k] and xor it with previous check
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t = (gflog[t] + acc_input_wlog[jj])%(M-1);
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t = gfexp[t];
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chk ^=t;
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}
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}
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y[k] = chk;
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}
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#ifdef QRA_DEBUG
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// verify that the encoder accumulator is terminated to 0
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// (we designed the code this way so that Iex = 1 when Ia = 1)
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kk = a*k;
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for (j=0;j<a;j++) {
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jj = kk+j;
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if (acc_input_idx[jj]<0)
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continue;
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t = x[acc_input_idx[jj]];
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if (t) {
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// multiply input by weight[k] and xor it with previous check
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t = (gflog[t] + acc_input_wlog[jj])%(M-1);
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t = gfexp[t];
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chk ^=t;
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}
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}
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return (chk==0);
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#else
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return 1;
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#endif // QRA_DEBUG
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}
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}
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static void qra_ioapprox(float *src, float C, int nitems)
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{
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// In place approximation of the modified bessel function I0(x*C)
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// Computes src[k] = Io(src[k]*C) where Io() is the modified Bessel function of first kind and order 0
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float v;
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float vsq;
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while (nitems--) {
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v = src[nitems]*C;
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// rational approximation of log(Io(v))
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vsq = v*v;
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v = vsq*(v+0.039f)/(vsq*.9931f+v*2.6936f+0.5185f);
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if (v>80.f) // avoid floating point exp() overflows
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v=80.f;
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src[nitems] = (float)exp(v);
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}
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}
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float qra_mfskbesselmetric(float *pix, const float *rsq, const int m, const int N, float EsNoMetric)
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{
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// Computes the codeword symbols intrinsic probabilities
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// given the square of the received input amplitudes.
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// The input vector rqs must be a linear array of size M*N, where M=2^m,
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// containing the squared amplitudes (rp*rp+rq*rq) of the input samples
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// First symbol amplitudes should be stored in the first M positions,
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// second symbol amplitudes stored at positions [M ... 2*M-1], and so on.
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// Output vector is the intrinsic symbol metric (the probability distribution)
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// assuming that symbols are transmitted using a M-FSK modulation
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// and incoherent demodulation.
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// As the input Es/No is generally unknown (as it cannot be exstimated accurately
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// when the codeword length is few tens symbols) but an exact metric requires it
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// we simply fix it to a predefined EsNoMetric value so that the metric is what
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// expected at that specific value.
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// The metric computed in this way is optimal only at this predefined Es/No value,
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// nevertheless it is usually better than a generic parameter-free metric which
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// makes no assumptions on the input Es/No.
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// returns the estimated noise standard deviation
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int k;
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float rsum = 0.f;
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float sigmaest, cmetric;
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const int M = 1<<m;
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const int nsamples = M*N;
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// compute total power and modulus of input signal
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for (k=0;k<nsamples;k++) {
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rsum = rsum+rsq[k];
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pix[k] = (float)sqrt(rsq[k]);
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}
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rsum = rsum/nsamples; // average S+N
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// IMPORTANT NOTE: in computing the noise stdev it is assumed that
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// in the input amplitudes there's no strong interference!
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// A more robust estimation can be done evaluating the histogram of the input amplitudes
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sigmaest = (float)sqrt(rsum/(1.0f+EsNoMetric/M)/2); // estimated noise stdev
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cmetric = (float)sqrt(2*EsNoMetric)/sigmaest;
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for (k=0;k<N;k++) {
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// compute bessel metric for each symbol in the codeword
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qra_ioapprox(PD_ROWADDR(pix,M,k),cmetric,M);
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// normalize to a probability distribution
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pd_norm(PD_ROWADDR(pix,M,k),m);
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}
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return sigmaest;
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}
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#ifdef QRA_DEBUG
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void pd_print(int imsg,float *ppd,int size)
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{
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int k;
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printf("imsg=%d\n",imsg);
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for (k=0;k<size;k++)
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printf("%7.1e ",ppd[k]);
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printf("\n");
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}
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#endif
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#define ADDRMSG(fp, msgidx) PD_ROWADDR(fp,qra_M,msgidx)
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#define C2VMSG(msgidx) PD_ROWADDR(qra_c2vmsg,qra_M,msgidx)
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#define V2CMSG(msgidx) PD_ROWADDR(qra_v2cmsg,qra_M,msgidx)
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#define MSGPERM(logw) PD_ROWADDR(qra_pmat,qra_M,logw)
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#define QRACODE_MAX_M 256 // Maximum alphabet size handled by qra_extrinsic
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int qra_extrinsic(const qracode *pcode,
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float *pex,
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const float *pix,
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int maxiter,
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float *qra_v2cmsg,
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float *qra_c2vmsg)
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{
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const int qra_M = pcode->M;
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const int qra_m = pcode->m;
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const int qra_V = pcode->V;
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const int qra_MAXVDEG = pcode->MAXVDEG;
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const int *qra_vdeg = pcode->vdeg;
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const int qra_C = pcode->C;
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const int qra_MAXCDEG = pcode->MAXCDEG;
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const int *qra_cdeg = pcode->cdeg;
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const int *qra_v2cmidx = pcode->v2cmidx;
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const int *qra_c2vmidx = pcode->c2vmidx;
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const int *qra_pmat = pcode->gfpmat;
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const int *qra_msgw = pcode->msgw;
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// float msgout[qra_M]; // buffer to store temporary results
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float msgout[QRACODE_MAX_M]; // we use a fixed size in order to avoid mallocs
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float totex; // total extrinsic information
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int nit; // current iteration
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int nv; // current variable
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int nc; // current check
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int k,kk; // loop indexes
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int ndeg; // current node degree
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int msgbase; // current offset in the table of msg indexes
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int imsg; // current message index
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int wmsg; // current message weight
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int rc = -1; // rc>=0 extrinsic converged to 1 at iteration rc (rc=0..maxiter-1)
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// rc=-1 no convergence in the given number of iterations
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// rc=-2 error in the code tables (code checks degrees must be >1)
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// rc=-3 M is larger than QRACODE_MAX_M
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if (qra_M>QRACODE_MAX_M)
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return -3;
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// message initialization -------------------------------------------------------
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// init c->v variable intrinsic msgs
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pd_init(C2VMSG(0),pix,qra_M*qra_V);
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// init the v->c messages directed to code factors (k=1..ndeg) with the intrinsic info
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for (nv=0;nv<qra_V;nv++) {
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ndeg = qra_vdeg[nv]; // degree of current node
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msgbase = nv*qra_MAXVDEG; // base to msg index row for the current node
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// copy intrinsics on v->c
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for (k=1;k<ndeg;k++) {
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imsg = qra_v2cmidx[msgbase+k];
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pd_init(V2CMSG(imsg),ADDRMSG(pix,nv),qra_M);
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}
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}
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// message passing algorithm iterations ------------------------------
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for (nit=0;nit<maxiter;nit++) {
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// c->v step -----------------------------------------------------
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// Computes messages from code checks to code variables.
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// As the first qra_V checks are associated with intrinsic information
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// (the code tables have been constructed in this way)
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// we need to do this step only for code checks in the range [qra_V..qra_C)
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// The convolutions of probability distributions over the alphabet of a finite field GF(qra_M)
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// are performed with a fast convolution algorithm over the given field.
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//
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// I.e. given the code check x1+x2+x3 = 0 (with x1,x2,x3 in GF(2^m))
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// and given Prob(x2) and Prob(x3), we have that:
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// Prob(x1=X1) = Prob((x2+x3)=X1) = sum((Prob(x2=X2)*Prob(x3=(X1+X2))) for all the X2s in the field
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// This translates to Prob(x1) = IWHT(WHT(Prob(x2))*WHT(Prob(x3)))
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// where WHT and IWHT are the direct and inverse Walsh-Hadamard transforms of the argument.
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// Note that the WHT and the IWHF differs only by a multiplicative coefficent and since in this step
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// we don't need that the output distribution is normalized we use the relationship
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// Prob(x1) =(proportional to) WH(WH(Prob(x2))*WH(Prob(x3)))
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// In general given the check code x1+x2+x3+..+xm = 0
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// the output distribution of a variable given the distributions of the other m-1 variables
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// is the inverse WHT of the product of the WHTs of the distribution of the other m-1 variables
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// The complexity of this algorithm scales with M*log2(M) instead of the M^2 complexity of
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// the brute force approach (M=size of the alphabet)
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for (nc=qra_V;nc<qra_C;nc++) {
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ndeg = qra_cdeg[nc]; // degree of current node
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if (ndeg==1) // this should never happen (code factors must have deg>1)
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return -2; // bad code tables
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msgbase = nc*qra_MAXCDEG; // base to msg index row for the current node
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// transforms inputs in the Walsh-Hadamard "frequency" domain
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// v->c -> fwht(v->c)
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for (k=0;k<ndeg;k++) {
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imsg = qra_c2vmidx[msgbase+k]; // msg index
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np_fwht(qra_m,V2CMSG(imsg),V2CMSG(imsg)); // compute fwht
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}
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// compute products and transform them back in the WH "time" domain
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for (k=0;k<ndeg;k++) {
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// init output message to uniform distribution
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pd_init(msgout,pd_uniform(qra_m),qra_M);
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// c->v = prod(fwht(v->c))
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// TODO: we assume that checks degrees are not larger than three but
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// if they are larger the products can be computed more efficiently
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for (kk=0;kk<ndeg;kk++)
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if (kk!=k) {
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imsg = qra_c2vmidx[msgbase+kk];
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pd_imul(msgout,V2CMSG(imsg),qra_m);
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}
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// transform product back in the WH "time" domain
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// Very important trick:
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// we bias WHT[0] so that the sum of output pd components is always strictly positive
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// this helps avoiding the effects of underflows in the v->c steps when multipling
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// small fp numbers
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||
|
|
msgout[0]+=1E-7f; // TODO: define the bias accordingly to the field size
|
||
|
|
|
||
|
|
np_fwht(qra_m,msgout,msgout);
|
||
|
|
|
||
|
|
// inverse weight and output
|
||
|
|
imsg = qra_c2vmidx[msgbase+k]; // current output msg index
|
||
|
|
wmsg = qra_msgw[imsg]; // current msg weight
|
||
|
|
|
||
|
|
if (wmsg==0)
|
||
|
|
pd_init(C2VMSG(imsg),msgout,qra_M);
|
||
|
|
else
|
||
|
|
// output p(alfa^(-w)*x)
|
||
|
|
pd_bwdperm(C2VMSG(imsg),msgout, MSGPERM(wmsg), qra_M);
|
||
|
|
|
||
|
|
} // for (k=0;k<ndeg;k++)
|
||
|
|
|
||
|
|
} // for (nc=qra_V;nc<qra_C;nc++)
|
||
|
|
|
||
|
|
// v->c step -----------------------------------------------------
|
||
|
|
for (nv=0;nv<qra_V;nv++) {
|
||
|
|
|
||
|
|
ndeg = qra_vdeg[nv]; // degree of current node
|
||
|
|
msgbase = nv*qra_MAXVDEG; // base to msg index row for the current node
|
||
|
|
|
||
|
|
for (k=0;k<ndeg;k++) {
|
||
|
|
// init output message to uniform distribution
|
||
|
|
pd_init(msgout,pd_uniform(qra_m),qra_M);
|
||
|
|
|
||
|
|
// v->c msg = prod(c->v)
|
||
|
|
// TODO: factor factors to reduce the number of computations for high degree nodes
|
||
|
|
for (kk=0;kk<ndeg;kk++)
|
||
|
|
if (kk!=k) {
|
||
|
|
imsg = qra_v2cmidx[msgbase+kk];
|
||
|
|
pd_imul(msgout,C2VMSG(imsg),qra_m);
|
||
|
|
}
|
||
|
|
|
||
|
|
#ifdef QRA_DEBUG
|
||
|
|
// normalize and check if product of messages v->c are null
|
||
|
|
// normalize output to a probability distribution
|
||
|
|
if (pd_norm(msgout,qra_m)<=0) {
|
||
|
|
// dump msgin;
|
||
|
|
printf("warning: v->c pd with invalid norm. nit=%d nv=%d k=%d\n",nit,nv,k);
|
||
|
|
for (kk=0;kk<ndeg;kk++) {
|
||
|
|
imsg = qra_v2cmidx[msgbase+kk];
|
||
|
|
pd_print(imsg,C2VMSG(imsg),qra_M);
|
||
|
|
}
|
||
|
|
printf("-----------------\n");
|
||
|
|
}
|
||
|
|
#else
|
||
|
|
// normalize the result to a probability distribution
|
||
|
|
pd_norm(msgout,qra_m);
|
||
|
|
#endif
|
||
|
|
// weight and output
|
||
|
|
imsg = qra_v2cmidx[msgbase+k]; // current output msg index
|
||
|
|
wmsg = qra_msgw[imsg]; // current msg weight
|
||
|
|
|
||
|
|
if (wmsg==0) {
|
||
|
|
pd_init(V2CMSG(imsg),msgout,qra_M);
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
// output p(alfa^w*x)
|
||
|
|
pd_fwdperm(V2CMSG(imsg),msgout, MSGPERM(wmsg), qra_M);
|
||
|
|
}
|
||
|
|
|
||
|
|
} // for (k=0;k<ndeg;k++)
|
||
|
|
} // for (nv=0;nv<qra_V;nv++)
|
||
|
|
|
||
|
|
// check extrinsic information ------------------------------
|
||
|
|
// We assume that decoding is successful if each of the extrinsic
|
||
|
|
// symbol probability is close to ej, where ej = [0 0 0 1(j-th position) 0 0 0 ]
|
||
|
|
// Therefore, for each symbol k in the codeword we compute max(prob(Xk))
|
||
|
|
// and we stop the iterations if sum(max(prob(xk)) is close to the codeword length
|
||
|
|
// Note: this is a more restrictive criterium than that of computing the a
|
||
|
|
// posteriori probability of each symbol, making a hard decision and then check
|
||
|
|
// if the codeword syndrome is null.
|
||
|
|
// WARNING: this is tricky and probably works only for the particular class of RA codes
|
||
|
|
// we are coping with (we designed the code weights so that for any input symbol the
|
||
|
|
// sum of its weigths is always 0, thus terminating the accumulator trellis to zero
|
||
|
|
// for every combination of the systematic symbols).
|
||
|
|
// More generally we should instead compute the max a posteriori probabilities
|
||
|
|
// (as a product of the intrinsic and extrinsic information), make a symbol by symbol hard
|
||
|
|
// decision and then check that the syndrome of the result is indeed null.
|
||
|
|
|
||
|
|
totex = 0;
|
||
|
|
for (nv=0;nv<qra_V;nv++)
|
||
|
|
totex += pd_max(V2CMSG(nv),qra_M);
|
||
|
|
|
||
|
|
if (totex>(1.*(qra_V)-0.01)) {
|
||
|
|
// the total maximum extrinsic information of each symbol in the codeword
|
||
|
|
// is very close to one. This means that we have reached the (1,1) point in the
|
||
|
|
// code EXIT chart(s) and we have successfully decoded the input.
|
||
|
|
rc = nit;
|
||
|
|
break; // remove the break to evaluate the decoder speed performance as a function of the max iterations number)
|
||
|
|
}
|
||
|
|
|
||
|
|
} // for (nit=0;nit<maxiter;nit++)
|
||
|
|
|
||
|
|
// copy extrinsic information to output to do the actual max a posteriori prob decoding
|
||
|
|
pd_init(pex,V2CMSG(0),(qra_M*qra_V));
|
||
|
|
return rc;
|
||
|
|
}
|
||
|
|
|
||
|
|
void qra_mapdecode(const qracode *pcode, int *xdec, float *pex, const float *pix)
|
||
|
|
{
|
||
|
|
// Maximum a posteriori probability decoding.
|
||
|
|
// Given the intrinsic information (pix) and extrinsic information (pex) (computed with qra_extrinsic(...))
|
||
|
|
// compute pmap = pex*pix and decode each (information) symbol of the received codeword
|
||
|
|
// as the symbol which maximizes pmap
|
||
|
|
|
||
|
|
// Returns:
|
||
|
|
// xdec[k] = decoded (information) symbols k=[0..qra_K-1]
|
||
|
|
|
||
|
|
// Note: pex is destroyed and overwritten with mapp
|
||
|
|
|
||
|
|
const int qra_M = pcode->M;
|
||
|
|
const int qra_m = pcode->m;
|
||
|
|
const int qra_K = pcode->K;
|
||
|
|
|
||
|
|
int k;
|
||
|
|
|
||
|
|
for (k=0;k<qra_K;k++) {
|
||
|
|
// compute a posteriori prob
|
||
|
|
pd_imul(PD_ROWADDR(pex,qra_M,k),PD_ROWADDR(pix,qra_M,k),qra_m);
|
||
|
|
xdec[k]=pd_argmax(NULL, PD_ROWADDR(pex,qra_M,k), qra_M);
|
||
|
|
}
|
||
|
|
}
|