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@@ -0,0 +1,643 @@
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/*
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* remez.c - Parks-McClellan algorithm for FIR filter design (C version)
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*
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* Copyright (C) 1995,1998 Jake Janovetz
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* Copyright (C) 1998-2005 Atari800 development team (see DOC/CREDITS)
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*
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* This file is part of the Atari800 emulator project which emulates
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* the Atari 400, 800, 800XL, 130XE, and 5200 8-bit computers.
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*
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* Atari800 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 2 of the License, or
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* (at your option) any later version.
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*
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* Atari800 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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*
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* You should have received a copy of the GNU General Public License
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* along with Atari800; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
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*/
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#include <stdio.h>
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#include <stdlib.h>
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#include <math.h>
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#include "remez.h"
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#ifdef ASAP /* external project, see http://asap.sf.net */
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#include "asap_internal.h"
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#else
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#include "log.h"
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#include "util.h"
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#endif
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#define NEGATIVE 0
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#define POSITIVE 1
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#define Pi 3.1415926535897932
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#define Pi2 6.2831853071795865
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#define GRIDDENSITY 16
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#define MAXITERATIONS 40
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/*******************
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* CreateDenseGrid
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*=================
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* Creates the dense grid of frequencies from the specified bands.
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* Also creates the Desired Frequency Response function (D[]) and
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* the Weight function (W[]) on that dense grid
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*
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*
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* INPUT:
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* ------
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* int r - 1/2 the number of filter coefficients
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* int numtaps - Number of taps in the resulting filter
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* int numband - Number of bands in user specification
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* double bands[] - User-specified band edges [2*numband]
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* double des[] - Desired response per band [numband]
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* double weight[] - Weight per band [numband]
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* int symmetry - Symmetry of filter - used for grid check
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*
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* OUTPUT:
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* -------
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* int gridsize - Number of elements in the dense frequency grid
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* double Grid[] - Frequencies (0 to 0.5) on the dense grid [gridsize]
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* double D[] - Desired response on the dense grid [gridsize]
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* double W[] - Weight function on the dense grid [gridsize]
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*******************/
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static void CreateDenseGrid(int r, int numtaps, int numband, double bands[],
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const double des[], const double weight[],
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int *gridsize, double Grid[],
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double D[], double W[], int symmetry)
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{
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int i, j, k, band;
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double delf, lowf, highf;
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delf = 0.5 / (GRIDDENSITY * r);
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/* For differentiator, hilbert,
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* symmetry is odd and Grid[0] = max(delf, band[0]) */
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if (symmetry == NEGATIVE && delf > bands[0])
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bands[0] = delf;
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j = 0;
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for (band = 0; band < numband; band++) {
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Grid[j] = bands[2 * band];
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lowf = bands[2 * band];
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highf = bands[2 * band + 1];
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k = (int) ((highf - lowf) / delf + 0.5); /* .5 for rounding */
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for (i = 0; i < k; i++) {
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D[j] = des[band];
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W[j] = weight[band];
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Grid[j] = lowf;
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lowf += delf;
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j++;
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}
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Grid[j - 1] = highf;
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}
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/* Similar to above, if odd symmetry, last grid point can't be .5
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* - but, if there are even taps, leave the last grid point at .5 */
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if ((symmetry == NEGATIVE) &&
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(Grid[*gridsize - 1] > (0.5 - delf)) &&
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(numtaps % 2))
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{
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Grid[*gridsize - 1] = 0.5 - delf;
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}
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}
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/********************
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* InitialGuess
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*==============
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* Places Extremal Frequencies evenly throughout the dense grid.
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*
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*
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* INPUT:
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* ------
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* int r - 1/2 the number of filter coefficients
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* int gridsize - Number of elements in the dense frequency grid
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*
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* OUTPUT:
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* -------
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* int Ext[] - Extremal indexes to dense frequency grid [r+1]
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********************/
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static void InitialGuess(int r, int Ext[], int gridsize)
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{
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int i;
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for (i = 0; i <= r; i++)
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Ext[i] = i * (gridsize - 1) / r;
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}
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/***********************
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* CalcParms
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*===========
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*
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*
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* INPUT:
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* ------
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* int r - 1/2 the number of filter coefficients
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* int Ext[] - Extremal indexes to dense frequency grid [r+1]
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* double Grid[] - Frequencies (0 to 0.5) on the dense grid [gridsize]
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* double D[] - Desired response on the dense grid [gridsize]
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* double W[] - Weight function on the dense grid [gridsize]
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*
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* OUTPUT:
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* -------
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* double ad[] - 'b' in Oppenheim & Schafer [r+1]
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* double x[] - [r+1]
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* double y[] - 'C' in Oppenheim & Schafer [r+1]
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***********************/
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static void CalcParms(int r, const int Ext[], const double Grid[],
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const double D[], const double W[],
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double ad[], double x[], double y[])
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{
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int i, j, k, ld;
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double sign, xi, delta, denom, numer;
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/* Find x[] */
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for (i = 0; i <= r; i++)
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x[i] = cos(Pi2 * Grid[Ext[i]]);
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/* Calculate ad[] - Oppenheim & Schafer eq 7.132 */
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ld = (r - 1) / 15 + 1; /* Skips around to avoid round errors */
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for (i = 0; i <= r; i++) {
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denom = 1.0;
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xi = x[i];
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for (j = 0; j < ld; j++) {
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for (k = j; k <= r; k += ld)
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if (k != i)
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denom *= 2.0 * (xi - x[k]);
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}
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if (fabs(denom) < 0.00001)
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denom = 0.00001;
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ad[i] = 1.0 / denom;
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}
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/* Calculate delta - Oppenheim & Schafer eq 7.131 */
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numer = denom = 0;
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sign = 1;
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for (i = 0; i <= r; i++) {
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numer += ad[i] * D[Ext[i]];
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denom += sign * ad[i] / W[Ext[i]];
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sign = -sign;
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}
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delta = numer / denom;
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sign = 1;
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/* Calculate y[] - Oppenheim & Schafer eq 7.133b */
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for (i = 0; i <= r; i++) {
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y[i] = D[Ext[i]] - sign * delta / W[Ext[i]];
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sign = -sign;
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}
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}
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/*********************
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* ComputeA
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*==========
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* Using values calculated in CalcParms, ComputeA calculates the
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* actual filter response at a given frequency (freq). Uses
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* eq 7.133a from Oppenheim & Schafer.
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*
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*
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* INPUT:
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* ------
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* double freq - Frequency (0 to 0.5) at which to calculate A
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* int r - 1/2 the number of filter coefficients
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* double ad[] - 'b' in Oppenheim & Schafer [r+1]
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* double x[] - [r+1]
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* double y[] - 'C' in Oppenheim & Schafer [r+1]
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*
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* OUTPUT:
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* -------
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* Returns double value of A[freq]
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*********************/
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static double ComputeA(double freq, int r, const double ad[],
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const double x[], const double y[])
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{
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int i;
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double xc, c, denom, numer;
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denom = numer = 0;
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xc = cos(Pi2 * freq);
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for (i = 0; i <= r; i++) {
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c = xc - x[i];
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if (fabs(c) < 1.0e-7) {
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numer = y[i];
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denom = 1;
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break;
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}
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c = ad[i] / c;
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denom += c;
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numer += c * y[i];
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}
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return numer / denom;
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}
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/************************
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* CalcError
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*===========
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* Calculates the Error function from the desired frequency response
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* on the dense grid (D[]), the weight function on the dense grid (W[]),
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* and the present response calculation (A[])
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*
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*
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* INPUT:
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* ------
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* int r - 1/2 the number of filter coefficients
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* double ad[] - [r+1]
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* double x[] - [r+1]
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* double y[] - [r+1]
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* int gridsize - Number of elements in the dense frequency grid
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* double Grid[] - Frequencies on the dense grid [gridsize]
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* double D[] - Desired response on the dense grid [gridsize]
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* double W[] - Weight function on the desnse grid [gridsize]
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*
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* OUTPUT:
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* -------
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* double E[] - Error function on dense grid [gridsize]
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************************/
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static void CalcError(int r, const double ad[],
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const double x[], const double y[],
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int gridsize, const double Grid[],
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const double D[], const double W[], double E[])
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{
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int i;
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double A;
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for (i = 0; i < gridsize; i++) {
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A = ComputeA(Grid[i], r, ad, x, y);
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E[i] = W[i] * (D[i] - A);
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}
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}
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/************************
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* Search
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*========
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* Searches for the maxima/minima of the error curve. If more than
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* r+1 extrema are found, it uses the following heuristic (thanks
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* Chris Hanson):
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* 1) Adjacent non-alternating extrema deleted first.
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* 2) If there are more than one excess extrema, delete the
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* one with the smallest error. This will create a non-alternation
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* condition that is fixed by 1).
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* 3) If there is exactly one excess extremum, delete the smaller
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* of the first/last extremum
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*
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*
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* INPUT:
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* ------
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* int r - 1/2 the number of filter coefficients
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* int Ext[] - Indexes to Grid[] of extremal frequencies [r+1]
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* int gridsize - Number of elements in the dense frequency grid
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* double E[] - Array of error values. [gridsize]
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* OUTPUT:
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* -------
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* int Ext[] - New indexes to extremal frequencies [r+1]
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************************/
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static void Search(int r, int Ext[], int gridsize, const double E[])
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{
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int i, j, k, l, extra; /* Counters */
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int up, alt;
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int *foundExt; /* Array of found extremals */
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/* Allocate enough space for found extremals. */
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foundExt = (int *) Util_malloc((2 * r) * sizeof(int));
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k = 0;
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/* Check for extremum at 0. */
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if (((E[0] > 0.0) && (E[0] > E[1])) ||
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((E[0] < 0.0) && (E[0] < E[1])))
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foundExt[k++] = 0;
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/* Check for extrema inside dense grid */
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for (i = 1; i < gridsize - 1; i++) {
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if (((E[i] >= E[i - 1]) && (E[i] > E[i + 1]) && (E[i] > 0.0)) ||
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((E[i] <= E[i - 1]) && (E[i] < E[i + 1]) && (E[i] < 0.0)))
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foundExt[k++] = i;
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}
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/* Check for extremum at 0.5 */
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j = gridsize - 1;
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if (((E[j] > 0.0) && (E[j] > E[j - 1])) ||
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((E[j] < 0.0) && (E[j] < E[j - 1])))
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foundExt[k++] = j;
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/* Remove extra extremals */
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extra = k - (r + 1);
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||||
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while (extra > 0) {
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if (E[foundExt[0]] > 0.0)
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up = 1; /* first one is a maxima */
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else
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up = 0; /* first one is a minima */
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l = 0;
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||||
alt = 1;
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for (j = 1; j < k; j++) {
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if (fabs(E[foundExt[j]]) < fabs(E[foundExt[l]]))
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l = j; /* new smallest error. */
|
||||
if ((up) && (E[foundExt[j]] < 0.0))
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up = 0; /* switch to a minima */
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else if ((!up) && (E[foundExt[j]] > 0.0))
|
||||
up = 1; /* switch to a maxima */
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||||
else {
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||||
alt = 0;
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||||
break; /* Ooops, found two non-alternating */
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||||
} /* extrema. Delete smallest of them */
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||||
} /* if the loop finishes, all extrema are alternating */
|
||||
|
||||
/* If there's only one extremal and all are alternating,
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||||
* delete the smallest of the first/last extremals. */
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||||
if ((alt) && (extra == 1)) {
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||||
if (fabs(E[foundExt[k - 1]]) < fabs(E[foundExt[0]]))
|
||||
l = foundExt[k - 1]; /* Delete last extremal */
|
||||
else
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||||
l = foundExt[0]; /* Delete first extremal */
|
||||
}
|
||||
|
||||
/* Loop that does the deletion */
|
||||
for (j = l; j < k; j++) {
|
||||
foundExt[j] = foundExt[j+1];
|
||||
}
|
||||
k--;
|
||||
extra--;
|
||||
}
|
||||
|
||||
/* Copy found extremals to Ext[] */
|
||||
for (i = 0; i <= r; i++) {
|
||||
Ext[i] = foundExt[i];
|
||||
}
|
||||
|
||||
free(foundExt);
|
||||
}
|
||||
|
||||
|
||||
/*********************
|
||||
* FreqSample
|
||||
*============
|
||||
* Simple frequency sampling algorithm to determine the impulse
|
||||
* response h[] from A's found in ComputeA
|
||||
*
|
||||
*
|
||||
* INPUT:
|
||||
* ------
|
||||
* int N - Number of filter coefficients
|
||||
* double A[] - Sample points of desired response [N/2]
|
||||
* int symm - Symmetry of desired filter
|
||||
*
|
||||
* OUTPUT:
|
||||
* -------
|
||||
* double h[] - Impulse Response of final filter [N]
|
||||
*********************/
|
||||
static void FreqSample(int N, const double A[], double h[], int symm)
|
||||
{
|
||||
int n, k;
|
||||
double x, val, M;
|
||||
|
||||
M = (N - 1.0) / 2.0;
|
||||
if (symm == POSITIVE) {
|
||||
if (N % 2) {
|
||||
for (n = 0; n < N; n++) {
|
||||
val = A[0];
|
||||
x = Pi2 * (n - M) / N;
|
||||
for (k = 1; k <= M; k++)
|
||||
val += 2.0 * A[k] * cos(x * k);
|
||||
h[n] = val / N;
|
||||
}
|
||||
}
|
||||
else {
|
||||
for (n = 0; n < N; n++) {
|
||||
val = A[0];
|
||||
x = Pi2 * (n - M)/N;
|
||||
for (k = 1; k <= (N / 2 - 1); k++)
|
||||
val += 2.0 * A[k] * cos(x * k);
|
||||
h[n] = val / N;
|
||||
}
|
||||
}
|
||||
}
|
||||
else {
|
||||
if (N % 2) {
|
||||
for (n = 0; n < N; n++) {
|
||||
val = 0;
|
||||
x = Pi2 * (n - M) / N;
|
||||
for (k = 1; k <= M; k++)
|
||||
val += 2.0 * A[k] * sin(x * k);
|
||||
h[n] = val / N;
|
||||
}
|
||||
}
|
||||
else {
|
||||
for (n = 0; n < N; n++) {
|
||||
val = A[N / 2] * sin(Pi * (n - M));
|
||||
x = Pi2 * (n - M) / N;
|
||||
for (k = 1; k <= (N / 2 - 1); k++)
|
||||
val += 2.0 * A[k] * sin(x * k);
|
||||
h[n] = val / N;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/*******************
|
||||
* isDone
|
||||
*========
|
||||
* Checks to see if the error function is small enough to consider
|
||||
* the result to have converged.
|
||||
*
|
||||
* INPUT:
|
||||
* ------
|
||||
* int r - 1/2 the number of filter coeffiecients
|
||||
* int Ext[] - Indexes to extremal frequencies [r+1]
|
||||
* double E[] - Error function on the dense grid [gridsize]
|
||||
*
|
||||
* OUTPUT:
|
||||
* -------
|
||||
* Returns 1 if the result converged
|
||||
* Returns 0 if the result has not converged
|
||||
********************/
|
||||
|
||||
static int isDone(int r, const int Ext[], const double E[])
|
||||
{
|
||||
int i;
|
||||
double min, max, current;
|
||||
|
||||
min = max = fabs(E[Ext[0]]);
|
||||
for (i = 1; i <= r; i++) {
|
||||
current = fabs(E[Ext[i]]);
|
||||
if (current < min)
|
||||
min = current;
|
||||
if (current > max)
|
||||
max = current;
|
||||
}
|
||||
if (((max - min) / max) < 0.0001)
|
||||
return 1;
|
||||
return 0;
|
||||
}
|
||||
|
||||
/********************
|
||||
* REMEZ_CreateFilter
|
||||
*=======
|
||||
* Calculates the optimal (in the Chebyshev/minimax sense)
|
||||
* FIR filter impulse response given a set of band edges,
|
||||
* the desired reponse on those bands, and the weight given to
|
||||
* the error in those bands.
|
||||
*
|
||||
* INPUT:
|
||||
* ------
|
||||
* int numtaps - Number of filter coefficients
|
||||
* int numband - Number of bands in filter specification
|
||||
* double bands[] - User-specified band edges [2 * numband]
|
||||
* double des[] - User-specified band responses [numband]
|
||||
* double weight[] - User-specified error weights [numband]
|
||||
* int type - Type of filter
|
||||
*
|
||||
* OUTPUT:
|
||||
* -------
|
||||
* double h[] - Impulse response of final filter [numtaps]
|
||||
********************/
|
||||
|
||||
void REMEZ_CreateFilter(double h[], int numtaps, int numband, double bands[],
|
||||
const double des[], const double weight[], int type)
|
||||
{
|
||||
double *Grid, *W, *D, *E;
|
||||
int i, iter, gridsize, r, *Ext;
|
||||
double *taps, c;
|
||||
double *x, *y, *ad;
|
||||
int symmetry;
|
||||
|
||||
if (type == REMEZ_BANDPASS)
|
||||
symmetry = POSITIVE;
|
||||
else
|
||||
symmetry = NEGATIVE;
|
||||
|
||||
r = numtaps / 2; /* number of extrema */
|
||||
if ((numtaps % 2) && (symmetry == POSITIVE))
|
||||
r++;
|
||||
|
||||
/* Predict dense grid size in advance for memory allocation
|
||||
* .5 is so we round up, not truncate */
|
||||
gridsize = 0;
|
||||
for (i = 0; i < numband; i++) {
|
||||
gridsize += (int) (2 * r * GRIDDENSITY *
|
||||
(bands[2 * i + 1] - bands[2 * i]) + .5);
|
||||
}
|
||||
if (symmetry == NEGATIVE) {
|
||||
gridsize--;
|
||||
}
|
||||
|
||||
/* Dynamically allocate memory for arrays with proper sizes */
|
||||
Grid = (double *) Util_malloc(gridsize * sizeof(double));
|
||||
D = (double *) Util_malloc(gridsize * sizeof(double));
|
||||
W = (double *) Util_malloc(gridsize * sizeof(double));
|
||||
E = (double *) Util_malloc(gridsize * sizeof(double));
|
||||
Ext = (int *) Util_malloc((r + 1) * sizeof(int));
|
||||
taps = (double *) Util_malloc((r + 1) * sizeof(double));
|
||||
x = (double *) Util_malloc((r + 1) * sizeof(double));
|
||||
y = (double *) Util_malloc((r + 1) * sizeof(double));
|
||||
ad = (double *) Util_malloc((r + 1) * sizeof(double));
|
||||
|
||||
/* Create dense frequency grid */
|
||||
CreateDenseGrid(r, numtaps, numband, bands, des, weight,
|
||||
&gridsize, Grid, D, W, symmetry);
|
||||
InitialGuess(r, Ext, gridsize);
|
||||
|
||||
/* For Differentiator: (fix grid) */
|
||||
if (type == REMEZ_DIFFERENTIATOR) {
|
||||
for (i = 0; i < gridsize; i++) {
|
||||
/* D[i] = D[i] * Grid[i]; */
|
||||
if (D[i] > 0.0001)
|
||||
W[i] = W[i] / Grid[i];
|
||||
}
|
||||
}
|
||||
|
||||
/* For odd or Negative symmetry filters, alter the
|
||||
* D[] and W[] according to Parks McClellan */
|
||||
if (symmetry == POSITIVE) {
|
||||
if (numtaps % 2 == 0) {
|
||||
for (i = 0; i < gridsize; i++) {
|
||||
c = cos(Pi * Grid[i]);
|
||||
D[i] /= c;
|
||||
W[i] *= c;
|
||||
}
|
||||
}
|
||||
}
|
||||
else {
|
||||
if (numtaps % 2) {
|
||||
for (i = 0; i < gridsize; i++) {
|
||||
c = sin(Pi2 * Grid[i]);
|
||||
D[i] /= c;
|
||||
W[i] *= c;
|
||||
}
|
||||
}
|
||||
else {
|
||||
for (i = 0; i < gridsize; i++) {
|
||||
c = sin(Pi * Grid[i]);
|
||||
D[i] /= c;
|
||||
W[i] *= c;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/* Perform the Remez Exchange algorithm */
|
||||
for (iter = 0; iter < MAXITERATIONS; iter++) {
|
||||
CalcParms(r, Ext, Grid, D, W, ad, x, y);
|
||||
CalcError(r, ad, x, y, gridsize, Grid, D, W, E);
|
||||
Search(r, Ext, gridsize, E);
|
||||
if (isDone(r, Ext, E))
|
||||
break;
|
||||
}
|
||||
#ifndef ASAP
|
||||
if (iter == MAXITERATIONS) {
|
||||
Log_print("remez(): reached maximum iteration count. Results may be bad.");
|
||||
}
|
||||
#endif
|
||||
|
||||
CalcParms(r, Ext, Grid, D, W, ad, x, y);
|
||||
|
||||
/* Find the 'taps' of the filter for use with Frequency
|
||||
* Sampling. If odd or Negative symmetry, fix the taps
|
||||
* according to Parks McClellan */
|
||||
for (i = 0; i <= numtaps / 2; i++) {
|
||||
if (symmetry == POSITIVE) {
|
||||
if (numtaps % 2)
|
||||
c = 1;
|
||||
else
|
||||
c = cos(Pi * (double) i / numtaps);
|
||||
}
|
||||
else {
|
||||
if (numtaps % 2)
|
||||
c = sin(Pi2 * (double) i / numtaps);
|
||||
else
|
||||
c = sin(Pi * (double) i / numtaps);
|
||||
}
|
||||
taps[i] = ComputeA((double) i / numtaps, r, ad, x, y) * c;
|
||||
}
|
||||
|
||||
/* Frequency sampling design with calculated taps */
|
||||
FreqSample(numtaps, taps, h, symmetry);
|
||||
|
||||
/* Delete allocated memory */
|
||||
free(Grid);
|
||||
free(W);
|
||||
free(D);
|
||||
free(E);
|
||||
free(Ext);
|
||||
free(taps);
|
||||
free(x);
|
||||
free(y);
|
||||
free(ad);
|
||||
}
|
||||
Reference in New Issue
Block a user