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705 lines
17 KiB
C
705 lines
17 KiB
C
/**************************************************************************
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* Parks-McClellan algorithm for FIR filter design (C version)
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*-------------------------------------------------
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* Copyright (c) 1995,1998 Jake Janovetz (janovetz@uiuc.edu)
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*
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* This library is free software; you can redistribute it and/or
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* modify it under the terms of the GNU Library General Public
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* License as published by the Free Software Foundation; either
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* version 2 of the License, or (at your option) any later version.
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*
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* This library 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 GNU
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* Library General Public License for more details.
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*
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* You should have received a copy of the GNU Library General Public
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* License along with this library; if not, write to the Free
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* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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*
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*************************************************************************/
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#include "remez.h"
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#include <math.h>
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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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void CreateDenseGrid(int r, int numtaps, int numband, double bands[],
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double des[], double weight[], int *gridsize,
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double Grid[], double D[], double W[],
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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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/*
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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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*/
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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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{
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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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{
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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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/*
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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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*/
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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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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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void CalcParms(int r, int Ext[], double Grid[], double D[], 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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/*
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* Find x[]
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*/
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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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/*
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* Calculate ad[] - Oppenheim & Schafer eq 7.132
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*/
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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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{
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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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{
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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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/*
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* Calculate delta - Oppenheim & Schafer eq 7.131
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*/
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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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{
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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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/*
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* Calculate y[] - Oppenheim & Schafer eq 7.133b
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*/
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for (i=0; i<=r; i++)
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{
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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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double ComputeA(double freq, int r, double ad[], double x[], 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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{
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c = xc - x[i];
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if (fabs(c) < 1.0e-7)
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{
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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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void CalcError(int r, double ad[], double x[], double y[],
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int gridsize, double Grid[],
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double D[], 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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{
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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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void Search(int r, int Ext[],
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int gridsize, 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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/*
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* Allocate enough space for found extremals.
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*/
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foundExt = (int *)malloc((2*r) * sizeof(int));
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k = 0;
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/*
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* Check for extremum at 0.
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*/
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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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/*
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* Check for extrema inside dense grid
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*/
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for (i=1; i<gridsize-1; i++)
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{
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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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/*
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* Check for extremum at 0.5
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*/
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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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/*
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* Remove extra extremals
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*/
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extra = k - (r+1);
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while (extra > 0)
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{
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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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{
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if (fabs(E[foundExt[j]]) < fabs(E[foundExt[l]]))
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l = j; /* new smallest error. */
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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))
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up = 1; /* switch to a maxima */
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else
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{
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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 */
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/*
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* 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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*/
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if ((alt) && (extra == 1))
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{
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if (fabs(E[foundExt[k-1]]) < fabs(E[foundExt[0]]))
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l = foundExt[k-1]; /* Delete last extremal */
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else
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l = foundExt[0]; /* Delete first extremal */
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}
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for (j=l; j<k; j++) /* Loop that does the deletion */
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{
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foundExt[j] = foundExt[j+1];
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}
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k--;
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extra--;
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}
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for (i=0; i<=r; i++)
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{
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Ext[i] = foundExt[i]; /* Copy found extremals to Ext[] */
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}
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free(foundExt);
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}
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/*********************
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* FreqSample
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*============
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* Simple frequency sampling algorithm to determine the impulse
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* response h[] from A's found in ComputeA
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*
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*
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* INPUT:
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* ------
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* int N - Number of filter coefficients
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* double A[] - Sample points of desired response [N/2]
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* int symmetry - Symmetry of desired filter
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*
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* OUTPUT:
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* -------
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* double h[] - Impulse Response of final filter [N]
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*********************/
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void FreqSample(int N, double A[], double h[], int symm)
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{
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int n, k;
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double x, val, M;
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M = (N-1.0)/2.0;
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if (symm == POSITIVE)
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{
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if (N%2)
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{
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for (n=0; n<N; n++)
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{
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val = A[0];
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x = Pi2 * (n - M)/N;
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for (k=1; k<=M; k++)
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val += 2.0 * A[k] * cos(x*k);
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h[n] = val/N;
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}
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}
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else
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{
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for (n=0; n<N; n++)
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{
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val = A[0];
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x = Pi2 * (n - M)/N;
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for (k=1; k<=(N/2-1); k++)
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val += 2.0 * A[k] * cos(x*k);
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h[n] = val/N;
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}
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}
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}
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else
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{
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if (N%2)
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{
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for (n=0; n<N; n++)
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{
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val = 0;
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x = Pi2 * (n - M)/N;
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for (k=1; k<=M; k++)
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val += 2.0 * A[k] * sin(x*k);
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h[n] = val/N;
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}
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}
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else
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{
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for (n=0; n<N; n++)
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{
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val = A[N/2] * sin(Pi * (n - M));
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x = Pi2 * (n - M)/N;
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for (k=1; k<=(N/2-1); k++)
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val += 2.0 * A[k] * sin(x*k);
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h[n] = val/N;
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}
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}
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}
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}
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/*******************
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* isDone
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*========
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* Checks to see if the error function is small enough to consider
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* the result to have converged.
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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 coeffiecients
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* int Ext[] - Indexes to extremal frequencies [r+1]
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* double E[] - Error function on the dense grid [gridsize]
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*
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* OUTPUT:
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* -------
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* Returns 1 if the result converged
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* Returns 0 if the result has not converged
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********************/
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short isDone(int r, int Ext[], double E[])
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{
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int i;
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double min, max, current;
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min = max = fabs(E[Ext[0]]);
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for (i=1; i<=r; i++)
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{
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current = fabs(E[Ext[i]]);
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if (current < min)
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min = current;
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if (current > max)
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max = current;
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}
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if (((max-min)/max) < 0.0001)
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return 1;
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return 0;
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}
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/********************
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* remez
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*=======
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* Calculates the optimal (in the Chebyshev/minimax sense)
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* FIR filter impulse response given a set of band edges,
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* the desired reponse on those bands, and the weight given to
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* the error in those bands.
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*
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* INPUT:
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* ------
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* int numtaps - Number of filter coefficients
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* int numband - Number of bands in filter specification
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* double bands[] - User-specified band edges [2 * numband]
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* double des[] - User-specified band responses [numband]
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* double weight[] - User-specified error weights [numband]
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* int type - Type of filter
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*
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* OUTPUT:
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* -------
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* double h[] - Impulse response of final filter [numtaps]
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********************/
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void remez(double h[], int numtaps,
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int numband, double bands[], double des[], double weight[],
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int type)
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{
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double *Grid, *W, *D, *E;
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int i, iter, gridsize, r, *Ext;
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double *taps, c;
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double *x, *y, *ad;
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int symmetry;
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if (type == BANDPASS)
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symmetry = POSITIVE;
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else
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symmetry = NEGATIVE;
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r = numtaps/2; /* number of extrema */
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if ((numtaps%2) && (symmetry == POSITIVE))
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r++;
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/*
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* Predict dense grid size in advance for memory allocation
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* .5 is so we round up, not truncate
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*/
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gridsize = 0;
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for (i=0; i<numband; i++)
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{
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gridsize += (int)(2*r*GRIDDENSITY*(bands[2*i+1] - bands[2*i]) + .5);
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}
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if (symmetry == NEGATIVE)
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{
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gridsize--;
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}
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/*
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* Dynamically allocate memory for arrays with proper sizes
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*/
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Grid = (double *)malloc(gridsize * sizeof(double));
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D = (double *)malloc(gridsize * sizeof(double));
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W = (double *)malloc(gridsize * sizeof(double));
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E = (double *)malloc(gridsize * sizeof(double));
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Ext = (int *)malloc((r+1) * sizeof(int));
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taps = (double *)malloc((r+1) * sizeof(double));
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x = (double *)malloc((r+1) * sizeof(double));
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y = (double *)malloc((r+1) * sizeof(double));
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ad = (double *)malloc((r+1) * sizeof(double));
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/*
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* Create dense frequency grid
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*/
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CreateDenseGrid(r, numtaps, numband, bands, des, weight,
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&gridsize, Grid, D, W, symmetry);
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InitialGuess(r, Ext, gridsize);
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/*
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* For Differentiator: (fix grid)
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*/
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if (type == DIFFERENTIATOR)
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{
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for (i=0; i<gridsize; i++)
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{
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/* D[i] = D[i]*Grid[i]; */
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if (D[i] > 0.0001)
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W[i] = W[i]/Grid[i];
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}
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}
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/*
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* For odd or Negative symmetry filters, alter the
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* D[] and W[] according to Parks McClellan
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*/
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if (symmetry == POSITIVE)
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{
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if (numtaps % 2 == 0)
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{
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for (i=0; i<gridsize; i++)
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{
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c = cos(Pi * Grid[i]);
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D[i] /= c;
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W[i] *= c;
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}
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}
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}
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else
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{
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if (numtaps % 2)
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{
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for (i=0; i<gridsize; i++)
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{
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c = sin(Pi2 * Grid[i]);
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D[i] /= c;
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W[i] *= c;
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}
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}
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else
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{
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for (i=0; i<gridsize; i++)
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{
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c = sin(Pi * Grid[i]);
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D[i] /= c;
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W[i] *= c;
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}
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}
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}
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/*
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* Perform the Remez Exchange algorithm
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*/
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for (iter=0; iter<MAXITERATIONS; iter++)
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{
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CalcParms(r, Ext, Grid, D, W, ad, x, y);
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CalcError(r, ad, x, y, gridsize, Grid, D, W, E);
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Search(r, Ext, gridsize, E);
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if (isDone(r, Ext, E))
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break;
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}
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if (iter == MAXITERATIONS)
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{
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printf("Reached maximum iteration count.\nResults may be bad.\n");
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}
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CalcParms(r, Ext, Grid, D, W, ad, x, y);
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/*
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* Find the 'taps' of the filter for use with Frequency
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* Sampling. If odd or Negative symmetry, fix the taps
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* according to Parks McClellan
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*/
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for (i=0; i<=numtaps/2; i++)
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{
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if (symmetry == POSITIVE)
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{
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if (numtaps%2)
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c = 1;
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else
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c = cos(Pi * (double)i/numtaps);
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}
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else
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{
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if (numtaps%2)
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c = sin(Pi2 * (double)i/numtaps);
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else
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c = sin(Pi * (double)i/numtaps);
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}
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taps[i] = ComputeA((double)i/numtaps, r, ad, x, y)*c;
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}
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/*
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* Frequency sampling design with calculated taps
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*/
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FreqSample(numtaps, taps, h, symmetry);
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/*
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* Delete allocated memory
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*/
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free(Grid);
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free(W);
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free(D);
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free(E);
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free(Ext);
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free(x);
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free(y);
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free(ad);
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}
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