322 lines
9.1 KiB
C
322 lines
9.1 KiB
C
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/*---------------------------------------------------------------------------*\
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FILE........: lsp.c
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AUTHOR......: David Rowe
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DATE CREATED: 24/2/93
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This file contains functions for LPC to LSP conversion and LSP to
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LPC conversion. Note that the LSP coefficients are not in radians
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format but in the x domain of the unit circle.
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\*---------------------------------------------------------------------------*/
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/*
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Copyright (C) 2009 David Rowe
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All rights reserved.
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This program is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License version 2.1, as
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published by the Free Software Foundation. This program is
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distributed in the hope that it will be useful, but WITHOUT ANY
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WARRANTY; without even the implied warranty of MERCHANTABILITY or
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FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with this program; if not, see <http://www.gnu.org/licenses/>.
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*/
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#include "defines.h"
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#include "lsp.h"
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#include <math.h>
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#include <stdio.h>
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#include <stdlib.h>
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/*---------------------------------------------------------------------------*\
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Introduction to Line Spectrum Pairs (LSPs)
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------------------------------------------
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LSPs are used to encode the LPC filter coefficients {ak} for
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transmission over the channel. LSPs have several properties (like
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less sensitivity to quantisation noise) that make them superior to
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direct quantisation of {ak}.
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A(z) is a polynomial of order lpcrdr with {ak} as the coefficients.
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A(z) is transformed to P(z) and Q(z) (using a substitution and some
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algebra), to obtain something like:
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A(z) = 0.5[P(z)(z+z^-1) + Q(z)(z-z^-1)] (1)
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As you can imagine A(z) has complex zeros all over the z-plane. P(z)
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and Q(z) have the very neat property of only having zeros _on_ the
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unit circle. So to find them we take a test point z=exp(jw) and
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evaluate P (exp(jw)) and Q(exp(jw)) using a grid of points between 0
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and pi.
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The zeros (roots) of P(z) also happen to alternate, which is why we
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swap coefficients as we find roots. So the process of finding the
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LSP frequencies is basically finding the roots of 5th order
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polynomials.
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The root so P(z) and Q(z) occur in symmetrical pairs at +/-w, hence
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the name Line Spectrum Pairs (LSPs).
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To convert back to ak we just evaluate (1), "clocking" an impulse
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thru it lpcrdr times gives us the impulse response of A(z) which is
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{ak}.
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\*---------------------------------------------------------------------------*/
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/*---------------------------------------------------------------------------*\
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FUNCTION....: cheb_poly_eva()
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AUTHOR......: David Rowe
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DATE CREATED: 24/2/93
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This function evalutes a series of chebyshev polynomials
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FIXME: performing memory allocation at run time is very inefficient,
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replace with stack variables of MAX_P size.
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\*---------------------------------------------------------------------------*/
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static float
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cheb_poly_eva(float *coef,float x,int order)
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/* float coef[] coefficients of the polynomial to be evaluated */
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/* float x the point where polynomial is to be evaluated */
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/* int order order of the polynomial */
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{
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int i;
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float *t,*u,*v,sum;
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float T[(order / 2) + 1];
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/* Initialise pointers */
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t = T; /* T[i-2] */
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*t++ = 1.0;
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u = t--; /* T[i-1] */
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*u++ = x;
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v = u--; /* T[i] */
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/* Evaluate chebyshev series formulation using iterative approach */
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for(i=2;i<=order/2;i++)
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*v++ = (2*x)*(*u++) - *t++; /* T[i] = 2*x*T[i-1] - T[i-2] */
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sum=0.0; /* initialise sum to zero */
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t = T; /* reset pointer */
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/* Evaluate polynomial and return value also free memory space */
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for(i=0;i<=order/2;i++)
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sum+=coef[(order/2)-i]**t++;
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return sum;
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}
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/*---------------------------------------------------------------------------*\
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FUNCTION....: lpc_to_lsp()
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AUTHOR......: David Rowe
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DATE CREATED: 24/2/93
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This function converts LPC coefficients to LSP coefficients.
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\*---------------------------------------------------------------------------*/
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int lpc_to_lsp (float *a, int order, float *freq, int nb, float delta)
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/* float *a lpc coefficients */
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/* int order order of LPC coefficients (10) */
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/* float *freq LSP frequencies in radians */
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/* int nb number of sub-intervals (4) */
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/* float delta grid spacing interval (0.02) */
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{
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float psuml,psumr,psumm,temp_xr,xl,xr,xm = 0;
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float temp_psumr;
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int i,j,m,flag,k;
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float *px; /* ptrs of respective P'(z) & Q'(z) */
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float *qx;
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float *p;
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float *q;
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float *pt; /* ptr used for cheb_poly_eval()
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whether P' or Q' */
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int roots=0; /* number of roots found */
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float Q[order + 1];
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float P[order + 1];
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flag = 1;
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m = order/2; /* order of P'(z) & Q'(z) polynimials */
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/* Allocate memory space for polynomials */
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/* determine P'(z)'s and Q'(z)'s coefficients where
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P'(z) = P(z)/(1 + z^(-1)) and Q'(z) = Q(z)/(1-z^(-1)) */
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px = P; /* initilaise ptrs */
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qx = Q;
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p = px;
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q = qx;
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*px++ = 1.0;
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*qx++ = 1.0;
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for(i=1;i<=m;i++){
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*px++ = a[i]+a[order+1-i]-*p++;
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*qx++ = a[i]-a[order+1-i]+*q++;
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}
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px = P;
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qx = Q;
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for(i=0;i<m;i++){
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*px = 2**px;
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*qx = 2**qx;
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px++;
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qx++;
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}
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px = P; /* re-initialise ptrs */
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qx = Q;
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/* Search for a zero in P'(z) polynomial first and then alternate to Q'(z).
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Keep alternating between the two polynomials as each zero is found */
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xr = 0; /* initialise xr to zero */
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xl = 1.0; /* start at point xl = 1 */
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for(j=0;j<order;j++){
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if(j%2) /* determines whether P' or Q' is eval. */
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pt = qx;
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else
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pt = px;
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psuml = cheb_poly_eva(pt,xl,order); /* evals poly. at xl */
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flag = 1;
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while(flag && (xr >= -1.0)){
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xr = xl - delta ; /* interval spacing */
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psumr = cheb_poly_eva(pt,xr,order);/* poly(xl-delta_x) */
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temp_psumr = psumr;
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temp_xr = xr;
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/* if no sign change increment xr and re-evaluate
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poly(xr). Repeat til sign change. if a sign change has
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occurred the interval is bisected and then checked again
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for a sign change which determines in which interval the
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zero lies in. If there is no sign change between poly(xm)
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and poly(xl) set interval between xm and xr else set
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interval between xl and xr and repeat till root is located
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within the specified limits */
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if(((psumr*psuml)<0.0) || (psumr == 0.0)){
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roots++;
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psumm=psuml;
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for(k=0;k<=nb;k++){
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xm = (xl+xr)/2; /* bisect the interval */
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psumm=cheb_poly_eva(pt,xm,order);
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if(psumm*psuml>0.){
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psuml=psumm;
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xl=xm;
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}
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else{
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psumr=psumm;
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xr=xm;
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}
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}
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/* once zero is found, reset initial interval to xr */
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freq[j] = (xm);
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xl = xm;
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flag = 0; /* reset flag for next search */
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}
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else{
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psuml=temp_psumr;
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xl=temp_xr;
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}
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}
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}
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/* convert from x domain to radians */
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for(i=0; i<order; i++) {
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freq[i] = acosf(freq[i]);
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}
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return(roots);
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}
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/*---------------------------------------------------------------------------*\
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FUNCTION....: lsp_to_lpc()
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AUTHOR......: David Rowe
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DATE CREATED: 24/2/93
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This function converts LSP coefficients to LPC coefficients. In the
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Speex code we worked out a way to simplify this significantly.
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\*---------------------------------------------------------------------------*/
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void lsp_to_lpc(float *lsp, float *ak, int order)
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/* float *freq array of LSP frequencies in radians */
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/* float *ak array of LPC coefficients */
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/* int order order of LPC coefficients */
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{
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int i,j;
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float xout1,xout2,xin1,xin2;
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float *pw,*n1,*n2,*n3,*n4 = 0;
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float freq[order];
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float Wp[(order * 4) + 2];
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/* convert from radians to the x=cos(w) domain */
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for(i=0; i<order; i++)
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freq[i] = cosf(lsp[i]);
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pw = Wp;
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/* initialise contents of array */
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for(i=0;i<=4*(order/2)+1;i++){ /* set contents of buffer to 0 */
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*pw++ = 0.0;
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}
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/* Set pointers up */
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pw = Wp;
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xin1 = 1.0;
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xin2 = 1.0;
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/* reconstruct P(z) and Q(z) by cascading second order polynomials
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in form 1 - 2xz(-1) +z(-2), where x is the LSP coefficient */
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for(j=0;j<=order;j++){
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for(i=0;i<(order/2);i++){
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n1 = pw+(i*4);
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n2 = n1 + 1;
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n3 = n2 + 1;
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n4 = n3 + 1;
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xout1 = xin1 - 2*(freq[2*i]) * *n1 + *n2;
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xout2 = xin2 - 2*(freq[2*i+1]) * *n3 + *n4;
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*n2 = *n1;
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*n4 = *n3;
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*n1 = xin1;
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*n3 = xin2;
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xin1 = xout1;
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xin2 = xout2;
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}
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xout1 = xin1 + *(n4+1);
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xout2 = xin2 - *(n4+2);
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ak[j] = (xout1 + xout2)*0.5;
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*(n4+1) = xin1;
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*(n4+2) = xin2;
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xin1 = 0.0;
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xin2 = 0.0;
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}
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}
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