最近在写一些数据处理的程序。经常需要对数据进行平滑处理。直接用FIR滤波器或IIR滤波器都有一个启动问题，滤波完成后总要对数据掐头去尾。因此去找了些简单的数据平滑处理的方法。

在一本老版本的《数学手册》中找到了几个基于最小二乘法的数据平滑算法。将其写成了C 代码，测试了一下，效果还可以。这里简单的记录一下，算是给自己做个笔记。

算法的原理很简单，以五点三次平滑为例。取相邻的5个数据点，可以拟合出一条3次曲线来，然后用3次曲线上相应的位置的数据值作为滤波后结果。简单的说就是 Savitzky-Golay 滤波器 。只不过Savitzky-Golay 滤波器并不特殊考虑边界的几个数据点，而这个算法还特意把边上的几个点的数据拟合结果给推导了出来。

不多说了，下面贴代码。首先是线性拟合平滑处理的代码. 分别为三点线性平滑、五点线性平滑和七点线性平滑。

void linearSmooth3 ( double in[], double out[], int N )
{
    int i;
    if ( N < 3 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 5.0 * in[0] + 2.0 * in[1] - in[2] ) / 6.0;

        for ( i = 1; i <= N - 2; i++ )
        {
            out[i] = ( in[i - 1] + in[i] + in[i + 1] ) / 3.0;
        }

        out[N - 1] = ( 5.0 * in[N - 1] + 2.0 * in[N - 2] - in[N - 3] ) / 6.0;
    }
}

void linearSmooth5 ( double in[], double out[], int N )
{
    int i;
    if ( N < 5 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 3.0 * in[0] + 2.0 * in[1] + in[2] - in[4] ) / 5.0;
        out[1] = ( 4.0 * in[0] + 3.0 * in[1] + 2 * in[2] + in[3] ) / 10.0;
        for ( i = 2; i <= N - 3; i++ )
        {
            out[i] = ( in[i - 2] + in[i - 1] + in[i] + in[i + 1] + in[i + 2] ) / 5.0;
        }
        out[N - 2] = ( 4.0 * in[N - 1] + 3.0 * in[N - 2] + 2 * in[N - 3] + in[N - 4] ) / 10.0;
        out[N - 1] = ( 3.0 * in[N - 1] + 2.0 * in[N - 2] + in[N - 3] - in[N - 5] ) / 5.0;
    }
}

void linearSmooth7 ( double in[], double out[], int N )
{
    int i;
    if ( N < 7 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 13.0 * in[0] + 10.0 * in[1] + 7.0 * in[2] + 4.0 * in[3] +
                  in[4] - 2.0 * in[5] - 5.0 * in[6] ) / 28.0;

        out[1] = ( 5.0 * in[0] + 4.0 * in[1] + 3 * in[2] + 2 * in[3] +
                  in[4] - in[6] ) / 14.0;

        out[2] = ( 7.0 * in[0] + 6.0 * in [1] + 5.0 * in[2] + 4.0 * in[3] +
                  3.0 * in[4] + 2.0 * in[5] + in[6] ) / 28.0;

        for ( i = 3; i <= N - 4; i++ )
        {
            out[i] = ( in[i - 3] + in[i - 2] + in[i - 1] + in[i] + in[i + 1] + in[i + 2] + in[i + 3] ) / 7.0;
        }

        out[N - 3] = ( 7.0 * in[N - 1] + 6.0 * in [N - 2] + 5.0 * in[N - 3] +
                      4.0 * in[N - 4] + 3.0 * in[N - 5] + 2.0 * in[N - 6] + in[N - 7] ) / 28.0;

        out[N - 2] = ( 5.0 * in[N - 1] + 4.0 * in[N - 2] + 3.0 * in[N - 3] +
                      2.0 * in[N - 4] + in[N - 5] - in[N - 7] ) / 14.0;

        out[N - 1] = ( 13.0 * in[N - 1] + 10.0 * in[N - 2] + 7.0 * in[N - 3] +
                      4 * in[N - 4] + in[N - 5] - 2 * in[N - 6] - 5 * in[N - 7] ) / 28.0;
    }
}

然后是利用二次函数拟合平滑。

void quadraticSmooth5(double in[], double out[], int N)
{
    int i;
    if ( N < 5 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 31.0 * in[0] + 9.0 * in[1] - 3.0 * in[2] - 5.0 * in[3] + 3.0 * in[4] ) / 35.0;
        out[1] = ( 9.0 * in[0] + 13.0 * in[1] + 12 * in[2] + 6.0 * in[3] - 5.0 *in[4]) / 35.0;
        for ( i = 2; i <= N - 3; i++ )
        {
            out[i] = ( - 3.0 * (in[i - 2] + in[i + 2]) +
                      12.0 * (in[i - 1] + in[i + 1]) + 17 * in[i] ) / 35.0;
        }
        out[N - 2] = ( 9.0 * in[N - 1] + 13.0 * in[N - 2] + 12.0 * in[N - 3] + 6.0 * in[N - 4] - 5.0 * in[N - 5] ) / 35.0;
        out[N - 1] = ( 31.0 * in[N - 1] + 9.0 * in[N - 2] - 3.0 * in[N - 3] - 5.0 * in[N - 4] + 3.0 * in[N - 5]) / 35.0;
    }
}


void quadraticSmooth7(double in[], double out[], int N)
{
    int i;
    if ( N < 7 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 32.0 * in[0] + 15.0 * in[1] + 3.0 * in[2] - 4.0 * in[3] -
                  6.0 * in[4] - 3.0 * in[5] + 5.0 * in[6] ) / 42.0;

        out[1] = ( 5.0 * in[0] + 4.0 * in[1] + 3.0 * in[2] + 2.0 * in[3] +
                  in[4] - in[6] ) / 14.0;

        out[2] = ( 1.0 * in[0] + 3.0 * in [1] + 4.0 * in[2] + 4.0 * in[3] +
                  3.0 * in[4] + 1.0 * in[5] - 2.0 * in[6] ) / 14.0;
        for ( i = 3; i <= N - 4; i++ )
        {
            out[i] = ( -2.0 * (in[i - 3] + in[i + 3]) +
                       3.0 * (in[i - 2] + in[i + 2]) +
                      6.0 * (in[i - 1] + in[i + 1]) + 7.0 * in[i] ) / 21.0;
        }
        out[N - 3] = ( 1.0 * in[N - 1] + 3.0 * in [N - 2] + 4.0 * in[N - 3] +
                      4.0 * in[N - 4] + 3.0 * in[N - 5] + 1.0 * in[N - 6] - 2.0 * in[N - 7] ) / 14.0;

        out[N - 2] = ( 5.0 * in[N - 1] + 4.0 * in[N - 2] + 3.0 * in[N - 3] +
                      2.0 * in[N - 4] + in[N - 5] - in[N - 7] ) / 14.0;

        out[N - 1] = ( 32.0 * in[N - 1] + 15.0 * in[N - 2] + 3.0 * in[N - 3] -
                      4.0 * in[N - 4] - 6.0 * in[N - 5] - 3.0 * in[N - 6] + 5.0 * in[N - 7] ) / 42.0;
    }
}

最后是三次函数拟合平滑。

/**
 * 五点三次平滑
 *
 */
void cubicSmooth5 ( double in[], double out[], int N )
{

    int i;
    if ( N < 5 )
    {
        for ( i = 0; i <= N - 1; i++ )
            out[i] = in[i];
    }

    else
    {
        out[0] = (69.0 * in[0] + 4.0 * in[1] - 6.0 * in[2] + 4.0 * in[3] - in[4]) / 70.0;
        out[1] = (2.0 * in[0] + 27.0 * in[1] + 12.0 * in[2] - 8.0 * in[3] + 2.0 * in[4]) / 35.0;
        for ( i = 2; i <= N - 3; i++ )
        {
            out[i] = (-3.0 * (in[i - 2] + in[i + 2])+ 12.0 * (in[i - 1] + in[i + 1]) + 17.0 * in[i] ) / 35.0;
        }
        out[N - 2] = (2.0 * in[N - 5] - 8.0 * in[N - 4] + 12.0 * in[N - 3] + 27.0 * in[N - 2] + 2.0 * in[N - 1]) / 35.0;
        out[N - 1] = (- in[N - 5] + 4.0 * in[N - 4] - 6.0 * in[N - 3] + 4.0 * in[N - 2] + 69.0 * in[N - 1]) / 70.0;
    }
    return;
}

void cubicSmooth7(double in[], double out[], int N)
{
    int i;
    if ( N < 7 )
    {
        for ( i = 0; i <= N - 1; i++ )
        {
            out[i] = in[i];
        }
    }
    else
    {
        out[0] = ( 39.0 * in[0] + 8.0 * in[1] - 4.0 * in[2] - 4.0 * in[3] +
                  1.0 * in[4] + 4.0 * in[5] - 2.0 * in[6] ) / 42.0;
        out[1] = ( 8.0 * in[0] + 19.0 * in[1] + 16.0 * in[2] + 6.0 * in[3] -
                  4.0 * in[4] - 7.0* in[5] + 4.0 * in[6] ) / 42.0;
        out[2] = ( -4.0 * in[0] + 16.0 * in [1] + 19.0 * in[2] + 12.0 * in[3] +
                  2.0 * in[4] - 4.0 * in[5] + 1.0 * in[6] ) / 42.0;
        for ( i = 3; i <= N - 4; i++ )
        {
            out[i] = ( -2.0 * (in[i - 3] + in[i + 3]) +
                       3.0 * (in[i - 2] + in[i + 2]) +
                      6.0 * (in[i - 1] + in[i + 1]) + 7.0 * in[i] ) / 21.0;
        }
        out[N - 3] = ( -4.0 * in[N - 1] + 16.0 * in [N - 2] + 19.0 * in[N - 3] +
                      12.0 * in[N - 4] + 2.0 * in[N - 5] - 4.0 * in[N - 6] + 1.0 * in[N - 7] ) / 42.0;
        out[N - 2] = ( 8.0 * in[N - 1] + 19.0 * in[N - 2] + 16.0 * in[N - 3] +
                      6.0 * in[N - 4] - 4.0 * in[N - 5] - 7.0 * in[N - 6] + 4.0 * in[N - 7] ) / 42.0;
        out[N - 1] = ( 39.0 * in[N - 1] + 8.0 * in[N - 2] - 4.0 * in[N - 3] -
                      4.0 * in[N - 4] + 1.0 * in[N - 5] + 4.0 * in[N - 6] - 2.0 * in[N - 7] ) / 42.0;
    }
}

上面的代码经过了简单的测试，可以放心使用。

最近还有人向我要9点线性平滑和11点线性平滑的系数。我简单算了算，把系数列在了后面。不过我觉得这两组系数意义不大，与其这样做还不如设计一个FIR 滤波器或者 SG 滤波器。

9点线性平滑

yy[0]=(17*y[0] + 14*y[1] + 11*y[2] + 8*y[3] + 5*y[4] + 2*y[5] – y[6] – 4*y[7] – 7*y[8])/45;
yy[1]=(56*y[0] + 47*y[1] + 38*y[2] + 29*y[3] + 20*y[4] + 11*y[5] + 2*y[6] – 7*y[7] – 16*y[8])/180;
yy[2]=(22*y[0] + 19*y[1] + 16*y[2] + 13*y[3] + 10*y[4] + 7*y[5] + 4*y[6] + y[7] – 2*y[8])/90;
yy[3]=(32*y[0] + 29*y[1] + 26*y[2] + 23*y[3] + 20*y[4] + 17*y[5] + 14*y[6] + 11*y[7] + 8*y[8])/ 180;

yy[4]=(y[0] + y[1] +  y[2] + y[3] + y[4] + y[5] + y[6] + y[7] + y[8])/9;

yy[5]=(8*y[0] + 11*y[1] + 14*y[2] + 17*y[3] + 20*y[4] + 23*y[5] + 26*y[6] + 29*y[7] + 32*y[8])/ 180;
yy[6]=(-2*y[0] + y[1] + 4*y[2] + 7*y[3] + 10*y[4] + 13*y[5] + 16*y[6] + 19*y[7] + 22*y[8])/90;
yy[7]=(-16*y[0] – 7*y[1] +  2*y[2] + 11*y[3] + 20*y[4] + 29*y[5] + 38*y[6] + 47*y[7] + 56*y[8])/ 180;
yy[8]=(-7*y[0] – 4*y[1] – y[2] + 2*y[3] + 5*y[4] + 8*y[5] + 11*y[6] + 14*y[7] + 17*y[8])/45;

11点线性平滑

yy[0]=0.3181818181818182*
      y[0] + 0.2727272727272727*y[1] + 0.22727272727272727*y[2] +
    0.18181818181818182*y[3] + 0.13636363636363635*y[4] +
    0.09090909090909091*y[5] + 0.045454545454545456*y[6] +
    1.6152909428804953*^-17*y[7] – 0.045454545454545456*y[8] –
    0.09090909090909091*y[9] – 0.13636363636363635*y[10]

yy[1]=0.27272727272727276*
      y[0] + 0.23636363636363633*y[1] + 0.19999999999999998*y[2] +
    0.16363636363636364*y[3] + 0.12727272727272726*y[4] +
    0.09090909090909091*y[5] + 0.05454545454545455*y[6] +
    0.018181818181818202*y[7] – 0.01818181818181818*y[8] –
    0.054545454545454536*y[9] – 0.09090909090909088*y[10]

yy[2]=0.2272727272727273*
      y[0] + 0.19999999999999996*y[1] + 0.17272727272727267*y[2] +
    0.14545454545454542*y[3] + 0.11818181818181815*y[4] +
    0.09090909090909091*y[5] + 0.06363636363636363*y[6] +
    0.03636363636363638*y[7] + 0.009090909090909094*y[8] –
    0.01818181818181816*y[9] – 0.0454545454545454*y[10]

yy[3]=0.18181818181818188*
      y[0] + 0.16363636363636355*y[1] + 0.1454545454545454*y[2] +
    0.12727272727272723*y[3] + 0.10909090909090904*y[4] +
    0.0909090909090909*y[5] + 0.07272727272727272*y[6] +
    0.054545454545454564*y[7] + 0.036363636363636376*y[8] +
    0.01818181818181823*y[9] + 8.326672684688674*^-17*y[10]

yy[4]=0.13636363636363644*
      y[0] + 0.12727272727272718*y[1] + 0.11818181818181811*y[2] +
    0.10909090909090903*y[3] + 0.09999999999999995*y[4] +
    0.0909090909090909*y[5] + 0.08181818181818182*y[6] +
    0.07272727272727275*y[7] + 0.06363636363636364*y[8] +
    0.05454545454545459*y[9] + 0.04545454545454555*y[10]

yy[5]=0.090909090909091*
      y[0] + 0.09090909090909077*y[1] + 0.09090909090909083*y[2] +
    0.09090909090909083*y[3] + 0.09090909090909084*y[4] +
    0.0909090909090909*y[5] + 0.09090909090909091*y[6] +
    0.09090909090909094*y[7] + 0.09090909090909093*y[8] +
    0.090909090909091*y[9] + 0.09090909090909102*y[10]

yy[6]=0.04545454545454558*
      y[0] + 0.0545454545454544*y[1] + 0.06363636363636352*y[2] +
    0.07272727272727264*y[3] + 0.08181818181818173*y[4] +
    0.0909090909090909*y[5] + 0.1*y[6] + 0.10909090909090911*y[7] +
    0.11818181818181821*y[8] + 0.12727272727272737*y[9] +
    0.13636363636363652*y[10]

yy[7]=1.1102230246251565*^-\
16*y[0] + 0.01818181818181802*y[1] + 0.03636363636363624*y[2] +
    0.05454545454545445*y[3] + 0.07272727272727264*y[4] +
    0.0909090909090909*y[5] + 0.10909090909090909*y[6] +
    0.12727272727272732*y[7] + 0.1454545454545455*y[8] +
    0.16363636363636372*y[9] + 0.181818181818182*y[10]

yy[8]=-0.0454545454545453*
      y[0] – 0.018181818181818354*y[1] + 0.009090909090908955*y[2] +
    0.03636363636363624*y[3] + 0.06363636363636355*y[4] +
    0.09090909090909088*y[5] + 0.11818181818181818*y[6] +
    0.1454545454545455*y[7] + 0.17272727272727273*y[8] +
    0.2000000000000001*y[9] + 0.22727272727272746*y[10]

yy[9]=-0.09090909090909072*
      y[0] – 0.05454545454545473*y[1] – 0.018181818181818327*y[2] +
    0.01818181818181805*y[3] + 0.05454545454545444*y[4] +
    0.09090909090909088*y[5] + 0.12727272727272726*y[6] +
    0.16363636363636366*y[7] + 0.2*y[8] + 0.23636363636363647*y[9] +
    0.27272727272727293*y[10]

yy[10]=-0.1363636363636362*
      y[0] – 0.09090909090909116*y[1] – 0.04545454545454561*y[2] –
    1.6653345369377348*^-16*y[3] + 0.04545454545454533*y[4] +
    0.09090909090909088*y[5] + 0.13636363636363635*y[6] +
    0.18181818181818188*y[7] + 0.2272727272727273*y[8] +
    0.27272727272727293*y[9] + 0.3181818181818184*y[10]