记录 C 语言下的数字滤波器及其应用示例。
需要包含的头文件
需要包含的头文件如下所示。
#include <stdio.h>
#include <stdlib.h>
#include <malloc.h>
#include <math.h>随机数生成
测试过程中需要用随机数生成噪声，生成随机数代码如下所示。
/*********************************************************************************************************
* 函数名称： Uniform
* 函数功能： 生成均匀分布的随机数
* 输入参数： a：区间下限
* b：区间上限
* seed：随机数种子
* 输出参数： void
* 返 回 值： 随机数
* 创建日期： 2023年08月10日
* 注 意：
*********************************************************************************************************/
double Uniform(double a, double b, long int* seed)
{
double t;
*seed = 2045 * (*seed) + 1;
*seed = *seed - (*seed / 1048576) * 1048576;
t = (*seed) / 1048576.0;
t = a + (b - a) * t;
return t;
}
/*********************************************************************************************************
* 函数名称： Gauss
* 函数功能： 生成正态分布的随机数
* 输入参数： mean：正态分布的均值μ
* sigma：正态分布的均方差σ
* seed：随机数种子
* 输出参数： void
* 返 回 值： 随机数
* 创建日期： 2023年08月10日
* 注 意：
*********************************************************************************************************/
double Gauss(double mean, double sigma, long int* seed)
{
int i;
double x, y;
for (x = 0, i = 0; i < 12; i++)
{
x += Uniform(0.0, 1.0, seed);
}
x = x - 6.0;
y = mean + x * sigma;
return y;
}维纳（Wiener）数字滤波
维纳（Wiener）数字滤波器实现代码如下所示。
/*********************************************************************************************************
* 函数名称： Levin
* 函数功能： 求解一般托布利兹方程组的莱文森算法
* 输入参数： t：双精度实型一维数组，长度为 n。存放对称托布利兹矩阵的元素 t0,t1,...,tn-1。
* b：双精度实型一维数组，长度为 n。存放方程组右端的常数常量。
* n：整形变量。方程组的阶数。
* x：双精度实型一维数组，长度为 n。存放方程组的解。
* 输出参数： void
* 返 回 值： 本函数的返回值若小于 0，则说明方程是病态的。
* 创建日期： 2023年09月20日
* 注 意：
*********************************************************************************************************/
int Levin(double* t, double* b, int n, double* x)
{
int i, j, k;
double a, beta, q, c, h, *y, *s;
s = malloc(n * sizeof(double));
y = malloc(n * sizeof(double));
a = t[0];
if ((fabs(a) + 1.0) == 1.0)
{
free(s);
free(y);
return (-1);
}
y[0] = 1.0;
x[0] = b[0] / a;
for (k = 1; k < n; k++)
{
beta = 0.0;
q = 0.0;
for (j = 0; j < k; j++)
{
beta += y[j] * t[j + 1];
q += x[j] * t[k - j];
}
if ((fabs(a) + 1.0) == 1.0)
{
free(s);
free(y);
return (-1);
}
c = -beta / a;
s[0] = c * y[k - 1];
y[k] = y[k - 1];
if (k != 1)
{
for (i = 1; i < k; i++)
{
s[i] = y[i - 1] + c * y[k - i - 1];
}
}
s[k] = y[k - 1];
a += c * beta;
if ((fabs(a) + 1.0) == 1.0)
{
free(s);
free(y);
return (-1);
}
h = (b[k] - q) / a;
for (i = 0; i < k; i++)
{
x[i] += h * s[i];
y[i] = s[i];
}
x[k] = h * y[k];
}
free(s);
free(y);
return (1);
}
/*********************************************************************************************************
* 函数名称： Wiener
* 函数功能： 维纳数字滤波
* 输入参数： rxx：双精度实型一维数组，长度为（p+1）。信号 x(n) 的自相关函数 rxx(i)。
* rdx：双精度实型一维数组，长度为（p+1）。信号 d(n) 与 x(n) 的互相关函数 rdx(i)。
* p ：整型变量。维纳滤波器的阶数。
* h ：双精度实型一维数组，长度为（p+1）。维纳滤波器的系数。
* e ：双精度实型变量。维纳滤波器的最小均方误差。
* x ：双精度实型一维数组，长度为 n。存放输入信号 x(i)。
* y ：双精度实型一维数组，长度为 n。存放输出信号 y(i)。
* n ：整形变量。输入信号的长度。
* 输出参数： void
* 返 回 值： void
* 创建日期： 2023年09月21日
* 注 意：
*********************************************************************************************************/
void Wiener(double* rxx, double* rdx, int p, double* h, double* e, double* x, double* y, int n)
{
int i, k;
double sum;
Levin(rxx, rdx, p + 1, h);
sum = 0.0;
for (i = 0; i <= p; i++)
{
sum += rdx[i] * h[i];
}
*e = rdx[0] - sum;
for (k = 0; k < n; k++)
{
y[k] = 0.0;
for (i = 0; i <= p; i++)
{
if ((k - i) >= 0)
{
y[k] += h[i] * x[k - i];
}
}
}
}
维纳（Wiener）滤波器的使用如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
int i, k, n, p;
long seed;
double e;
static double rxx[4000], rdx[4000];
static double h[4000], x[4000], y[4000], s[4000];
FILE* fp;
//获取原始数据
n = 2000;
for (i = 0; i < n; i++)
{
s[i] = g_arrEcgWave[i];
}
//添加白噪声
seed = 157l;
for (i = 0; i < n; i++)
{
x[i] = s[i] + 0.1 * Gauss(0.0, 1.0, &seed);
}
//生成互相关函数
p = 2000;
for (k = 0; k <= p; k++)
{
rdx[k] = 0.0;
for (i = 0; i < (n - k); i++)
{
rdx[k] += s[i] * s[i + k];
}
rdx[k] = rdx[k] / n;
}
//生成自相关函数
rxx[0] = rdx[0] + 1.0;
for (i = 1; i <= p; i++)
{
rxx[i] = rdx[i];
}
//滤波
Wiener(rxx, rdx, p, h, &e, x, y, n);
printf("The Minimum MSE Error = %lf\n", e);
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("wieners.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%lf,%lf,%lf\n", i, x[i], s[i], y[i]);
}
fclose(fp);
return 0;
}实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
最小均方（LMS）自适应数字滤波
实现代码如下所示。
/*********************************************************************************************************
* 函数名称： LMS
* 函数功能： 最小均方（LMS）自适应数字滤波
* 输入参数： x ：双精度实型一维数组，长度为 n。输入信号。
* d ：双精度实型一维数组，长度为 n。理想输出信号。
* y ：双精度实型一维数组，长度为 n。实际输出信号。
* n ：整型变量。输入信号的长度。
* w ：双精度实型一维数组，长度为 m。自适应滤波器的加权系数。
* m ：整型变量。自适应滤波器的长度（阶数-1）。
* mu：双精度实型变量。收敛因子。
* 输出参数： void
* 返 回 值： void
* 创建日期： 2023年09月22日
* 注 意：
*********************************************************************************************************/
void LMS(double* x, double* d, double* y, int n, double* w, int m, double mu)
{
int i, k;
double e;
for (i = 0; i < m; i++)
{
w[i] = 0.0;
}
for (k = 0; k < m; k++)
{
y[k] = 0.0;
for (i = 0; i <= k; i++)
{
y[k] += x[k - i] * w[i];
}
e = d[k] - y[k];
for (i = 0; i <= k; i++)
{
w[i] += 2.0 * mu * e * x[k - i];
}
}
for (k = m; k < n; k++)
{
y[k] = 0.0;
for (i = 0; i < m; i++)
{
y[k] += x[k - i] * w[i];
}
e = d[k] - y[k];
for (i = 0; i < m; i++)
{
w[i] += 2.0 * mu * e * x[k - i];
}
}
}测试代码如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
int i, m, n;
long seed;
double mu;
static double d[4000], x[4000], y[4000], w[4000];
FILE* fp;
//获取理想输出信号
n = 2000;
for (i = 0; i < n; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加噪声
seed = 13579l;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * Gauss(0.0, 1.0, &seed);
}
//LMS 滤波
m = 50;
mu = 0.0005;
LMS(x, d, y, n, w, m, mu);
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("lms.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%lf,%lf,%lf\n", i, x[i], d[i], y[i]);
}
fclose(fp);
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
归一化（LMS）自适应数字滤波
实现代码如下所示。
/*********************************************************************************************************
* 函数名称： NLMS
* 函数功能： 归一化（LMS）自适应数字滤波
* 输入参数： x ：双精度实型一维数组，长度为 n。开始时存放输入信号，最后存放实际输出信号。
* d ：双精度实型一维数组，长度为 n。理想输出信号。
* n ：整型变量。输入信号的长度。
* w ：双精度实型一维数组，长度为 m。自适应滤波器的加权系数。
* m ：整型变量。自适应滤波器的长度（阶数-1）。
* mu ：双精度实型变量。学习因子，0 < mu < 1。
* sigma2：双精度实型变量。输入信号的功率估值 σ^2。
* a ：双精度实型变量。遗忘因子，0 <= a <= 1。
* px ：双精度实型一维数组，长度为 m。在分块处理时，用于保存输入信号的过去值。
* 输出参数： void
* 返 回 值： void
* 创建日期： 2023年09月23日
* 注 意：
*********************************************************************************************************/
void NLMS(double* x, double* d, int n, double* w, int m, double mu, double sigma2, double a, double* px)
{
int i, k;
double e, tmp;
for (k = 0; k < n; k++)
{
px[0] = x[k];
x[k] = 0.0;
for (i = 0; i < m; i++)
{
x[k] += px[i] * w[i];
}
e = d[k] - x[k];
sigma2 = a * px[0] * px[0] + (1.0 - a) * sigma2;
tmp = 2 * mu / (m * sigma2);
for (i = 0; i < m; i++)
{
w[i] += tmp * e * px[i];
}
for (i = (m - 1); i >= 1; i--)
{
px[i] = px[i - 1];
}
}
}测试代码如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
int i, m, n;
double a, mu, sigma2;
long seed;
static double d[4000], x[4000], y[4000], w[4000], px[4000];
FILE* fp;
//获取理想输出信号
for (i = 0; i < 2000; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加噪声
seed = 13579l;
n = 2000;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * Gauss(0.0, 1.0, &seed);
y[i] = x[i];
}
//LMS 滤波
m = 51;
mu = 0.2;
sigma2 = 0.2;
a = 0;
for (i = 0; i < m; i++){w[i] = 0.0;}
for (i = 0; i < m; i++){px[i] = 0.0;}
NLMS(y, d, n, w, m, mu, sigma2, a, px);
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("nlms.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%10.7lf,%10.7lf,%10.7lf\n", i, x[i], d[i], y[i]);
}
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
递推最小二乘（RLS）自适应数字滤波
实验代码如下所示。
/*********************************************************************************************************
* 函数名称： RLS
* 函数功能： 递推最小二乘（RLS）自适应数字滤波
* 输入参数： x：双精度实型一维数组，长度为 n。开始时存放输入信号，最后存放实际输出信号。
* d：双精度实型一维数组，长度为 n。理想输出信号。
* n：整型变量。输入信号的长度。
* w：双精度实型一维数组，长度为 m。自适应滤波器的加权系数。
* m：整型变量。自适应滤波器的长度（阶数-1）。
* r：双精度实型变量。遗忘因子，0 < r <= 1。
* 输出参数： void
* 返 回 值： void
* 创建日期： 2023年09月23日
* 注 意：
*********************************************************************************************************/
void RLS(double* x, double* d, int n, double* w, int m, double r)
{
int i, j, k;
double a, s, *g, *u, *px, *p;
g = malloc(m * sizeof(double));
u = malloc(m * sizeof(double));
px = malloc(m * sizeof(double));
p = malloc(m * m * sizeof(double));
for (i = 0; i < m; i++)
{
for (j = 0; j < m; j++)
{
p[i * m + j] = 0.0;
}
}
for (i = 0; i < m; i++)
{
p[i * m + i] = 1.0e+8;
}
for (i = 0; i < m; i++)
{
px[i] = 0.0;
}
for (k = 0; k < n; k++)
{
px[0] = x[k];
for (j = 0; j < m; j++)
{
u[j] = 0.0;
for (i = 0; i < m; i++)
{
u[j] = u[j] + (1 / r) * p[j * m + i] * px[i];
}
}
s = 1.0;
for (i = 0; i < m; i++)
{
s = s + u[i] * px[i];
}
for (i = 0; i < m; i++)
{
g[i] = u[i] / s;
}
x[k] = 0.0;
for (i = 0; i < m; i++)
{
x[k] = x[k] + w[i] * px[i];
}
a = d[k] - x[k];
for (i = 0; i < m; i++)
{
w[i] = w[i] + g[i] * a;
}
for (j = 0; j < m; j++)
{
for (i = 0; i < m; i++)
{
p[j * m + i] = (1 / r) * p[j * m + i] - g[j] * u[i];
}
}
for (i = (m - 1); i >= 1; i--)
{
px[i] = px[i - 1];
}
}
free(g);
free(u);
free(px);
free(p);
}测试代码如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
int i, m, n;
long seed;
static double r, w[4000], d[4000], x[4000], y[4000];
FILE* fp;
//获取理想输出信号
for (i = 0; i < 2000; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加噪声
seed = 13579l;
n = 2000;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * Gauss(0.0, 1.0, &seed);
y[i] = x[i];
}
// RLS 滤波
m = 4;
r = 1.0;
for (i = 0; i < m; i++){ w[i] = 0.0;}
RLS(y, d, n, w, m, r);
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("rls.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%10.7lf,%10.7lf,%10.7lf\n", i, x[i], d[i], y[i]);
}
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
滑动平均滤波
实现代码如下。
/*********************************************************************************************************
* 函数名称： MovAverageFilter
* 函数功能： 滑动平均滤波
* 输入参数： filter：双精度实型一维数组，长度为 oder。滤波缓冲区。
* order ：整型变量。滤波阶数。
* dat ：双精度实型变量。新的采样数据
* 输出参数： void
* 返 回 值： 滤波值
* 创建日期： 2023年09月20日
* 注 意：
*********************************************************************************************************/
double MovAverageFilter(double* filter, int order, double dat)
{
int i;
double sum;
for (i = 0; i < order - 1; i++)
{
filter[i] = filter[i + 1];
}
filter[order - 1] = dat;
sum = 0;
for (i = 0; i < order; i++)
{
sum = sum + filter[i];
}
sum = sum / order;
return sum;
}测试代码如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
#define FILTER_ORDER 8
static double s_arrFilter[FILTER_ORDER] = {0};
int i, n;
long seed;
static double d[4000], x[4000], y[4000];
FILE* fp;
//获取理想输出信号
n = 2000;
for (i = 0; i < 2000; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加噪声
seed = 13579l;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * Gauss(0.0, 1.0, &seed);
}
//滑动平均滤波
for (i = 0; i < n; i++)
{
y[i] = MovAverageFilter(s_arrFilter, FILTER_ORDER, x[i]);
}
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("MovAverageFilter.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%10.7lf,%10.7lf,%10.7lf\n", i, x[i], d[i], y[i]);
}
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
心电信号添加 50Hz 工频干扰，利用滑动平均滤波滤除工频干扰示例如下。2kHz 采样率下滤波阶数为 40，500Hz 采样率下滤波阶数为 10，其它采样率可以根据这个规律设置滤波阶数。
int main(void)
{
extern const double g_arrEcgWave[4000];
#define FILTER_ORDER 40
static double s_arrFilter[FILTER_ORDER] = { 0 };
int i, n;
long seed;
static double d[4000], x[4000], y[4000];
double pi, time;
FILE* fp;
//获取理想输出信号
n = 2000;
for (i = 0; i < 2000; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加 50Hz 工频干扰
seed = 13579l;
pi = 3.1415926535;
time = 0;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * sin(time * 2 * pi / (1.0 / 50.0));
time = time + 0.0005;
}
//滑动平均滤波
for (i = 0; i < n; i++)
{
y[i] = MovAverageFilter(s_arrFilter, FILTER_ORDER, x[i]);
}
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("MovAverageFilter.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%10.7lf,%10.7lf,%10.7lf\n", i, x[i], d[i], y[i]);
}
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
中值滤波
实现代码如下。
/*********************************************************************************************************
* 函数名称： MedianFilter
* 函数功能： 中值滤波
* 输入参数： filter1：双精度实型一维数组，长度为 oder。滤波缓冲区。
* filter2：双精度实型一维数组，长度为 oder。滤波缓冲区。
* order ：整型变量。滤波阶数。
* dat ：双精度实型变量。新的采样数据
* 输出参数： void
* 返 回 值： 滤波值
* 创建日期： 2023年09月23日
* 注 意：
*********************************************************************************************************/
double MedianFilter(double* filter1, double* filter2, int order, double dat)
{
int i, j;
double swap;
//将最新的数据保存到缓冲区 1
for (i = 0; i < order - 1; i++)
{
filter1[i] = filter1[i + 1];
}
filter1[order - 1] = dat;
//将数据拷贝到缓冲区 2
for (i = 0; i < order; i++)
{
filter2[i] = filter1[i];
}
//重新排序
for (i = 0; i < order; i++)
{
for (j = i + 1; j < order; j++)
{
if (filter2[j] > filter2[i])
{
swap = filter2[j];
filter2[j] = filter2[i];
filter2[i] = swap;
}
}
}
//返回中位值
return filter2[order / 2];
}
测试代码如下所示。其中心电波形数据为模拟器标准数据，脉率值为 60BPM，采样率为 2kHz，在文章的末尾给出。
int main(void)
{
extern const double g_arrEcgWave[4000];
#define FILTER_ORDER 8
static double s_arrFilter1[FILTER_ORDER] = { 0 };
static double s_arrFilter2[FILTER_ORDER] = { 0 };
int i, n;
long seed;
static double d[4000], x[4000], y[4000];
FILE* fp;
//获取理想输出信号
n = 2000;
for (i = 0; i < 2000; i++)
{
d[i] = g_arrEcgWave[i];
}
//获取输入数据，加噪声
seed = 13579l;
for (i = 0; i < n; i++)
{
x[i] = g_arrEcgWave[i] + 0.1 * Gauss(0.0, 1.0, &seed);
}
//滑动平均滤波
for (i = 0; i < n; i++)
{
y[i] = MedianFilter(s_arrFilter1, s_arrFilter2, FILTER_ORDER, x[i]);
}
//输出波形数据，按照原始信号、参考信号、滤波后信号顺序
fp = fopen("MedianFilter.csv", "w");
for (i = 0; i < n; i++)
{
fprintf(fp, "%d,%10.7lf,%10.7lf,%10.7lf\n", i, x[i], d[i], y[i]);
}
return 0;
}最终实验结果如下所示。蓝色为参考信号，绿色为输入信号，黄色为输出信号。
心电波形数据
本文中使用到的心电波形数据如下所示。
/*
* 心电波形数据
* 采样率 2kHz
* 脉率值 60BPM
* 单位 mV
*/
const double g_arrEcgWave[4000] =
{
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0.16,0.16,0.16,0.16,0.16,0.16,0.17,0.17,0.17,0.17,0.16,0.16,0.16,0.16,0.16,0.16,0.16,0.16,0.15,0.15,0.15,0.15,
0.15,0.15,0.15,0.15,0.14,0.14,0.14,0.14,0.13,0.14,0.13,0.13,0.13,0.13,0.13,0.13,0.12,0.12,0.12,0.12,0.12,0.12,
0.12,0.12,0.12,0.11,0.12,0.12,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,
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