一阶数字高通滤波器的设计与实现

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一阶数字高通滤波器的设计与实现

引用

CSDN

https://blog.csdn.net/weixin_38604759/article/details/139160079

本文主要介绍了数字一阶高通滤波器的计算公式推导、算法实现及仿真。文章通过Python和C语言两种编程语言实现了数字一阶高通滤波器，并通过仿真和实际硬件测试验证了其性能。文章内容完整，包含了理论推导、代码实现和实验验证等多个方面，具有较高的技术含量和实用价值。

1. 计算公式推导

一阶高通滤波器的差分方程为：

$$
y(n) = b_0 x(n) + b_1 x(n-1) + a_0 y(n-1)
$$

其中，$x(n)$为输入信号，$y(n)$为输出信号，$b_0$、$b_1$和$a_0$为滤波器系数。系数的计算公式如下：

$$
b_0 = \frac{1}{\tan(\frac{\omega_c}{2}T_s) + 1}
$$

$$
b_1 = -b_0
$$

$$
a_0 = \frac{1 - \tan(\frac{\omega_c}{2}T_s)}{\tan(\frac{\omega_c}{2}T_s) + 1}
$$

其中，$\omega_c$为截止频率，$T_s$为采样周期。

2. 算法实现及仿真

Python实现

import numpy as np
import matplotlib.pyplot as plt

# 参数设置
Wc = 2 * np.pi * 30
Tsw = 0.00314
halfdigiW = np.tan(Wc / 2 * Tsw)
b10 = 1 / (halfdigiW + 1)
b11 = -b10
a10 = (1 - halfdigiW) / (halfdigiW + 1)

# 信号生成
x = np.linspace(-np.pi, np.pi, 2000)
data = np.zeros_like(x)
data1 = np.zeros_like(x)
y1 = np.zeros_like(x)

for i in range(len(x)):
    data[i] = np.sin(2 * np.pi * x[i]) + 0.5 * np.sin(30 * 2 * np.pi * x[i])
    data1[i] = 0.5 * np.sin(130 * 2 * np.pi * x[i])
    y1[i] = b10 * data[i] + b11 * data[i - 1] + a10 * y1[i - 1]

y1[0] = 0
y1[1] = 0

# 波形绘制
plt.subplot(2, 1, 1)
plt.plot(x, data, label='sig+noise')
plt.plot(x, y1, 'r', label='first order HP')
plt.subplot(2, 1, 2)
plt.plot(x, data1, label='sig')
plt.plot(x, y1, 'r', label='first order HP')
plt.grid()
plt.legend()
plt.show()

C语言实现

#ifndef _MHPF1W_F_H_
#define _MHPF1W_F_H_
#include <stdint.h>

struct MHpf1W_F {
    struct {
        void (*Set)(struct MHpf1W_F *self, float cutFreq, int samFreq);
        void (*VaryCutFreq)(struct MHpf1W_F *self, float cutFreq);
        float cutFreq;
        float samFreq;
    } Init;
    struct {
        int (*In)(struct MHpf1W_F *self, int Xn);
        int out_y;
    } Prd;
    struct {
        float Ts;
        int a0, b0, b1;
        int Xn_1, Yn_1;
    } pri;
};

void MHpf1W_F_Create(struct MHpf1W_F *self);
#endif

#include "MHpf1W_F.h"
#include <string.h>
#include <math.h>

static const float PI = 3.1415926535897932384626f;
#define MID(a, min, max) (a = (a < min) ? min : (a < max) ? a : max)
#define Q15M(a, b) ((a * b) >> 15)

static void _Update(struct MHpf1W_F *self) {
    float halfdigiW, tgAnaWT;
    halfdigiW = PI * self->Init.cutFreq * self->pri.Ts;
    tgAnaWT = tan(halfdigiW);
    self->pri.b0 = 1 / (tgAnaWT + 1) * 32768;
    self->pri.b1 = -self->pri.b0;
    self->pri.a0 = ((1 - tgAnaWT) / (tgAnaWT + 1)) * 32768;
    self->pri.Xn_1 = 0;
    self->pri.Yn_1 = 0;
}

static void InitSet(struct MHpf1W_F *self, float cutFreq, int samFreq) {
    self->Init.cutFreq = MID(cutFreq, 0.0f, samFreq * 0.5f);
    self->Init.samFreq = samFreq;
    self->pri.Ts = 1.0f / self->Init.samFreq;
    _Update(self);
}

static void InitVaryCutF(struct MHpf1W_F *self, float cutFreq) {
    self->Init.cutFreq = cutFreq;
    _Update(self);
}

static int PrdIn(struct MHpf1W_F *self, int Xn) {
    self->Prd.out_y =
        Q15M(self->pri.b0, Xn) +
        Q15M(self->pri.b1, self->pri.Xn_1) +
        Q15M(self->pri.a0, self->pri.Yn_1);
    self->pri.Yn_1 = self->Prd.out_y;
    self->pri.Xn_1 = Xn;
    return self->Prd.out_y;
}

void MHpf1W_F_Create(struct MHpf1W_F *self) {
    memset(self, 0, sizeof(struct MHpf1W_F));
    self->Init.Set = InitSet;
    self->Init.VaryCutFreq = InitVaryCutF;
    self->Prd.In = PrdIn;
}

3. 仿真结果分析

Python仿真结果

输入信号是幅值为1、频率为1Hz的低频信号加上幅值为0.5、频率为30Hz的高频信号，采样时间为0.003s。从仿真波形可以看出，当截止频率fc为0.5Hz时，滤波后的波形有微小的衰减作用，但几乎和原波形一致。随着截止频率的增加，对低频的滤除效果越来越强，高频信号逐渐接近其本身波形。当截止频率高于高频信号频率时，高频信号也会被滤掉。当截止频率大于等于采样频率的一半时，输出变为一条直线。