如何用C语言写4阶巴特沃斯高通滤波器，要求输入输出均为长整型

巴特沃斯高通滤波器是一种数字滤波器，用于滤除低频信号并保留高频信号。4阶的巴特沃斯高通滤波器具有更陡峭的通带边缘和更窄的过渡带，可以更好地滤除低频信号。下面是一个用C语言实现4阶巴特沃斯高通滤波器的示例代码，输入输出均为长整型：

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define M_PI 3.14159265358979323846

long b[5]; // 分子系数
long a[5]; // 分母系数

long x[5]; // 输入信号
long y[5]; // 输出信号

void butterworth_highpass_filter_4th_order(long fc, int fs, double Q)
{
    double w0 = 2 * M_PI * fc / fs;
    double alpha = sin(w0) / (2 * Q);
    double cosw0 = cos(w0);
    double alpha2 = alpha * alpha;
    double beta = sqrt(alpha2 + cosw0 * cosw0 - 2 * alpha * cosw0);
    double gamma = alpha2 + cosw0 * beta + alpha;

    b[0] = (long)(alpha2 * pow(2, 30) / gamma);
    b[1] = (long)(2 * alpha2 * pow(2, 30) / gamma);
    b[2] = (long)(alpha2 * pow(2, 30) / gamma);
    b[3] = (long)(-2 * alpha2 * pow(2, 30) / gamma);
    b[4] = (long)(alpha2 * pow(2, 30) / gamma);

    a[0] = pow(2, 30) + (long)(alpha * pow(2, 30));
    a[1] = (long)(-2 * pow(2, 30) * cosw0);
    a[2] = pow(2, 30) - (long)(alpha * pow(2, 30));
    a[3] = (long)(-2 * pow(2, 30) * cosw0);
    a[4] = pow(2, 30) - (long)(alpha2 * pow(2, 30) / gamma);

    double gain = 1 / (double)a[0];

    b[0] = (long)(b[0] * gain);
    b[1] = (long)(b[1] * gain);
    b[2] = (long)(b[2] * gain);
    b[3] = (long)(b[3] * gain);
    b[4] = (long)(b[4] * gain);
    a[1] = (long)(-a[1]);
    a[3] = (long)(-a[3]);
    a[4] = (long)(-a[4]);

    x[0] = x[1] = x[2] = x[3] = x[4] = 0;
    y[0] = y[1] = y[2] = y[3] = y[4] = 0;
}

long butterworth_highpass_filter_4th_order_apply(long input)
{
    x[4] = x[3];
    x[3] = x[2];
    x[2] = x[1];
    x[1] = x[0];
    x[0] = input;

    y[4] = y[3];
    y[3] = y[2];
    y[2] = y[1];
    y[1] = y[0];
    y[0] = (long)((b[0] * x[0] + b[1] * x[1] + b[2] * x[2] + b[3] * x[3] + b[4] * x[4] - a[1] * y[1] - a[3] * y[3] - a[4] * y[4]) / (double)a[0]);

    return y[0];
}

int main()
{
    long fc = 1000; // 截止频率为1kHz
    int fs = 8000; // 采样率为8kHz
    double Q = 0.707; // 质因数Q为0.707

    butterworth_highpass_filter_4th_order(fc, fs, Q);

    // 生成测试信号，1s的8000Hz正弦波
    int num_samples = 8000;
    long test_signal[num_samples];
    for (int i = 0; i < num_samples; i++) {
        test_signal[i] = (long)(sin(2 * M_PI * i * 1000 / fs) * pow(2, 30));
    }

    // 对测试信号进行滤波
    long filtered_signal[num_samples];
    for (int i = 0; i < num_samples; i++) {
        filtered_signal[i] = butterworth_highpass_filter_4th_order_apply(test_signal[i]);
    }

    // 将结果输出到文件
    FILE *fp = fopen("filtered_signal.txt", "w");
    if (fp == NULL) {
        printf("Failed to open file\n");
        return 1;
    }

    for (int i = 0; i < num_samples; i++) {
        fprintf(fp, "%ld\n", filtered_signal[i]);
    }

    fclose(fp);

    return 0;
}

在这个示例中，我们首先定义了一个 `butterworth_highpass_filter_4th_order` 函数，用于计算4阶巴特沃斯高通滤波器的分子和分母系数。然后，我们定义了一个 `butterworth_highpass_filter_4th_order_apply` 函数，用于对输入信号进行滤波。在 `main` 函数中，我们首先生成了一个1秒钟的正弦波作为测试信号，然后对其进行滤波，并将结果输出到文件中。

需要注意的是，在这个示例中，我们将输入信号和输出信号都乘以了 $2^{30}$，以避免在计算过程中出现浮点运算误差。在使用时，需要根据具体的需求进行修改。