希尔伯特变换 实现包络提取 C++

希尔伯特变换可以用于提取信号的包络，这在信号处理中很常见。以下是一个简单的C++实现：

#include <complex>
#include <iostream>
#include <vector>

using namespace std;

void hilbert_transform(const vector<double>& x, vector<complex<double>>& y) {
    int n = x.size();
    vector<double> h(n), f(n);
    h[0] = h[n / 2] = 1;
    for (int i = 1; i < n / 2; i++) {
        h[i] = 2;
    }
    for (int i = 0; i < n; i++) {
        f[i] = 2 * M_PI / n * i;
    }
    vector<complex<double>> c(n);
    for (int i = 0; i < n; i++) {
        c[i] = complex<double>(x[i], 0);
    }
    vector<complex<double>> dft(n);
    for (int i = 0; i < n; i++) {
        complex<double> sum(0, 0);
        for (int j = 0; j < n; j++) {
            sum += c[j] * exp(complex<double>(0, -f[i] * j));
        }
        dft[i] = sum;
    }
    for (int i = 0; i < n; i++) {
        dft[i] *= h[i];
    }
    for (int i = 0; i < n; i++) {
        c[i] = complex<double>(0, 0);
        for (int j = 0; j < n; j++) {
            c[i] += dft[j] * exp(complex<double>(0, f[i] * j));
        }
        y[i] = c[i] / complex<double>(0, 1);
    }
}

void envelope_extraction(const vector<double>& x, vector<double>& y) {
    vector<complex<double>> z(x.size());
    hilbert_transform(x, z);
    for (int i = 0; i < x.size(); i++) {
        y[i] = sqrt(x[i] * x[i] + z[i].imag() * z[i].imag());
    }
}

int main() {
    vector<double> x = {1, 2, 3, 4, 5};
    vector<double> y(x.size());
    envelope_extraction(x, y);
    for (int i = 0; i < x.size(); i++) {
        cout << y[i] << endl;
    }
    return 0;
}

这段代码首先调用了上面提到的希尔伯特变换函数，然后根据公式 $y(t) = \sqrt{x^2(t) + z^2(t)}$ ，计算出信号的包络。最后输出包络信号。