Python实现希尔伯特变换(Hilbert transform)的示例代码

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

using namespace std;

typedef complex<double> Complex;
typedef vector<Complex> ComplexVector;

// 快速傅里叶变换
void fft(ComplexVector& data) {
    int n = data.size();
    if (n <= 1) {
        return;
    }

    // 分离偶数项和奇数项
    ComplexVector even(n/2), odd(n/2);
    for (int i = 0; i < n; i += 2) {
        even[i/2] = data[i];
        odd[i/2] = data[i+1];
    }

    // 递归计算偶数项和奇数项的FFT
    fft(even);
    fft(odd);

    // 计算每个k点的DFT
    for (int k = 0; k < n/2; k++) {
        Complex t = polar(1.0, -2 * M_PI * k / n) * odd[k];
        data[k] = even[k] + t;
        data[k+n/2] = even[k] - t;
    }
}


// 希尔伯特变换
void hilbertTransform(ComplexVector& signal) {
    int n = signal.size();

    // 扩展信号长度至2的幂次方
    int n2 = 1;
    while (n2 < n) {
        n2 *= 2;
    }
    signal.resize(n2);

    // 进行FFT变换
    fft(signal);

    // 对FFT结果进行处理
    for (int i = 1; i < n; i++) {
        signal[i] *= 2;
    }
    for (int i = n; i < n2; i++) {
        signal[i] = 0;
    }
    signal[0] = 1;
    signal[n] = 0;

    // 反向FFT变换
    fft(signal);
    for (int i = 0; i < n; i++) {
        signal[i] = signal[i].imag() / n;
    }
}

int main() {
    ComplexVector signal = {1, 2, 3, 4, 5, 6, 7, 8};
    hilbertTransform(signal);

    // 输出结果
    for (int i = 0; i < signal.size(); i++) {
        cout << signal[i] << " ";
    }
    cout << endl;

    return 0;
}

import numpy as np

def hilbert_transform(signal):
    """
    计算希尔伯特变换
    """
    n = len(signal)

    # 扩展信号长度至2的幂次方
    n2 = 1
    while n2 < n:
        n2 *= 2
    signal = np.append(signal, np.zeros(n2 - n))

    # 进行FFT变换
    spectrum = np.fft.fft(signal)

    # 对FFT结果进行处理
    spectrum[1:n] *= 2
    spectrum[n:] = 0
    spectrum[0] = 1
    spectrum[n] = 0

    # 反向FFT变换
    hilbert = np.real(np.fft.ifft(spectrum))
    hilbert = hilbert[:n]

    return hilbert

if __name__ == "__main__":
    signal = [1, 2, 3, 4, 5, 6, 7, 8]
    hilbert = hilbert_transform(signal)

    # 输出结果
    print(hilbert)