How to implement Halbert transform using Python?

1. What is the Hilbert transform?

The Hilbert transform was initially only defined for periodic functions (that is, functions on a circle). In this case, it is the same as the Hilbert transform. Special kernel convolution. More commonly, however, for functions defined on the real straight line R (the boundary of the upper half-plane), the Hilbert transform is convolved with a Cauchy kernel. The Hilbert transform is closely related to the Parley-Wiener theorem, which is another method that links the holomorphic function in the upper half-plane and the Fourier transform of the function on the real line. kind of result.

2. Implementation principles and code examples in VC

Hilbert transform can be implemented in VC through fast Fourier transform (FFT).

The following is a simple C code to implement the Hilbert transform, which requires the use of the standard library of C 11 and above. First we need to implement an FFT function, and then use the FFT function to implement the Hilbert transform.

#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;
}

In the above code, we first implement a fast Fourier transform function fft, and then use FFT to calculate the Hilbert transform in the hilbertTransform function. In the calculation process of the Hilbert transform, we first extended the length of the signal, then performed the FFT transform, then processed the FFT results according to the formula of the Hilbert transform, and finally performed the inverse FFT transform to obtain The final Hilbert transform result.

In the above code, we use the complex type complex and the vector type vector to conveniently process signals and FFT results. In practical applications, we can read the input signal from a file or obtain it from real-time collected data, and then call the hilbertTransform function to perform Hilbert transformation to obtain the transformed signal.

3. Use Python code to implement

Hilbert transformation can also be easily implemented using Python. The following is a sample code that uses the numpy library to implement the Hilbert transform:

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)

In the above code, we first extend the input signal to a power length of 2, and then use the numpy.fft.fft function. FFT transformation, process the FFT result, and finally use the numpy.fft.ifft function to perform the reverse FFT transformation to obtain the Hilbert transform result.

It should be noted that since the results returned by the numpy.fft.fft function are arranged from small to large according to the frequency of the FFT transformation, and the Hilbert transformation is performed in the time domain, so we Certain processing of the FFT results is required to obtain the correct Hilbert transform results. In the above code, we perform a series of processing on the FFT results, including multiplying the amplitude of the non-zero frequency part by 2, setting the frequencies outside the non-zero frequency part to zero, and changing the values ​​of the DC component and Nyquist frequency component respectively. Set to 1 and 0 to get the correct Hilbert transform result.

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