信号分析

フーリエ変換の関数。便利。

from scipy.fftpack import fft
 
def get_fft_values(y_values, T, N, f_s):
    f_values = np.linspace(0.0, 1.0/(2.0*T), N//2)
    fft_values_ = fft(y_values)
    fft_values = 2.0/N * np.abs(fft_values_[0:N//2])
    return f_values, fft_values

Since our signal is sampled at a rate f_s of 100 Hz, the FFT will return the frequency spectrum up to a frequency of f_s / 2 = 50 Hz. The higher your sampling rate is, the higher the maximum frequency is FFT can calculate.

In the get_fft_values function above, the scipy.fftpack.fft function returns a vector of complex valued frequencies. Since they are complex valued, they will contain a real and an imaginary part. The real part of the complex value corresponds with the magnitude, and the imaginary part with the phase of the signal. Since we are only interested in the magnitude of the amplitudes, we use np.abs() to take the real part of the frequency spectrum.

The FFT of an input signal of N points, will return an vector of N points. The first half of this vector (N/2 points) contain the useful values of the frequency spectrum from 0 Hz up to the Nyquist frequency of f_s / 2. The second half contains the complex conjugate and can be disregarded since it does not provide any useful information.

おまけ

以下の２つの手法とフーリエ変換を用いてピーク値を利用して、feature extract する。その後、機械学習に入れる。

こうすることで、位相に関係なく repeated value の 特徴が抽出できる。

from scipy.signal import welch
 
# Power Spectral Density.

def get_psd_values(y_values, T, N, f_s):
    f_values, psd_values = welch(y_values, fs=f_s)
    return f_values, psd_values

# 自己相関
def autocorr(x):
    result = np.correlate(x, x, mode='full')
    return result[len(result)//2:]
 
def get_autocorr_values(y_values, T, N, f_s):
    autocorr_values = autocorr(y_values)
    x_values = np.array([T * jj for jj in range(0, N)])
    return x_values, autocorr_values

http://ataspinar.com/2018/12/21/a-guide-for-using-the-wavelet-transform-in-machine-learning/