Power Spectrum Transform Preprocessor¶

This preprocessor computes the power spectrum of a given signal, using the 1D Discrete Fourier Transform (DFT). The power spectrum is the squared magnitude of the Fourier transform of the signal. We first compute the DFT with the following:

\[ X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i k n / N}, \quad k = 0, \ldots, N-1. \]

where \(N\) is the number of samples and \(k\) is the frequency index.

We can then compute the power spectrum as:

\[ P_k = |X_k|^2, \quad k = 0, \ldots, N-1. \]

The power spectrum is useful for identifying important features in the signal that may not be visible in the time domain.

The spectral density can be computed as:

\[ S_k = \frac{2}{f_s N} P_k, \quad k = 0, \ldots, N-1. \]

The spectral density is the power spectrum normalized by the number of samples and sampling frequency. The spectral density is useful for comparing signals with different sampling rates and number of samples.

where \(f_s\) is the sampling frequency.

Limitations¶

The input signal must be real-valued.

The transform is sensitive to noise and outliers.

Bases: Preprocess

Power Spectrum using a 1D DFT.

`call(signal, nfft=None, normfft='backward', where=lambda : not np.isnan(x))` ¶

Compute the power spectrum on the values in x. This uses a 1D DFT to compute the power spectrum. See autonfeat.preprocess.functional.dft for more information.

Parameters:

Name	Type	Description	Default
`signal`	`ndarray`	The array to compute the power spectrum of.	required
`nfft`	`Union[int, int_]`	The number of points to use for the FFT (1D DFT). If `None`, the length of `x` is used. Default is `None`.	`None`
`normfft`	`str`	The normalization mode to use when computng the FFT (1D DFT). Default is 'backward'. See `autonfeat.preprocess.functional.dft` for more information. Options include: 'backward': The backward transform is scaled by `1/n`. 'ortho': The forward and backward transforms are scaled by `1/sqrt(n)`. 'forward': The forward transform is not scaled.	`'backward'`
`where`	`Callable[[Union[int, float, int_, float_]], Union[bool, bool_]]`	A function that takes a value and returns `True` or `False`. Default is `lambda x: not np.isnan(x)` i.e. a measurement is valid if it is not a `NaN` value.	`lambda : not numpy.isnan(x)`

Returns:

Type	Description
`ndarray`	The power spectrum of `x`.

Examples¶

We define as signal as \(f(t) = 2 \sin(2 \pi t) + \sin(10 \cdot 2 \pi t)\) for \(t \in [1, 5]\) with a sampling rate of 100 samples per second. We then find the 1D DFT, power spectrum and spectral density of the signal.

Transform Signal¶

import numpy as np
from autonfeat.preprocess import DFTPreprocessor, PowerSpectrumPreprocessor

start_time = 1  # Start time in seconds
end_time = 5    # End time in seconds
sampling_rate = 100  # Number of samples per second
num_samples = int((end_time - start_time) * sampling_rate)

# Signal = 5 x sin(2 x pi x t) + sin(10 x 2 x pi x t)
time = np.linspace(start_time, end_time, num_samples)
freqs = np.fft.fftfreq(num_samples, 1 / sampling_rate)
signal = 5 * np.sin(2 * np.pi * time) + np.sin(10 * 2 * np.pi * time)

# Preprocess and transform signal
dft_preprocessor = DFTPreprocessor()
power_spectrum_preprocessor = PowerSpectrumPreprocessor()
freq_spectrum = dft_preprocessor(signal=signal)
power_spectrum = power_spectrum_preprocessor(signal=signal)
spectral_density = (2 / len(freqs)) * (power_spectrum ** 2)

Visualize Transform¶

We then visualize the signal, its Fourier transform, the power spectrum and the spectral density.

import matplotlib.pyplot as plt

# Plot results
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(10, 6))

# Plot signal
ax1.plot(time, signal)
ax1.set_xlabel("Time (s)")
ax1.set_ylabel("f(t)")
ax1.set_title("Signal")

# Plot frequency spectrum
ax2.plot(freqs, np.abs(freq_spectrum))
ax2.set_xlabel("Frequency (Hz)")
ax2.set_ylabel("| FFT(f(x)) |")
ax2.set_title("Frequency Spectrum")

# Plot power spectrum
ax3.plot(freqs, power_spectrum)
ax3.set_xlabel("Frequency (Hz)")
ax3.set_ylabel("Power")
ax3.set_title("Power Spectrum")

# Plot spectral density
ax4.plot(freqs, spectral_density)
ax4.set_xlabel("Frequency (Hz)")
ax4.set_ylabel("Spectral Density")
ax4.set_title("Spectral Density")

plt.tight_layout()
plt.show()

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