Spectral analysis¶

Spectral signal analysis is the field concerned with the estimation of the distribution of a signal (more exactly its power) in the frequency domain.

The Power Spectrum Density (PSD) S_X of X_n is defined as the squared discrete time Fourier transform of X_n

$\forall f \in \mathbb{R}, \qquad S_X = \left|{\sum_n{X_n e^{-2\pi j f n}}}\right|^2$

Under suitable technical conditions, this is equal to the discrete time Fourier transform of the autocorrelation function for stationary signals in the wide sense:

$\forall f \in \mathbb{R}, \qquad S_X = \sum_n{\gamma(n) e^{-2\pi j f n}}$

This is called the power spectrum density because integrating it over the frequency domain gives you the average power of X and because it can be proved that S_X is always positive for any f.

Spectral density estimation¶

Since in practice, we only have finite signals, we need method to estimate the PSD. talkbox implements various methods for PSD estimation, which are classified in two broad classes:

non-parametric (general methods, estimate the PSD directly from the signal)

parametric (use an underlying signal model, for example AR; is less general, but generally more efficient in some sense if applicable).

Non-parametric estimation¶

Periodogram¶

The raw periodogram is a relatively straightforward estimator of the PSD. The raw periodogram I of a signal of length N is defined as:

$I(f) \triangleq \frac{{|\sum_n{x[n] e^{-2\pi j k f/f_s}}|}^2}{fs N}$

where f_s is the sampling rate. In practice, the periodogram can only be computed on a frequency grid; the most commonly used grid is k/N (normalized frequencies) for k in [0, ..., N-1]. With this grid, the sum becomes a simple DFT which can be computed using the FFT.

Examples¶

As a first example, let’s generate a simple sinusoid signal embedded into white noise:

import numpy as np
import matplotlib.pyplot as plt
from scikits.talkbox.spectral.basic import periodogram
fs = 1000
x = np.sin(2 * np.pi * 0.15 * fs * np.linspace(0., 0.3, 0.3 * fs))
x += 0.1 * np.random.randn(x.size)
px, fx = periodogram(x, nfft=16384, fs=fs)
plt.plot(fx, 10 * np.log10(px))

Plotting the log periodogram then gives:

The number of points used for the FFT has been set high to highlight the lobe, artefact of the rectangular window.

scikits.talkbox.spectral.basic.periodogram(x, nfft=None, fs=1)¶

Compute the periodogram of the given signal, with the given fft size.

Parameters:

x : array-like

input signal

nfft : int

size of the fft to compute the periodogram. If None (default), the length of the signal is used. if nfft > n, the signal is 0 padded.

fs : float

Sampling rate. By default, is 1 (normalized frequency. e.g. 0.5 is the Nyquist limit).

Returns:

pxx : array-like

The psd estimate.

fgrid : array-like

Frequency grid over which the periodogram was estimated.

Notes

Only real signals supported for now.

Returns the one-sided version of the periodogram.

Discrepency with matlab: matlab compute the psd in unit of power / radian / sample, and we compute the psd in unit of power / sample: to get the same result as matlab, just multiply the result from talkbox by 2pi

Examples

Generate a signal with two sinusoids, and compute its periodogram:

>>> fs = 1000
>>> x = np.sin(2 * np.pi  * 0.1 * fs * np.linspace(0, 0.5, 0.5*fs))
>>> x += np.sin(2 * np.pi  * 0.2 * fs * np.linspace(0, 0.5, 0.5*fs))
>>> px, fx = periodogram(x, 512, fs)

Parametric estimation¶

ar method¶

TODO: To be implemented

Useful complements¶

A random signal $X$ is said to be (strictly) stationary if its distribution does not depend on the time. For time-discrete signals $X_n$ of distribution $F_{X_n}$ , this is written:

$\forall n, k, \qquad F_{X_n} = F_{X_{n+k}}$

Other stationary concepts can be defined, when only some moments of the signal do not depend on time. A widely used concept is weakly stationary signals. A signal is said to be weakly stationary (or stationary in the wide sense) if its means and covariance do not depend on time, and its covariance function $\gamma(n, k)$ depends only on the time difference n-k:

$\forall n, k, \qquad \gamma(n, k) = \mathbb{E}[(X[n]-m)(X[k] -m)] = \gamma(n-k)$

Spectral analysis¶

Spectral density estimation¶