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2016 | OriginalPaper | Chapter

10. Non-Parametric Spectral Methods

Author : Silvia Maria Alessio

Published in: Digital Signal Processing and Spectral Analysis for Scientists

Publisher: Springer International Publishing

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Abstract

This chapter deals with obtaining a good estimate of the power spectrum of a random signal on the basis of a finite number of samples of a typical realization of the underlying random process—one among the infinite sequences that the process can generate when we measure it. The simplest approach to spectral estimation, i.e., the periodogram, turns out to perform poorly: the variance of the estimate is high and does not decrease with increasing length of the data record—it is not a consistent estimate of the power spectrum. The search for a stable and consistent spectral estimate leads to the methods of Bartlett and Welch, and to the Blackman-Tukey method. We will also present statistical tests used judge the significance of any peak detected in a spectrum. A description of the multitaper method (MTM) and a brief account of the estimation of the cross-spectrum of two random signals will be followed by a discussion about the use of FFT for practical computation of spectral estimates and about the different normalization schemes adopted in literature for the power spectrum.

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previous chapter Statistical Approach to Signal Analysis
next chapter Parametric Spectral Methods
Footnotes
1
In this frame we use the two terms indifferently, even if the triangular window and the Bartlett window are actually two different sequences: the former has no zeros at its edges, while the latter includes two zero samples at the beginning and end. In the present case, there must be no zeros at the edges; therefore in principle we should call this a “triangular window”.
 
2
Note that the terms “input” and “output” here refer to a power spectrum estimation algorithm: a signal having a given \(\mathrm {SNR}_{\mathrm {i}}\) enters the algorithm; in the spectral estimate, the signal-to-noise ratio is modified by a factor depending on the shape and length of the taper and is transformed into \(\mathrm {SNR}_{\mathrm {o}}\). Looking at the latter, we can evaluate the capability of the estimator to detect the imprint of the sinusoid/complex exponential buried in noise.
 
3
For \(N\rightarrow \infty \) the spectral window would become a periodic impulse train, i.e., a train of Dirac \(\delta \), but for a finite N the bias can vanish only asymptotically.
 
4
Note that this choice for defining resolution is coherent with what we stated about the periodogram: there we talked about the possibility of defining resolution through the bandwith of the spectral window \(W_{B,2N-1}\) at 6 dB, but since \(w_{B, 2N-1}[n]=(1/N) w_{R,N}[n] *w_{R,N}[n]\), so that \(W_{B,2N-1}(\mathrm {e}^{\mathrm {j}\omega })\propto \left| W_R(\mathrm {e}^{\mathrm {j}\omega })_{B,N}\right| ^2\), this is equivalent to using for the periodogram the bandwidth at 3 dB for \(W_R(\mathrm {e}^{\mathrm {j}\omega })\).
 
5
Recall that here we are speaking of non-causal windows, having a real transform. Note that Fig. 5.​5 shows transform squared magnitudes and therefore does not allow to verify the above statement.
 
6
The convolution with a Dirac \(\delta \) would leave the function unaltered.
 
7
It may be more intuitive to refer to the discrete trigonometric expansion of a signal (Sect. 3.​7): the two Gaussian variables in this case are the expansion coefficients \(a_k\) and \(b_k\); the former is related to a cosine of \(\omega _k n\) and the latter to a sine, and sine and cosine are mutually in phase quadrature. Therefore \(a_k^2+b_k^2= \left| X[k]\right| ^2\) has a \(\chi ^2\) distribution with \(\nu =2\). Note that this conclusion is independent of the sequence length, because the sinusoidal sequences used as the expansion basis are infinitely long.
 
8
The data is seasonal climatic anomalies of sea surface temperature (SST) in \(^{\circ }\)C, spatially averaged over the so-called NINO3 region of the Pacific Ocean. This region is comprised between latitudes of 50\(^{\circ }\) South and 50\(^{\circ }\) North and between longitudes of 90\(^{\circ }\) West and 150\(^{\circ }\) West, and is particularly representative for the study of climatic variability at interannual time scale, and more precisely for the study of the El Niño—La Niña phenomena. The data used here spans the years 1871–2012 and were drawn from monthly NINO3 data available at http://​www.​esrl.​noaa.​gov/​psd/​gcos_​wgsp/​Timeseries/​Data/​nino3.​long.​data.
The term “climatic anomalies” means that the data has been centered by subtracting from each seasonal sample the mean value of the corresponding season, computed over the whole record; for instance: spring of 1900 minus average over all springs. This makes the data free from the annual temperature cycle, the dominant presence of which in the spectrum would obscure weaker cyclicities.
 
9
The Rayleigh-Ritz method is a classical variational method for finding approximate solutions of differential equations, whose exact solutions are hard to find. The method was first used by Lord Rayleigh in 1870 to solve the vibration problem of organ pipes closed on one end and open at the other. However, the approach did not receive much recognition by the scientific community. Nearly 40 years later, due to the publication of two papers by Ritz, the method came to be called the Ritz method. To recognize the contributions of both authors, the theory was later renamed the Rayleigh-Ritz method.
 
10
The median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. The median smoothing filter computes the median of PSD values falling inside a small frequency bin centered on a given frequency and assigns the result to this central frequency.
 
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Metadata
Title
Non-Parametric Spectral Methods
Author
Silvia Maria Alessio
Copyright Year
2016
Book
Digital Signal Processing and Spectral Analysis for Scientists

Print ISBN: 978-3-319-25466-1

Electronic ISBN: 978-3-319-25468-5

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-25468-5_10