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2022 | OriginalPaper | Chapter

3. Multitaper Spectral Estimation

Author : Jonathan Kirby

Published in: Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere

Publisher: Springer International Publishing

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Abstract

With discretely sampled and truncated data, a power spectrum computed simply by squaring and summing the real and imaginary components of the Fourier transform provides a biased estimate of the true power spectrum. And while windowing with a single taper decreases spectral leakage, it does not reduce the variance of the spectrum. This chapter introduces a class of multiple tapers called discrete prolate spheroidal sequences, which are designed to reduce leakage by solving the so-called concentration problem. Furthermore, they form an orthogonal set meaning that their spectrum estimates are independent, allowing for averaging and an associated reduction of variance.

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previous chapter The Fourier Transform

next chapter The Continuous Wavelet Transform

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A more detailed proof can be found in Percival and Walden (1993).

For those readers unsure of, or rusty on, eigenvectors and eigenvalues, see the Appendix to this chapter.

A Toeplitz matrix is a matrix in which the values of each descending diagonal from left to right are constant.

The concept of orthogonality is expanded upon in the Appendix to this chapter.

The Kronecker delta, \(\delta _{jk}\), is very similar to the Dirac delta function; it has the value 1 if \(j=k\), and 0 when \(j\ne k\).

Strictly speaking, W is the half-bandwidth, as the total bandwidth extends from \(-W\) to \(+W\). Nevertheless, W is typically referred to as the ‘bandwidth’. Note also that the eigenfunctions, \(U_k\), can actually have multiple ‘central lobes’, as shown in the right-hand panels of Fig. 3.3.

Note that, from Eq. 3.18, we have \(|U_k(f)|^2 = |V_k(f)|^2\), so that Fig. 3.3 is also showing the spectra of the eigenfunctions.

A signal with a ‘red’ power spectrum has high power in its long-wavelength harmonics, and low power in its short-wavelength harmonics. Fractally distributed data are examples of this. Taking this further, ‘blue’ spectra possess low-power long-wavelength harmonics and high-power short-wavelength harmonics; ‘white noise’ has equal power at all wavelengths.

Other schemes are possible, such as annuli of equal area, or exponentially decreasing width.

That is not to say that non-orthogonal coordinate systems are never useful; while having limited applicability, they are sometimes very useful when solving certain problems.

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Title: Multitaper Spectral Estimation
Author: Jonathan Kirby
Publisher: Springer International Publishing
Book: Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere
Print ISBN: 978-3-031-10860-0

Electronic ISBN: 978-3-031-10861-7

Copyright Year: 2022
DOI: https://doi.org/10.1007/978-3-031-10861-7_3