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6 - The Power Spectrum

Published online by Cambridge University Press:  03 February 2022

Timothy DelSole
Affiliation:
George Mason University, Virginia
Michael Tippett
Affiliation:
Columbia University, New York
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Summary

This chapter introduces the power spectrum. The power spectrum is the Fourier Transform of the autocovariance function, and the autocovariance function is the (inverse) Fourier Transform of the power spectrum. As such, the power spectrum and autocovariance function offer two complementary but mathematically equivalent descriptions of a stochastic process. The power spectrum quantifies how variance is distributed over frequencies and is useful for identifying periodic behavior in time series. The discrete Fourier transform of a time series can be summarized in a periodogram, which provides a starting point for estimating power spectra. Estimation of the power spectrum can be counterintuitive because the uncertainty in periodogram elements does not decrease with increasing sample size. To reduce uncertainty, periodogram estimates are averaged over a frequency interval called the bandwidth. Trends and discontinuities in time series can lead to similar low-frequency structure despite very different temporal characteristics. Spectral analysis provides a particularly insightful way to understand the behavior of linear filters.

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Publisher: Cambridge University Press
Print publication year: 2022

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  • The Power Spectrum
  • Timothy DelSole, George Mason University, Virginia, Michael Tippett, Columbia University, New York
  • Book: Statistical Methods for Climate Scientists
  • Online publication: 03 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781108659055.007
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  • The Power Spectrum
  • Timothy DelSole, George Mason University, Virginia, Michael Tippett, Columbia University, New York
  • Book: Statistical Methods for Climate Scientists
  • Online publication: 03 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781108659055.007
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Power Spectrum
  • Timothy DelSole, George Mason University, Virginia, Michael Tippett, Columbia University, New York
  • Book: Statistical Methods for Climate Scientists
  • Online publication: 03 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781108659055.007
Available formats
×