Blackman–Tukey transformation

The Blackman–Tukey transformation (or Blackman–Tukey method) is a

power spectra without requiring large Fourier transforms."^[1] It was published by Ralph Beebe Blackman

and John Tukey in 1958.

Background

Transformation

In

discrete Fourier Transform motivated researchers to reduce the number of calculations required, resulting in the (now obsolete) Blackman–Tukey method based on the Wiener-Khinchin theorem.^[2]

Statistical estimation

samples of the population in probability (group of subset). In time series analysis, discrete

data obtained as a function of time is usually the only type of data available, instead of samples of population or group of subsets taken simultaneously.

Difficulty is commonly avoided using an

periodic at all portions of time.^{[clarification needed}

]

Blackman–Tukey transformation method

The method is fully described in Blackman and Tukey's 1958 journal publications republished as their 1959 book "The measurement of power spectra, from the point of view of communications engineering"^[3] and is outlined by the following procedures:

Calculate the autocorrelation function with the data

Apply a suitable window function, and finally

Compute a discrete Fourier transform (now done with FFT) of the data to obtain the power density spectrum

Autocorrelation makes the wave smoothed rather than averaging several waveforms.^{[clarification needed]} This function is set to window, the corresponding waveform toward its extremes.^{[clarification needed]} Computation gets faster if more data is
spectral estimation.^{[clarification needed}
]

References

^ Cooley, James. "The Re-Discovery of the Fast Fourier Transform Algorithm" (PDF). web.cs.dal.ca. Archived from the original (PDF) on 2012-12-24. However, we had a previous collaboration in 1953 when Tukey was a consultant at John Von Neuman's computer project at the Institute for Advanced Study in Princeton, New Jersey, where I was a programmer. I programmed for him what later became the very popular Blackman-Tukey method of spectral analysis [5]. The important feature of this method was that it gave good smoothed statistical estimates of power spectra without requiring large Fourier transforms. Thus, our two collaborations were first on a method for avoiding large Fourier transforms since they were so costly and then a method for reducing the cost of the Fourier transforms.

FFT
and fast computers, power density spectral estimation was almost never done as described in the last section. Rather the onerous computational load led scientists, as far as possible, to reduce the number of calculations required. The so-called Blackman-Tukey method... ... The Blackman-Tukey estimate is based upon ... and the choice of suitable window weights...A large literature grew up devoted to the window choice. Again, one trades bias against variance through the value M, which one prefers greatly to minimize. The method is now obsolete because the ability to generate the Fourier coefficients directly permits much greater control over the result. The bias discussion of the Blackman-Tukey method is particularly tricky, as is the determination of ν. Use of the method should be avoided except under those exceptional circumstances when for some reason only R~(τ) is known.

^ Blackman, R. B.; Tukey, J. W. (1958). The Measurement of Power Spectra, from the point of view of Communications Engineering (new Dover 1959 edition, an unabridged and corrected republication of Part I and Part II of "The Measurement of Power Spectra from the Point of View of Communications Engineering" which originally appeared in the January 1958 and March 1958 issues of Volume XXXVII of the Bell System Technical Journal. ed.). American Telephone and Telegraph Company. Retrieved 2022-04-11.

External links

"Blackman–Tukey Correlogram and Cross-Spectrum". spectraworks.com. Retrieved 2014-08-25.

David Meko (10 February 2013). "Spectral Analysis -- Smoothed Periodogram Method" (PDF). Archived from the original (PDF) on 17 September 2003. Retrieved 2014-08-25.

"random – What is the distinction between ergodic and stationary? – Signal Processing Stack Exchange". dsp.stackexchange.com. Retrieved 2014-08-25.

Nick Kingsbury (31 October 2005). "Connexions module: m11103 | Ergodicity" (PDF). Retrieved 2014-08-25.

Donna Williams (15 January 2004). "Understanding FFT Windows" (PDF). Archived from the original (PDF) on 13 February 2015. Retrieved 2014-08-25.

Peter Cheung (20 February 2011). "Signal Transmission through LTI Systems" (PDF). Retrieved 2014-08-25.

"Windowing effect on spectral leakage and phase – Newsreader – MATLAB Central".
Mathworks MATLAB Newsgroup. 2009-11-18. Archived from the original
on 2016-03-04. Retrieved 2014-08-25.

Santhanam, Balu. "Ergodic Processes" (PDF). Archived from the original (PDF) on 2012-01-06.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Blackman–Tukey_transformation&oldid=1195721468"

[1] Cooley, James. "The Re-Discovery of the Fast Fourier Transform Algorithm" (PDF). web.cs.dal.ca. Archived from the original (PDF) on 2012-12-24. However, we had a previous collaboration in 1953 when Tukey was a consultant at John Von Neuman's computer project at the Institute for Advanced Study in Princeton, New Jersey, where I was a programmer. I programmed for him what later became the very popular Blackman-Tukey method of spectral analysis [5]. The important feature of this method was that it gave good smoothed statistical estimates of power spectra without requiring large Fourier transforms. Thus, our two collaborations were first on a method for avoiding large Fourier transforms since they were so costly and then a method for reducing the cost of the Fourier transforms.

[2] FFT
and fast computers, power density spectral estimation was almost never done as described in the last section. Rather the onerous computational load led scientists, as far as possible, to reduce the number of calculations required. The so-called Blackman-Tukey method... ... The Blackman-Tukey estimate is based upon ... and the choice of suitable window weights...A large literature grew up devoted to the window choice. Again, one trades bias against variance through the value M, which one prefers greatly to minimize. The method is now obsolete because the ability to generate the Fourier coefficients directly permits much greater control over the result. The bias discussion of the Blackman-Tukey method is particularly tricky, as is the determination of ν. Use of the method should be avoided except under those exceptional circumstances when for some reason only R~(τ) is known.

[3] Blackman, R. B.; Tukey, J. W. (1958). The Measurement of Power Spectra, from the point of view of Communications Engineering (new Dover 1959 edition, an unabridged and corrected republication of Part I and Part II of "The Measurement of Power Spectra from the Point of View of Communications Engineering" which originally appeared in the January 1958 and March 1958 issues of Volume XXXVII of the Bell System Technical Journal. ed.). American Telephone and Telegraph Company. Retrieved 2022-04-11.

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