 |
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
|
The Blackman–Tukey transformation (orBlackman–Tukey method) is a digital signal processing method to transform data from the time domain to the frequency domain. It was originally programmed around 1953 by James Cooley for John TukeyatJohn von Neumann's Institute for Advanced Study as a way to get "good smoothed statistical estimatesofpower spectra without requiring large Fourier transforms."[1] It was published by Ralph Beebe Blackman and John Tukey in 1958.
Insignal processing, transformation from the time domain to another domain, such as the frequency domain, is used to focus on the details of a waveform. Many of the waveform's details can be analyzed much more easily in a domain other than the original. Different methods exist to do transformation from time domain to frequency domain; the most prominent is the Fourier transform, which the Blackman–Tukey method uses. Prior to the advent of fast computers and the 1965 rediscovery of the fast Fourier transform, the large number of computations necessary for the 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 is used to determine the expected value(s) of statistical expected values of statistical quantities. Statistical estimation also tries to find the expected values. The expected values are those values that we expect among the random values, derived from 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 ergodic process, that changes with time and probability gets involved with it, and it's not always periodic at all portions of time.[clarification needed]
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:
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 correlated and if memory capacity of the system increases then overlap save sectioning technique would be applied.[clarification needed] If the autocorrelation function in Blackman–Tukey is computed using FFT, then it will name fast correlation method for spectral estimation.[clarification needed]
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.
Prior to the advent of the 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.
