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

digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner ar ...

method to transform

data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

from the

time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the ...

to the

frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a ...

. It was originally programmed around 1953 by

James Cooley James William Cooley (1926 – June 29, 2016) was an American mathematician. Cooley received a B.A. degree in 1949 from Manhattan College, Bronx, NY, an M.A. degree in 1951 from Columbia University, New York, NY, and a Ph.D. degree in 1961 in ...

for

John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

as a way to get "good smoothed statistical estimates of power spectra without requiring large

Fourier transforms A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

." It was published by Ralph Beebe Blackman and John Tukey in 1958.

Background

Transformation

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, transformation from the

to another

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...

, such as the

, is used to focus on the details of a

waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electron ...

. Many of the waveform's details can be analyzed much easier in another domain than the original. Different methods exist to do transformation from time domain to frequency domain; the most prominent is the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

, which the Blackman–Tukey method uses. Prior to the advent of fast computers and the 1965 rediscovery of the

fast Fourier Transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...

, the large number of computations necessary for the

discrete Fourier Transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

motivated researchers to reduce the number of calculations required, resulting in the (now obsolete) Blackman–Tukey method based on the Wiener-Khinchin theorem.

Statistical estimation

Statistical estimation Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...

is used to determine the

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

(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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

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 In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies t ...

'' process, that changes with time and probability gets involved with it, and it's not always periodic at all portions of time.

Blackman–Tukey transformation method

The method is fully described in

Blackman Blackman, Originally ''Blæcmann'', is an Old English name meaning "''dark man''". The name was once given to Danish Vikings who settled in southern Scotland. It is also listed in the genealogy of the kings of Bernicia. Early on, it was commonly u ...

and

Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

's 1958 journal publications republished as their 1959 book "The measurement of power spectra, from the point of view of communications engineering" and is outlined by the following procedures: # Calculate the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

function with the data # Apply a suitable

window function In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the in ...

, and finally # Compute a

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

(now done with

FFT A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...

) of the data to obtain the power density spectrum Autocorrelation makes the wave smoothed rather than averaging several waveforms. This function is set to window, the corresponding waveform toward its extremes. Computation gets faster if more data is

correlated In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statisti ...

and if memory capacity of the system increases then overlap save sectioning technique would be applied. If the autocorrelation function in Blackman–Tukey is computed using FFT, then it will name fast correlation method for

spectral estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the s ...

References

External links

* * * * * * * * Electrical engineering {{electronics-stub