time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

, Bartlett's method (also known as the method of averaged periodograms), is used for estimating

power spectra The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...

. It provides a way to reduce the variance of the

periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most ...

in exchange for a reduction of resolution, compared to standard

s. A final estimate of the spectrum at a given frequency is obtained by averaging the estimates from the periodograms (at the same frequency) derived from non-overlapping portions of the original series. The method is used in

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

, and applied

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. Common applications of Bartlett's method are frequency response measurements and general spectrum analysis. The method is named after M. S. Bartlett who first proposed it.

Definition and procedure

Bartlett’s method consists of the following steps: # The original N point data segment is split up into K (non-overlapping) data segments, each of length M # For each segment, compute the

by computing the

discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...

(DFT version which does not divide by M), then computing the squared magnitude of the result and dividing this by M. # Average the result of the

s above for the K data segments. ## The averaging reduces the variance, compared to the original N point data segment. The end result is an array of power measurements vs. frequency "bin".

Related methods

*The

Welch method Welch's method, named after Peter D. Welch, is an approach for spectral density estimation. It is used in physics, engineering, and applied mathematics for estimating the power of a signal at different frequencies. The method is based on the co ...

: this is a method that uses a modified version of Bartlett’s method in which the portions of the series contributing to each periodogram are allowed to overlap. *Periodogram smoothing.

Definition and procedure

Related methods

References

Further reading