Welch's method, named after Peter D. Welch, is an approach for spectral density estimation. It 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 ...

for estimating the

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

of a

signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...

at different

frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

. The method is based on the concept of using

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 ...

spectrum estimates, which are the result of converting a signal from the time domain 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 signa ...

. Welch's method is an improvement on the standard

spectrum estimating method and on Bartlett's method, in that it reduces noise in the estimated

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, ...

in exchange for reducing the frequency resolution. Due to the noise caused by imperfect and finite data, the noise reduction from Welch's method is often desired.

Definition and procedure

The Welch method is based on Bartlett's method and differs in two ways: # The signal is split up into overlapping segments: the original data segment is split up into L data segments of length M, overlapping by D points. ## If D = M / 2, the overlap is said to be 50% ## If D = 0, the overlap is said to be 0%. This is the same situation as in the Bartlett's method. # The overlapping segments are then windowed: After the data is split up into overlapping segments, the individual L data segments have a window applied to them (in the time domain). ## Most

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 inte ...

s afford more influence to the data at the center of the set than to data at the edges, which represents a loss of information. To mitigate that loss, the individual data sets are commonly overlapped in time (as in the above step). ## The windowing of the segments is what makes the Welch method a "modified"

. After doing the above, the

is calculated 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- ...

, and then computing the squared magnitude of the result. The individual

s are then averaged, which reduces the variance of the individual power measurements. The end result is an array of power measurements vs. frequency "bin".

Related approaches

Other overlapping windowed Fourier transforms include: *

Modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped transform, lapped: it is designed to be performed on consecutive blocks of a larger ...

Short-time Fourier transform The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divi ...

References

* * * {{Citation , last1 =Proakis , first1 =John G. , last2 =Manolakis , first2 =Dimitri G. , title =Digital Signal Processing: Principles, Algorithms and Applications , place =Upper Saddle River, NJ , publisher =Prentice-Hall , year =1996 , edition =3 , pages
910–913
, language =English , id =sAcfAQAAIAAJ , isbn =9780133942897 , url =https://archive.org/details/digitalsignalpro00proa/page/910 Frequency-domain analysis Digital signal processing Waves

Definition and procedure

Related approaches

See also

References