Welch's method, named after
Peter D. Welch, is an approach for
spectral density estimation.
It is used in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
,
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, and applied
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
for estimating the
power of a
signal
A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology.
In ...
at different
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
.
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 mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
. Welch's method is an improvement on the standard
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 estimating method and on
Bartlett's method, in that it reduces noise in the estimated
power spectra 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. Typically, window functions are symmetric around ...
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"
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 ...
.
After doing the above, 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 ...
is calculated by computing 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 ...
, and then computing the squared magnitude of the result, yielding power spectrum estimates for each segment. The individual
spectrum estimates 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: it is designed to be performed on consecutive blocks of a larger dataset, where s ...
*
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 divide ...
See also
*
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). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
*
Power spectrum
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...
*
Spectral density estimation
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