In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, and applied mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Welch's method, named after P.D. Welch, is used for estimating the power

Electric power

Electric power is the rate at which electric energy is transferred by an electric circuit. The SI unit of power is the watt.-Circuits:Electric power, like mechanical power, is represented by the letter P in electrical equations...

 of a signal

Signal (electrical engineering)

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

 at different frequencies

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

: that is, is is an approach to spectral density estimation

Spectral density estimation

In statistical signal processing, the goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal...

. The method is based on the concept of using periodogram

Periodogram

The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...

 spectrum estimates, which are the result of converting a signal from the time domain to the frequency domain. Welch's method is an improvement on the standard periodogram

Periodogram

The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...

 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:

1. 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.
   1. If D = M / 2, the overlap is said to be 50%
   2. If D = 0, the overlap is said to be 0%. This is the same situation as in the Bartlett's method.
2. 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).
   1. Most window function
      Window function
      In signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
      
      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).
   2. The windowing of the segments is what makes the Welch method a "modified" periodogram
      Periodogram
      The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...
      
      .

After doing the above, the periodogram

Periodogram

The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...

 is calculated by computing the discrete Fourier transform

Discrete Fourier transform

In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...

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

Periodogram

The periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898 as in the following quote:...

s are then time-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
  Modified discrete cosine transform
  The modified discrete cosine transform is a Fourier-related transform based on the type-IV discrete cosine transform , with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset,...
  
  
• Short-time Fourier transform
  Short-time Fourier transform
  The short-time Fourier transform , or alternatively short-term Fourier transform, 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....