Welch's 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 concept of using periodogram 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 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. 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 functions 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. After doing the above, the periodogram is calculated by computing the discrete Fourier transform, and then computing the squared magnitude of the result. The individual periodograms are then averaged, which reduces the variance of the individual power measurements. The end result is an array of power measurements vs. frequency "bin".

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Spectral density 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 signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Welch's method

Bartlett's method

In time series analysis, Bartlett's method (also known as the method of averaged periodograms), is used for estimating power spectra. It provides a way to reduce the variance of the periodogram in exchange for a reduction of resolution, compared to standard periodograms. 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, engineering, and applied mathematics.