Welch method

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In physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating power spectra.

The Welch method, based on Barlett's procedure, splits a set of data into smaller sets of data and calculates the modified periodogram (the power spectrum) of each set, which produces an array of power measurements vs. frequency "bin".

The modified periodogram is calculated by applying a window function to the time-domain data, computing the discrete Fourier transform, and then computing the squared magnitude of the result.
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.

The individual periodograms are then time-averaged, which reduces the variance of the individual power measurements.

This method is used by MATLAB's pwelch command to calculate spectral density.

References

"The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms", IEEE Transactions on Audio Electroacoustics, Volume AU-15 (June 1967), pages 70-73.

Oppenheim, A.V., and R.W. Schafer, Digital Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1975, pp 548-554.

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