Welch Periodogram — pwelch • oce

Compute periodogram using the Welch (1967) method. This is somewhat analogous to the Matlab function of the same name, but it is not intended as a drop-in replacement.

Usage

pwelch(
  x,
  window,
  noverlap,
  nfft,
  fs,
  spec,
  demean = FALSE,
  detrend = TRUE,
  plot = TRUE,
  debug = getOption("oceDebug"),
  ...
)

Arguments

x: a vector or timeseries to be analyzed. If x is a timeseries, then it there is no need to fs, and doing so will result in an error if it does not match the value inferred from x.
window: optional numeric vector specifying a window to be applied to the timeseries subsamples. This is ignored if spec is provided. Otherwise, if window is provided, then it must either be of the same length as nfft or be of length 1. In the first case, the vector is multiplied into the timeseries subsample, and the length of window must equal nfft is that is supplied. In the second then window is taken to be the number of sub-intervals into which the time series is to be broken up, with a hamming window being used for each sub-interval. If window is not specified and nfft is given, then the window is constructed as a hamming window with length nfft. And, if neither window nor nfft are specified, then x will be broken up into 8 portions.
noverlap: number of points to overlap between windows. If not specified, this will be set to half the window length.
nfft: length of FFT. See window for how nfft interacts with that argument.
fs: frequency of time-series. If x is a time-series, and if fs is supplied, then time-series is altered to have frequency fs.
spec: optional function to be used for the computation of the spectrum, to allow finer-grained control of the processing. If provided, spec must accept a time-series as its first argument, and must return a list containing the spectrum in spec and the frequency in freq. Note that no window will be applied to the data after subsampling, and an error will be reported if window and spec are both given. An error will be reported if spec is given but nfft is not given. Note that the values of demean, detrend and plot are ignored if spec is given. However, the ... argument is passed to spec.
demean, detrend: logical values that can control the spectrum calculation, in the default case of spec. These are passed to spectrum() and thence spec.pgram(); see the help pages for the latter for an explanation.
plot: logical, set to TRUE to plot the spectrum.
debug: a flag that turns on debugging. Set to 1 to get a moderate amount of debugging information, or to 2 to get more.
...: optional extra arguments to be passed to spectrum(), or to spec, if the latter is given.

Value

pwelch returns a list mimicking the return value from spectrum(), containing frequency freq, spectral power spec, degrees of freedom df, bandwidth bandwidth, etc.

Details

First, x is broken up into chunks, overlapping as specified by noverlap. These chunks are then multiplied by the window, and then passed to spectrum(). The resulting spectra are then averaged, with the results being stored in spec of the return value. Other entries of the return value mimic those returned by spectrum().

It should be noted that the actions of several parameters are interlocked, so this can be a complex function to use. For example, if window is given and has length exceeding 1, then its length must equal nfft, if the latter is also provided.

Bugs

Both bandwidth and degrees of freedom are just copied from the values for one of the chunk spectra, and are thus incorrect. That means the cross indicated on the graph is also incorrect.

Historical notes

2021-06-26: Until this date, pwelch() passed the subsampled timeseries portions through detrend() before applying the window. This practice was dropped because it could lead to over-estimates of low frequency energy (as noticed by Holger Foysi of the University of Siegen), perhaps because detrend() considers only endpoints and therefore can yield inaccurate trend estimates. In a related change, demean and detrend were added as formal arguments, to avoid users having to trace the documentation for spectrum() and then spec.pgram(), to learn how to remove means and trends from data. For more control, the spec argument was added to let users sidestep spectrum() entirely, by providing their own spectral computation functions.

References

Welch, P. D., 1967. 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, AU-15, 70–73.

Author

Dan Kelley

Examples

library(oce)
Fs <- 1000
t <- seq(0, 0.296, 1 / Fs)
x <- cos(2 * pi * t * 200) + rnorm(n = length(t))
X <- ts(x, frequency = Fs)
s <- spectrum(X, spans = c(3, 2), main = "random + 200 Hz", log = "no")
w <- pwelch(X, plot = FALSE)
lines(w$freq, w$spec, col = "red")
w2 <- pwelch(X, nfft = 75, plot = FALSE)
lines(w2$freq, w2$spec, col = "green")
abline(v = 200, col = "blue", lty = "dotted")

cat("Checking spectral levels with Parseval's theorem:\n")
#> Checking spectral levels with Parseval's theorem:
cat("var(x)                              = ", var(x), "\n")
#> var(x)                              =  1.560333 
cat("2 * sum(s$spec) * diff(s$freq[1:2]) = ", 2 * sum(s$spec) * diff(s$freq[1:2]), "\n")
#> 2 * sum(s$spec) * diff(s$freq[1:2]) =  1.551254 
cat("sum(w$spec) * diff(s$freq[1:2])     = ", sum(w$spec) * diff(w$freq[1:2]), "\n")
#> sum(w$spec) * diff(s$freq[1:2])     =  0.8124954 
cat("sum(w2$spec) * diff(s$freq[1:2])    = ", sum(w2$spec) * diff(w2$freq[1:2]), "\n")
#> sum(w2$spec) * diff(s$freq[1:2])    =  0.8130717 
# co2
par(mar = c(3, 3, 2, 1), mgp = c(2, 0.7, 0))
s <- spectrum(co2, plot = FALSE)
plot(log10(s$freq), s$spec * s$freq,
    xlab = expression(log[10] * Frequency), ylab = "Power*Frequency", type = "l"
)
title("Variance-preserving spectrum")
pw <- pwelch(co2, nfft = 256, plot = FALSE)
lines(log10(pw$freq), pw$spec * pw$freq, col = "red")