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.
pwelch(
x,
window,
noverlap,
nfft,
fs,
spec,
demean = FALSE,
detrend = TRUE,
plot = TRUE,
debug = getOption("oceDebug"),
...
)
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
.
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.
number of points to overlap between windows. If not specified, this will be set to half the window length.
length of FFT. See window
for how nfft
interacts with
that argument.
frequency of time-series. If x
is a time-series, and if
fs
is supplied, then time-series is altered to have frequency
fs
.
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
.
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.
logical, set to TRUE
to plot the spectrum.
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.
pwelch
returns a list mimicking the return value from spectrum()
,
containing frequency freq
, spectral power spec
, degrees of
freedom df
, bandwidth bandwidth
, etc.
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.
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.
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.
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.
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) = 1.399508
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]) = 1.452406
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]) = 0.6915331
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]) = 0.7083285
# 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")