R: Welch periodogram

pwelch {oce} | R Documentation

## Welch periodogram

### Description

Compute periodogram using the Welch (1967) method. First, `x` is broken up into chunks, overlapping as specified by `noverlap`. These chunks are then detrended with `detrend()`, 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()`.

### Usage

    pwelch(
      x,
      window,
      noverlap,
      nfft,
      fs,
      spectrumtype,
      esttype,
      plot = TRUE,
      debug = getOption("oceDebug"),
      ...
    )

### Arguments

`x` | a vector or timeseries to be analyzed. If a timeseries, then there is no need to specify `fs`.
`window` | window specification, either a single value giving the number of windows to use, or a vector of window coefficients. If not specified, then 8 windows are used, each with a Hamming (raised half-cosine) window.
`noverlap` | number of points to overlap between windows. If not specified, this will be set to half the window length.
`nfft` | length of FFT. This cannot be given if `window` is given, and the latter is a single integer.
`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`.
`spectrumtype` | not used (yet)
`esttype` | not used (yet)
`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()`. Unless specified in this list, `spectrum()` is called with `plot=FALSE` to prevent plotting the separate spectra, and with `taper=0`, which is not needed with the default Hanning window. However, the other defaults of `spectrum()` are used, e.g. `detrend=TRUE`.

### Value

List mimicking the return value from `spectrum()`, containing frequency `freq`, spectral power `spec`, degrees of freedom `df`, bandwidth `bandwidth`, etc.

### 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.

### Author(s)

Dan Kelley

### 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.

### 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")
    cat("var(x)                              = ", var(x), "\n")
    cat("2 * sum(s$spec) * diff(s$freq[1:2]) = ", 2 * sum(s$spec) * diff(s$freq[1:2]), "\n")
    cat("sum(w$spec) * diff(s$freq[1:2])     = ", sum(w$spec) * diff(w$freq[1:2]), "\n")
    cat("sum(w2$spec) * diff(s$freq[1:2])    = ", sum(w2$spec) * diff(w2$freq[1:2]), "\n")
    ## 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')

[Package oce version 1.2-0 Index]