Welch: averaging to tame variance
Welch's method splits the record into (usually overlapping) segments, tapers each, computes a periodogram, and averages them. Averaging K approximately independent segments drops the variance by roughly 1/K while the estimate stays (asymptotically) unbiased apart from the window's leakage.
\hat{S}_W(f) = \frac{1}{K}\sum_{k=1}^{K}\frac{1}{U N_s}\left|\sum_{n=0}^{N_s-1} w_n\, x_n^{(k)}\, e^{-i2\pi f n}\right|^2, \qquad U=\tfrac{1}{N_s}\sum_n w_n^2Welch's averaged, windowed periodogram; U normalizes the window's power.
Welch tames the periodogram's noisiness by chopping the signal into overlapping windowed segments, computing a periodogram for each, and averaging. Averaging K noisy estimates cuts the variance by roughly K — trading frequency resolution for a smoother, more trustworthy spectrum.
- \hat{S}_W(f)
- Welch's averaged spectral estimate.
- K
- The number of segments averaged.
- w_n
- The window applied to each segment to reduce spectral leakage.
- U
- A normalizer for the window's power.
The resolution you buy and the resolution you lose
Welch's variance reduction is not free: chopping a T-second record into K segments of length T/K coarsens the frequency resolution to \Delta f \approx K/T. That is the bias–variance dial from Guide 1, now with a knob you can turn. For a fixed record you are choosing between one sharp-but-grassy spectrum and many smooth-but-blurry looks averaged together.
Multitaper: averaging without throwing away data
Welch discards resolution by shortening segments. Multitaper estimation, due to Thomson, keeps the full length T and instead averages several orthogonal tapers applied to the same segment. The tapers are the discrete prolate spheroidal sequences (DPSS, or Slepian sequences): the unique set of sequences maximally concentrated within a chosen frequency half-bandwidth W. Each taper yields a nearly independent 'eigenspectrum,' and their average is the multitaper estimate.
\hat{S}_{MT}(f) = \frac{1}{K}\sum_{k=0}^{K-1}\left|\sum_{n=0}^{N-1} v_n^{(k)}\, x_n\, e^{-i2\pi f n}\right|^2Multitaper estimate: the average of K orthogonal Slepian eigenspectra.
Instead of splitting the data (and losing resolution), multitaper multiplies the whole record by several specially designed orthogonal windows (tapers), takes a spectrum from each, and averages them. You get variance reduction without throwing away any data.
- \hat{S}_{MT}(f)
- The multitaper spectral estimate.
- v_n^{(k)}
- The k-th Slepian taper — an orthogonal window.
- K
- The number of tapers averaged.
- x_n
- The data samples.
Time–bandwidth: the one number that controls everything
The design parameter is the time–bandwidth product NW (often written p), which fixes the spectral half-bandwidth W over which power is deliberately smoothed. Given NW, the number of well-concentrated tapers you may average is K = 2NW - 1. Larger NW means more tapers, hence lower variance, but a wider smoothing bandwidth 2W, hence coarser resolution and more bias near sharp peaks. This is the bias–variance tradeoff reduced to a single dial.
W = \frac{NW}{N\,\Delta t} = \frac{NW}{T}, \qquad K = 2\,NW - 1 \quad\Rightarrow\quad NW=4 \;\Rightarrow\; K=7Half-bandwidth and usable taper count; e.g. NW = 4 gives 7 tapers smoothing ±4/T Hz.
A single dial — the time–bandwidth product NW — controls the whole multitaper trade-off: a larger NW smooths over a wider frequency band and lets you use more tapers (less variance), at the cost of frequency sharpness.
- NW
- The time–bandwidth product — the master dial.
- W
- The resulting half-bandwidth of frequency smoothing.
- T
- The record duration.
- K = 2NW-1
- The number of usable, well-concentrated tapers.
Choosing NW=4 over a record gives K=7 tapers, smoothing about \pm 4/T Hz.
- Pick the spectral resolution (half-bandwidth W) your science needs.
- Set NW = W\cdot T from your record length T.
- Use K = 2NW-1 tapers (drop the last one or two if their concentration eigenvalues fall well below 1).
Practical multitaper: error bars and line components
Because a multitaper estimate is an average of K eigenspectra, you get an internal uncertainty estimate essentially for free: a jackknife over tapers yields confidence intervals on \log \hat{S}. Thomson's harmonic F-test uses the eigenspectra to test for a pure sinusoid at each frequency — invaluable for identifying and removing line noise or SSVEP components without notching out real neural power.