Windowing, leakage and zero-padding

Why this matters

Every spectrum you compute is the spectrum of a finite piece of signal, and cutting a piece out changes the spectrum. A pure tone becomes a peak with skirts; a strong low-frequency rhythm smears across the band above it; a 60 Hz line leaks into the gamma band. Windows control the leakage, at a price in resolution, and zero-padding, which many people use to “increase resolution”, does no such thing. Thirty minutes here prevents a class of over-interpretation that appears in published spectra with some regularity.

Concepts

Leakage: the cost of a finite segment

Taking N samples out of a longer signal is the same as multiplying the signal by a rectangular window that is one inside the segment and zero outside. Multiplication in time is convolution in frequency, so the spectrum you compute is the true spectrum smeared by the window’s own transform. For the rectangular window that transform is a sinc function: a main lobe two bins wide (from the first zero on one side to the first zero on the other) flanked by side lobes whose first is only about 13 dB below the peak and which decay slowly, at about 6 dB per octave (TODO(confirm) the side-lobe figures the author wants quoted; these are the standard textbook values).

Consequences you can see in the widget:

• A tone whose frequency falls exactly on a bin (an integer number of cycles in the segment) produces a single clean line, because the sinc’s zeros land on every other bin.
• A tone between bins spreads across many bins; its energy leaks. Its peak is lower, its neighbours are higher, and the side lobes extend far from it.
• A large low-frequency component (drift, delta) leaks upward into every band, raising the floor under the alpha and beta peaks; a strong 60 Hz line leaks downward into the gamma band. Leakage runs both ways.

Real EEG never has its rhythms on bins, so leakage is always present; the question is only how much and how far it spreads.

Windows: trading main-lobe width for side-lobe height

A window that tapers smoothly to zero at the segment edges removes the sharp discontinuity that the rectangular window creates, and its transform has much lower side lobes; the price is a wider main lobe. The standard choices:

(TODO(confirm) the numerical side-lobe levels and decay rates in this table before promotion; they are standard values from the window-design literature, not measured on this site.)

The rule that follows: use Hann (or another smooth window) whenever you estimate a PSD, because the leakage from the aperiodic low-frequency power otherwise dominates the spectrum; use a rectangular window only when you know the content is on-bin or when you are analysing a transient and cannot afford to taper it away. Multitaper methods (L4.3) use several orthogonal tapers at once to get low leakage and low variance.

Tapering epoch edges

Tapering does more than reduce leakage in a spectrum. An epoch cut out of a continuous recording has a jump at each edge, from the data’s value to whatever comes next (zero, or the next epoch). A filter (L1.5) or a time-frequency transform (L4.2) sees that jump as a broadband event and rings around it. Tapering the edges removes the jump, at the cost of de-weighting the data near the edges; that is why Welch overlaps its segments, and why time-frequency analyses keep the events of interest away from the epoch edges.

Zero-padding: interpolation, not resolution

Appending zeros to a segment before the FFT (the nfft argument in scipy.signal.welch and numpy.fft.rfft) increases the number of bins, so the plotted spectrum has more points and looks smoother. That is all it does. The resolution of a spectrum is set by the length of the data, 1 / T (L1.2); the zeros add no information, and two tones closer together than 1 / T remain one peak however many zeros you append. What zero-padding does do is interpolate the underlying continuous spectrum of the windowed segment onto a finer grid, which is useful for one thing: reading the location of a peak more precisely than the bin spacing, because the true peak of a leaked tone may fall between bins. Used for that, with the caveat stated, it is a good tool; used to claim resolution, it is the pitfall below.

The check is simple and the widget makes it: pad a segment by a factor of four and see the curve smooth out; then add a second tone 0.3 Hz from the first with a 2-s segment and see that padding never separates them, while doubling the segment length does.

The data behind this lesson

• The widget’s real trace is the same 60-s ds-eegbci O1 recording as L1.3 (160 Hz; ODC-By 1.0), plus a synthetic pure tone at a frequency you choose, deliberately off-bin.
• The figure is generated by data/scripts/make_figures.py from a synthetic 10.3 Hz tone (no dataset); its JSON sidecar carries the parameters.

Explore

Set the tone between two bins. Compare rectangular, Hann and Hamming: watch the peak widen and the skirts fall. Raise the zero-padding and see the curve smooth without the peak narrowing. Then put a second tone 0.3 Hz away and confirm that only a longer segment separates them.

Practice

Section 2 of the shared L1.3 notebook computes the spectrum of an off-bin tone with each window, plots the windows and their transforms in decibels, and repeats a two-tone example with and without zero-padding and with a doubled segment length.

Exercises