Reading a Spectrum Without Fooling Yourself

The DFT does not analyse the signal you recorded. It analyses the infinite signal you would get by repeating your window end to end, forever. That is not an interpretation, it is what the basis functions are: sinusoids with a whole number of cycles in the window, which is the only way they can be periodic in it.

If your tone happens to complete a whole number of cycles inside the window, the repetition is seamless and everything works. If it does not, the wrap-around creates a discontinuity that was never in your data, and the transform faithfully reports the broad spectrum of that artificial jump.

Key idea: Leakage is not measurement noise and not a bug. It is the correct spectrum of a signal you did not intend to analyse: your window, tiled.

Take a 64-point DFT of a pure sine, with no window applied, and move the tone by half a bin.

Exactly on a bin, the spectrum is one clean spike and everything else is floating-point dust. Half a bin off, the energy smears across the entire spectrum: the neighbour is barely lower than the peak, and a bin twelve away, a fifth of the way across the whole band, still sits only 29 dB down.

Nothing about the signal changed. A pure sine is a pure sine. The entire difference is where it landed relative to an arbitrary grid, and in real data you never get to choose.

The discontinuity comes from the ends of the window not matching. So multiply the signal by a shape that falls smoothly to zero at both ends before transforming. Whatever the wrap does now, it does it to nothing.

import numpy as np

n = np.arange(64)
x = np.sin(2*np.pi*8.5*n/64)          # deliberately between bins

raw = np.abs(np.fft.rfft(x))
win = np.abs(np.fft.rfft(x * np.hanning(64)))

Measured on that same off-bin tone, the level twelve bins away drops from -29.4 dB unwindowed to -73.3 dB with a Hann window. That is a factor of about 160 in amplitude, for one extra multiply per sample.

Definition: A window is a taper applied before the transform. Not applying one is still a choice: it is the rectangular window, and it is the worst-behaved of the family.

The rectangular window's sidelobes sit only 13.3 dB down, so a strong tone leaks enough to bury a weak one 20 dB below it anywhere in the spectrum. Blackman pushes them to 58 dB down, a factor of over 800 in amplitude.

These figures match the reference table in Harris's 1978 paper on windows for harmonic analysis, which is still the standard citation, but they are computed here rather than quoted, so you can reproduce them in four lines.

Suppressing sidelobes costs resolution. The taper effectively shortens the window, and a shorter window has a wider main lobe.

So the choice is a real trade, not a ranking. Two tones three bins apart are resolvable with a rectangular window and merged into one hump by Blackman. A weak tone next to a strong one is invisible under a rectangular window and plain under Blackman.

The last column is the third effect: a tone between bin centres reads low, by up to 3.9 dB unwindowed. If you are measuring amplitudes rather than just spotting peaks, that error is larger than most of the ones people worry about.

The confusion is understandable, because the picture improves in a visible way. What improved is the sampling of the curve, not the curve. The only cure for insufficient resolution is a longer recording.

One transform of a whole recording tells you what frequencies are present and never when. The fix is mechanical: cut the signal into short overlapping chunks, window each one, transform each one, and stack the results into an image. That is the short-time Fourier transform, and the image is a spectrogram.

from scipy import signal
f, t, Zxx = signal.stft(x, fs=48000, nperseg=1024, noverlap=768)

Overlap matters and is easy to skip. The window tapers samples near its edges to nearly zero, so without overlap those samples barely contribute to any frame, and a transient landing on a boundary can vanish. Advancing by a quarter or a half of the window length is the usual choice.

Choosing the chunk length sets both resolutions at once, in opposite directions. A short window pins down when and blurs what; a long window does the reverse.

This is not an artefact of the algorithm. It is a theorem: the time spread and frequency spread of any signal obey

$\Delta t \cdot \Delta f \geq \frac{1}{4\pi}$

the same inequality that appears in quantum mechanics as the position-momentum uncertainty principle, for the same mathematical reason.

In practice: Pick the window from the question. Musical pitch at 48 kHz wants 2048 samples or more, about 43 ms, buying roughly 23 Hz resolution. Locating drum hits wants 256 samples, about 5 ms, and accepts 190 Hz. A spectrogram that looks wrong is usually a spectrogram whose window was chosen by default.

Spectra are plotted logarithmically because the interesting range is enormous. Between a loud passage and the noise floor of a decent recording there can be six orders of magnitude, and on a linear axis everything below the top few percent is a flat line on the bottom.

$L = 20 \log_{10}\!\left(\frac{A}{A_{\text{ref}}}\right)$

A few conversions worth memorising: 6 dB is a factor of 2 in amplitude, 20 dB is a factor of 10, and 60 dB is a factor of 1000.

Gotcha: The factor is 20 for amplitude and 10 for power, because power goes as amplitude squared. Getting this wrong doubles or halves every figure you report, and it is the most common error in the field. If a colleague's numbers are exactly twice yours, this is why.

Before believing a feature you see in a spectrum, check whether the analysis put it there.

1. Was a window applied? If not, every peak has sidelobes 13 dB down, and a small bump near a large peak is probably leakage.
2. Is the feature narrower than the main lobe? Nothing real can be. If it is, you are looking at the interpolation from zero-padding.
3. Does it move when you change the window length? Real components stay put. Artefacts of the analysis shift.
4. Is the peak amplitude trustworthy? Not unless it sits on a bin centre or you have accounted for scalloping loss.
5. Could it be an alias? A feature at a suspiciously arbitrary frequency may be something above Nyquist folded down, and no amount of spectral care will fix that after the fact.

Almost every misread spectrum comes from one of these five, and all five are checkable in a minute.