Spectral Leakage in Practice | DSP-101

The Measurement Problem

In theory the DFT of a pure sinusoid is a single non-zero bin. In practice it almost never is: the energy of one tone smears across many neighbouring bins, an effect called spectral leakage. The previous lesson traced leakage to its cause — a finite observation window multiplies the signal by a rectangular gate, and multiplication in time is convolution in frequency, so every true spectral line is replaced by a shifted copy of the window’s own spectrum. This lesson is about what that does to a measurement: how leakage corrupts the amplitude you read off a bin, how to make it vanish when you can, and how to bound it when you cannot.

The stakes are practical. An engineer reading a spectrum analyser needs to know whether a peak sits at −40 dB because the tone is genuinely that weak, because leakage from a strong neighbour has raised the noise floor, or because the tone fell between two bins and was measured almost 4 dB low. All three are leakage phenomena, and each has a different fix.

Coherent Sampling: When Leakage Vanishes

There is one case where a rectangular window produces no leakage at all. The window’s spectrum — a Dirichlet (periodic sinc) kernel — has zero crossings at every bin except the one it is centred on. So if a tone’s frequency lands exactly on a DFT bin, every other bin sits on one of those zeros and the tone appears as a single clean line. This alignment is called coherent sampling, and it occurs precisely when the frequency is an integer number of whole cycles over the observation window:

Coherent-Sampling Condition

f_0 = k\,\frac{f_s}{N}, \quad k \in \{0, 1, \dots, N-1\}

A tone at f₀ lands exactly on bin k — and leaks into no other bin — when f₀ is an integer multiple of the bin spacing f_s/N, i.e. when a whole number of cycles fits in the N-sample record.

Coherent sampling is why instrument-calibration labs lock the signal generator and the digitiser to one reference clock and choose N so the test tone is bin-aligned: it removes leakage at the source rather than correcting for it afterward. But it only works when you control the signal. A tone arriving from the outside world — a carrier, a vibration, a heartbeat — will almost always fall between bins, and then leakage is unavoidable and must be managed with a window.

Scalloping Loss: The Picket-Fence Effect

The DFT reports the spectrum only at its bin centres — it samples the underlying continuous spectrum at N discrete points, like viewing a scene through a picket fence. A tone that lands between two bins is measured on the skirt of the window’s main lobe, not at its peak, so its amplitude reads low. The worst case is a tone exactly halfway between two bins.

For a rectangular window that worst-case shortfall is 3.92 dB — the reported amplitude is only 64% of the true value — large enough to matter in any amplitude measurement. This deterministic, frequency-dependent under-reading is scalloping loss (also called picket-fence loss). It is not noise; it is a bias you can predict from where the tone falls relative to the bins. Windows with a broader, flatter main lobe scallop less: a Hann window drops the worst case to 1.42 dB, and dedicated flat-top windows are engineered to hold it under 0.1 dB — which is why they are the default for amplitude-accurate measurement even though their wide main lobe ruins frequency resolution.

Processing Gain and Equivalent Noise Bandwidth

Leakage also governs how a windowed DFT measures noise and power, and this is the bridge to the power spectral density of Module 9, Lesson 1. A window is a filter, and each DFT bin passes not only its own frequency but a band of noise around it, shaped by the window’s main lobe and sidelobes. The width of the ideal rectangular filter that would pass the same noise power is the window’s equivalent noise bandwidth (ENBW):

Equivalent Noise Bandwidth

\text{ENBW} = N\,\frac{\sum_{n=0}^{N-1} w[n]^2}{\left(\sum_{n=0}^{N-1} w[n]\right)^2}

ENBW, in units of DFT bins, depends only on the window coefficients w[n]. A rectangular window has ENBW = 1 bin (the narrowest possible); every tapering window is wider.

ENBW is what lets you turn a windowed magnitude spectrum into a calibrated power spectral density in watts per hertz — the unit Module 9 uses but does not derive. A single line’s squared magnitude is a power; dividing by the sampling rate and the window’s energy converts it to a density:

Calibrated PSD from a Windowed DFT

\hat{S}_{xx}(f_k) = \frac{2\,\lvert X_w[k]\rvert^2}{f_s \sum_{n=0}^{N-1} w[n]^2}

The modified periodogram: |X_w[k]|² normalised by f_s and the window energy Σw[n]² gives the one-sided PSD in power per hertz (interior bins). Without the Σw[n]² term the estimate would be biased by whichever window you chose.

The same ENBW that calibrates the noise floor is why a low-leakage window costs sensitivity to weak tones: spreading a bin’s noise over 2 bins of bandwidth raises the noise floor by 10 log₁₀(2) ≈ 3 dB relative to the rectangular window. Choosing a window is therefore always a three-way trade among frequency resolution, amplitude accuracy, and the ability to see a small signal next to a large one.

Choosing a Window

The classic reference is Fredric Harris’s 1978 survey, which tabulates these figures of merit for dozens of windows. The five below span the useful range from “no window” to “extreme sidelobe suppression”:

Window Figures of Merit (Harris, 1978)

Window	−3 dB main lobe (bins)	Peak sidelobe (dB)	Scalloping loss (dB)	ENBW (bins)
Rectangular	0.89	−13	3.92	1.00
Hann	1.44	−32	1.42	1.50
Hamming	1.30	−43	1.75	1.36
Blackman	1.68	−58	1.10	1.73
Blackman-Harris (4-term)	1.90	−92	0.83	2.00

Read the table as one trade-off swept top to bottom: as the window tapers harder, the peak sidelobe plunges (−13 dB to −92 dB, so a weak tone becomes visible beside a strong one) and scalloping loss shrinks — but the main lobe widens and the ENBW grows, so frequency resolution and noise sensitivity get worse. The Blackman-Harris row is the low-leakage extreme: its −92 dB sidelobes can reveal a tone 90 dB below its neighbour, at the cost of a main lobe more than twice the rectangular width.

Practical Strategies

If you control the source

Sample Coherently

Lock generator and digitiser to one clock and choose N so the tone is bin-aligned. Leakage disappears with no window penalty — the calibration-lab approach.

For accurate amplitude

Use a Flat-Top Window

Scalloping loss under 0.1 dB wherever the tone falls. Wide main lobe, so poor for resolving close tones — but the right choice when the level must be right.

Weak tone by a strong one

Use Low Sidelobes

Blackman-Harris, or a Kaiser window with high β. −90 dB sidelobes keep a strong carrier from burying its weak neighbour, trading resolution for dynamic range.

For absolute levels

Correct for the Window

Divide out the window’s coherent gain (Σw[n]) and its ENBW before quoting a power or a density — otherwise the reading is biased by the window you happened to pick.

All of this assumed the DFT itself was affordable to compute. Module 6 lifts that assumption: the Fast Fourier Transform reduces the DFT’s O(N²) cost to O(N log N), which is what makes real-time spectral analysis — and everything in this lesson — practical.

Key Takeaways

Spectral leakage is the finite window’s fingerprint: a true line is replaced by a copy of the window’s spectrum, spreading one tone’s energy across many bins.
Coherent sampling — a tone at exactly f₀ = k·f_s/N — produces zero leakage, because every other bin lands on a zero of the window spectrum. It only works when you control the signal.
Scalloping loss is a predictable amplitude under-reading for tones between bins: 3.92 dB worst-case for a rectangular window, 1.42 dB for Hann, under 0.1 dB for a flat-top.
Equivalent noise bandwidth (ENBW = N·Σw²/(Σw)²) is the width of the rectangular filter passing the same noise power; it calibrates a windowed DFT into a PSD in W/Hz.
Every window trades frequency resolution against sidelobe level, amplitude accuracy, and noise sensitivity; Harris’s 1978 table quantifies the trade for each.
Practical rule: sample coherently if you can; flat-top for amplitude; Blackman-Harris for dynamic range; always correct for coherent gain and ENBW before quoting absolute power.