 Spectral leakage bias in harmonic distortion measurements

ID: spectral-leakage-bias-in-harmonic-distortion-measurements

Spectral leakage bias in harmonic distortion measurements occurs when a finite record spreads a strong component into bins interpreted as distinct harmonics. A waveform can have zero physical total harmonic distortion while a naive FFT-bin ratio reports otherwise. Confining a sinusoid to a finite interval creates spectral sidelobes; the pinned primary guide derives this leakage from the finite-time Fourier transform.^[ref]

For an exact discrete calculation, take

x_{m} = A cos (2 π ν m / N + ϕ)

, where

ν

is the number of cycles in the record. With the unscaled forward discrete Fourier transform, a finite geometric sum gives

X_{k} = \frac{A}{2} {e^{i ϕ} D_{N} (ν - k) + e^{- i ϕ} D_{N} (- ν - k)},

where

D_{N} (q) = \sum_{m = 0}^{N - 1} e^{2 πi q m / N} = e^{iπ q (N - 1) / N} \frac{s i n π q}{s i n ( π q / N )}

whenever the denominator is nonzero, with the finite-sum limit elsewhere. Integer-cycle coherent sampling makes the unwanted integer bins cancel; a noninteger cycle count generally does not. The negative-frequency contribution also matters because the input is real.

For an original pure-tone example, let

A = 1

ϕ = 0

N = 128

and

ν = 5.3

. Its true higher-harmonic amplitudes are zero. Yet choosing the nearest bins

5

11

, and

16

for the fundamental, second and third harmonic gives

\frac{∣ X _{11} ∣ ^{2} + ∣ X _{16} ∣ ^{2}}{∣ X _{5} ∣} \approx 0.06581.

A naive result would be about

6.58%

distortion from this undistorted sinusoid. This is an exactly specified record calculation, not an instrument's measured distortion floor.

A window reduces some sidelobes at the cost of a broader main lobe and a changed amplitude normalization. Its coherent gain of a window must be treated consistently, and one cannot assume that one selected bin captures a noncoherent tone's full amplitude. Zero padding samples the same finite-record spectrum more densely; it does not restore missing observation time or remove leakage.

A measurement can instead synchronize the source and acquisition for an integer number of cycles, fit a justified sinusoidal model, or use a validated window/band-integration method. The finite-record phasor extraction with coherent sampling gives a particularly transparent exact case. Calibrated receiver response and actual harmonic coverage remain necessary, as in the primary Anritsu apparatus.^[ref] Power spectral density and squared-amplitude spectrum outputs also have different units; an arbitrary square root of a density bin is not a calibrated harmonic amplitude.^[ref]

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