Harmonic (wave, $f_n=n{f}_1$)

A harmonic component is a sinusoid whose frequency is a positive integer multiple of a specified fundamental frequency: $f_n=n{f}_1$. The fundamental itself is the first harmonic component. In a periodic Fourier series with period $T=1/f_1$,the constant $a_0$ is a DC component, and the positive-frequency sinusoidal terms are the harmonic components. A component may have zero amplitude; a missing fundamental does not require the spacing of the remaining components to change. An overtone is a component above the lowest mode, but physical overtones need not be exact integer multiples.

For an original generation model, apply $v_{\mathrm{in}}(t)=1\mathrm{V}\cos (2\pi 1000t)$ to a memoryless nonlinear device with $v_{\mathrm{out}}=v_{\mathrm{in}}+\beta v_{\mathrm{in}}^2$ and $\beta =0.2{\mathrm{V}}^{-1}$. The identity ${\cos }^{2}x=(1+\cos 2x)/2$ givesThe nonlinearity produces both a DC offset and a second harmonic component; they are different parts of the frequency spectrum. The second harmonic component's peak amplitude is $0.1\mathrm{V}$, while the fundamental's is $1\mathrm{V}$.

A linear time-invariant device in sinusoidal steady state cannot create this extra frequency from a single sinusoidal input; a nonlinear response, a different input component, or an unexcluded transient needs to be considered. In an actual measurement, the source's own harmonic components and the receiver's frequency response must also be accounted for. Anritsu's primary measurement procedure separately acquires the fundamental and selected harmonic components, with measurement calibration covering their frequencies.^[ref] A discrete Fourier transform can instead separate a suitably sampled record; its bin amplitudes still depend on normalization and record length.^[ref]

Total harmonic distortion measures the root-sum-square amplitude of the selected higher harmonic components relative to the fundamental amplitude. For nonzero $A_1$ and one common amplitude convention,Peak amplitudes or root mean square amplitudes give the same ratio when used consistently for all sinusoids. The finite upper order $M$, measurement bandwidth, and treatment of noise must be specified. The primary Anritsu procedure distinguishes this fundamental-normalized definition from normalization by the full harmonic component sum.^[ref]

For an original example, let a waveform have fundamental peak amplitude $2\mathrm{V}$, second harmonic component $0.06\mathrm{V}$, and third harmonic component $0.08\mathrm{V}$, with no other components. ThenA logarithmic amplitude-ratio report is $20{\log }_{10}(0.05)\approx -26.02\mathrm{dB}$. The phases do not enter this RMS sum because distinct integer harmonic components are orthogonal over a full period.

If the denominator instead includes the fundamental and all included harmonic components, excluding DC, the resulting ratio isabout $4.994\%$ in the example. The two conventions are close at low distortion but differ markedly when higher harmonic components are large; only the latter is bounded by one. A quoted percentage without its denominator convention is incomplete. Broadband noise is a different contribution and is not silently included in the displayed THD formula.

A motivating apparatus uses a clean sinusoidal source, the device under test, and calibrated receivers tuned to the fundamental and harmonic component frequencies. Anritsu's worked system calibrates the receiver over those frequencies before combining separate measurements.^[ref] With an FFT-based instrument, spectral leakage bias in harmonic distortion measurements can misassign a strong fundamental to harmonic component bins; aliasing and insufficient receiver bandwidth can also hide or relocate components. Small THD therefore describes a stated test and analysis, not every possible waveform or operating level.

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 $x_m=A\cos (2\pi \nu m/N+\varphi )$, where $\nu$ is the number of cycles in the record. With the unscaled forward discrete Fourier transform, a finite geometric sum giveswherewhenever 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 $A=1$, $\varphi =0$, $N=128$ and $\nu =5.3$. Its true higher-harmonic amplitudes are zero. Yet choosing the nearest bins $5$, $11$, and $16$ for the fundamental, second and third harmonic givesA 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]