= Harmonic
{disambiguate=wave}
{title2=$f_n=nf_1$}
{wiki=Harmonic}
= Harmonic component
{synonym}
A <harmonic component> is a <sinusoid> whose <frequency> is a positive integer multiple of a specified <fundamental frequency>: $f_n=nf_1$. The fundamental itself is the first <harmonic component>. In a periodic <Fourier series> with period $T=1/f_1$,
$$
x(t)=a_0+\sum_{n\ge1}A_n\cos(2\pi nf_1t+\phi_n),
$$
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 $v_{\rm in}(t)=1\,\mathrm V\cos(2\pi\,1000t)$ to a memoryless nonlinear device with $v_{\rm out}=v_{\rm in}+\beta v_{\rm in}^2$ and $\beta=0.2\,\mathrm{V^{-1}}$. The identity $\cos^2x=(1+\cos2x)/2$ gives
$$
v_{\rm out}(t)=\{0.1+\cos(2\pi\,1000t)+0.1\cos(2\pi\,2000t)\}\,\mathrm V.
$$
The 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>.<Total Harmonic Distortion Measurements by Anritsu>[Printed pp. 2–4, definitions, calibration, and multichannel setup]{ref} A <discrete Fourier transform> can instead separate a suitably sampled record; its bin <amplitudes> still depend on normalization and record length.<SciPy 1.16.2 documentation>[scipy.fft.fft, Parameters: norm and Notes: forward transform]{ref}
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