 Harmonic (wave)

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]