Total harmonic distortion

To understand a system with an input and an output, such as an audio amplifier, we start with an ideal system where the transfer function is linear and time-invariant. When a sinusoidal signal of frequency ω passes through a non-ideal, non-linear device, additional content is added at multiples nω (harmonics) of the original frequency. THD is a measure of that additional signal content not present in the input signal.

When the main performance criterion is the ″purity″ of the original sine wave (in other words, the contribution of the original frequency with respect to its harmonics), the measurement is most commonly defined as the ratio of the RMS amplitude of a set of higher harmonic frequencies to the RMS amplitude of the first harmonic, or fundamental, frequency[1][2][4][5][6][7][8][9]

${\displaystyle \mathrm {THD_{F}} \,=\,{\frac {\sqrt {V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots }}{V_{1}}}}$

where V_n is the RMS value of the nth harmonic voltage and V₁ is the RMS value of the fundamental component.

In practice, the THD_F is commonly used in audio distortion specifications (percentage THD); however, THD is a non-standardized specification and the results between manufacturers are not easily comparable. Since individual harmonic amplitudes are measured, it is required that the manufacturer disclose the test signal frequency range, level and gain conditions, and number of measurements taken. It is possible to measure the full 20–20 kHz range using a sweep (though distortion for a fundamental above 10 kHz is inaudible).

Measurements for calculating the THD are made at the output of a device under specified conditions. The THD is usually expressed in percent or in dB relative to the fundamental as distortion attenuation.

A variant definition uses the fundamental plus harmonics as the reference, though usage is discouraged:[4][10][11]

${\displaystyle \mathrm {THD_{R}} \,=\,{\frac {\sqrt {V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots }}{\sqrt {V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+\cdots }}}\,=\,{\frac {\mathrm {THD_{F}} }{\sqrt {1+\mathrm {THD} _{\mathrm {F} }^{2}}}}}$

These can be distinguished as THD_F (for "fundamental"), and THD_R (for "root mean square").[12][13] THD_R cannot exceed 100%. At low distortion levels, the difference between the two calculation methods is negligible. For instance, a signal with THD_F of 10% has a very similar THD_R of 9.95%. However, at higher distortion levels the discrepancy becomes large. For instance, a signal with THD_F 266% has a THD_R of 94%.[4] A pure square wave with infinite harmonics has THD_F of 48.3%,[1][14][15] or THD_R of 43.5%.[16][17]

Some use the term "distortion factor" as a synonym for THD_R,[18] while others use it as a synonym for THD_F.[19][20]

The International Electrotechnical Commission (IEC) also defines another term total harmonic factor for the "ratio of the RMS value of the harmonic content of an alternating quantity to the RMS value of the quantity" using a different equation.[21]

THD+N means total harmonic distortion plus noise. This measurement is much more common and more comparable between devices. It is usually measured by inputting a sine wave, notch filtering the output, and comparing the ratio between the output signal with and without the sine wave:[22]

${\displaystyle \mathrm {THD\!\!+\!\!N} ={\frac {\displaystyle \sum _{n=2}^{\infty }{\text{harmonics}}+{\text{noise}}}{\text{fundamental}}}}$

Like the THD measurement, this is a ratio of RMS amplitudes,[7][23] and can be measured as THD_F (bandpassed or calculated fundamental as the denominator) or, more commonly, as THD_R (total distorted signal as the denominator).[24]

A meaningful measurement must include the bandwidth of measurement. This measurement includes effects from ground loop power line hum, high-frequency interference, intermodulation distortion between these tones and the fundamental, and so on, in addition to harmonic distortion. For psychoacoustic measurements, a weighting curve is applied such as A-weighting or ITU-R BS.468, which is intended to accentuate what is most audible to the human ear, contributing to a more accurate measurement. A-weighting is a rough way to estimate the frequency sensitivity of every persons' ears, as it doesn't take into account the non-linear behavior of the ear.[25] The loudness model proposed by Zwicker includes these complexities. The model is described in the German standard DIN45631[26]

For a given input frequency and amplitude, THD+N is reciprocal to SINAD, provided that both measurements are made over the same bandwidth.

The distortion of a waveform relative to a pure sinewave can be measured either by using a THD analyzer to analyse the output wave into its constituent harmonics and noting the amplitude of each relative to the fundamental; or by cancelling out the fundamental with a notch filter and measuring the remaining signal, which will be total aggregate harmonic distortion plus noise.

Given a sinewave generator of very low inherent distortion, it can be used as input to amplification equipment, whose distortion at different frequencies and signal levels can be measured by examining the output waveform.

There is electronic equipment both to generate sinewaves and to measure distortion; but a general-purpose digital computer equipped with a sound card can carry out harmonic analysis with suitable software. Different software can be used to generate sinewaves, but the inherent distortion may be too high for measurement of very low-distortion amplifiers.

For many standard signals, the above criterion may be calculated analytically in a closed form.[1] For example, a pure square wave has THD_F equal to

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{8}}-1\,}}\approx \,0.483\,=\,48.3\%}$

The sawtooth signal possesses

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{6}}-1\,}}\approx \,0.803\,=\,80.3\%}$

The pure symmetrical triangle wave has THD_F of

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{4}}{96}}-1\,}}\approx \,0.121\,=\,12.1\%}$

For the rectangular pulse train with the duty cycle μ (called sometimes the cyclic ratio), the THD_F has the form

${\displaystyle \mathrm {THD_{F}} \,(\mu )={\sqrt {{\frac {\mu (1-\mu )\pi ^{2}\,}{2\sin ^{2}\pi \mu }}-1\;}}\,,\qquad 0<\mu <1}$

and logically, reaches the minimum (≈0.483) when the signal becomes symmetrical μ=0.5, i.e. the pure square wave.[1] Appropriate filtering of these signals may drastically reduce the resulting THD. For instance, the pure square wave filtered by the Butterworth low-pass filter of the second-order (with the cutoff frequency set equal to the fundamental frequency) has THD_F of 5.3%, while the same signal filtered by the fourth-order filter has THD_F of 0.6%.[1] However, analytic computation of the THD_F for complicated waveforms and filters often represents a difficult task, and the resulting expressions may be quite laborious to obtain. For example, the closed-form expression for the THD_F of the sawtooth wave filtered by the first-order Butterworth low-pass filter is simply

${\displaystyle \mathrm {THD_{F}} \,=\,{\sqrt {{\frac {\,\pi ^{2}}{3}}-\pi \coth \pi \,}}\,\approx \,0.370\,=\,37.0\%}$

while that for the same signal filtered by the second-order Butterworth filter is given by a rather cumbersome formula[1]

${\displaystyle \mathrm {THD_{F}} \,={\sqrt {\pi \,{\frac {\;\cot {\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}-\cot ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth {\dfrac {\pi }{\sqrt {2\,}}}-\cot {\dfrac {\pi }{\sqrt {2\,}}}-\coth {\dfrac {\pi }{\sqrt {2\,}}}\;}{{\sqrt {2\,}}\left(\!\cot ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}+\coth ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}\!\right)}}\,+\,{\frac {\,\pi ^{2}}{3}}\,-\,1\;}}\;\approx \;0.181\,=\,18.1\%}$

Yet, the closed-form expression for the THD_F of the pulse train filtered by the pth-order Butterworth low-pass filter is even more complicated and has the following form

${\displaystyle \mathrm {THD_{F}} \,(\mu ,p)=\csc \pi \mu \,\cdot \!{\sqrt {\mu (1-\mu )\pi ^{2}-\,\sin ^{2}\!\pi \mu \,-\,{\frac {\,\pi }{2}}\sum _{s=1}^{2p}{\frac {\cot \pi z_{s}}{z_{s}^{2}}}\prod \limits _{\scriptstyle l=1 \atop \scriptstyle l\neq s}^{2p}\!{\frac {1}{\,z_{s}-z_{l}\,}}\,+\,{\frac {\,\pi }{2}}\,\mathrm {Re} \sum _{s=1}^{2p}{\frac {e^{i\pi z_{s}(2\mu -1)}}{z_{s}^{2}\sin \pi z_{s}}}\prod \limits _{\scriptstyle l=1 \atop \scriptstyle l\neq s}^{2p}\!{\frac {1}{\,z_{s}-z_{l}\,}}\,}}}$

where μ is the duty cycle, 0<μ<1, and

${\displaystyle z_{l}\equiv \exp {\frac {i\pi (2l-1)}{2p}}\,,\qquad l=1,2,\ldots ,2p}$

see[1] for more details.

• Audio system measurements
• Signal-to-noise ratio
• Timbre

• Conversion: Distortion attenuation in dB to distortion factor THD in %
• Swept Harmonic Distortion Measurements
• Harmonic Distortion Measurements in the Presence of Noise