Definitions

# Total harmonic distortion

The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the Fundamental frequency. Lesser THD, for example, allows the components in a loudspeaker, amplifier or microphone or other equipment to make a violin sound like a violin when played back, and not a cello or simply a distorted noise.

## Explanation

In most cases, the transfer function of a system is linear and time-invariant. When a signal passes through a non-linear device, additional content is added at the harmonics of the original frequencies. THD is a measurement of the extent of that distortion.

The measurement is most commonly the ratio of the sum of the powers of all harmonic frequencies above the fundamental frequency to the power of the fundamental:


mbox{THD} = {sum{mbox{harmonic powers}} over mbox{fundamental frequency power}} = {{P_2 + P_3 + P_4 + cdots + P_n} over P_1}

Other calculations for amplitudes, voltages, currents, and so forth are equivalent. For a voltage signal, for instance, the ratio of the squares of the RMS voltages is equivalent to the power ratio:

$mbox\left\{THD\right\} = \left\{\left\{V_2^2 + V_3^2 + V_4^2 + cdots + V_n^2\right\} over V_1^2\right\}$

In this calculation, Vn means the RMS voltage of harmonic n, where n=1 is the fundamental harmonic. One can also calculate THD using all harmonics (n=∝):


mbox{THD} = {{V_{RMS}^2 - V_1^2} over V_1^2}

Other definitions may be used. Many authors define THD as an amplitude ratio rather than a power ratio. This results in a definition of THD which is the square root of that given above. For example in terms of voltages the definition would be:

$mbox\left\{THD\right\} = \left\{sqrt\left\{V_2^2 + V_3^2 + V_4^2 + cdots + V_n^2\right\} over V_1\right\}$

This latter definition is commonly used in audio distortion (percentage THD) specifications. It is unfortunate that these two conflicting definitions of THD (one as a power ratio and the other as an amplitude ratio) are both in common usage. Fortunately if the THD is expressed in dB then both definitions are equivalent. This is not the case if the THD is expressed as a percentage. The power THD can be higher than 100% and is known as IEEE, but for audio measurements 100% is preferred as maximum, thus the IEC version is used (Rohde & Schwartz, Bruel and Kjær use it).


mbox{THD} = {{V_{RMS}^2 - V_1^2} over V_{RMS}^2}

A measurement must also specify how it was measured. Measurements for calculating the THD are made at the output of a device under specified conditions. The THD is usually expressed in percent as distortion factor or in dB as distortion attenuation. A meaningful measurement must include the number of harmonics included.

## THD+N

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

mbox{THD+N} = {sum{mbox{harmonic powers}} + mbox{noise power} over mbox{total output power}}

A meaningful measurement must include the bandwidth of measurement. This measurement includes effects from intermodulation distortion, interference, and so on, instead of just harmonic distortion.

For a given input frequency and amplitude, THD+N is equal to SINAD, provided the bandwidth for the noise measurement is the same for both (the Nyquist bandwidth).