## Harmonic Distortion

published: Mar-18-2017 ###### harmonic distortion

There are different kinds of distortion. The distortion discussed here is harmonic distortion.
A pure sinewave does consist only of the fundamental wave. There should be no harmonics. Harmonics are multiples of the fundamental frequency.

Lets use a 1kHz sinewave in the examples below. The 1st harmonic is the fundamental frequency thus 1kHz in this example. The 2nd harmonic is 2kHz, the 3rd harmonic is 3x that of the fundamental so 3kHz, the 4th = 4x the fundamental, the 5th 5x etc. etc…
When the fundamental would be 440Hz the 2nd harmonic would be 880Hz, the 3rd = 1,320Hz etc.

A pure sinewave is hard to find in music. All instruments have harmonics by themselves in certain ratios. For this reason harmonic distortion may not be very audible with certain instruments because added harmonics could well be masked by harmonics already there in the original music.

In any case, the lower the distortion the better the sound reproduction (quality). Where amplifiers and DAC’s could have distortion levels at levels well below 0.001% most headphones may well have harmonic distortion well over a few percent. As long as the harmonics are 2nd order and maybe 3rd order in relatively low amplitudes it is easy to live with that and may not sound as ‘harsh’ sound for lower frequencies. Between 300Hz and 3kHz distortion levels should be as low as possible as in that range distortion may actually be perceived as ‘harsh/gritty’ sound.

Distortion levels can be made visible using an FFT (Fast Fourier Transform) plot.
This plot shows the fundamental (here 1kHz) which is referenced to 0dB here. In reality the level is 93dB SPL in the (simulated) plot below. Above we see the fundamental (1st harmonic) at 93dB SPL. This is set as a reference and now set as 0dB in the plot. When the pure undistorted 1kHz sinewave  is generated by the FFT program and amplified the tone should still be pure and have NO added harmonics at all.  Thus there should only be one ‘pole’ visible at 1kHz when the signal is viewed as the input signale of the headphone.In reality there will be very small amplitude harmonics present but these are well below the 100dB noise floor of the microphone circuit.

Non linearities in the headphone itself create the extra harmonics at higher frequencies. In this case it is about the HD800 on my test rig.
The 2nd harmonic is at 2kHz and reaches -53dB relative so basically is 53dB softer than the fundamental. The 3rd harmonic is at 3kHz and -75dB relative opposite the fundamental. The 4th and 5th are at -80dB. In reliality they may well be lower in amplitude because they are at the noise floor of the measurement system. With open headphones sounds around it are also picked up by the microphone and the microphone itself and the microphone pre-amp also raise the noise floor.

Mostly only a THD (Total Harmonic Distortion) number is given at obtained at 1kHz and usually around 90dB SPL. The THD is the total ENERGY of all ‘poles’ added together and then converted to either dB’s or a percentage. In this case the total energy of the 3rd to n-th distortion products will only slightly increase the amount of dB’s so the THD will be about equal (or a tiny bit higher) than the 2nd harmonic amplitude. A -53dB distortion product is around 0.22% at 1kHz. Most headphones perform best around 1kHz anyway and generally MUCH MUCH worse at lower and higher frequencies so it gladly used by manufacturers to specify distortion. Tyll (Innerfidelity) has measured the HD800 as well at 1kHz and measured it at around 0.2% too. Sennheiser specifies 0.02% at 102dB SPL.
It should be noted that the mic in my testrig is notorious for having higher 2nd harmonic products as well so the 0.2% may well be the limit of the used microphone ?

To show the distortion at different frequencies one would have to make similar plots at different frequencies. It would be easier to show them in one plot. Below a distortion plot with a frequency range of 45Hz to 30kHz. The lowest amplitude floor is raised on purpose here to show the noise floor of the system. In plots on this website signals below 35dB SPL are not visible. The reason is 2 fold. 1: the noise of the measurement system is higher than the actual signal there and thus not relevant as one looks at the noise floor and not the actual harmonics. Signals below 35dB SPL are not audible when peak levels  of 90dB SPL are present at the same time. They are masked… we cannot hear them when they are close in frequency to higher amplitudes. The SPL at 1kHz is around 92dB. The distortion products are below 40dB around 1kHz. The plot below is measured on a different occasion but is basically the same except the fundamental was 1dB higher (around 93dB) but runs up to 30kHz. The plot above doesn’t show the fundamental which is higher than the 80dB level.
This plot shows the harmonic distortion at all frequencies. The FFT above only at 1kHz.

The reason the harmonics traces are stopping before 30kHz is due to the fundamental stopping at 30kHz. This means that when the fundamental is at 15kHz the 2nd harmonic is 30kHz (the max it will measure). The same goes for the 3rd harmonic. when the fundamental reaches 10kHz the 3rd harmonic is already 30kHz. Etc for the 4th and 5th etc. which ‘end’ at 7.5kHz and 6kHz.
To make the graph more ‘readable’ the harmonic traces are ‘shifted’ to the left till the harmonics that belong to the fundamental are exactly below that fundamental. Above the 2 different plots (FFT and harmonics) in one picture. When you look at the amplitudes of the harmonics at 1kHz they will match the ones in the FFT which is only 1kHz. The FFT plot is not an actual measurement b.t.w., the harmonics plot is.
Take into account that the dB scale on the harmonics plot is in dB SPL where the small inlay FFT plot is in relative dB’s opposite the fundamental set at 0dB.

Because people are more used to seeing distortion numbers in percent instead of a signal level distance in dB below the same plot but the dB scale is now in percent relative to the fundamental at EACH frequency. So 50Hz as well as 1kHz and all other frequencies are each ‘normalised’ to 100%. For this reason the traces look different in amplitude compared to the dB plot above. Plots on this website (when shown) usually display the dB scale and not the percentage scale. Simply because our ears aren’t linear (percentage = linear) but logarithmic and thus closer to a dB scale.

There is also Intermodulation distortion but this isn’t measured. Intermodulation distortion are side products of 2 or more frequencies where unwanted signals are added that are additions and subtractions of these frequencies.
Unlike harmonic distortion this type of distortion products is not related to frequencies in the music itself so there is less chance of masking 