Harmonic Distortion Measurement: The effects of sampling rate and stimulus frequency on the measured harmonic frequency (including THD and Rub & Buzz)
Author(s): Steve F. Temme
Harmonic distortion measurements can be confusing to interpret, and it’s important to understand the data that you are looking at, particularly the relationship between the stimulus frequency and the measured harmonic frequency. This brief guide explains the relationships between sampling rate, stimulus frequency and measured harmonics for both normalized and conventional harmonic distortion measurements.
Fig 1 shows the relationship between the fundamental and its harmonics. Note that the frequency of the second harmonic is at twice the frequency of the fundamental, the 3rd harmonic is at three times the frequency of the fundamental, the 4th at 4 times, and so on. Distortion measurements are relative, and involve comparing the level of the harmonic to the level of the fundamental. A test signal will typically sweep up to 20kHz to measure harmonics at a range of frequencies.
Conventional harmonic distortion
Total Harmonic Distortion (THD) is the percentage of the total signal that is affected by distortion due to the harmonics signals, according to IEC. It is the power sum of all the harmonics excluding the fundamental, divided by the power sum of all the harmonics, including the fundamental.
(IEC method; the IEEE method simply uses H1 as the denominator)
Conventional distortion curves are plotted under the corresponding excitation frequency of the measured fundamental. For example, if the fundamental is 1kHz, the second harmonic at 2kHz is compared in level to the fundamental at 1kHz and plotted at 1kHz.
For conventional harmonic distortion, the maximum frequency of a harmonic distortion product is limited by the sampling rate of the soundcard.
2.1 is a default set in SoundCheck and ensures that you are operating in the linear range of the anti-alias filter.
For example, with a sound card set to 48kHz sampling rate, the alias frequency is 22,800Hz, so the maximum measurable frequency for the 2nd harmonic is 11,400Hz (7600Hz for the 3rd harmonic, 5700Hz for the 4th and 4560Hz for the 5th). Any total distortion measurements will only include the harmonic distortion products that can be measured up to that frequency. For example, using the sound card above, if you are calculating distortion using the 2nd through 5th harmonics by sweeping to 20kHz, the total harmonic distortion calculated at 5kHz will include the 2nd through 4th harmonics but not the 5th. Below 4.56kHz it will include all selected distortion products.
There are only two ways that you can practically change this to increase your maximum measured frequency:
- Increase the sampling rate of your soundcard, if it is set to below the maximum. In SoundCheck, this is done in the hardware setup.
- Use a soundcard or audio interface with a higher sampling rate. See the audio interface page on Listen’s website to learn about sampling rates of Listen-supported audio interfaces.
Normalized harmonic distortion (THD or Rub and Buzz)
Normalized harmonic distortion is calculated using the harmonics at the actual measured frequency of their signals using the equation below:
In normalized distortion measurements, each measured harmonic is compared to the fundamental at the actual measured frequency of the harmonic. For example, with a fundamental at 1kHz, the second harmonic is at 2kHz and is compared in level to the fundamental at 2kHz and then plotted at 1kHz. Similarly, the 3rd harmonic, which will be at 3kHz, is compared to the level of the fundamental at 3kHz and included in the distortion calculation for 1kHz.
For normalized harmonic distortion (THD or Rub and Buzz) the maximum frequency of the distortion product is limited by the maximum frequency of the stimulus.
The maximum frequency of a harmonic distortion product is determined by the ratio of the stimulus frequency and the harmonic order being measured. Specifically:
For example, with a stimulus sweeping up to 20kHz, the 2nd harmonic distortion product can be measured to 10kHz, the 3rd to 6.7kHz, the 4th to 5kHz, the 5th to 4kHz and so on. This means that a THD result will be recorded up to 10kHz, but it will only include all of the 2nd to 5th harmonic distortion products up to 4kHz.
To make a normalized distortion measurement that includes up to the 10th harmonic, you need to measure the fundamental at the measured frequency of the 10th harmonic, which will be 10 times the fundamental frequency. This means that with a sweep going up to 20kHz, the maximum fundamental that can be measured to the 10th harmonic will be 2kHz, so the 10th harmonic component of the distortion will only be included up to 2kHz.
When making THD measurements, it is usually good enough to measure the 2nd through 5th harmonics, because the higher level harmonics will be insignificant in level compared to the lower harmonics (see fig. 1). Sweeping to 20kHz means that all distortion components would be included up to 4kHz, and above that the higher order harmonics become dropped.
Rub and Buzz measurements are usually made using higher order harmonics, often 10th-15th order. This effectively makes the range where all components are included quite small, and it’s important to be aware when interpreting results that at frequencies above 1kHz you start to lose distortion components.
For a detailed explanation of how distortion measurements are graphed, please see Steve Temme’s paper entitled How to Graph Distortion Measurements.
Note: Remember that the frequency range of your microphone or transducer may also limit your measured harmonics. If measurements of higher order harmonics over a large frequency range are important, you need to ensure that all components in your setup will support this.