Sound Modulation Metrics: Fluctuation Strength and Roughness
The twin metrics of Fluctuation Strength and Roughness quantify the amount of modulation present in a sound. When a sound level rises and falls over time, it is said to be modulated.
Unlike decibels, which only quantifies the absolute level of sound, these two sound metrics quantify the following aspects of a sound (Figure 1) into a single number:
The more noticeable the modulation, the higher the values of Fluctuation Strength or Roughness.
Depending on the number of modulations per second present in the sound, either the Fluctuation Strength or Roughness metric might be appropriate:
It should be noted that 20 Hz is not a ‘hard stop’ between the two metrics, it is a gradual transition. The differences between Fluctuation Strength and Roughness are summarized in Figure 2.
Fluctuation Strength can be used to quantify low frequency modulations, like the droning of propeller planes, the rumble of an exhaust, or lugging of an electric motor.
Roughness can be used to quantify high frequency modulations, like the buzzing of an electronic razor, the rapid blade passing noise of a fan, or the ‘sewing machine noise’ of fuel injectors.
What causes modulation of sound?
There are two main reasons why the level of sound may rise and fall over time:
Frequency based modulation occurs when two tones of similar magnitude and different frequency are played simultaneously. Over time, the difference in frequency causes the phase between the two tones to vary as shown in Figure 3. Sometimes the tones are in phase, at other times they are out of phase.
For example, if a 100 Hertz tone and a 120 Hertz tone of the same amplitude were played at the same time (i.e., summed), they would create a 20 Hz amplitude modulation as shown in Figure 4. The 100 Hertz tone has an amplitude of 2 Pascals of pressure, as does the 120 Hertz tone:
The frequency difference (120 Hz minus 100 Hz in this case) between the two sinusoidal tones becomes the modulation frequency (20 Hertz in this case).
This modulation frequency is based on the difference in frequency, and not dependent on the actual frequencies involved:
Modulation frequency versus actual frequencies are shown in Figure 5.
Any frequencies can create a modulation. Don’t confuse modulation rate (often expressed in Hertz) with the frequencies creating the modulation.
For example, if hearing a 4 Hz modulation (i.e., four modulations per second), it does not mean that there is a 4 Hz component in the sound. A four Hertz modulation could be created by:
A four Hertz sound is outside the range of human hearing (20 Hz to 20,000 Hz)! However, a four Hertz modulation of audible sound is NOT outside of human hearing.
For two tones to create a modulation, they typically have to be within the frequency range of one critical band.
Fluctuation Strength describes modulation frequencies below 20 Hz. Because the sound varies slowly over time (below 20 modulations per second), a listener can hear each individual rise and fall in the sound.
Human hearing investigations show that a 4 Hz modulation frequency is more noticeable than other modulation frequencies. This becomes more and more noticeable the closer the modulation frequency is to four times per second, with all else being equal:
In typical human speech, syllables occur about 4 times per second.
Fluctuation Strength is expressed in units of Vacil. This comes from the Latin word vacillātus, and is a short form of the English word vacillate.
The reference time signal for Fluctuation Strength of one vacil value is shown in Figure 6. A reference sound of 1 vacil corresponds to a 1 KHz tone of 60 dB with a 100 % amplitude modulation of 4Hz.
Note that it takes at least one modulation before the vacil value stabilizes, an important consideration when calculating this metric. For a steady state signal, it is important to use the metric values from stabilized portion.
Modulation can be created by multiple tones, not just by two tones. In Figure 7, multiple tones are present, which causes a modulation that “comes and goes” over time.
In this case, the Fluctuation Strength metric rises and falls over time in sync with the modulation that “comes and goes”.
Roughness is used to describe modulations which occur more than 20 times a second up to 300 times per second. When modulations occur more than 20 times a second, the human ear cannot distinguish individual modulations.
With modulations of 20 to 150 times per second, there is a sensation of a stationary, but rough tone. At higher modulation frequencies (about 150 to 300 times per second), listeners often describe a sensation of hearing three separate tones.
The unit used to describe roughness is the "asper" as shown in Figure 8. Everything else being equal, roughness produces the highest aspers when the modulation frequency is 70 Hz, versus any other modulation frequency.
A reference sound of one asper is produced by a 70 Hz, 100% modulated, 1 kHz tone of 60 dB.
Example: Electric Motor ‘Lugging’ Sound and Fluctuation Strength
Electric motors (Figure 9) are often used in many applications, from moving a seat forward and backward in a car, opening and closing a sunroof, or raising and lowering a platform.
These systems (sunroof, seat, platform, etc.) are inspected at the factory before shipment for any flaws or issues. When a seat is not quite installed properly on its track, sound metrics like fluctuation strength can detect the problem. Because the seat does not move smoothly on the track, the electric motor makes a modulated “lugging” or “warbling” sound as it tries to overcome the extra friction.
This distinct sound is obvious to a listener, but not always obvious from visual inspection. When looking directly at the time recording of the data, the lugging noise is not apparent. A modulation metric which keys in on the ‘lugging’ sound is required.
A comparison between a normal electric motor, which contains no lugging sound, and an electric motor with the lugging sound, is shown in Figure 10.
Despite having similar sound pressures, the fluctuation strength values of the two signals are very different, with the motor containing the lugging sound having significantly higher fluctuation strength.
Example: Exhaust Note Sound and Roughness
The exhaust system of cars, trucks and motorcycles are carefully tuned to convey certain brand images. For example, a sports car exhaust system is typically tuned to create modulation to convey “sportiness”. A luxury car is tuned with less modulation to convey a smooth, refined image.
There is less order content in a luxury exhaust system than a sport car as shown in Figure 11.
The amount of modulation that an exhaust system produces can be controlled by the exhaust system design. This is done by controlling the length of the exhaust piping as shown in Figure 12.
If the piping is of equal length, the modulation is less. With piping of unequal length, the modulation is more. The roughness sound metric indicates the difference between the ‘luxury’ and ‘sporty’ configuration as shown in Figure 13.
The ‘sporty’ sound has a higher Roughness (red curve), while the ‘Luxury’ sound has a lower Roughness (blue curve).
To calculate the metrics Fluctuation Strength and Roughness metrics in LMS Test.Lab, turn on the Sound Metrics license. From the main LMS Test.Lab menu, select “Tools -> Add-ins -> Sound Quality Metrics” as shown in Figure 14.
The license “Sound Quality Metrics” adds the ability to calculate modulation metrics to the “Signature Throughput Processing” worksheet. In “Section Settings” a new tab is added called “Modulation Metrics” where Roughness and Fluctuation Strength can be selected as shown below in Figure 15.
In throughput processing, a frame and an increment are used to calculate either Fluctuation Strength or Roughness over time:
For more on the relationship between Frame and Increment, see the LMS Test.Lab Throughput processing knowledge base article.
A reference book with more information about Roughness and Fluctuation Strength is Psycho-acoustics: Facts and Models by Eberhard Zwicker and Hugo Fastl. It was first published by Springer Berlin Heidelberg in 1990, and has several updates since that time.