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# Sound Modulation Metrics: Fluctuation Strength and Roughness

Siemens Phenom

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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:

• Modulation Frequency (fmod) – The number of rises and falls (i.e., modulations) in the sound per second
• Modulation Level (ΔL) – The perceived magnitude level change over time

The more noticeable the modulation, the higher the values of Fluctuation Strength or Roughness.

Figure 1: The amplitude modulation of a sound is described by a frequency and level

Depending on the number of modulations per second present in the sound, either the Fluctuation Strength or Roughness metric might be appropriate:

• Fluctuation Strength - Describes sounds with 20 modulations per second or less
• Roughness - Describes sounds with modulations between 20 and 300 times per second

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.

Figure 2: Table of Fluctuation Strength and Roughness

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:

1. Amplitude - The amplitude, or level, of the sound might rise and fall over time, as if someone was moving the volume knob on a radio up and down. This can happen even if the frequency content of the signal remains constant.
2. Frequency - Multiple frequency tones present in the sound constructively and destructively interfere with each other causing the modulation.

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.

Figure 3: When listened to simultaneously, a 100 Hertz tone and 120 Hertz tone will constructively and destructively interfere with each other

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:

• When the two tones are 180 degrees out of phase, the amplitude sum becomes zero
• When the tones are in phase, the amplitude sum becomes 4 Pascal
• At other times, the sum is between 0 and 4 Pascals

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).

Figure 4: The modulation rate (20 per second) is the difference of the two tones (100 and 120 Hz)

This modulation frequency is based on the difference in frequency, and not dependent on the actual frequencies involved:

• For tones of 100 Hz and 120 Hz of equal magnitude, the modulation frequency is 20 Hz
• For 1000 Hz and 1020 Hz tones, the modulation frequency is also 20 Hz

Modulation frequency versus actual frequencies are shown in Figure 5.

Figure 5: Different frequencies can create the same modulation rate

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:

• Two tones at 200 and 204 Hz
• Two tones at 850 and 854 Hz
• Two tones at 1000 and 1004 Hz, etc.

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

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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:

1. 200 and 220 Hz tones (20 Hz difference) – Modulation noticeable
2. 200 and 208 Hz tones (8 Hz difference) – Perceived modulation more than #1, less than #3
3. 200 and 204 Hz tones (4 Hz difference) – Most noticeable modulation of 1, 2, 3, and 4
4. 200 and 201 Hz tones (1 Hz difference) – Less noticeable than #3

In typical human speech, syllables occur about 4 times per second.

Vacil Unit

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.

Figure 6: Reference sound signal for one Vacil (green line, right scale) and corresponding value (blue line, left scale)

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.

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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.

Figure 7: Time history with intermittent modulation due to multiple tones (green) and corresponding fluctuation strength vacil value (blue)

In this case, the Fluctuation Strength metric rises and falls over time in sync with the modulation that “comes and goes”.

Roughness

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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.

Asper Unit

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.

Figure 8: Reference 70 Hz modulated Asper sound signal (green line, right scale) and corresponding value one Asper value (red line, left scale)

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

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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.

Figure 9: Electric Motor

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.

Figure 10: Comparison of the electric motor noise with and without ‘lugging noise’ and corresponding fluctuation strength

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

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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.

Figure 11: Left – Colormap of ‘Luxury’ exhaust with few orders, Right – ‘Sporty’ exhaust with many orders

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.

Figure 12: Exhaust system with equal length runners

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.

Figure 13: Roughness metric shows higher roughness (red) for ‘Sporty’ sound (with high modulation) and lower roughness (blue) for ‘Luxury’ sound (with lower modulation)

The ‘sporty’ sound has a higher Roughness (red curve), while the ‘Luxury’ sound has a lower Roughness (blue curve).

Simcenter Testlab

To calculate the metrics Fluctuation Strength and Roughness metrics in Simcenter Testlab (formerly called LMS Test.Lab), turn on the Sound Metrics license.

From the main Simcenter Testlab menu, select “Tools -> Add-ins -> Sound Quality Metrics” as shown in Figure 14. If using token licensing, the Sound Quality Metric add-in occupies 33 tokens.

Figure 14: Choose “Tools -> Add-ins -> Sound Quality Metrics”

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.

Figure 15: Modulation Metrics menu in Simcenter Testlab

In throughput processing, a frame and an increment are used to calculate either Fluctuation Strength or Roughness over time:

• Frame – Each frame time segment produces one Roughness or Fluctuation Strength value. For example, a frame could be one second and produce one ‘asper’ or one ‘vacil’ value. A frame of at least 200 milliseconds is recommended for the roughness calculation.
• Increment – The increment is used in conjunction with the frame and time duration of the recording. Processing a recording of 20 seconds, with a frame of 1 second and increment of 0.5 seconds, will produce 21 values spaced 0.5 seconds apart.  This is equivalent to a 50% overlap when processing the data.

For more on the relationship between Frame and Increment, see the Simcenter Testlab 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.