1. EE599-020 Audio Signals and Systems
Psychoacoustics (Loudness)
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Weber’s Law
A difference limen (DL) or just-noticeable difference (JND) is the smallest perceived change in a quantity.
Weber’s Law states that the JND in I (intensity change)is proportional to I (i.e. I/ I is constant).
This implies that a constant in dB for a measure of intensity change would indicate the JNDs for the auditory system for all intensity levels.
For broadband sounds(white noise) greater than 20 dB the DL is 0.6dB. For narrowband sounds (tones) the DL decreases slightly with increasing intensity.

3. Phons
A phon is not a measure of loudness, but a frequency compensated decibel scale. Tones with the same loudness as a 1000 Hz tone at N dB are N phons.

4. Loudness
Loudness is a sensation proportional to the sound pressure of a mechanical wave. The sensation is quantified in units of Sones expressed as:
The international standard chose 1 Sone = loudness level of a 1000 Hz tone at 40 dB or 40 phons.

5. Loudness Example
For 2 broadband sounds of intensity I1 and I2 determined how much greater I2 must be in order for it to be perceived as twice as loud as I1. Indicate result in dB, also indicate the amplitude scaling this corresponds to.
Answer: 10dB and

6. Loudness Homework(1)
Assume a DL corresponds to .6 dB in sound intensity. Compute the DL in terms of a ratio of sones.
Determine the corresponding amplitude scaling for the DL. Create and play sound in Matlab that plays Gaussian white noise at a reference level for 2 seconds followed by a test white noise sound at half the DL level (in dB). Indicate whether it sounded louder or not. If not increase the sound by .2 dB until you can hear the difference. Do the same for a tone at 1000 Hz. Indicate your estimate DL in dB for both cases.

7. Loudness Homework (2)
Verify that a 60 Hz hum at 50 dB has a loudness level of 20 phons. (Hint: Don’t use equal loudness curves in text)
What is its corresponding loudness level in sones?

8. Duration and Loudness
For sounds less than about 200 ms, Loudness increases with duration. A model derived for detection threshold levels in dB as a function of time is based on a leaky integration model:
where L0() is the detection threshold for a very long sound, and  is the time constant dependent on frequency (160 ms for 125 Hz and down to 52 ms for 4000 Hz).
Rule of thumb: Loudness increase by 10dB for every factor of 10 increase of duration up to about 200 ms.

9. Homework(3)
Plot curves for L0(t)- L0() for 16ms < t < 1024 ms for  = 52, 83, and 160 ms on a log scale for the x-axis (use “semilogx” in Matlab, L is already on a log scale). Compare to a straight line of –10dB per decade, which represents perfect integration.