Prof. Dr. -Ing. Gerald Schuller
Jupyter Notebook: Renato Profeta

Applied Media Systems Group
Technische Universität Ilmenau

Psychoacoustics¶

Block Diagram of a Perceptual Audio Encoder¶

Structure of the Human Ear¶

Quelle: Ars Auditus; http://www.dasp.uni-wuppertal.de/index.php?id=57, 2010

• eardrum – transforms sound wave into vibrations
• ossicular bones - transfer the mechanical vibrations to the cochlea
• cochlear structure - induces traveling waves along the length of the basilar membrane
• neural receptors - connected along the length of the basilar membrane
  • convert these traveling into chemical and electrical signals

Cochlea¶

• cochlea of a 5 month old fetus:
  • spiral-shaped, fluid-filled structure
  • contains the coiled basilar membrane
• blue arrow $\rightarrow$ oval window
• yellos arrow $\rightarrow$ round window

Organ of Corti¶

• organ of corti of a guinea pig

• white bar = 20 μm

• $\approx$ 3500 IHC and $\approx$ 12000 OHC at humans.

• hair cells convert fluid motion into electrical impulses in auditory nerve.

Preprocessing of Sound in the Peripheral System¶

• frequency selectivity of the basilar membrane

Source: http://cochlearimplanthelp.com/journey/choosing-a-cochlear-implant/electrodes-and-channels/

- traveling wave envelopes occur in response to an acoustic tone complex containing e.g. sinusoids of 400 Hz, 1600 Hz and 6400 Hz - peak responses for each sinusoid are localized along the membrane surface, with each peak occurring at a particular distance from the oval window (cochlear "input")
Source: Yuli You "Audio Coding Theory and Applications"

Information Processing in the Auditory System¶

• basilar membrane as a filter bank
• bank of highly overlapping bandpass filters
• the magnitude responses are asymmetric and nonlinear (level dependent)
• non-uniform bandwidth, and the bandwidths increase with increasing frequency

Sound Perception¶

Frequency and Level Range of Human Hearing¶

Threshold in Quiet or the Absolute Threshold¶

• Hearing threshold of 100 persons with normal hearing for sine tones (50% curve is the median).

• Approximations:

$$\large \dfrac{L_{T_q}}{dB} = 3.64 \left( \frac{f}{kHz} \right)^{-0.8} - \exp \left( -0.6\left(\dfrac{f}{kHz}-3.3\right)^2\right)

• 10^{-3}\left(\dfrac{f}{kHz}\right)^4

$$

Hearing Threshold and Age¶

• Average pure-tone audiograms in dB Hearing Loss in (a) men and (b) women grouped by their age in decades (the parameter is age group in years). The extended high-frequency range is zoomed for clarity.

• Pure-tone threshold standard deviation of all participants as a function of frequency (the parameter is age in 10-year groups).

  
  

Loudness¶

• Loudness LeveL:

  • Loudness N: psychological concept to describe the magnitude of an auditory sensation, the loudness of a sound (measured in 'sone')
  • loudness level $L_N$ of a sound is measured in 'phon'
  • $L_N$ of a sound is the sound pressure of a 1 kHz tone which is as loud as the sound
  • 1 sone is equivalent to 40 phons, which is defined as the loudness level of a pure 1 kHz tone at $L_N$ 40 dB SPL.

• Equal-Loudness Level Contours:

  • Equal loudness contours of pure tone sin a free sound field.
  • The parameter is expressed in loudness level, $L_N$, and loudness, N. Can be observed:
    • The sensitivity of the human ear -a function of frequency
    • The most sensitive to sounds around 2–4 kHz

• Loudness Scale:
  • aim: double the number of units on this scale means magnitude of sensation is doubled
  
  $\rightarrow$ relation between loudness level $L_N$ and the loudness N (rule of thumb): $$2 \cdot N \hat{=} L_N + 10 phon$$ - one potential experiment:
  listen to a sound with $L_{N_1}$ 1and then adjust the same sound until
  $N_2 = 2 \cdot N_1$, then compare $L_{N_1}$ and $L_{N_2}$

Critical Bands¶

Frequency Grouping in Human Hearing¶

• Different interpretations that produce the same segmentation:
  • Constant distance in the Cochlea
  • By using tones under the threshold in quiet, their intensity add up in a critical band and are now audible
  • Tones in a critical band above the threshold in quiet: their energy adds up
• Formula for the width of the critical bands:
  • for frequencies < 500 Hz: Constant 100Hz width
  • for frequencies > 500 Hz: 0.2*frequency

: Zwicker, Fastl “Psychoacoustics Facts and Models”, p.159

• Critical bandwidth as a function of frequency, that quantifies the cochlear filter passbands.
• Approximations for low and high frequency ranges are indicated by broken lines.

Excursus - Critical Bands and Loudness¶

• Spectral effects -influence of bandwidth:
  • bandwidth of the signals plays an important role
  • sound level also influence loudness level
  • $\rightarrow$ total sound intensity (SPL) have to be constant to measure loudness as function of bandwidth
  • $\rightarrow$ critical bandwidth

Bark Scale¶

• Critical-band concept used in many models and hypothesis
  

$\rightarrow$ unit was defined leading to so-called critical-band rate scale

• scale ranging from 0 –24, unit "Bark"
• relation between z and f is important for understanding many characteristics of human ear

• Critical-band concept used in many models and hypotheses
  

$\rightarrow$ unit was defined leading to so-called critical-band rate scale

• scale ranging from 0 –24, unit "Bark"(after Zwicker)
• One Bark corresponds to one critical band
• Attempt to approximate critical bands with formulas:

Critical Bandrate z:

$$\large \dfrac{z}{Bark} = 13 \arctan \left( 0.76 \cdot \dfrac{f}{kHz} \right) + 3.5 \cdot \arctan \left( \dfrac{f}{7.5kHz}\right)^2$$

Critical Bandwidth:

$$\large \Delta f_b = 25 + 75 \left( 1 + 1.4 \left( \dfrac{f}{kHz} \right)^2 \right)^{0.69}$$

Masking¶

• data compression:

  • exploitation of perception in critical bands with reference to the threshold in quiet is not enough

• Basic principle:

  • a test signal, called a maskee is placed at the center frequency of the critical bandwidth.
  • one masking signal, called masker (equal power and distance from maskee).
  • If the $P_{maskee}$ is weak relative to the total power of the maskers $\rightarrow$ the test signal is not audible $\rightarrow$ test signal is masked
  • In order for the test signal to become audible, its power has to be raised to above a certain level – masking threshold.

Masking of Pure Tones by Noise -Broad-Band Noise¶

• Broad-band noise:
  • white noise from 20 Hz 20 - 20kHz.

Zwicker, Fastl "Psychoacoustics - Facts and Models", 2nd Edition, 1999.

• masking threshold for pure tones masked by broad band noise of different levels.
• uniform masking noise (UMN) by equalization of the 10 dB per decade slope.

Masking of Pure Tones by Noise -Narrow-Band Noise¶

• narrow-band noise:

  • noise with a bandwidth equal or smaller than critical bandwidth
  
  Zwicker, Fastl "Psychoacoustics - Facts and Models", 2nd Edition, 1999.
  

• threshold of pure tones masked by narrow-band noise for different centre frequencies.
• difference between maximum of masked threshold and test tone level.

• dependence of masked threshold on level of narrow-band noise.
• dips at higher levels $\rightarrow$ nonlinear effects (difference noise caused by interactions between test tone and noise)

Masking of Pure Tones by Low-Pass or High-Pass Noise¶

Masking of Pure Tones by Pure Tone¶

• pure tone:
  • single frequency

• 1 kHz masking tone with level of 80 dB.

• threshold for 'detection of anything'

• Difficulties:

  • beats (hatching)
  • masker and difference tone (stippling)

Masking of Pure Tone by Complex Tones¶

• complex tone:
  • fundamental tone with its harmonics

Zwicker, Fastl "Psychoacoustics - Facts and Models", 2nd Edition, 1999.

• threshold of pure tones masked by a complex tone with 200 Hz fundamental frequency and nine harmonics.

Tonality¶

• Tonality index $\alpha$:
  • noisy signal: $\alpha =0$
  • tonal signal: $\alpha =1$
• System theory:
  • Sharp spectral lines = Signal is periodic = Signal is predictable.
  • Approximation: If the signal is predictable then it should be periodic.
  • Therefore we can use prediction to approximate if a signal is tonal (by periodicity).
• Example:

Masking - Spreading Function¶

Calculating the Masking Threshold¶

• Comparison of the signal level to Masking Threshold:

$$\large \dfrac{O_f(i)}{dB} = \alpha (14.5+i) + (1+\alpha)\cdot \alpha_v$$ $$\large \alpha_v = -2 - 2.05 \arctan \left( \dfrac{f}{4kHz} \right) - 0.75 \arctan \left( \dfrac{f^2}{2.56 kHz^2} \right)$$

where $\alpha \dots$ Tonality index, $\alpha_v \dots$ Noise Coefficient

• Approxitamtion:

$$\large \dfrac{O_f(i)}{dB} = \alpha(14.5+i) + (1 - \alpha)\cdot 5.5$$

• Simultaneous Masking Threshold

$$\large T(f) = 10^{\frac{L_s(f)- O_f(i)}{10}}$$

where $L_s(f) \dots$ Sound Pressure Level, $O_f{i} \dots$ Distance to Masking Threshold

In-Band Making¶

Zolzer, "Digital Audio Signal Processig"

Masking Neighboring Bands¶

• spread of masking due to the non-linearity of auditory filters
• resulting masking threshold = sum of power of neighbouring spreading functions-
• here: value at intersection of neighbouringspreading functions taken

Zolzer, "Digital Audio Signal Processig"
$$\large S_1 = 27 \cdot \dfrac{dB}{Bark}$$ $$\large S_2 = 24 + 0.23 \left( \dfrac{f}{kHz} \right)^{-1} - 0.2 \cdot \dfrac{L_s(f)}{dB} \dfrac{dB}{Bark}$$

Temporal Masking Effects¶

• Post-Masking: corresponds to decay in the effect of the masker $\rightarrow$ expected

• Pre-Masking: appears during time before masker is switched on:

  • Quick build-up time for loud maskers
  • Slower build-up time for faint test sounds

• Frequency resolution $\leftrightarrow$ Blurringing time

• Frequency resolution in the ear $\rightarrow$ Masking in time

• Because of in-ear fast processing between quiet to loud signals, we get Pre-Echoes

  • Pre-Masking: 1-5 ms

  • Post-Masking: ~100ms

  Zwicker. Fastl "Psychoacoustics Facts and Models"