In the field of room acoustics, it is well known that reverberation can be characterized statistically in a particular region of the time-frequency domain (after the transition time and above Schroeder's frequency). Since the 1950s, various formulas have been established, focusing on particular aspects of reverberation: exponential decay over time, correlations between frequencies, correlations between sensors at each frequency, and time-frequency distribution. In this paper, the author introduces a stochastic reverberation model, which permits us to retrieve all these well-known results within a common mathematical framework. To the best of the author's knowledge, this is the first time that such a unification work is presented. The benefits are multiple: several formulas generalizing the classical results are established that jointly characterize the spatial, temporal, and spectral properties of late reverberation.

Topics
Microphones, Room acoustics, Sound source perception, Schroeder frequency, Linear filters, Signal processing, Spectral phenomena and properties, Radiometry, Statistical mechanics models, Stochastic processes
1.
Allen
,
J. B.
, and
Berkley
,
D. A.
 (
1979
). “
Image method for efficiently simulating small-room acoustics
,”
J. Acoust. Soc. Am.
 
65
(
4
),
943
–
950
.
https://doi.org/10.1121/1.382599
Google Scholar
Crossref
Search ADS
 
2.
Balian
,
R.
, and
Bloch
,
C.
 (
1970
). “
Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surface
,”
Ann. Phys.
 
60
(
2
),
401
–
447
.
https://doi.org/10.1016/0003-4916(70)90497-5
Google Scholar
Crossref
Search ADS
 
3.
Baskind
,
A.
 (
2003
). “
Modèles et méthodes de description spatiale de scènes sonores: Application aux enregistrements binauraux” (“Models and methods of spatial description of sound scenes: Application to binaural recordings”)
, Ph.D. thesis,
Université Pierre et Marie Curie (UPMC)
, Paris, France.
Google Scholar
 
4.
Chiu
,
S. N.
,
Stoyan
,
D.
,
Kendall
,
W. S.
, and
Mecke
,
J.
 (
2013
). “
Point processes I—The Poisson point process
,” in
Stochastic Geometry and Its Applications
, 3rd ed. (
Wiley
,
Hoboken, NJ
), Chap. 2.
Google Scholar
Crossref
Search ADS
 
5.
Cohen
,
L.
 (
1989
). “
Time-frequency distributions-a review
,”
Proc. IEEE
 
77
(
7
),
941
–
981
.
https://doi.org/10.1109/5.30749
Google Scholar
Crossref
Search ADS
 
6.
Cook
,
R. K.
,
Waterhouse
,
R. V.
,
Berendt
,
R. D.
,
Edelman
,
S.
, and
Thompson
,
M. C.
, Jr. (
1955
). “
Measurement of correlation coefficients in reverberant sound fields
,”
J. Acoust. Soc. Am.
 
27
(
6
),
1072
–
1077
.
https://doi.org/10.1121/1.1908122
Google Scholar
Crossref
Search ADS
 
7.
Cremer
,
L.
,
Müller
,
H. A.
, and
Schultz
,
T. J.
 (
1982
). “
Statistical room acoustics
,” in
Principles and Applications of Room Acoustics
(
Applied Science Publishers
,
London, United Kingdom
), Vol.
1
, Chap. (ii).
Google Scholar
 
8.
Elko
,
G. W.
 (
2001
). “
Spatial coherence functions for differential microphones in isotropic noise fields
,”
Microphone Arrays
 
61
–
85
.
Google Scholar
 
9.
Jacobsen
,
F.
, and
Roisin
,
T.
 (
2000
). “
The coherence of reverberant sound fields
,”
J. Acoust. Soc. Am.
 
108
(
1
),
204
–
210
.
https://doi.org/10.1121/1.429457
Google Scholar
Crossref
Search ADS
PubMed
 
10.
Jot
,
J.-M.
,
Cerveau
,
L.
, and
Warusfel
,
O.
 (
1997
). “
Analysis and synthesis of room reverberation based on a statistical time-frequency model
,” in
AES Convention
 103 
(Audio Engineering Society
,
New York
), pp.
4629
–
4658
.
Google Scholar
 
11.
Joyce
,
W. B.
 (
1975
). “
Sabine's reverberation time and ergodic auditoriums
,”
J. Acoust. Soc. Am.
 
58
(
3
),
643
–
655
.
https://doi.org/10.1121/1.380711
Google Scholar
Crossref
Search ADS
 
12.
Kuttruff
,
H.
 (
2014
).
Room Acoustics
, 5th ed. (
CRC Press
,
Boca Raton, FL
), pp.
1
–
374
.
Google Scholar
 
13.
Maa
,
D.-Y.
 (
1939
). “
Distribution of eigentones in a rectangular chamber at low frequency range
,”
J. Acoust. Soc. Am.
 
10
(
3
),
235
–
238
.
https://doi.org/10.1121/1.1915981
Google Scholar
Crossref
Search ADS
 
14.
Moorer
,
J. A.
 (
1979
). “
About this reverberation business
,”
Comp. Music J.
 
3
(
2
),
13
–
28
.
https://doi.org/10.2307/3680280
Google Scholar
Crossref
Search ADS
 
15.
Polack
,
J. D.
 (
1988
). “
La transmission de l'énergie sonore dans les salles” (“The transmission of sound energy in rooms”)
, Ph.D. thesis,
Université du Maine
, Le Mans, France.
Google Scholar
 
16.
Polack
,
J.-D.
 (
1992
). “
Modifying chambers to play billiards: The foundations of reverberation theory
,”
Acta Acust. Acust.
 
76
(
6
),
256
–
272
.
Google Scholar
 
17.
Polack
,
J.-D.
 (
1993
). “
Playing billiards in the concert hall: The mathematical foundations of geometrical room acoustics
,”
Appl. Acoust.
 
38
(
2
),
235
–
244
.
https://doi.org/10.1016/0003-682X(93)90054-A
Google Scholar
Crossref
Search ADS
 
18.
Polack
,
J.-D.
 (
2007
). “
The relationship between eigenfrequency and image source distributions in rectangular rooms
,”
Acta Acust. Acust.
 
93
,
1000
–
1011
.
Google Scholar
 
19.
Schroeder
,
M. R.
 (
1962
). “
Frequency-correlation functions of frequency responses in rooms
,”
J. Acoust. Soc. Am.
 
34
(
12
),
1819
–
1823
.
https://doi.org/10.1121/1.1909136
Google Scholar
Crossref
Search ADS
 
20.
Schroeder
,
M. R.
 (
1987
). “
Statistical parameters of the frequency response curves of large rooms
,”
J. Audio Eng. Soc.
 
35
(
5
),
299
–
306
.
Google Scholar
 
21.
Schroeder
,
M. R.
, and
Kuttruff
,
K. H.
 (
1962
). “
On frequency response curves in rooms. Comparison of experimental, theoretical, and Monte Carlo results for the average frequency spacing between maxima
,”
J. Acoust. Soc. Am.
 
34
(
1
),
76
–
80
.
https://doi.org/10.1121/1.1909022
Google Scholar
Crossref
Search ADS
 
22.
Schultz
,
T.
 (
1971
). “
Diffusion in reverberation rooms
,”
J. Sound Vib.
 
16
(
1
),
17
–
28
.
https://doi.org/10.1016/0022-460X(71)90392-0
Google Scholar
Crossref
Search ADS
 
23.
Vincent
,
E.
, and
Campbell
,
D. R.
 (
2008
). “
Roomsimove
,” GNU Public License, http://homepages.loria.fr/evincent/software/Roomsimove_1.4.zip (Last viewed March 26, 2019).
Google Scholar
 
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2019
Acoustical Society of America
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