Publisher’s version / Version de l'éditeur:
Journal of the Acoustical Society of America, 72, 1, pp. 196-199, 1982-07
READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE.
https://nrc-publications.canada.ca/eng/copyright
Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la
première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.
Questions? Contact the NRC Publications Archive team at
PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.
NRC Publications Archive
Archives des publications du CNRC
This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.
Access and use of this website and the material on it are subject to the Terms and Conditions set forth at
Note on the independent sampling of mean-square pressure in
reverberant sound fields
Chu, W. T.
https://publications-cnrc.canada.ca/fra/droits
L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.
NRC Publications Record / Notice d'Archives des publications de CNRC:
https://nrc-publications.canada.ca/eng/view/object/?id=f95f143b-72b2-4388-929b-806500e900bd https://publications-cnrc.canada.ca/fra/voir/objet/?id=f95f143b-72b2-4388-929b-806500e900bdN21d
National Research
Conseil national
$
Council Canada
de recherches Canada
e . 2
NOTE ON THE INDEPENDENT SAMPLING OF MEAN-SQUARE PRESSURE I N REVERBERANT SOUND FIELDS
by W.T. Chu
Reprinted from
Journal of the Acoustical Society of America Vol. 72, No. 1, July 1982
p. 196
-
199DBR Paper No. 1056
Division of Building Research
Price $1 .OO OTTAWA
A N A L Y Z E D
hr ' c 4 r s - t tBLDG.
RES.
L I B R A R Y
82-
11-
05
NRCC 20663On t r a i t e d a n s l e p r 6 s e n t document du c o n c e p t d e pr6l'evement d'gchan- t i l l o n s i n d g p e n d a n t s d e l a p r e s s l o n q u a d r a t i q u e moyenne d a n s l e s champs a c o u s t i q u e s r g v e r b g r s s d u s ii d e s s o n s p u r s e t on d 6 d u i t l e s r 6 s u l t a t s d e Lubman s u r l a f o n c t i o n d e cova- r i a n c e d e l a p r e s s i o n q u a d r a t i q u e mo y enne
.
Note on the independent sampling of mean-square pressure in
reverberant
~d
fields
W. T. Chu
Division of Building Research, National Research Council of Canada. Ottawa, Canada K I A OR6
(Received 15 October 198 1; accepted for publication 26 March 1982)
A discussion of the concept of independent sampling of mean-square pressure in pure-to reverberant sound fields is presented together with a derivation of Lubman's result on th covariance function of the mean-square pressure.
PACS numbers: 43.55.Br, 43.85.Dj rRODUCTlON
n = tan- l ( sin ~
Oi/Z
cos Oi).The need for spatial averaging of the mean-square pres- sur SOU roa bee ing -A" I ci14 cre ton ed i spa tial one
e
3
in connection with laboratory measurements ofmd power and sound transmission loss in reverberation Ims is well known. Several papers'-8 in the Journal have ,n devoted to discussion of various types of spatial averag-
.
The concept of independent sampling can be accepteddily for such incoherent reverberant sound fields as are ated by narrow-band random noise excitation. For pure- e excitation, however, the reverberant sound fields creat-
--
m rooms of fixed geometry are coherent9 and the applica-bility of this concept needs to be clarified. The main objective of this Note is to present a more consistent picture of the subject and to provide a derivation of Lubman's result on the tial covariance of the mean-square pressure that is essen- to many of the previous discussions and to the present
IEVERBERANT FIELD UNDER PURE-TONE CITATION
Rooms with fixed geometry will be considered. At fre-
quencies above the Schroeder's large room frequency, lo
f,
= 2000(T&JV)1'2, (1)where T, is the reverberation time (s) and V, room volume
(m3), a large number of room modes will be excited to vibrate
at the source frequency. Using the free-wave model,4 the reverberant sound field can be idealized as consisting of a large number of plane waves with equal amplitudes but dif- ferent phases. At any point in the room sufficiently far from the source and any reflecting surfaces the directions of prop- agation of the plane-wave trains are assumed to be uniformly distributed in all directions. The acoustic pressure at such a location can be written as
.,
In other words, the sound pressures at all points in space vary sinusoidally in time, but with different amplitudes and
phases. As a result, the temporal mean-square pressure
3
shows large variation with position. For a practical estimate
of the space-averaged value of
3
it is advisable to obtain afinite number of independent samples of
3.
To ensure inde-pendence, a necessary condition is that the spatial covar- iance between paired samples is negligibly small. As the esti- mation process involves spatial averaging of 3 , it is appropriate to assume that the random process describing the field is stationary and ergodic with respect to position. Thus ensemble average can be replaced by spatial average and different statistics of the field are independent of posi- tion and orientation. The spatial covariance of the mean- square pressure can be defined as
where ( ) denotes spatial averaging, r = Jr,
-
r11,
and (3)
represents the space-averaged value of
3.
Lubman2 presented a result for the normalized covar- iance function without proof. Only recently Jacobsen" pro- vided a derivation of Lubman's result for a pure-tone field using Hilbert transform technique. A more conventional ap- proach is possible, however, and will be outlined in Sec. 11. First, it is necessary to summarize several second-order sta- tistics of this field.
A. Probability density of mean-square pressure Using the free-wave model mentioned in the previous section, Waterhouse4 has derived the probability density
(2) function for the normalized mean-square pressure. If I is
defined by
where q5i is the phase of the ith plane wave and w the driving
frequency of the source. {
4,.
) is considered to be a set of N I =3
(rV(?),
(5)statistically independent random variables uniformly dis- the probability density function
P
(I ) istributed on (0,277). Note that Eq. (2) can be rewritten as P ( I ) = e - I . (6)
p(r, t ) = A cos(ot
+
a),where
A =
(x
cos+
(x
sin mi):(3)
B. Joint probability density of instantaneous pressure
In Eq. (2), if N is, large, it follows from the central-limit
theorem'' that the instantaneous pressurep is Gaussian dis-
tributed for any specific time, t = ti. For a pair of points that
a 6 La 611vusll up oint probabil-
ity density is the product of two Gaussian density functions. For other pairs of carrelated points it is possible that their joint probability density may not be bivariate Gaussian, al- though they are Gaussian distributed individually.I3
To ensure that the joint probability density is biva: Gaussian a computer simulation of the reverberant field been performed using the free-wave model. If the assu
...,
tion that the directions of the plane waves are uniformly distributed is incorporated into Eq. (2), the instantaneous pressures at two adiacent locations can be written as, follow- ing Jacobsen's wo ted their j riate lhas Imn- irk," cos [wt, M L ~('1, f l ) =C
2
j = 1 i = 1 (7) =2
,i
u s [at,+
#,(r,,-
kr cos-
, j = l r = lI
0 X d probabilit: 1 2 y density f u ~ ~. . nctions of tr - ..FIG. 2. Comparison of standardize
formed variables with Gaussian function. Combination of 30 waves 1
equal amplitudes but random phases. 2500 pail
ans-
where L = positive integer value of (N?r/2M )sin( j?r/M ), r = Ir, - r,l, and k is the wavenumber ( = @/speed of sound). To generate Eq. (7), the zenith angle (O<B<v) has been divided into M equal increments. The factor (Nv/
2M )sin( jv/M) represents the number of incident plane waves distributed over the ring element, as shown in Fig. 1.
The spatial variation of the amplitude of the pair of points at any instant of time was simulated on the computer by gener- ating different sets of #i(rl, j) using a routine that produces uniformly distributed random numbers.
It is rather difficult to test the normality of a two-di- mensional joint-probability density function. A simpler ap- proach is to rely on the fact that two random variables hav- ing a bivariate Gaussian probability density function can be transformed into two independent Gaussian random varia- bles by a rotation of coordinates (see, for example, Daven- port and RootI4) and to test the normality of probability density functions of the transformed variables individually. Such an approach was taken by the author in his computer simulation. A total of 2500 pairs of points were generated for the probability density computation. A representative com- parison of the density functions of the transformed variables
rs of points
.
.
..
were used.
.
.
with the standard Gaussian probability aenslty 1s show1 Fig. 2. A chi-square test applied to these density functi using 27 class intervals of equal width showed that the pothesis of normality was acceptable at the 0.05 level of nificance. It is therefore concluded that the joint probabi density is bivariate Gaussian for any pair of points at ; instant of time in the free-wave model of a pure-tone re1 berant sound field.I in ons hy- 0.- v1g- lity any {er-
C. Space-time cross correlation of pressurc
Using Eq. (7), the space-time cross correla sure can be written as
B rtion of PI t1b(r29 t,)) M M L L =
C
2
C
2
( u s [ u t 1+
#,(r19 m ) ] m = 1 j = 1 I = 1 i = 1 (8) the It can be demonstrated that the main contril:quadruple summation comes from terms with m = j r
1 = i. Hence ( ~ ( r l , tlb(r29 t2))
l M L
w(tl
-
t,)+
kr cosConverting the summation over discrete zenith angles b to continuous integration,
W l , tlb(r29 t2))
ack
=
(5)
1
[w(t, - t,)+
kr cos B ]sin B dB= (N/2) [sin(kr)/kr] cos [w(t, - t,)]
.
(9)The procedure is very similar to that given by Jacob- sen," but the interpretation is different. Although the nor- malized correlation of Eq. (9) with zero-time difference has d
FIG. 1. Geometry of ray path of incident plane wave on a microphone pair.
same fon given by Cook et a1.,I5 it has minimal ctical value. Equation (9) only serves as an intermediate to the final solution of the covariance of the mean- are pressure. The formulation derived by Cook et al. is incoherent reverberant fields9 (e.g., narrow-band noise itation) and is valid for any fixed pair of points in space.
: results can be utilized for the investigation of certain
ects of "diffusion" and to provide information about the
uctures b~ d in reverberation rn as that the . [ s i n l k O r ) l k O r ] 1 A V E R A G E D O V E R 113 O C T A V E B A N D W I D T H 1 ror exc Tht 0-,-. , auk' aco cha ~ t s to stn eing teste
ZOVARIANCE OF MEAN-SQUARE PRESSURE
As has been stated before, to ensure independent sarn-
~g of
3
it is necessary that the spatial covariance betweenred samples be negligibly small. Hence the form of this
ariance has to be determined. The mean-square pressure sually defined as 1
=
= lim,
l,
d ( r l , t )dt. 7- m ( 10) yu1, pail cov is u KorFIG. A l . Effect of bandwidth on normal mce functio
the number of positions is large. A similar situation results if
averaging over modal pattern and frequency is applied by the use of a rotating diffuser. The case of narrow-band noise excitation can also be considered as averaging over frequen- cy. It is necessary, however, to ensure that there are no sig- nificant changes to the form of the covariance function as a result of frequency averaging. It is shown in the Appendix that the form of the covariance function changes insignifi- cantly when averaged over a bandwidth of 1/3 octave. As
den
has been demonstrated in Sec. IB, the joint probability .sity of pressure is bivariate Gaussian. Thus the covar- :e of pressure squared can be written as (see, for example, venport and RootI4),
bl, t,lP2(r2, t2)) = b2(r1, fl)) (p2(r2, t2))
+
2b(r,, t,lp(r2, t2D2. ( 12)~stituting Eqs. (12) and (1 1) in Eq. (4), it follows that
ianc Da7
(n21
APPENDIX: EFFECT OF FREQUENCY AVERAGING
Under narrow-band noise excitation the normalized co- variance function of the mean-square pressure can be ob- tained by averaging Eq. (1 5) over frequency; i.e.,
w
Sut
R (r) = 2 lim
-
JTr
f 1 ~ ( r 2 , h))2dtd dl2. (13)T-r- T2 0 0
By expanding the integrand into a Taylor series about the
mean wavenumber [k, = (k,
+
kl)/2] it can be shown thatUsi ng
Eq.
(9) for the space-time cross correlation of pressure,I easy to show that
R (r) = (N/2)2[sin(kr)/kr]2. ( 14)
ne normalized covariance is
(r) = [sin(kr)/krI2. (15)
'his result was first obtained by L ~ b m a n . ~
DISCUSSION
The normalized
whereF (kor) = [sin(ky)/kOrl2andF " andF " " are thesecond
and fourth derivatives of F, respectively.
(R
(r)) has beencomputed for the case of (k2
-
k,) equal to 1/3 octave wide.Figure (Al) shows a comparison of
(2
(r)) /. with [sin(kor)/kOrl2. The conclusion is that no significant changes have oc-
curred as a result of frequency averaging. covariance of the mean-square pres-
e
3
is diminishing beyond kr = T , according to Eq. (1 5).us if
3
is measured at points half a wavelength or more-,.~rt, the samples can be considered to be uncorrelated. They cannot, however, be regarded as independent because
the probability density of
3
is not Gaussian. They are ex-on en ti ally distributed as indicated by Eq. (6).
The condition of independent sampling can be ap- roached if some additional averaging schemes are incorpo- ated. For example, averaging over many independent source positions (at least half a wavelength apart) is a possi- bility. As has been shown by Waterhouse: the probability
density function of the averaged
3
approaches normal ifsun Thl
ana
'M. R. Schroeder, "Effect of frequency and space averaging on 1
mission responses of multimode media," J. Acoust. Soc. Am. 46, (1969).
'D. Lubman, "Spatial averaging in a diffuse sound field," J. Acoust. Soc. Am. 46,532-534 (1969).
'M. R. Schroeder, "Spatial averaging in a diffuse sound field and the equi- valent number of independent measurements," J. Acoust. Soc. Am. 46, the trans- ,277-283
n. waternouse, spauru propernes of reveroeranr sound fields," J. Acoust. Soc. Am. 43, 14361444 (1968).
'R. Waterhouse, "Spatial properties of reverberant sound fields," 6th Int. Congr. Acoust., Tokyo, Japan, Paper E-5-l(1968).
6R. Waterhouse and D. Lubman, "Discrete versus continuous space aver- aging in a reverberant sound field," J. Acoust. Soc. Am. 48, 1-5 (1970). 'D. Lubrnan, R. Waterhouse, and C. Chien, "Effectiveness of continuous
spatial averaging in a diffuse sound field," J. Acoust. Soc. Am. 53, 650- 659 (1973).
'C. Chien, R. Waterhouse, and D. Lubman, "Spherical averaging in a dif- fuse sound field," J. Acoust. Soc. Am. 57,972-975 (1975).
9W. T. Chu, "Comments on the coherent and incoherent nature of a rever-
berant sound field," J. Acoust 50c. Am. 69, 171S1715 (198
'OM. R. Schroeder, "Frequency autocorrelation functions of f ~ ~ ~ ~ i l l ~ l ~ 1 sponses in rooms," J. Acoust. Soc. Am. 34, 1819-1823 (1962).
"F. Jacobsen, "The diffuse sound field," Acoust. Lab., Tech. Univ. Dr mark, Rep. No. 27 (1979).
I2J. H. Laning, Jr., and R. H. Battin,Random P r o c e s in Automatic Ca trol (McGraw-W, New York, 1956).
'3Reference 12, p. 85.
I4W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory Random Signals and Noise (McGraw-Hill, New York, 1958), p. 168, "R K. G m k , R. V. Waterhouse, R. D. Berendt, S. Edelman, and M. C.
Thompson, Jr., "Measurement of correlation coefficients in reverberant sound fields," J. Acoust. Soc. Am. 27, 1072-1077 (1955).
This publication i s being d i s t r i b u t e d by the Diviaion of Building R e s e a r c h of the National R e s e a r c h Council of Canada. I t should not be reproduced in whole o r in p a r t without p e r m i s s i o n of the original publisher. The Di- vision would be glad to b e of a s s i s t a n c e i n obtaining s u c h p e r m i s s i o n .
Publications of the Division m a y b e obtained by m a i l - ing the a p p r o p r i a t e r e m i t t a n c e ( a Bank, E x p r e s s , o r P o s t Office Money O r d e r , o r a cheque, m a d e payable t o t h e R e c e i v e r G e n e r a l of Canada, c r e d i t NRC) t o the National R e s e a r c h Council of Canada, Ottawa. KIA OR&. S t a m p s a r e not acceptable.
A l i s t of a l l publications of the Division i s available and m a y b e obtained f r o m the Publications Section, Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, Ottawa. KIA OR6.
