Statistical properties of energy spectral response as a function of direct-to-reverberant energy ratio

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Publisher’s version / Version de l'éditeur:

Journal of the Acoustical Society of America, 68, 4, pp. 1208-1210, 1980-10

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Statistical properties of energy spectral response as a function of

direct-to-reverberant energy ratio

Chu, W. T.

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National Research

Conseil national

0.950

I

$

Council Canada

de

recherches Canada

o p .

2

TLDG

-

STATISTICAL PROPERTIES

OF

ENERGY SPI

_AL

RESPONSE AS A FUNCTION OF DIRECTeTC

REVERBERANT ENERGY RATIO

by

W.T.

Chu

ANALYZED

Reprinted

from

Journal, Acoustical Society of America

VoL 68, No.

4,

October

1980

p. 1208 1210

DBR

Paper No. 950

Dividon of Building Research

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RESUME

On a d6rivE des expressions analytiques de propri6t6s statistiques de la courbe spectrale dans une salle de rgverbgration, en fonction du rapport de 116nergie

directe 3 l'gnergie de r6verbbration. On compare avec

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Statistical properties of energy spectral respa

function of direct-to-reverberant energy ratioa)

-r - tior ber the m e r a t rea W. T. Chu

Division of Building Research, National Research Council of Cant (Received 15 May 1980; accepted for publication 4 July 1980)

zda. Ottawa , Canada KIA OR6

Analytical expressions for statistical properties of the spectral response curve in a reverberation room have been derived as a function of direct-to-reverberant energy ratio. Comparison with results obtained by other numerical methods is given.

PAC! 5 numbers: 43.55.B~ 43.45.Dn

r

ION

. . . .

In a recent article,' Jetzt proposed a new method for measuring the critical distance of a room, based on computing the standard deviation of the spectral r e - snonse from its mean. Applying a Monte Carlo simula-

i with 20000 samples taken a t each direct-to-rever-

,ant energy ratio, he obtained "theoretical" values of standard deviation of the spectral response from its an a s a function of direct-to-reverberant energy io. These values can, however, be obtained more d i l y by analytical analysis of the spectral response For pure-tone excitation, Dieste1213 has shown that the probability distribution of sound p r e s s u r e at different locations in a room is related to the ratio of the direct-to-reverberant sound. Bowers and Lubman4 gave corresponding expressions for sound p r e s s u r e levels. These results might well be adopted for the present investigation. As no detailed derivation was given in these r e p o r t s , it seemed instructive to derive the results from f i r s t principles.

I. THEORY

Consider a fixed speaker-microphone pair in a r e v e r - beration room with the speaker emitting a pure-tone

signal of frequency w. The instantaneous sound pres-

s u r e picked up by the microphone can be written as

where

p,

and p, represent the direct and the reverberant sound field, respectively. The mean-square pres- s u r e is given by

where the overbar denotes the time average value. If the direct field and the reverberant field a r e statistically independent, the cross-product t e r m will be zero. For pure-tone excitation, however, the reverberant

field i s also a coherent field and

p,

and p r a r e definitely correlated.

For the direct field,

p , ( w , t ) = A c o s w t , ( 3 )

where the phase angle has been s e t to z e r o a s the ori- gin of reference for other phase angles. For the reverberant field

p . ( w , t ) = ~ g c o s ( w t + a , ) , (4)

1.1

assuming it to be composed of many plane traveling waves of equal amplitude crisscrossing in space. This results in a s e t of n phase angles ai distributed at random over the range 0 to 2a, and a fixed value of p:(wl). At a different source frequency there will be

another s e t of random phase angles, assumed to be statistically independent of the f i r s t s e t , and another value p:(w2). Although this s e e m s to be a rather simple representation of the reverberant sound field, i t does provide correct results for many statistical quantities, as verified by a more detailed analysis using the eigenmode approach.= This provides confidence in this mod- e l for the present analysis.

Substituting Eqs. (3) and (4) in Eq. (2) gives

Waterhouse5 has shown that both C n c o s a i and C n s i n a i a r e normally distributed variables with mean value z e r o and variance n/2; in addition, they a r e statistically independent.

Let

a'After this work was completed, it was learned that an identi- So that

cal result had been reported independently by K. J. Ebeling

in another Letter to the Editor in this issue U. Acoust. Soc. p2((w) = (B'/~)(u

+

V )

Am. 68, 1206-1207 (1980)l. It was felt, however, that the

somewhat different derivation presented here provides a suf- Using the general rules f o r probability distribution:

ficiently different view of the subject to warrant the apparent it can be shown that the probability density functions,

duplication. P , for u and v a r e

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An equat: and Lubn ion identj nan4 with? .cal to Eq gut proof ~y Bowers

,.

(15) wa

, and whc

?n s given b D = 0, for v 3 0 , c.. - .. , n The me given b. urve i s tl an value of the spectral response c

Y , , = (B2/2)((u)+ (v))

,

r's resul' t8 also. r h i s a g r e e s with Schroede:

(P')

where (

It i s c l e a r that the spectral response curves a r e dij ferent f& different locations of this speaker-microphone pair of fixed separation. The statistical prope ties of the different spectral response curves, how- e v e r , a r e independent of location of the speaker-mic rophone pair provided they a r e not close to the boun- d a r i e s of the room. F o r a different separation of the p a i r , there will be a different value for U ; tl

bility density function is s t i l l given by Eq. (1 :u) and (v) can be computed using Eq. (7), ide.,

The result i s

he proba-

,5).

Define

II. RESULTS AND DISCUSSION

Once the probability density functiurl u r r l l G

p r e s s u r e level of the s p e c t r a l response curv it is an easy matter to compute the standard from the mean using

in.. nf C h a 8 sound

s o that

F r o m Eqs. (3) and (4),

No serious attempts have been made to obtain analyti solutions for Eq. (16) for m = 1 and 2. Numerical intc gration was used to solve for the mean and mean- square values of y for the standard deviation of y f r o the mean. Results a r e tabulated in Table I for differ1 direct-to-reverberant energy ratio. They compare very well with results given by Jetzt,' who asserted that the direct and reverberant p r e s s u r e s a r e indepe dent, although such an assumption was in fact not uti ized in the Monte Carlo simulation. Thus his " c o m p ~ ted", curve f o r the standard deviation of the spectral response applies, in fact, to c a s e s with pure-tone ex citation. c a l I

-

111 ent and (F:)w =nB2/2 .

For a fixed separation of speaker and microphone,

3 :

can be considered a s independent of w . _{Using the}

usual definition for direct-to-reverberant energy ratio,

D =F;/(F;), = A 2 / n ~ 2 , (11)

Equation (9) can therefore be rewritten as

I = (U

+

v)/n(l

+

D)

.

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AS

C

C O S ~ ~ and

C

s i n a i a r e statistically independent, u and v can also be considered as independent. Knowing the probability density functions for u and v , that for I can readily be derived7 and the result i s

TABLE I. Comparison of standard deviation of spectral response from its mean between current results and those of Jetzt (Ref. 1).

Direct-to-reverberant _Standard_{deviation (dB)} energy ratio

(dB) Jetzt's results Current results P(1) = X Z , { ~ [ D ( ~ + D ) I ] ~ ~ ~ } , f o r I a 0 , (13)

-18 .O 5.57 5.55

for I< 0 , -12 .O 5.57 5.55

-6 .O 5.53 5.51

where I, is the modified Bessel function. Equation (13) o .O 5.19 5.08

- .-

can also be derived from the result given by Diestel.'13 3.1 4.35 4.33

In t e r m s of sound p r e s s u r e level, i.e., defining

where k = ln(10)/10, one can also show that

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The author would like to th lbman foi

g his unpublished report4 a. merous d

Ins concerning statistical 1-UUILI acuustics.

rhis pape ~ilding RE ---I-

--

ank D. LI nd for nu

---

!r is a 'contribution from the Divisic !search, National Research Council n w a . ar~d i s ~ u b l i s h e d with the a m r o v a l of 1

rector of the Divii

'J,. J. Jetzt, "Critical (

the sound energy spec 1204-1211 (1979). sion. r supply- liscus

-

-n. cr. UiesLel, --Lur ~ o n a l l a u s b r e i ~ ~ ~ ~ 111 L c r r c m u l t a a L Rtlumen," Acustica 12, 113-118 (1962).

'H. G. Diestel, "Probability distribution of sinusoidal

pressure in a room," J. Acoust. Soc. Am. 35, 2019-

I (1963).

'H. Bowers and D. Lubman, '?)eci ;ing in rev ant rooms," LTVR. Tech. Report ~ T R - 1 3 0 ( V. Waterhouse, "Statistical pi- f reverbel sound fields," J. Acoust. Soc. Am. 43, 1436-1444 (1961

%. T. Chu, '<Computer studies of reverberant sound fie be1 averag ; 0-71200/ operties o men sound 2022 erber- 11968). :ant .

..

rectangular rooms: eigenmode model," J. Acoust. Soc Suppl. 1 65, S52 (1979).

'H. Cramdr, Mathematical Methods of Statistics (Prin leasurement of rooms from U. P., Princeton, N J , 1946), Chap. 15.

Dnse," J. Acoust. Soc. Am. 65, 8 ~Schroeder, <'Die statistischen Parameter d e r Frel . zkuwen von grossen Raumen," Acustica 4, 594-600

5 1.

Ids In

.

Am.

quen- (1954).

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