APPLICATION OF FAST HARTLEY TRANSFORM TO ACOUSTIC INTENSITY MEASUREMENT

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APPLICATION OF FAST HARTLEY TRANSFORM TO ACOUSTIC INTENSITY MEASUREMENT

W. Gan

To cite this version:

W. Gan. APPLICATION OF FAST HARTLEY TRANSFORM TO ACOUSTIC INTEN- SITY MEASUREMENT. Journal de Physique Colloques, 1990, 51 (C2), pp.C2-721-C2-724.

�10.1051/jphyscol:19902167�. �jpa-00230470�

Colloque C2, supplement au no2, Tome 51, Fevrier 1990 l e r Congres FranCais d'acoustique 1990

APPLICATION OF FAST HARTLEY TRANSFORM TO ACOUSTIC INTENSITY MEASUREMENT W.S. GAN

Acoustical Services (1989) Pte Ltd, 29 Telok Ayer S t r e e t , Singapore 0104. Republic o f Singapore

A b s t r a c t

-

We propose t h e a p p l i c a t i o n of f a s t H a r t l e y transform (FHT) 30 a c o u s t i c i n t e n s i t y measurement t o r e p l a c e f a s t F o u r i e r t r a n s f orm

(FFT). The FHT has s p e c i a l a d v a t a g e of c a l c u l a t i n g a c o u s t i c i n t e n - s i t y because i t i s real-valued.

1

-

INTRODUCTION

The two-microphone a c o u s t i c i n t e n s i t y technique i s now a s t a n d a r d method of measuring a c o u s t i c i n t e n s i t y f o r both t h e n e a r f i e l d a s well a s t h e f a r - f i e l d . During t h e l a s t decade, t h e development of t h e f a s t F o u r i e r trmls- form (EFT) s i g n a l p r o c e s s o r , t h e improvements of microphones ,aid t h e per- f e c t i o n of e l e c t r o n i c technology enable t h e commercialisation o f a r e l i a b l e a c o u s t i c i n t e n s i t y meter s u i t a b l e f o r both t he laboratory and f i e l d measure- m n t s of sound i n t e n s i t y . I n t h i s p a p e r , t h e f a s t H a r t l e y transform (FHT) i s a p p l i e d t o a c o u s t i c i n t e n s i t y measurement t o r e p l a c e f a s t F o u r i e r transform (Pm). The H a r t l e y transform of a real-valued f u n c t i o n i s a l s o real-valued and i t s e v a l u a t i o n does n o t involve complex f u n c t i o n and t h u s have a speed advantage over FFT. I n a d d i t i o n , using H a r t l e y transform, one has t h e ad- vantage of having t h e same formula f o r forward and i n v e r s e transform and t e n d s t o r e q u i r e l e s s memory space s i n c e complex numbers a r e n o t used and

a s e p a r a t e i n v e r s i o n p r o g r ~ m i s n o t required.

2

-

PROPERTIES OF HARTLEY TRANSNRM

Given t h e r e a l d a t a s e t f ( X ) , t h e d i s c r e t e H a r t l e y t r a n s f o r n R(s) i s d e f i n e d a s follows:

N- l

H(s) = 1 / N f ( X ) c a s (2Tsx/N)

x=o

where c a s

0(=

c o s d + s i n

d .

The d i s c r e t e F o u r i e r transform F ( s ) i s d e f i n e d a s

N - l

F ( s ) = V N f (X) exp

L

^{-12 PX/N}

1

x=o

and t h e well-known f n c t o r i n g r e l a t i o n s h i p s F ( s ) = F e ( s )

+

F,( S) e r p

C-

i 2 F s/ N

I

F(s3m) = F e ( s )

-

F O ( s ) exp

C-

i 2 T s / ~ ) where W = 2m

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902167

C2-722 COLLOQUE DE PHYSIQUE

a r e used t o determine F ( s ) . ' i n t h e f a s t Fourier analysis where t h e F o u r i e r transforms over t h e even and odd d a t a , r e s p e c t i v e l y , a r e denoted by Fe(s) and Fo(s). From (1) and ( 2 ) , t h e FFT can be obtained from t h e FHT using t h e conversion /l/

F(.) = I H ( S )

+

iR(-s)]

/

( l

+

i ) (53

Thus, using (3) and (5), t h e R a r t l e y transform c a r be separated a s follows:

H ( s ) = He(s)

+

Ro(s). cos (27i-s/N)

+

R,(-S) s i n (2Ts/l?) (6) H(-s) = He(-S)

+

H,(-S) cos (2Ts/N)

-

Ho(s) s i n ( 2 r s / N )

(7)

3

-

D&KCVA(PION OF EXPRESSIONS FOR ACOUSTIC INTENSITY /2/

The most common i n t e n s i t y aeasurement technique uses two c l o s e l y spaced pre- s s u r e microphones t o o b t a i n t h e sound pre'ssure from t h e ,average, and t h e p a r t i c l e v e l o c i t y from f he d i f f erence of t h e two measured sound pressures.

I n t h e frequency domain, t h e complex i n t e n s i t y can be expressed i n terms of t h e auto-spectra and cross-spectra between t h e two microphones

-2

where Gll(&) and G22(Cc)) a r e t h e p r e s s u r e auto-spectral d e n s i t i e s a t

-

t h e microphone l o c a t i o n s , G12(&) i s t h e complex c r o s s - s p e c t r a l d e n s i t y and b r i s t h e d i s t a n c e between t h e microphones. Both i n t e n s i t y e s t i m a t e s oan be obtained from

and Q ( W ) = I m

1

^{~ ~ ==' (1 q}[ G 2 2 ( u )

-

^Gll(O)

]

2 ~ p ~ '

The p o t e n t i a l energy d e n s i t y is' s i m i l a r l y obtained from t h e s p a t i a l p r e s s u r e average

The k i n e t i c energy estimate i s derived from t h e p a r t i c l e v e l o c i t y which i s obtained from t h e pressure g r a d i e n t

and T(Q) =

2C Gll((cl )

+

G22(G1 ) ~ 2 R e ( G * ~ ( 4 1 ) The above Eqs. ( 9 ) t o (14) show t h a t a l l q u a n t i t i e s of i n t e r e s t can be

obtained from auto- and c r o s s - s p e c t r a l d e n s i t i e s between two c l o s e l y spaced p r e s s u r e sensors.

4

-

APPLICA'PLur;' up HBR'PLEY TRANSFORM TO ACOUSTIC INTEVSITY F i r s t t h e power spectrum can be obtained from (5) a s

avoided a s well a s the necessity of ever converting the Hartley transform t o the Fourier transform. Then t h e auto-spectra can be obtained by Hartley transforming t h e power spectrum. A s a t e s t of t h e theory, t h e two-dimensional s p e c t r a and a u t o c c r r e l a t i o n o f a two-dimensional Rect function, Rect(x)

-

Rect(y) were performed. As expected, t h e power spectrum was sincZu ~ i n c %

(Sinc X Zi s i n ~ x / 5 ? X ) , and t h e a u t o c o r r e l a t i o n was a two-dimensional t r a i n g l e function r\ (X) A (y)

.

^{Thi S w i l l} give a two-dimensional B-Scan image of the acoustic i n t e n s i t y . The common two-microphone method only gives a one-dimensional A-Seem image of the acoustic i n t e n s i t y .

5 -

^A WEB EXTELtSION TO !iON-LINEm ACOUSTIC INTEITSITY One s t a r t s w i t h t h e inhomogeneous Helmholtz equation

whe?e c = speed of sound, Q' = sound source and

S=

sound f i e l d . To solve f o r uonlinear acoustic i n t e n s i t y , t h e nonlinear sound source has t o be used.

LighthTll /3/ defined t h e source Q' a s v2,0

,

where

P

i s t h e mass d e n s i t y and rearrange thew ave equation a s

where T ;

,

⁼ L i g h t h i l l ' ^S s t r e s s tensor.

A J

I n (17), t h e l i n e a r sound f i e l d described by the left-hand s i d e i s driven by the non-linear t e r m s on the right-hand side. To i l l u s t r a t e nonlinear

sound sources, one uses t h e model t h a t a small region of a homogeneous gas i s i n low Mach number t u r b u l e n t motion and t h i s i s t h e nonlinear region.

This model has relevance t o the n o i s e produced by subsonic a i r j e t s . It i s Reynolds s t r e s s element of L i g h t h i l l ' s s t r e s s t e n s o r t h a t generates t h a t sound and L i g h t h i l l ' s formula f o r the density f i e l d f o r ^this nonlinear sound source i s given by

where y = p o s i t i o n ordinate and u i , u j = sound v e l o c i t y . L i g h t h i l l showed t h a t a t low Mach number t h e sources

+- ^p

Uluj and p i i = P Jui

[

, +

a r e i d e n t i c a l so t h a t t h e sound source can a l t e r n a t i v e l y be thought of a s t h e product of the pressure f l u c t u a t i o n and t h e s t r a i n r a t e i n t h e d i r e c t i o n of radiation2

(20) i s t h e expression f o r t h e nonlinear acoustic i n t e n s i t y f o r low Mach number case. ( v = volume).

Prom (20), one can see t h a t nonlinear acoustic i n t e n s i t y involves l a r g e matrices. A s t h e s i z e of t h e matrix becomes l a ~ g e r , a u t o c o r r e l a t i o n i n t h e

s p a t i a l domain r e q a i r e s longer time than a u t o c o r r e l a t i o n i n %he frequency domain. Hence one decides t o use a u t o c o r r e l a t i o n i n t h e frequency domain f o r t h e nonlinear acoustic i n t e n s i t y . Here a key advantage of t h e Hartley t r a n s - form f o r l a r g e a u t o c o r r e l a t i o n i s t h e use of t h e same formula f o r

C2-724 COLLOQUE DE PHYSIQUE

transforming and inverse transforming.

/l/ Bracewell, R.N.

,

The Hartley Transform, Clarendon P r e s s , O x f a r d, 1986.

/2/ Elko, G.W., Proceedings of t h e 2ud I p t e s n a t i o n a l Congress on Acoustic I n t e n s i t y , S e n l i s , France, (1985) 69.

/3/

L i g h t h i l l , M.J., Proc. Roy. Soc. (1952)

564.