MODULATION CHARACTERISTICS FOR PARAMETRIC RECEIVING ARRAYS

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MODULATION CHARACTERISTICS FOR PARAMETRIC RECEIVING ARRAYS

James Truchard

To cite this version:

James Truchard. MODULATION CHARACTERISTICS FOR PARAMETRIC RECEIVING AR- RAYS. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-140-C8-145. �10.1051/jphyscol:1979825�.

�jpa-00219530�

MODULATION CHARACTERISTICS FOR PARAMETRIC RECEIVING ARRAYS James J. TRUCHARD

Applied Research Laboratories The University of Texas at Austin Austin, Texas 78712, U.S.A.

Résumé. - Lors d'études expérimentales de réseaux paramétriques de réception on a u t i l i s é un t r a i - tement du signal par démodulation de phase et d'amplitude. Les expressions théoriques des compo- santes de bande latérale aux fréquences somme et différence ont été obtenues à p a r t i r de l'équation d'onde du second ordre, due â Westervelt. Ces composantes ont été additionnées à la porteuse puis exprimées comme composantes de modulation de c e l l e - c i . Pour les expériences on a u t i l i s é un roseau paramétrique de 15 m, fonctionnant à 90 kHz, avec des fréquences de signal de 3 à 6 kHz. Les s i - gnaux d'entrée de l'hydrophone ont été f i l t r é s au moyen d'un f i l t r e de bande et ecrêtés pour s'as- surer de la suppression de toute modulation d'amplitude, et la modulation de phase du signal a été mesurée à l'aide d'un détecteur de phase.

Les diagrammes de rayonnements mesurés â la sortie du détecteur de phase restent inchangés lors du désalignement de la pompe ou de l'hydrophone dans le réseau, par contre les composantes de modulation d'amplitude sont considérablement modifiées. Ce résultat o f f r e un bon accord avec les calculs théoriques. La théorie et l'expérience montrent clairement que la modulation d'un son de fréquence élevée par une onde sonore de basse fréquence est essentiellement une modulation de phase.

Abstract. - Experiments have been conducted on the parametric receiving array using amplitude and phase demodulators for signal processing. Theoretical expressions were derived by finding the sum and difference frequency sideband components using the second-order wave equation originated by Westervelt. The sum and difference frequency components were added to the carrier and".then expressed in terms of modulation components of the carrier. Experiments were conducted using a 15 m parametric receiving array operating at 90 kHz. Signal frequencies in the renge from 3 to 6 kHz were used.

The hydrophone input signals were bandpass filtrered and then clipped to ensure that no amplitude modulation was left on the signal. A phase detector was used to observe the phase modulation of the signal. Beam patterns measured at the phase detector output were unchanged when either the pump or the hydrophone in the array was misaligned. On the other hand, the amplitude modulation compo- nents changed dramatically when the pump or the hydrophone was misaligned. This result agreed well with the theoretical expressions. The theory and the experiment clearly demonstrate that the modu-

lation of a high frequency sound wave by a low frequency sound wave is primarily a phase modulation.

1. INTRODUCTION. - In previous papers, /1-3/ the au- thor studied the sideband characteristics of the parametric receiving array signal. Theoretical ex- pressions were derived for the sum and the diffe- rence frequency components for several geometries for the parametric receiving array. The amplitude of each sideband signal was measured and compared to the theoretical expressions. The theory agreed well with the experimental results in both the case of aligned and misaligned transducers. The experi- ments, however, did not measure the relative phase between the carrier and the two sideband signals.

In this paper, the theory is formulated in a form which can be used to verify experimentally the phase

characteristics of the sideband signals. In order to accomplish this, the modulation processes are described in terms of amplitude and phase modula- tions. The theoretical results which were previously obtained in Ref. 2 are reworked into a form which gives the expression for the amplitude and the pha- se modulation caused by the modulation process.

Since the phase of the two sideband signals is related to the type of modulation, we have a means to experimentally measure the phase characteris- tics of the sideband signals by measuring the am- plitude and phase modulation components of the signal. In this paper, we describe some experi- ments which were conducted using amplitude and phase demodulators for the measurement of the modulation process. We find that expressions for- mulated in the previous paper, when expressed in terms of amplitude and phase modulation terms, do agree well with the actual experimental results.

2. MODULATION CHARACTERISTICS FOR CASES OF THE PARAMETRIC RECEIVING ARRAY. - We will now reconsi- der the parametric receiving array with an omnidi- rectional pump. The geometry for this case is gi- ven Fig. 1. The signal source is assumed to be far from the parametric receiver so that the low fre- quency waves are approximately planar in the vici- nity of the receiver. Then at the point (X.Y.Z)

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979825

JOURNAL DE PHYSIQUE c8-141

and a t time t, t h e sound waves can be represented i n complex form as

z

-I- x RECEIVER

F i g . 1.

-

Parametric Receiving Array w i t h a p o i n t Source Pump.

P1 1

P1 1 ^.:j exp I-(al-jki) L-jwltI (1)

and

p12 = P12 exp 1-(a2-jk2) ( x cose + Y s i n 0 ) - j w 2 t

I ,

⁽²⁾

where

2 2 2 1/2

L = ( x + y + z )

.

We found t h e second order pressure t o be

where

and

I n t h i s example, t h e c a r r i e r i s pll and t h e sidebands a r e p2(+) and p 2 ( - ) . If we sum t h e t h r e e s i g n a l s t o g e t h e r and consider o n l y the r e a l terms, we have

where

and

and we l e t a+

-

= s i n c e w+

-

^w1

^.

The pressure can then be expressed i n t h e form

[sinX

+

wlB cosX COSY

-

w2B sinX sinY I. (11) Next, we express t h e pressure i n terms o f am- p l it u d e and phase modulation components by making use o f t h e approximations

and

cos(wlB cosy) 1

,

since lw16 cosy1 << 1. Using the i d e n t i t y

[sin(C+D) = sinC cosD

+

cosC sinD], we can express the pressure as

[ s i n ( X

+

^{wlB cosy)}

-

w2B sinX sinY I

.

⁽¹²⁾

We can see t h a t t h e pressure has two compo- nents : one t h a t represents a phase modulation and one t h a t represents an amplitude modulation. We a l s o see t h a t t h e phase modulation i s p r o p o r t i o n a l t o u1 w h i l e t h e amplitude modulation i s propor- t i o n a l t o w2. Since w, >z w2 f o r t h e parametric r e c e i v i n g array, t h e phase modulation term w i l l dominate.

I n o u r n e x t example we consider a l i n e pump source and a p o i n t r e c e i v e r . The geometry f o r t h i s example i s schown i n F i g . 2.

Y b L P ^_I

Y

SIGNAL WAVE w 2

F i g . 2.

-

parametric Receiving Array w i t h a L i n e Source Pump.

The sideband pressure f o r the sum and d i f f e - rence frequency components was found t o be 1-3

where

A*

. ^{* $}

k2 s i n (8+g1) k 2 s i n e 1

-

^k, s i n e 1 , (14) and

6

i s t h e same as Eq. 6.

I n c l u d i n g t h e usual expression f o r t h e f a r f i e l d o f a l i n e array, t h e t o t a l pressure i n t h e f a r f i e l d o f t h e 1 in e source i s

-Pll expi-all] sin(kla s i n e ' ) p = --? s i n x

(

K1a s i n e t

)

Expanding the cosines and r e a r r a n g i n g we g e t -Pll exp[-all] sin(kla s i n e ' )

P=T,

[

kla s i n e 1 sinX

cosx COSY (16)

Remembering t h a t Bul << 1, u2 << ul, we can express t h i s i n t h e f o l l o w i n g approximate form :

exp [-alL ] sin(kla s i n e ' ) p = -P

11

7

kla s i n e '

where

and

Since t h e c a r r i e r frequency i s much g r e a t e r than t h e s i g n a l frequency, t h e f i r s t term i n Eq. (18) w i l l be many times l a r g e r than t h e second term.

The term sinaA+/aA+

-

sinaA-/aA- w i l l a l s o be v e r y small i f the r e c e i v e r i s on t h e a x i s o f t h e pump transducer. Therefore, f o r a l l p r a c t i c a l purposes, t h e second term i n Eq. (18) can be ignored. We then have t h e phase modulation component t o be

wlB(ka s i n e ' )

@ ( t ) = 2 s i n ( k a s i n e ' )

Both terms i n t h e amplitude modulation expression w i l l be small because, i n general,

sinaA+/aA+

-

sinaA-/aA- w i l l be a small term even when m u l t i p l i e d by ul. The second term i s propor-

t i o n a l t o w2 and w i l l be very small also. Conse- q u e n t l y , f o r most p r a c t i c a l cases o f parametric r e c e i v i n g arrays, t h e phase modulation component w i l l be s u b s t a n t i a l l y l a r g e r than t h e amplitude modulation component. The expression

sinaA+/aA+

+

sinaA-/aA- w i l l remain e s s e n t i a l l y constant as t h e angle o f t h e pump transducer i s changed s l i g h t l y . As a consequence, t h e beam p a t - t e r n f o r t h e phase modulation term w i l l remain e s s e n t i a l l y unchanged i f we m i s a l i g n t h e pump transducer. On t h e o t h e r hand, the. ampl i tude modu- l a t i o n component w i l l be s i g n i f i c a n t l y changed as we m i s a l i g n t h e pump transducer because i t i n - cludes a term which i s zero when t h e pump trans- ducer i s aligned, b u t nonzero when t h e pump t r a n s - ducer i s misaligned. As a consequence, t h e beam p a t t e r n s f o r t h e phase modulation term and t h e ampl i tude modulation term w i l l be s u b s t a n t i a l l y d i f f e r e n t when t h e pump transducer i s misaligned.

This c h a r a s t e r i s t i c can be e a s i l y v e r i f i e d experi- m e n t a l l y .

EXPERIMENTS.

-

A s e r i e s o f experiments was conduc- t e d t o v e r i f y t h e c h a r a c t e r i s t i c s which we have s t u d i e d i n t h e previous section. A p i c t o r i a l o f

JOURNAL DE PHYSIQUE c8-143

F i g . 3.

-

P i c t o r i a l f o r t h e P a n m e t r i c Receiving Array Experiment

the experiment i s shown i n F i g . 3. The parametric pump transducer. The r o l e s o f t h e two transducers r e c e i v i n g a r r a y i s described i n r e f . 3, The pump c o u l d be reversed so t h a t t h e l i n e transducer transducer, which i s e i t h e r a p o i n t source o r l i n e c o u l d be used e i t h e r as a pump o r as a r e c v e r . source, generated a 90 kHz c a r r i e r frequency which The a r r a y was suspended w i t h an I-beam which could i s received by a second transducer 15 m from t h e be r o t a t e d on a s h a f t . The e l e c t r o n i c apparatus

- ----

THEORETICAL EXPERIMENTAL

(a) PHASE DETECTOR OUTPUT (b) AMPLITUDE DETECTOR OUTPUT 4.

-

⁵ kHz Beam P a t t e r n w i t h t h e Pump Transducer Aligned w i t h t h e Receiver.

f o r these experiments was q u i t e d i f f e r e n t from t h a t used p r e v i o u s l y . Instead o f a band-reject c r y s t a l f i 1 te r

,

a phase demodulator and ampl i tude demodu- 1 a t o r were used.

Parametric r e c e i v i n g a r r a y beam patterms were made w i t h t h e phase demodulator and a r e shown

i n F i g . 4(a). The experimental r e s u l t s a r e compa- r e d w i t h the theory o f Eq. (18). I n Fig. 4(b) we have t h e corresponding example f o r the amplitude d e t e c t o r . When t h e pump transducer was aligned, no e s s e n t i a l d i f f e r e n c e was found between t h e beam p a t t e r n w i t h t h e phase d e t e c t o r and the beam pat- t e r n w i t h t h e amplitude d e t e c t o r . I n F i g . 5(a) -we show an example where t h e pump was misaligned so t h a t t h e response o f t h e pump s i g n a l was 3 dB down.

The corresponding beam p a t t e r n f o r t h e o u t p u t o f t h e amplitude d e t e c t o r i s shown i n Fig. 5(b). We see t h a t t h e beam p a t t e r n w i t h t h e phase demodula- t o r i s e s s e n t i a l l y unchanged w h i l e the beam p a t - t e r n f o r t h e amplitude demodulator i s q u i t e d i f f e - r e n t . This d i f f e r e n c e occurs because the ampl i tude modulation term as expressed i n Eq. (19) i n c l u d e s

an asymmetrical component which i s s i m i l a r i n amk, p l i t u d e t o t h e symmetrical component f o r our p a r t i - c u l a r example. The beam p a t t e r n w i t h a phase demo-

0'

d u l a t o r remains n e a r l y unchanged, w h i l e t h e beam p a t t e r n f o r t h e amplitude demodulator i s dramatical- 1y.changed when t h e pump transducer i s misaligned.

Some d i f f i c u l t y i n t h e experimental measure- ment o f t h e amplitude modulation occurred due t o a l a c k o f f l a t n e s s i n t h e hydrophone response. Since t h e phase modulation components a r e considerably l a r g e r than t h e amplitude modulation components, t h e l a c k o f f l a t n e s s i n t h e hydrophone response caused a small component o f t h e phase modulation t o appear as an amplitude modulation. E q u a l i z a t i o n o f t h e hydrophone response would e l i m i n a t e t h i s problem. This e f f e c t increases t h e symmetrical component i n t h e beam p a t t e r n .

CONCLUSIONS.

-

I n t h i s paper, we have d e r i v e d t h e expressions f o r t h e phase and amplitude modula- t i o n components o f a low frequency wave modulating a h i g h frequency c a r r i e r i n t h e example o f t h e pa- r a m e t r i c r e c e i v i n g array. We have v e r i f i e d these expressions w i t h experiments.

(a) PHASE DETECTOR OUTPUT (b) AMPLITUDE DETECTOR OUTPUT

F i g . 5.

-

⁵ kHz Beam P a t t e r n w i t h t h e Pump Transducer Rotated t o I t s 3 dB Down Point.

JOURNAL DE PHYSIQUE

ACKNOWLEDGMENTS

The author would l i c e t o express his appreciation programs. This work was sponsored i n part by the t o the many people a t Applied Research Laboratories Office of Naval Research and the Naval Sea Systems who assisted in the construction of equipment f o r Command.

the experiments and i n the development of computer REFERENCES

/1/ J.J. Truchard, "The Detection of a Low-Frequen- /3/ J.J. Truchard, "Parametric Acoustic Receiving cy Plane Wave with a Parametric Receiving Array. 11. Experiment1', J . Acoust. Soc. Am.

58,

Arraym1, Paper 2.12, presented a t the 1973 1146-1150.

Symposium on F i n i t e Amplitude Wave Effects in Fluids, Copenhagen, Denmark.

/2/ J . J . Truchard, "Parametric Acoustic Receivinq . -

Array. I.

heo or^",

J . Acoust. Soc. Am.

58, ^-

1141-1145 (1975).