modulation characteristics for parametric receiving …
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MODULATION CHARACTERISTICS FORPARAMETRIC 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�
JOURNAL DE PHYSIQUE Colloque C8, supplement au N° 11, tome 40, novembve 1979, page C8-140
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 sé un t r a i tement du signal par démodulation de phase et d'amplitude. Les expressions théoriques des composantes de bande latérale aux fréquences somme et différence ont été obtenues à par 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 ce l le -c i . Pour les expériences on a u t i l i sé 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 és au moyen d'un f i l t r e de bande et ecrêtés pour s'assurer 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 sort ie 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 offre 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 components 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 modulation 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 t ime t, the sound waves can be represented
i n complex form as z
-I- x RECEIVER
F ig . 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 n0)- jw2t I , (2)
where
2 2 2 1/2 L = ( x + y + z ) . We found the second order pressure t o be
where
and
I n t h i s example, the c a r r i e r i s pll and
the sidebands a re p2(+) and p2(- ) . If we sum the
th ree s igna ls together and consider on ly the r e a l terms, we have
where
and
and we l e t a+ = s ince w+ - - w1 .
The pressure can then be expressed i n the form
[sinX + wlB cosX COSY - w2B sinX sinY I. (11)
Next, we express the pressure i n terms o f am-
p l i t u d e and phase modulat ion components by making
use o f the 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 in(X + wlB cosy) - w2B sinX sinY I . (12)
We can see t h a t the pressure has two compo-
nents : one t h a t represents a phase modulat ion and one t h a t represents an amplitude modulation. We
a lso see t h a t the phase modulation i s p ropor t iona l
t o u1 w h i l e the ampli tude modulat ion i s propor-
t i o n a l t o w2. Since w, >z w2 f o r the parametr ic
rece iv ing array, the phase modulat ion term w i l l
dominate. I n our next example we consider a l i n e pump
source and a p o i n t receiver . The geometry f o r t h i s
example i s schown i n F ig . 2.
YbLP _I Y
SIGNAL WAVE w2
Fig. 2. - parametr ic Receiving Array w i t h a L ine Source Pump.
c8-142 J.J. TRUCHARD
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) k2 s ine1 - k, s ine1 , (14)
and 6 i s the same as Eq. 6.
Inc lud ing the usual expression f o r the f a r f i e l d o f
a l i n e array, the t o t a l pressure i n the f a r f i e l d
o f the 1 i n e source i s
-Pll expi-al l ] sin(kla s i n e ' ) p = --? sinx ( K1a s i n e t )
Expanding the cosines and rearranging we ge t
-Pll exp[-all] sin(kla s i n e ' ) P=T, [ kla s ine1 sinX
cosx COSY (16)
Remembering t h a t Bul << 1, u2 << ul, we can express t h i s i n the 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 ine '
where
and
Since the c a r r i e r frequency i s much greater than
t h e s igna 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 the second term.
The term sinaA+/aA+ - sinaA-/aA- w i l l a l so be very
small i f the rece ive r i s on the ax is o f the pump
transducer. Therefore, f o r a l l p r a c t i c a l purposes,
the second term i n Eq. (18) can be ignored. We
then have the phase modulat ion component t o be
wlB(ka s i n e ' )
@(t ) = 2 s in (ka s i n e ' )
Both terms i n the 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-
quent ly , f o r most p r a c t i c a l cases o f parametr ic
r e c e i v i n g arrays, the phase modulat ion 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 the amplitude
modulat ion 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 the angle o f the pump transducer i s
changed s l i g h t l y . As a consequence, the beam pat-
t e r n f o r the phase modulat ion term w i l l remain
e s s e n t i a l l y unchanged i f we misa l ign the pump
transducer. On the other 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 misa l ign the pump transducer because i t i n -
cludes a term which i s zero when the pump trans-
ducer i s al igned, b u t nonzero when the pump t rans-
ducer i s misal igned. As a consequence, the beam
pa t te rns f o r the phase modulat ion term and the
ampl i tude modulat ion 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 the 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-
menta l ly .
EXPERIMENTS. - A ser ies 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
studied i n the previous sect ion. A p i c t o r i a l o f
JOURNAL DE PHYSIQUE c8-143
F ig . 3. - P i c t o r i a l f o r the Panmet r i c Receiving Array Experiment
the experiment i s shown i n F ig. 3. The parametr ic pump transducer. The r o l e s o f the two transducers
rece iv ing a r ray i s described i n r e f . 3, The pump could be reversed so t h a t the 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 could be used e i t h e r as a pump o r as a rec ver . source, generated a 90 kHz c a r r i e r frequency which The a r ray was suspended w i t h an I-beam which could i s received by a second transducer 15 m from the be r o t a t e d on a sha 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 Pat te rn w i t h the Pump Transducer Al igned w i t h the Receiver.
~ 8 - 1 4 4 J . J . TRUCHARD
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 prev ious ly . Instead o f a band-reject c r y s t a l
f i 1 t e r , a phase demodulator and ampl i tude demodu-
1 a t o r were used.
Parametric rece iv ing a r ray beam patterms
were made w i t h the phase demodulator and a re shown
i n F ig. 4(a). The experimental r e s u l t s are compa-
red w i t h the theory o f Eq. (18). I n Fig. 4(b) we
have the corresponding example f o r the amplitude detector . When the pump transducer was al igned,
no essen t ia l d i f f e r e n c e was found between the beam
p a t t e r n w i t h the phase detector and the beam pat-
t e r n w i t h the amplitude detector . I n F ig. 5(a) -we
show an example where the pump was misal igned so
t h a t the response o f t h e pump s igna l was 3 dB down.
The corresponding beam p a t t e r n f o r the output o f
the amplitude de tec to r i s shown i n Fig. 5(b). We
see t h a t the beam pa t te rn w i t h the phase demodula-
t o r i s e s s e n t i a l l y unchanged wh i le the beam pat-
t e r n f o r the ampli tude demodulator i s q u i t e d i f fe-
r e n t . This d i f f e r e n c e occurs because the ampl i tude
modulat ion term as expressed i n Eq. (19) inc ludes
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 the symmetrical component fo 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'
du la to r remains near l y unchanged, w h i l e the beam
p a t t e r n f o r the amplitude demodulator i s dramatical-
1y.changed when the pump transducer i s misaligned. Some d i f f i c u l t y i n the experimental measure-
ment o f the amplitude modulat ion 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 the phase modulation components a re considerably
l a r g e r than the amplitude modulat ion components,
the lack o f f l a t n e s s i n the hydrophone response
caused a small component o f the phase modulat ion
t o appear as an amplitude modulation. Equal izat ion
o f the hydrophone response would e l im ina te t h i s
problem. This e f f e c t increases the symmetrical
component i n the beam pa t te rn .
CONCLUSIONS. - I n t h i s paper, we have der ived
the expressions f o r the phase and ampli tude modula-
t i o n components o f a low frequency wave modulat ing
a h igh frequency c a r r i e r i n the example o f the pa-
rametr ic rece iv ing 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
Fig. 5. - 5 kHz Beam Pat te rn w i t h the Pump Transducer Rotated t o I t s 3 dB Down Point.
JOURNAL DE PHYSIQUE
ACKNOWLEDGMENTS
The author would l i c e to express his appreciation programs. This work was sponsored in part by the
to the many people a t Applied Research Laboratories Office of Naval Research and the Naval Sea Systems
who assisted in the construction of equipment for Command.
the experiments and in 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 Finite 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).