The analysis of arrays using STARPAK

THE A N A L Y S I S OF A R R A Y S US I NG S T AR P A K M. J. Wi l m u t and W. W. W o l f e Royal Ro ad s M i l i t a r y C ollege FMO V i ctoria, B. C. VOS 1 BO ABSTRACT So m e of the m a i n f e a t u r e s and a typical a p p l i c a t i o n of S T A R P A K ( S i m u l a t i o n fo r T e s t i n g A r r a y Res po n se ) is presented. S TARPAK is a p a c k a g e of Fo r t r a n s u b r o u t i n e s d e s i g n e d to s t u d y the p e r f o r m a n c e of an a r b i t r a r y p l an a r ar r a y in a v a r i e t y of G a u s s i a n s i g n a l - n o i s e en v ir o n m e n t s . A r r a y d a t a w i t h the a p p r o p r i a t e s t a t i s t i c s are si m u l a t e d and then p r o c e s s e d using t he c o n v e n t i o n a l , optimal and B i e n v e n u tec h ni q ue s . S O M M A IR E Q u e l q u e s uns des tr a i t s p r i n c i p a u x ainsi q u ' u n e a pp l ication p r a t i q u e de "ST A RP A K" sont p r és entés. "STARPAK" est un e n s e m b l e de s o u s - r o u t i n e s en la ng u ag e "Fortran" qui a pour but d ' é t u d i e r le f o n c t i o n n e m e n t d'un g r o u p e q u e l c o n q u e de d é t e c t e u r s en p r é s e n c e d'un signal et d'un bruit gaussien. Les s i g n a u x r e c u e i l l i s par les d é t e c t e u r s de m ê m e que les do n né e s s t a t i s t i q u e s sont simulés, et, par la suite, a nalysés en se s e v a n t des t e c hn i qu e s: c o n v e n t i o n n e l , optimal et "Bienvenu". 1. Introduction The d e t e c t i o n of sinusoidal si g na l s in ocean a mbient no i se is an important p r o b l e m in u n d e r w a t e r a c ou s tics^. The a coustic e n e r g y r a d i a t e d by ships, s u b m a r i n e s and t o r p e d o e s are e x a m p l e s of such sinusoidal signals. The data a v a i l a b l e upon w h i c h to m a k e a d e c i s i o n (signal or no signal present) are often t h o s e r e c e i v e d at an a r r a y of h y d r o p h o n e s whi c h is located in the ocean. The d e t e c t i o n of s i g n a l s using a r r a y d a t a is a ver y c o m p l e x pr ob l em 2 and m a n y q u e s t i o n s can o n l y be a n s we r ed using numerical s i m u l a t i o n during t he analysis. S T A R P A K 3 is a p a c k a g e of F o r t r a n p r o g r am s useful in such a study. He r ei n w e d e s c r i b e the user f r i e n d l y i n pu t -o u tp u t f e a t u re s of STA R PA K and i ndicate h o w t h e p r o g r a m p a c k a g e m a y be e m p l o y e d to g e n e r a t e r a n d o m ar r ay data for an i m p o r t a n t cla s s of real w o r l d sinusoidal s i g n a l - o c e a n noi s e scenarios. S T A R P A K c o n t a i n s e f f i c i e n t a l g o r i t h m s for three m e t h o d s often s u g g e st e d for use in the d e t e c t i o n of sign a ls using arrays. F i n a l l y some c o m p a r i s o n s are m a d e of t h e s e p r o c e s s i n g t e ch niques. 2. Theory a) S t a t e m e n t of the P r o b l e m We a s su m e w e have a p l a n a r arr a y of n h y d r o p h o n e s located at a r b i t r a r y p o s i t i o n s in th e plane. The si gn a ls are sinusoidal and hen c e a fi rs t step in 19 t h e a n a l y s i s i s t o t r a n s f o r m t h e d a t a t o t h e f r e q u e n c y doma i n. In t h i s domai n s i g n a l s and n o i s e a r e assumed t o be c ompl ex G a u s s i a n v a r i a b l e s . The n o i s e i s model l d as he sum o f w h i t e , c y l i n d r i c a l and s p h e r i c a l n o i s e s w i t h a r b i t r a r y powers, s p h e r i c a l n o i s e i s n o i s e g e n e r a t e d by a l a r g e number o f d i s c r e t e s o u r c e s u n i f o - m l y d i s t r i b u t e d on a s p h e r e whose r a d i u s i s much 1a r g e r t h a n t h e dimensions of t h e a r r a y . C y l i n d r i c a l n o i s e i s n o i s e g e n e r a t e d by a l a r g e number o f d i s c r e t e s o u r c e s u n i f o r m l y d i s t r i b u t e d o v e r t h e s u r f a c e o f a c y l i n d e r whose r a d i u s i s much l a r g e r t h a n t h e d i m e n s i o n s o f t h e a r r a y and whose a x i s i s normal to the array plane. Whi t e n o i s e i s due t o a l a r g e number o f d i s c r e t e s o u r c e s whi ch a r e l o c a t e d c l o s e t o e a c h s e n s o r , and i s i n d e p e n d e n t f r om s e n s o r t o sensor. The s i g n a l s a r e p l a n e waves o f a s p e c i f i c power and d i r e c t i o n a r r i v i n g in t h e p l a n e o f t h e a r r a y . Our aim i s t o s i m u l a t e d a t a r e c e i v e d a t an a r r a y f o r t h e above s i g n a l n o i s e s c e n a r i o and t h e n p r o c e s s t h e d a t a w i t h a l g o r i t h m s s u i t a b l e f o r s i g n a l d e t e c t i o n i n s uc h s i t u a t i o n s . b) Da t a S i m u l a t i o n At e a c h h y d r o p h o n e t h e d a t a a r e z e r o mean G a u s s i a n . Hence t h e d a t a r e c e i v e d a t t h e a r r a y a r e c o m p l e t e l y d e c r i b e d by i t s c o v a r i a n c e m a t r i x Q. The f u n c t i o n a l f or m o f Q d e p e n d s on t h e s i g n a l - n o i s e s c e n a r i o ^ ^ . [_e t E be an n d i m e n s i o n a l v e c t o r o f i n d e p e n d e n t compl ex z e r o mean G a u s s i a n random v a r i a b l e s . S u b r o u t i n e s a r e a v a i l a b l e t o s i m u l a t e s uc h d a t a . Al s o s u p p o s e Q = U*U i s a Cholesky deco mp o s i ti on s of t h e c o v a r i a n c e m a t r i x . Here * d e n o t e s compl ex conjugate transpose. In s uc h a d e c o m p o s i t i o n U i s a l o we r t r i a n g u l a r m a t r i x . Then X = U*E i s a v e c t o r s a mp l e w i t h t h e r e q u i r e d s t a t i s t i c s . An i m p o r t a n t q u a n t i t y i n t h e f o l l o w i n g s e c t i o n i s t h e s a mp l e d c o v a r i a n c e m a t r i x , Q. It is d e f i n e d as t h e a v e r a g e o f a number o f XX* s a m p l e s . c) Array P r oc es s i n g The a d v a n t a g e s o f an a r r a y o v e r a s i n g l e s e n s o r a r e nu me r o u s . One o f t h e most i m p o r t a n t f e a t u r e s i s i t s d i r e c t i o n a l p r o p e r t y - whi ch e n a b l e s i t t o d i s c r i m i n a t e b e t we e n s i g n a l s a r r i v i n g f r um d i f f e r e n t d i r e c t i o n s . The d i r e c t i o n we a r e i n t e r e s t e d in a t a p a r t i c u l a r i n s t a n t i s c a l l e d t h e " l o o k d i r e c t i o n " . The c e n t r a l t a s k o f a r r a y p r o c e s s i n g i s t o i n v e s t i g a t e t e c h n i q u e s whi c h r e d u c e t h e e f f e c t due t o n o i s e f r o m " n o n - l o o k " d i r e c t i o n s . STARPAK e x a mi n e s t h r e e s uch methods. Al l r e q u i r e knowl e dge o f t h e s a mpl e d c o v a r i a n c e m a t r i x and o n e , t h e s i n g l e f r e q u e n c y v e r s i c i o f B i e n v e n u ' s d e t e c t i o n t e s t t e c h n i q u e ^ a s s ume s a p r i o r i knowl e dge o f t h e n o i s e o n l y c o v a r i a n c e m a t r i x , Q^. The c o n v e n t i o n a l ? ( B e ) , o p t i m a l ? (Bo) and Bi e n v e n u (Bb) beam o u t p u t s f o r d i r e c t i o n 0 a r e : Be ( 0) = C*D*QD0 C ( 1) a Bo( 0) = n 2 / ( C * D * Q - l D C) (2)a 0 S C*D*Q_1D C Bb{0) = ------- I — I■ c * d * q * - 1 q n q - 1 d0 c wher e C = [ l , . . . l ] * tion. ( 3} a and D_ i s t h e d i a g o n a l 20 s t e e r i n g m a t r i x i n t h e l ook d i r e c ­ A c o m p l e t e d e s c r i p t i o n of t h e s e p r o c e s s o r s and t h e i r p r o p e r t i e s is b e y o n d t h e s c o p e of t h i s p a p e r . H o w e v e r s o m e of t h e i r c h a r a c t e r i s t i c s are n o t e d here. In c o n v e n t i o n a l b e a m f o r m i n g t h e p h a s e s of t h e s e n s o r i n p u t s are a d j u s t e d so th at a s i g n a l f r o m t h e lo ok d i r e c t i o n a d d s c o h e r e n t l y . An o p t i m a l b e a m f o r m e r r e s u l t s w h e n w e p r o c e s s t h e d a t a so t h a t a c o n s t a n t sign al r e s p o n s e is m a i n t a i n e d in the lo ok d i r e c t i o n an d t h e p o w e r f r o m n o n - l o o k d i r e c t i o n s is m i n i m i z e d . An o p t i m a l b e a m f o r m e r is no t in g e n e r a l o p t i m u m f o r t h e d e t e c t i o n q u e s t i o n p o s e d he re . Th e B i e n v e n u s t a t i s t i c is b a s e d on t h e t h e o r y of h y p o t h e s i s t e s t i n g ? . If t h e d a t a ar e n o i s e o n l y Q w i l l in g e n e r a l be c l o s e t o ^ Q ^ and t h e b e a m o u t p u t c l o s e to one. In t h e c a s e w h e r e a sign al is p r e s e n t 0 will in g e n e r a l be d i f f e r e n t t h a n Q|\| an d t h e B i e n v e n u b e a m o u t p u t g r e a t e r t h a n o n e f o r t h e look d i r e c t i o n s containing signals. It is e a s i l y v e r i f i e d t h a t t h e a b o v e f o r m u l a e c a n be w r i t t e n as: B e (0) = |U D Q C | 2 0)b Bo (0 ) = n 2 / ( Y * Y ) w h e r e tl*Y = Dg C (2)b Bb(0 ) = ( Y * Y ) / ( Z * Z ) w h e r e Z = BX and and w h e r e 0X = D0 C (3 )b = B * B and Q = U*U are C h o l e s k y d e c o m p o s i t i o n s . U s i n g t h e s e c o n d set of e q u a t i o n s w e are a b l e to o b t a i n t h e b e a m o u t p u t s w i t h o u t e v a l u a t i n g t h e i n v e r s e of §, a v e r y d i f f i c u l t n u m e r i c a l p r o b l e m . In our i m p l e m e n t a t i o n o n c e Ô and Q m h a v e b e e n C h o l e s k y d e c o m p o s e d t h e b e a m o u t p u t s are c a l c u l a t e d b y c a r r y i n g o u t f o r w a r d and b a c k w a r d s u b s t i t u t i o n in t w o s y s t e m s of l i n e a r e q u a t i o n s and p e r f o r m i n g a n u m b e r of m a t r i x m u l t i p l i c a t i o n s . T h e r e s u l t is a f a s t and a c c u r a t e m e t h o d to o b t a i n t h e r e q u i r e d q u a n t i t i e s . 3. I m p l e m e n t a t i o n and E x a m p l e s T h e in p u t to S T A R P A K c o n s i s t s of an a r r a y g e o m e t r y and s i g n a l - n o i s e s c e n a r i o to be i n v e s t i g a t e d . As w e l l , an a pr i o r i n o i s e m a t r i x is r e q u i r e d fo r B i e n v e n u ' s m e t h o d . F i n a l l y , t h e n u m b e r of s a m p l e s to be a v e r a g e d in th e s a m p l e d c o v a r i a n c e m a t r i x is set. T h i s n u m b e r will be r e f e r r e d to as t h e n u m b e r of s a m p l e s a v e r a g e d in t h e f o l l o w i n g . A m e n u t y p e f o r m a t is u s e d to input t h e s e p a r a m e t e r s . A f t e r e x e c u t i o n w e o b t a i n t h e p r o c e s s o r b e a m o u t p u t s . S e e F i g u r e 1. OUTPUT INPU T 1. A r r a y G e o m e t r y ( n u m b e r of sensors, their location) 2. S i g n a l - N o i s e S c e n a r i o (signal d i r e c t i o n and p o w e r s , n o i s e m e c h a n i s m s and t h e i r p o w e r s ) 3. Conventional, op t i m a l and B i e n v e n u beam outputs versus angle. Other Parameters: i) a p r i o r i n o i s e c o v a r i a n c e m a t r i x if B i e n v e n u ’s m e t h o d is used; ii) n u m b e r of s a m p l e s in s a m p l e d c o v a r i an ce m a t r i x 21 F i g u r e 1. S c h e m a t i c of S T A R P A K We present one example to illustrate some of the main features of STARPAK. The reader can envision many more. Consider a 16-element equispaced linear array with interelement spacing d = ,4A where X is the wavelength of the signal. Imbedded in cylindrical noise of power 1. and white noise power .1 is one signal at 60° of power .4 (s i g n a l - t o - n o i se ratio -4. 4 1 d B ). We have assumed Q|\j is white and cylindrical of the appropriate powers. Typical results are shown in Figure 2 where the beam output for the three methods is plotted versus angle. All curves have been normalized to have a maximum value of one. Figures 2(b), (c) and (d) present examples when 32, 64 and 128 samples r e spectively have been averaged. We note the left-right ambiguity of linear arrays, that is the signal also appears at 300°. Two measures are useful when comparing processing methods when various numbers of samples are averaged. Signal beamwidth, BW, is defined as the width of the signal at its -3dB point and signal-to-bac kground noise ratio, SBN, as the peak signal level divided by the background noise. Signals processed by an ideal technique would have a BW of zero and SBN proportional to their signalto-noise ratios. In practice we expect BW to decrease and SBN increase as the number of samples averaged increases. Table I illustrates these trends for our data. As the number of samples averaged increases Q approaches Q and best performance is attained. This is referred to as the deterministic case. See Figure 2(a) and Table I for these values in our example. Number of Samples in Covariance Matrix Signal to Background Noise Ratio Beamwidth Be Bo Bb Be Bo Bb 32 10° 5° 28' 5.2 7 2.04 64 9° 4° 4' 5.4 5.6 2.75 128 10° 3.5° 2 5.3 5.62 4.18 10° 5° 5.76 6.98 7.31 (deterministic case) < 1 Be is Conventional, Bo is Optimal, and Bb is Bienvenu Processing Table I. Signal beamwidth and signal-to-bac kground noise ratio for the cases illustrated in Figure 2. Using these measures we observe that optimal processing is better than conventional. When the number of samples is at least 64 (at least four times the number of array elements) Bienvenu processing is preferred to optimal. It has an acceptable SBN ratio and superior resolution (smaller BW). Our e x p e r i ­ ence shows "this rule of thumb" holds in a wide variety of cases. When fewer than 64 samples are available for averaging Bienvenu processing is worse than optimal. This occurs due to the form of the Bienvenu statistic which is a quotient of two random variables as given in equation 3(a). Even small f l u c t u a ­ tions in both variables about their means cause large fluctuations in the o v e r ­ all statistic. 22 CONVENTIONAL _ BEAMPOWER ................ ...................... ...................... ...................... CONVENTIONAL OP TIMAL OPTIMAL O AN GL E D e te rm in is tic c o v a ria n c e m a tr ix . BEAMPOWER (a ) (d c g) AN GL E (b) F i g u r e 2. (d e g ) S am ple d c o v a r ia n c e m a t r i x u s i n g 32 s a m p le s . Beam o u t p u t v e r s u s a n g le f o r t e x t e x a m p le . 23 BIENVENU BEAMPOWER OPTIMAL CONVENTIONAL O ANGLE ------------ ---------------- ... ...................... -- CONVENTIONAL OPTI MAL (c ) (d e g ) S a m p le d c o v a r i a n c e m a t r i x u s i n g 64 s a m p le s . BEAMPOWER BIENVENU O AN GL E (d ) F i g u r e 2. lde g) S a m p le d c o v a r i a n c e m a t r i x u s i n g 128 sa m p les. Beam o u t p u t v e r s u s a n g le f o r t e x t e x a m p le . 24 4. Summary STARPAK i s a p o w e r f u l , v e r s a t i l e t o o l i n t h e s t u d y of an i m p o r t a n t c l a s s of r e a l w o r l d s i g n a l d e t e c t i o n p r o b l e ms u s i n g a r r a y s . The a l g o r i t h m s us e d in STARPAK a r e v e r y e f f i c i e n t . I t s u s e r f r i e n d l y i n p u t - o u t p u t f e a t u r e s make i t a c c e s s i b l e t o b o t h t h e n o v i c e and e x p e r i e n c e d r e s e a r c h e r . Beamwi dth and s i g n a l - t o - b a c k g r o u n d n o i s e r a t i o a r e two u s e f u l q u a n t i t i e s when c o mp a r i n g d e t e c t i o n c a p a b i l i t i e s o f a r r a y p r o c e s s i n g me t h o d s . Opt i mal proc essing is u s u a l l y b e t t e r than co nv en ti o n al. Bi e nve nu p r o c e s s i n g was s u p e r i o r t o o p t i m a l when t h e n o i s e f i e l d i s " a l m o s t " known e x a c t l y and a l a r g e number o f s a mp l e s a r e a v e r a g e d t o f or m t h e s a mpl ed c o v a r i a n c e m a t r i x . STARPAK i s a v a i l a b l e f r om t h e a u t h o r s . T h i s work was s u p p o r t e d by a g r a n t f r o m De f e n c e R e s e a r c h E s t a b l i s h m e n t P a c i f i c , V i c t o r i a . References 1. R. J . U r i c k , " P r i n c i p l e s o f Un d e r wa t e r Sound" , Mc Gr a w- Hi l l , 1983. 2. R. A. Monzi ngo and T. W. M i l l e r , Yor k, Wi l e y , 1980. 3. M. J . Wi l mut and W. W. Wo l f e , " S i m u l a t i o n f o r T e s t i n g A r r a y Re s pons e (STARPAK)", Royal Roads M i l i t a r y C o l l e g e , Se p t e mb e r 1983. 4. L. E. Br ennon and J . D. M a l l e t t , " E f f i c i e n t S i m u l a t i o n o f E x t e r n a l Noi s e I n c i d e n t on A r r a y s " , IEEE T r a n s . An t e n n a s and Pr o p . 7 4 0 - 1 , ( 1 9 7 6 ) . 5. " U n p a c k U s e r ' s Gu i d e " , Si am P h i l a d e l p h i a , 6. G. B i e n v e n u , " Un d e r wa t e r P a s s i v e D e t e c t i o n and S p a t i a l JASA 6 5 - 2 , 4 2 5 - 4 3 7 , ( 1 9 7 9 ) . 7. H. L. Van T r e e s , Wi l e y , ( 1 9 6 8 ) . " I n t r o d u c t i o n t o A d a p t i v e A r r a y s " , New "Detection, Estimation, 25 1979. Co h e r e n c e T e s t i n g " , and M o d u l a t i o n T h e o r y " , New York, Everything works together better when you keep it all in the family, especially when it’s the Brüel & Kjaer family. T h a t’s because in our large family of sound and vibration test instruments everything is designed to work together as a total system, from the exciter right through to the display m edium or data printer. So, when you need any instrumentation product from a single transducer to a complete system, check the Brüel & Kjaer catalog first, and keep your system all in the family. 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