space–time signal processing for moving antennae

Space–time signal processing for moving antennae

I.I. Gorban*

Institute of Mathematical Machines and Systems, National Academy of Sciences, 42 Ave Acad. Glushkov, Kiev 252187, Ukraine

Received 4 April 1996; accepted 19 July 1999

Abstract

For underwater acoustic systems that work in complicated conditions a new space–time signal processing (STSP) optimisation approachhas been developed. It considers, in complex, antenna’s motion, structure of the noise, and particularities of the medium. For coherent andstochastic signals STSP algorithms, matched with complicated antenna’s motion, multicomponent noises and non-uniform medium togetherwere produced and researched. It was found that in some cases complicated antenna’s motion led to rising of noise immunity of the STSPsystems. This effect may be used, in particular, in the systems with line arrays for alienating ambiguity in the estimation of signal direction.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Underwater acoustic systems; Space–time signal processing; Multicomponent noises

1. Introduction

In practice almost all underwater acoustic space–timesignal processing (STSP) systems are exploited in dynamicconditions. The antennae of these systems objects to the detec-tion or observation by them, and prevent the noise sourcescontinuously by changing their position in space. Sometimes,the motion is small in the interval of observation and thereforeit may be neglected without marked loss.However, more oftenthe motion is too significant to ignore it. Then the matchingof the signal processing with the motion is needed.

Many works were devoted to this problem. Differentquestions were researched, in particular, synthetic apertureprocessing in conditions of antenna’s motion in fixed direc-tion with constant velocity [1–4], synthetic aperture proces-sing when the antenna circulates in space [5], estimation andresolution capabilities of the STSP systems in case of anten-na’s motion with constant velocity [6–8], STSP with takinginto consideration the movement of the antenna and obser-ving object, both with constant velocities [9,10], beampattern stabilisation in conditions of complicated antenna’smotion [11]. Many important results were obtained.However some questions were not profoundly developed.One of them is STSP considered at the same time withantenna’s motion as other factors influencing the work ofSTSP systems.

The aim of this paper is to develop an approach to STSP

optimisation considering the complex antenna’s motion,noises, and medium. The next one is to work out andresearch optimum and near optimum STSP algorithms forsystems exploited in complicated conditions, in particular,when the antenna moves complicatedly in three dimensionswith variable velocity, angle rotations, and modifies theform, when noise has multicomponent structure, andmedium is not uniform. The research is based on theapproach developed for moving antennae in series ofauthor’s previous works in particular [12–19].

The paper is organised as follows: in Section 2, thegeneral theory of optimum STSP for mobile antennae isdeveloped. Coherent and stochastic signals are discussed.In Section 3, some simple examples of optimum STSP algo-rithms for mobile antennae in conditions of uncorrelatednoise, uniform and non-uniform mediums are given. Section4 is devoted to optimum STSP algorithms for mobile anten-nae in conditions of multicomponent noise. Section 5contains description of the methodology used to estimatethe noise immunity of STSP systems with moving antennae.Section 6 consists of description of some results ofcomparative computer calculations of noise immunity ofSTSP systems with mobile and static antennae. The mainresults are pointed in Section 7.

2. Maximum likelihood STSP for mobile antennae(general theory)

Let the antenna be arbitrarily moved (and modified) in

Advances in Engineering Software 31 (2000) 119–125

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* Fax: 138-044-266-2205.E-mail address:[email protected] (I.I. Gorban)

space and the signal on the space–time observation intervalV is modelled in complex form by

s�t; ~x; ~l s;F� � s�t; ~x; ~l s�ejF;

wheret is current time,~x is the space vector,~l s is the vectorof the estimated parameters andF is uniform random phaseon the interval [0, 2p).

In this case maximum likelihood STSP in a backgroundof additive Gaussian space–time correlated noise may bedescribed by integral

Q�~l s� �ZV

Zu�t; ~x�bp�t; ~x; ~l s� dt d~x; �1�

whereQ�~l s� is the result of the STSP,u�t; ~x� are the receivedoscillations,b�t; ~x; ~l s� is the function defining the structureof the STSP and is the solution of the following equationZV

ZKN�t1; t2; ~x1; ~x2�b�t2; ~x2; ~l s� dt2 d~x2 � s�t1; ~x1; ~l s�; �2�

the asterisk denotes complex conjugate,KN�t1; t2; ~x1; ~x2� isthe noise space–time correlation function.

Eqs. (1) and (2) differ from well known ones forstatic antennae by peculiarity of the space–time intervalV: in Eqs. (1) and (2) it depends not only on thegeometry of the antenna and time observation (as it isfor static antenna) but also on the parameters of the anten-na’s motion. If the movement is not fast and the time inter-val of noise correlation is not high, the antenna in short timeperiods may be approximately regarded as static and noiseelements from different periods as nearly uncorrelated. Thenapproximately

Q�~l s� �XMm�1

ZTm

Z~Xm

u�t; ~x�bpm�t; ~x; ~l s� dt d~x; �3�

where M is the quantity of the periods,Tm � ��m21�Tp;mTp� is the interval of themth time period,Tp is atime length of the periods,~Xm is the space interval of theobservation in themth period,bm�t; ~x; ~l s� is the solution ofthe equation known for static antennae:Z

Tm

Z~Xm

KN�t1; t2; ~x1; ~x2�bm�t2; ~x2; ~l s� dt2 d~x2 � sm�t1; ~x1; ~l s�

�t1 [ Tm; ~x1 [ ~Xm�; (4)

sm�t; ~x; ~l s� is the part of the signals�t; ~x; ~l s� in the space–time interval�Tm; ~Xm�:

The spectrum form of Eq. (3) is

Q�~l s� �XMm�1

Z∞

2 ∞

Z∞

2 ∞Um�f ; ~w�Bp

m�f ; ~w ; ~l s� df d~w ; �5�

where Um�f ; ~w� and Bpm�f ; ~w ; ~l s� are frequency wave

number spectrums corresponding to the functionsu�t; ~x�andbm�t; ~x; ~l s�; f is a frequency,~w is a vector of the spacefrequencies.

Analogous results were obtained for stationary Gaussian

stochastic signals described by the following correlationfunction:

Ks�t; ~x1; ~x2; ~l sx�

�Z∞

2 ∞gs�f �Gp

s�f ; ~x1; ~l sx�Gs�f ; ~x2; ~l sx� exp�j2pf t� df ;

where ~l sx is the vector of the estimated space parameters,gs�f � is power spectral density of the signal,Gs�f ; ~x; ~l sx� isGreen’s function of the signal. For not fast motion the maxi-mum likelihood STSP was found in the following approx-imation form:

Q�~l sx� �Z∞

2 ∞gs�f �

XMm�1

uZ∞

2 ∞Um�f ; ~w�Dp

m�f ; ~w ; ~l sx� d~w u2 df ;

�6�where Dm�f ; ~w ; ~l sx� is the wave number spectrum of thefunction Dm�f ; ~x; ~l sx� which is the solution of the equationZ

~Xm

Gp0N�f ; ~x1; ~x2�Dm�f ; ~x2; ~l sx� d~x2 � Gs�f ; ~x1; ~l sx�; �7�

G0N�f ; ~x1; ~x2� is the mutual power spectral density of thenoises.

It is clear from Eqs. (3)–(7) that for not fast antenna’smotion, maximum likelihood STSP consists of that which istypical for static antenna maximum likelihood space signalprocessing when taking into consideration on every timeperiod its form, place, and orientation in space, and thencoherent (for coherent signals) or noncoherent (for stochas-tic signals) summing and integration with weight obtainedresults.

3. Maximum likelihood STSP for mobile antennae inconditions of uncorrelated noise

Let the noise be uncorrelated in space and time processthat is described by power spectral densityg0: Coherentsignal spectrum on the intervalTm is modelled bySm�f ; ~x; ~l � � Sm�f ; ~l st�Gs�f ; ~x; ~l sx�; whereSm�f ; ~l st� is thespectrum of the time part of the signal,~l st is the time para-meters of the signal.

In this case maximum likelihood STSP algorithm may beobtained from Eqs. (4) and (5) in the following form:

Q�~l s� � 1=g0

Z∞

2 ∞

XMm�1

Qsm�f ; ~l sx�Spm�f ; ~l st� df ; �8�

whereQsm�f ; ~l sx� is the result of space processing in themthperiod:

Qsm�f ; ~l sx� �Z∞

2 ∞Um�f ; ~w�Gp

s�f ; ~w ; ~l sx� d~w ; �9�

Gps�f ; ~w ; ~l sx� is the wave number spectrum of the signal

Green’s function.The maximum likelihood STSP algorithm for stochastic

I.I. Gorban / Advances in Engineering Software 31 (2000) 119–125120

signal may be obtained from Eqs. (6) and (7) in the follow-ing form:


2 ∞gs�f �=g2

0

XMm�1

uQsm�f ; ~l sx�u2 df : �10�

From Eqs. (8)–(10) it follows that for coherent as wellas for stochastic signals the optimum STSP consists oftwo phases. The first phase described by Eq. (9) iscommon for both types of signals and is space proces-sing in every time period and frequency. On this phasethe peculiarities of the medium and current antenna’sform, position, and orientation in space is taken intoconsideration. The second phase described by Eqs. (8)and (10) is space–time processing. It consists of coherent(for coherent signal) or noncoherent (for stochastic signal)summing and integration with weight of the results obtainedin the first phase. On the second phase the spectrum of thesignal and differences in the results of space processing,which are caused by antenna’s motion are taken intoconsideration.

If the signal is a plane wave described by the vector~l sx�~nsx; where~nsx is the vector that determines the direction ofsignal propagation the signal processing is reduced. FromEq. (9) for such signal and uniform medium it followsQsm�f ; ~l sx� � Um�f ; f ~nsx�:

Hence, in this simplest case the first phase of STSPconsists of finding frequency-wave number spectrum corre-sponding to the estimated direction of signal propagation.This phase may be realised by forming in every frequencythe beam pattern that follows according to any changes inantenna’s form, position, and orientation in space and hasthe main lobe orientated in every time period in the directionof the signal. The second phase does not have somespecifics.

4. Maximum likelihood STSP for mobile antennae inconditions of local noises

Let the noise consist of uncorrelated and local compo-nents, and is described by the following mutual power spec-tral density:

G0N�f ; ~x1; ~x2� � g0d�~x2 2 ~x1�

1 gl�f �Gpl �f ; ~x1; ~l lx�Gl�f ; ~x2; ~l lx�;

where d�~x� is Dirac’s function,gl�f � and Gl�f ; ~x; ~l lx� arecorresponding power spectral density and Green’s functionof the local noise component,~l lx is the vector characterisedspace parameters of this component.

The maximum likelihood STSP algorithm may be found

from Eqs. (4) and (5) in the following form:

Q�~l s� � 1=g0

Z∞

2 ∞

XMm�1

�Qsm�f ; ~l sx�

2 bpm�f ; ~l sx; ~l lx�Qlm�f ; ~l lx��Sp

m�f ; ~l st� df ; �11�where

Qsm�f ; ~l sx� �Z∞


s�f ; ~w ; ~l sx� d~w ; �12�

Qlm�f ; ~l lx� �Z∞


l �f ; ~w ; ~l lx� d~w ; �13�

bm�f ; ~l sx; ~l lx�

�gl�f �=g0

Z~Xm

Gpl �f ; ~x; ~l lx�Gs�f ; ~x; ~l sx� d~x

1 1 gl�f �=g0

Z~Xm

Gpl �f ; ~x; ~l lx�Gl�f ; ~x; ~l lx� d~x

; �14�

Gl�f ; ~w ; ~l lx� is wave number spectrum corresponding to theGreen’s function of local noise component.

For stochastic signal the maximum likelihood STSP algo-rithm may be obtained from Eqs. (6) and (7) in the form:


2 ∞gs�f �=g2

0

XMm�1

uQsm�f ; ~l sx�

2 bpm�f ; ~l sx; ~l lx�Qlm�f ; ~l lx�u2 df : �15�

It is clear from Eqs. (11)–(15) that in conditions of localnoise the optimum STSP includes two channel spaceprocessing, one which matches with the signal and theother with local noise, both during every period when takinginto consideration space peculiarities of the antenna. Resultsobtained by these two channels subtracted from each other,with weighting depending on such factors as noise para-meters and current antenna’s form, its place, and orientationin space. After this there are coherent (for coherent signals)or noncoherent (for stochastic signals) summing and inte-gration with weight obtained results.

5. System noise immunity in conditions of antenna’smotion

For systems with moving antennae it has been researcheddistribution probability densities of output signals. It wasfound that distributions for systems with moving antennaethe same type as for systems with static ones. Therefore forcomparative estimation of system noise immunity in differ-ent conditions of antenna’s motion it is possible to usesignal-to-noise-ratio (SNR) that is often used for estimationof system noise immunity in static case. SNR was calculatedfor different STSP systems with mobile antennae. Referringto the system used in algorithm (11) the SNR was found in

I.I. Gorban / Advances in Engineering Software 31 (2000) 119–125 121

the following form:

g � 1g0

Z∞

2 ∞

XMm�1

uSm�f ; ~l st�u2Vms�f ; ~l sx�Rm�f ; ~l sx; ~l lx� df ;

�16�where

Vms�f ; ~l sx� �Z

~Xm

Gps�f ; ~x; ~l sx�Gs�f ; ~x; ~l sx� d~x; �17�

Rm�f ; ~l sx; ~l lx� � 1 1 gl�f �=g0Vml�f ; ~l lx��1 2 P2m�f ; ~l sx; ~l lx��

1 1 gl�f �=g0Vml�f ; ~l lx�;

Vml�f ; ~l lx� �Z

~Xm

Gpl �f ; ~x; ~l lx�Gl�f ; ~x; ~l lx� d~x;

Pm�f ; ~l sx; ~l lx� is the generalised partial beam pattern:

Pm�f ; ~l sx; ~l lx� �

Z~Xm

Gpl �f ; ~x; ~l lx�Gs�f ; ~x; ~l sx� d~x��Vms�f ; ~l sx�Vml�f ; ~l lx�

q��

��: �18�

For the system used in algorithm (15) the SNR was found inthe following form:

g � �Tp=2Z∞

2 ∞g2

s�f �=g20

XMm�1

V2ms�f ; ~l sx�R2

m�f ; ~l sx; ~l lx� df �1=2:

�19�From Eqs. (16)–(19) it follows that in case of uncorrelatednoise (when gl�f � � 0) for coherent signal with fixed

amplitudeAs

g � A2sTg0

1M

XMm�1

Vms�f ; ~l sx�" #

�20�

and for the stochastic signal

g � T2

Z∞

2 ∞g2

s�f �=g20

1M

XMm�1

V2ms�f ; ~l sx�

" #df

" #1=2

; �21�

whereT is time observation. It is clear from Eqs. (20) and(21) that in conditions of uncorrelated noise for both coher-ent, and stochastic signals SNR does not depend frommedium’s properties and changes in antenna position. Itdetermines by time observation and antenna’s middlevolume.

In conditions of correlated noise there are more complexdependencies. Analysis of Eqs. (16)–(19) shows that SNRchanges with variances of a direction of the local noisecomponent. If this component penetrates through the mainlobe of the generalise partial beam patterns or small sidelobes (when for allmPm�f ; ~l sx; ~l lx�p 1� SNR is the sameas for the system with static antenna. However if the localnoise component penetrates through a large lobes (when forsomemPm�f ; ~l sx; ~l lx� < 1 and for the rest them expressionis essentially less unit) SNR is higher. The degree dependson the parameters of medium and antenna rotation angle. Tolearn this dependence the model research were carried.Some results of the research are described in Section 6.

6. Model research

For the model research in conditions of complicated


Fig. 1. SNR for systems with static and moving line arrays as a function of the bearing of local noise component. The medium is uniform, the arrays contain 16uniformly spaced elements, the signal is located in the angle of 608, and the array rotation angle of the moving arrays is 208.

antenna’s motion a program package has been developed. Itwas used to study noise immunity of different STSP systemssome of which were intended for coherent and the other forstochastic signals, some of which considered the particula-rities of motion, noise, and medium, and the other ignoredthem. The calculation results for line horizontal array incase of two component noise is described by Eqs. (11)and (15) are presented in Figs. 1–3.

Figs. 1 and 2 are obtained for conditions of uniform and

nonuniform mediums correspondingly. The curves in bothfigures describe dependence SNR from bearing of localnoise component. The systems that worked in uniformmedium were tested with using the plane wave models forthe signal and the local noise component, and the systemsthat worked in nonuniform mediums were tested with usingGreen’s functions represented by three plane waves for thesignal and three plane waves for the local noise component.

The curves in Figs. 1a, 2a and 1b, 2b were obtained for


Fig. 2. SNR for systems with static and moving line arrays as a function of the bearing of local noise component. The medium is nonuniform, the arrays contain128 uniformly spaced elements, the signal is located in the angle of 208, and the array rotation angle of the moving arrays is 108.

Fig. 3. Advantage/disadvantage in SNR for different systems with moving arrays in comparing with the optimum system with static array. The noise ismulticomponent, the arrays contain 16 uniformly spaced elements, and the signal is located in the angle of 608.

STSP systems intended correspondingly for coherent andstochastic signals. The curves OSSA present the optimumsystems with static array that realise maximum likelihood ofSTSP and the rest of the curves of the systems realise differ-ent STSP in conditions of forward array’s motion with rota-tion. The curves OSMA were obtained for optimum systemswith moving array that were fully matched with array’smotion, noise and medium. The curves SIMAN correspondto the systems ignored as motion of the array as local noisecomponent. The curves SIN present the results for thesystems considered the fact of array’s motion by non-optimum manner. In these systems the local noise com-ponent was ignored and the main lobe of the generalisedbeam pattern followed according to any changes in array’sposition so that in every time period it was oriented in thesignal direction. The curves SINM in Fig. 2 present thesystems ignored local noise component and mediumparticularities.

The curves in Fig. 3a–d describe the advantage/disadvan-tage in SNR for different systems, that realise STSP inconditions of array’s motion. These systems are comparedwith optimum system with static array. The curves in thefigures are the functions of parametera that is ratio of arrayrotation angle to width of the large side lobe (or main lobe).Fig. 3a and c were obtained for coherent signal and Fig. 3band d for stochastic signal. Fig. 3a and b present results foruniform medium and Fig. 3c and d for the nonuniform one.

Analysis of calculations showed on some regulars

1. Ignoring the fact of antenna’s motion and local noisecomponent lead to loss in noise immunity (comparecurves SIMAN with curves OSMA in Figs. 1 and 2).The degree of loss depends on the ratio of antenna rota-tion angle to width of the main lobe of the beam pattern.With raising the angle the loss is fast increasing (seecurves SIMAN in Fig. 3a–d).

2. Noise immunity of optimum STSP systems with movingantenna sometimes is close to noise immunity of opti-mum systems with static antenna. It takes place whennoise is uncorrelated. The same is when noise is corre-lated and it contains local component, penetrated to thesystem through a main lobe or small side lobe of thebeam pattern (see curves OSMA and OSSA in Figs. 1and 2). If the local noise component penetrates to thesystem through large side lobes, the noise immunitiesof the compared systems are essentially different. Thesystem with moving antenna has higher noise immunitythan the system with static one. The advantage in SNRfast increases with raising the antenna rotation angle (seecurves OSMA in Fig. 3a–d).

3. Nonoptimum STSP system, in which antenna’s motion isconsidered but local noise component is ignored, hassome loss in noise immunity. When local noise compo-nent penetrates to the system through a main lobe orsmall side lobes, the loss often is not visible. However,when it penetrates through large side lobes the loss

becomes noticeable (compare curves SIN with OSMAin Figs. 1 and 2). In spite of losses, nonoptimum systemwith moving antenna has advantage in comparing withoptimum system with static antenna (compare curvesSIN and OSSA in Fig. 1 and 2).

4. The system, in which the antenna’s motion is consideredbut particularities of the medium are ignored, has loss innoise immunity. The degree is not high if the rays ormodes of the signal can not be resolved and it is essentialin contrary case (compare curves SIN, SINM, and OSMAin Fig. 2).

From these results it follows that in some cases compli-cated antenna’s motion plays positive role. STSP systemwith antenna complicatedly moving in space may haveessentially higher noise immunity than system with staticantenna. The effect of rising of noise immunity is connectedwith suppressing the large side lobes of the beam pattern.This effect may be used, in particular in the systems withline arrays. In such systems by using and consideringcomplicated array’s motion it is possible to suppress thelarge side lobes of the beam pattern and to alienate ambi-guity in estimation of the signal direction.

From presented results it also follows that to achieve highquality STSP it is important to consider all particularities ofthe antenna’s motion, the noise, and the medium. In compli-cated conditions when there are significant current changesin antenna’s position, there is complicated noise structure,and there is nonuniform medium, the most important toconsider the antenna’s motion. However, appropriately todo emphasis very good results may be obtained only if allthese particularities are taken into consideration in complex,together.

7. Conclusions

In this paper, we worked out STSP optimisation approachconsidered in complex antenna’s motion, structure of thenoise, and particularities of the medium. We developedoptimum and near optimum STSP algorithms matchedwith complicated antenna’s motion, multicomponentnoises, and nonuniform medium together. The noise immu-nity of different STSP systems was studied.

It was found that in some cases the systems with mobileantennae had essentially higher noise immunity than thesystems with static antennae. Rising the noise immunity isa result of complicated antenna’s motion. This effect may beefficiently used. One of the simplest example of its possibleusing is alienating ambiguity in estimation of the signaldirection in the systems with line arrays.

To achieve high quality STSP it is important to matchprocessing with all particularities of antenna’s motion,noise, and medium. Best results may be obtained if allthese particulars are considered together.


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