Multiple Antenna Array Signal Collecting and Processing

Introduction

The problem of detecting a random signal field in the presence ot" a random noise field has received various treatments in the literature. Despite certain limited analytical treatment this is an important problem that occurs in many areas, such as fading on long range communica­ tion and radar links, sonar, radio astronomy, e tc. The treatment addressed to this problem thus far consists almost exclusively of the deri­ vation of the optimal receiver structure. The early work was done for the scalar (one antenna) case by Price1 who only considered white noise and Middleton2 ,3 who considered colored noise. In these cases the derivations were performed distributed additive phase . 8,9 !:f(t) is assumed independent of y(t} with independent, Gaussian distributed orthogonal components. Thus .r_(t) has means (specular components)

*This research was supported in part by the Joint Services Electronics Program under Grant AFOSR 766-6 7E.

and covariance rna trices

where

Time Optimal Continuous Data Processing

The time optimal continuous data process­ ing for optimally detecting the above signal involves calculation of a test statistic t which is compared with a threshold and can be repre­ sented in a number of forms corresponding to different implementations of the decision equa­ tion. Two such forms are:6

1. The "phasing/ weighting/ combining­ estimator-correlator" form which is specified by

where g and QN are matrix functions defined by

and

Since (7) is a Wiener-Hopf integral equation, 6 then

is the usual optimal (minimum mean square error) estimate of .§..(v) under hypothesis H 1 · The test statistic is this optimal estimate correlated with a filtered version of the received signal.

where

is a filtered version of .r_(t), with the matrix func­ tion filter k being defined by

The above structures can also be expressed in simplified linear matrix operator notation used by Middleton and GroginskyS as

The block diagram illustrations of these two equivalent forms are indicated in Figure 1. For complex signals, the "T" superscript indicates complex conjugate as well as transpose so that from the filter-squarer form, it is evident that t is always positive. The above structures are preceded by a time delay (or phasing) network 1.. (or qJ) in Figure 1 to allow for the signal phasing (beam steering} requirement. Figure 2 illustrates the details of the structures in Figure 1 for n = 2 antenna coupling elements.

Characteristic Function Derivation

In order to evaluate system performance and specify how spatial sampling (array/distri­ buted aperture coupling} might enhance perform­ ance, it is necessary to find the probability density functions oft under all hypotheses in the problem structure (two in this case). This can be done by first calculating the character­ istic functions oft. The above decision equations (receiver structures) were derived by expand­ ing .r_(t) in a truncated vector Karhunen-Loeve series expansion 1 writing the log likelihood ratio in terms of this expansion 1 and then allowing the number of terms to become unbounded. 4 1 6 The mean of .r_(t) is assumed to be the same under both hypotheses in order to further general non­ threshold analytical calculation. This results in the test statistic

where 1. is a diagonal matrix with diagonal elements

and where (A .. } are the e igen values correspond­ ing to the ith lJigenfunctions in the Karhunen­ Loeve expansion under hypothsis H . I j = 0 1 1 · The characteristic function is then talculated as

Since .r. is complex, write

so that

Now let

where

Equation (20) is similar to the Fredholm determi­ nant7 and can be simplified in a similar manner. Then

Letting

and

where the se t of vector functions uk (t)} are complete and orthonormal. Note th<H h is simi­ lar to the resolvent kernel discussed in Courant-Hilbert 1 Chapter III .10 Then

The next step is to interpret H(r 1 t;u) and hence h(r I t;u) in terms of the space-time structure of the signal and noise fields. First it is assumed that the ck's represent a set of eigenvalues (not necessarily all positive). Then if

converges for some complete 1 orthonormal set of vector functions U (t)}, .P(r,t) will be a kernel having the eigenvafues c~ and the eigenfunctions (lk(t)}~ viz.

Then .H.{r 1 t;u) is a solution of the coupled system of integral equations

which can be verified by substituting the series expansions for.! and H in (28).

The problem is now one of relating .P to the signal and noise statistics by obtaining the (lk ( t)}. The .!Sy covariance functions have eigen­ va lue and eigenfunction sets defined by

(Note that the f,.'s and_se's are different under the two hypotheses H 0 a nd H1.) It can be shown that

by integratlng the product of (29) and (30) over tlme. Then.!! the eigenfunctions were the same under both hypotheses ~kl = .s12ko for all k),

so that

The next step is the solutlon of the integral equatlon (28) for .ff. This is facllltated by intro­ ducing the assumptions that the tlme (Ti' Tf) is long and that the processes are satisfactory so that

It can be shown that 6

so that Substituting (35) into (28), taking the Fourler transform, and solving for .ff(w;u) ll then yields

where j, g = 0,1 and jIg, and

Note that this is the complete solutlon for H . because of the above assumption of a long ~ser­ va tlon period. Then

so that

Optlmal Receiver Performance

The optlmal receiver performance is found by (a) evaluating equatlon (39) to obtain the characteristlc functlon M.t(w) of the test statlstlc .t, (b) inverse transforming M.t(w) to get thep.d.f. p(.t) of .t, and (c) integratlng p(.t) over the appro­ priate limlts to get the error probabllitles. Unfortunately this cannot be done in closed form for the general case. An exceptlon to this is the classical and useful threshold case which wlll now be developed for array coupling.

Threshold Case

The threshold conditlon is broadly defined as corresponding to a low SNR and the analytical condltion for this as sumptlon is

for g = 0 ,l. If (40) is satlsfied, then

(42)

2 where m. and crj are constants. Then J

which is seen to be the form of the characteristic function for a Gaussian random variable with mean mj and variance a?. It can be shown in a straightforward manner lhat the error p~obabilities with a Gaussian distributed test statistic 1- are

where 1- is the optimal threshold and 0

Middleton 2 has demonstrated that the test statistic becomes Gaussian in the scalar (one antenna) case under threshold conditions using sampling point analysis. The above results hold for continuous waveform processing and multiple antennas.

Optimal Threshold

The sys tern performance is then completely s pee lfied if a closed form express ion for the optimum threshold 1- can be found. The analy­ tical mechanics inv8\ved closely parallel those used to develop the characteristic function and only a general outline of the steps will be given here. In the Karhunen-Loeve derivation of the optimum test statistic, 6 the series expansion for the optimum threshold was found to be where T1 is a bias constant determined by the optimality criteria employed (Bayes, Neyman­ Pearson, etc.) and the t_'s are as previously defined. In order to obtain a closed form expres­ sion for this series it is convenient to assume that the noise also contains a white component of spectral intensity N 0 /2. This assumption is physically realistic because of receiver noise and as will be seen later, is also related to the deter­ mination of optimal antenna spacing. Then

and

Each summation is handled as before by defining a Dt(z), li 1 (r,t;u), and hi(r,t;u), and showing that lit is the solution of

where ,[ 1 is the covariance matrix of the non-white portion of the received voltage under hypothesis i, l = O,l. Assuming stationarity and a long observation time then yields

are the spectra of the non-white portions of the coupled voltages under the two hypotheses.

Optimal Antenna Dimensions

In order to maximize system performance it is necessary to simultaneously make Pp as small and PD as large as possible. This is done by maximizing the difference between the arguments in (44), or equivalently, by maximizing the square of the difference which is

and from the above characteristic function analysis,

so that

Singular De tee tion

At this point the phenomena of spatial s ingu­ lar detection can arise in the maximization of y. This phenomenon arises if receiver noise is neglicted, i.e., if all the noise is assumed to be received through the antennas. Singular detec­ tion occurs when the SN(w) matrix becomes singular 12 and two su;h cases need to be distin­ guished.

In the first case y is maximized for zero antenna spacings, i.e. , when the antennas are (ideally} located at the same point.l2 Later in this report it is shown that the Q.N matrix becomes singular for zero spacings so that y appears to become unbounded. The physical significance of this phenomenon is that for zero spacings the noises in the antennas are identical and the . optimum receiver structure phases and sums the antenna voltages in such a way as to completely eliminate the noise. Unfortunately the desired signals in the antennas are also identical and are also completely eliminated by the optimum receiver.

The second case involves plane wave noise yielding noises in the antennas as delayed ver­ sions of each other. Thus it is theoretically possible to completely eliminate the external noise without eliminating the signal provided the signal is not received from the s ame direction as the noise. In this case the performance is theoretically the same for all non-zero antenna spacings. The plane wave noise is therefore not considered further in regard to optimal antenna spacing.

The zero spacings solution is invalid as is also evident from the fact that for singular detection the threshold assumption no longer holds since the inequality in equation (40) is reversed. If the threshold assumption does not hold, then equation (55) for y is invalid and it is no longer evident that the optimum procedure is to maximize y.

In order to obtain physically meaningful results it is necessary to avoid the spatially singular detection situation. This is done by including an additive receiver noise whose statistical properties are independent of the antenna spacings. The receiver noise need not be white but may reasonably be assumed uncor­ related. This is the approach taken by Gaarderl3 '14 in maximizing the antenna gain function and challenges the conclusion of Martel and Mathews 12 who indicate that near zero spacings are optimal.

Problem Simplification

The insertion of receiver noise increases the complexity of the integrand in (55). Even in the most simple cases the exact evaluation of y becomes so cumbersome as to render a direct approach practically useless. It is therefore desirable to simplify the analytical task by finding some simpler expression which is maximal for the same antenna deployment that maximizes y. The cases under which this is possible will now be developed.

A considerable simplification results if max­ imizing the integrand is the same as maximizing the integral. The development which maximizes the integrand will be a function of w unless the integrand can be factored into the product of a function of only the deployment and a function of frequency alone. With physically realistic antenna cross correlation spectra such a factor­ ization is not pos slble. Thus the two maxim iza­ tions coincide only if the optimal deployment for the integral equals that for the integrand evaluated at a particular frequency, w 0 , perhaps the carrier frequency. This in turn occurs only when the integrand contains, as a factor, o(w- w 0 ). If the signal covariance matrix contains a common spectral term [S (w) =S (w) P (w)] then -s s -s

. 2 and the two maximizations coincide only if S (w) approximates a delta function. This will be fue case if the 3 db width of Ss (w) is much less than that of Tr[s-l(w) P (w) s-l(w) P (w)]. -N -s -N -s

In some cases it is also possible to use a simpler expression than the integrand in the maximization. If A. are the eigenvalues of A = s-l P , then 1 - -N -s

n It would be much simpler to maximize i~l At instead, but the two maximizations do not in general coincide. There are two obvious cases in which the maximizations do coincide, these being (1) when all the A's are equal, and (2) when

all the A's except one are zero.

If all the A's are equal, then (assuming A is normal) A is similar to a scalar multiple of the identity matrix and hence is itself a scalar mul­ tiple ofthe identity matrix. This in turn would require Q.N to be a scalar function of f.s, which is not possible here because of the assumed additive receiver noise. Thus the case in which the A's are all equal is physically unrealistic.

If all but one of the A's are zero, then A must have rank l and hence fs must have rank l.

This then is the plane wave signal case in which fs = l (a matrix, all of whose elements equal one) and y reduces to

is sometimes called the array gain function since it represents the SNR gain effected by the antenna array. The maximization of the antenna gain function by Gaarderl3 '14 is valid only for narrow band plane wave signals (requiring that s 2 (w) approximate a delta function). s

The integrand can also be simplified by further selection of the SN and P matrices. That this is the case might b-; suspe~ied because these matrices have the often convenient properties of being Hermitian and positive defin\te. The pro­ duct QNl fs ~s Hermitian only if Q.~ and fs commute. Q.N 1 and f.s will commute if <!nd only if they have the same eigenvectors, or equivalently if Q.N is a scalar function of f.s, or vice versa. (Note that these s ta temen ts are true only because Q.N and fs are Hermitian.) For the physically realizable cross covariance spectra considered in this report, the restriction that Q.N be a scalar function of fs requires that the additive receiver noise be uncorrelated and that the external noise and signal sources be of the same type.

Case Study

At this point it is desirable to investigate (l) specific functional forms for the signal and noise statistics and (2) the plane wave signal case without the benefit of the threshold assump­ tion. These will be considered as the case studies.

Physically Relevant Space-Time Correlation and Power Spectra Models

If physically meaningful conclusions are to be obtained, physically realistic space-time covariance functions must be employed. The system designer must first model the physical noise field environment and then use the model to obtain Q. (w) and hence fN(w). The procedures involvel and some representative results will now be cons ide red.

A treatment of this aspect has been given by Childersl5in terms of a vector antenna height function.h(w;8,cp) of the antennas used, which is useful when the coupling elements are dipoles. The voltage coupled in by an antenna is the dot product of its vector height function and the electric field vector of the incident radiation field. The cross power spectral density function between the signals coupled in by two antennas (Figure 3) is given by the quadratic form

where

is a power spectrum rna trix of the 8 and qJ com­ ponents of the electric vector of the incident field and where z is the separation between the antenna phase centers. Note that the auto power spectrum can be obtained from (60) by setting z=O and .h 1 = .h 2 . Note also that if hl = .h 2 , then QNLI has rank l and hence is singular for z = 0, illustrating the singular detection phenomenon.

It is readily apparent that various possible functional forms for SNlz(w) exist. Unfortunately except for a few special cases the resulting analytical forms of sN 12 (w) in terms of field, environment, and coupl1ng element variables, are usually very complex and lead to unwieldy math­ ematical complications. However, a number of special cases have been developed in the llteraturel6,l7,l8,l9,20,2l which result in reasonably simple analytical forms for SN 12 (w). Four such physical cases are now considered. The assumption that the noise is homogeneous lliNLI is independent of r) is common to all these cases and will not be separately stated for each case.

The first case is one cons ide red by Childers and Reedl6 in which dipole antennas and isotropic (QNLI is independent of 8 and tp), unpolarized (SN88 = SNtptp= SN and SN8tp= SNtp8 = 0) noise are assumed. The resulting expression for sN 12 (w) is then

where the dipole orientations are defined by (8 1 , tpl) and (8 2 , tp 2 ) . In the case of white noise (SNLI(w) = N 0 / 4n) this inverse transforms into a reasonably simple cross correlation function between antennas.

The remaining three cases involve isotropic antennas so that the quadratic form in (60) may be a scalar SN 6 (w;8,tp):

The second case assumes that the noise is iso­ tropic (SNLI is independent of 8 and tp) and results in the particularly simple expression

This is the case most frequently found in the literature.l7,l8,l9,20 The isotropic assumption is equivalent to the assumption that the random incident field is the result of a large number of independent random sources uniformly disturbed in a sphere centered around the antennas (See Figure 4). Such a situation is encountered when­ ever the antennas are enclosed in a large volume of random scattering elements (any elements whose size approaches operating wavelengths) and where there are no nearby (in terms of wavelength~ boundary surfaces )9, 20

The third case is also frequently found in the literaturel9, 20, 2l and involves non-isotropic noise such as obtained when the noise is gene­ rated by a large number of independent noise sources uniformly distributed on a large circular a rea of any plane passed through the array volume (See Figure 4). Such a situation is encountered when the random field is gu~ to scattering from a "distant" plane surface. 1 ' 0 The cross antenna power spectrum then has the form

where y is the angle between a llne connecting the antennas and the plane of the random sources.

The final case involves a directional random field of the form

so that the double integral in (64) factors into the product of two integrals. The function of f(8) can be arbitrary except that it should be normalized to make its integral equal to some convenient constant. The noise field will then have the shape f(e) in the e dimension and e-~!cos(cp-cpol I in the cp dimension. This latter form is that of a lobe centered at cp= 1L + cp 0 · This structure can be employed as a moael for the situation involving a distant random source (jamming, etc.) and a multipath environment so that the noise appears to be coming from an angular region of arbitrary dimensions (See Fig 4). Narrow lobe widths of the noise field correspond to large ~ values and for ~sin ~ 0 greater than about three, the rational cross antenna power spectrum

is obtained. The 3 db lobewidth of the noise field is determined by ~ and is

This spectrum agrees well with available experi­ mental data for point sources in multipath environment. (For example see the data given by Kurichara. 22)

The limiting case of (68) in which~ becomes unbounded corresponds to the plane wave noise field in which the noise comes from one particu­ lar angle. In this case the noises in the different antennas are identical (spatial coherence) except possibly for a delay. Atso, unless the signal field is of an identica l space-time correlation nature, the noise can be completely eliminated by properly phasing, weighting and combining the signals, leaving only the system noise introduced in the receiver front end. However if the signal field is exactly of this space-time form, no advantage can be obtained from space diversity since the spatial filtering of both s ignal and noise is identical.

Plane Wave (Point to Point Spatially Coherent) Signal Case

The trace operation in finding h . (w;u) can be performed exactly w ithout the threJhold assumption in the special case in which the s ignal in the various antennas is the same

(spatially coherent) except possibly for a delay, which can be phased as shown in Figure l. In this case all the elements of the S (w) matrix are -s the same; thus

where.!_ is a matrix, all of whose elements are l. It can be shown that

i" where n is the rank and A l represents the ij element of A- 1 . Thus, with (71)

where again j i' g. (Note that the summations are a result of the multiple antennas and have nothing to do with the Karhunen-Loeve expansion approach used earlier in the report.) The evalu­ ation of (72) is greatly simplifled if the additive noise is assumed to be white and uncorrelated from antenna to antenna, so that

This yields

so that

For the assumption that S (w) is rational, so that s

where the order of D exceeds that of N, then

and

can be evaluated by residue theory. As an example, for S(t) plane wave and Markovian,

The associated characteristic functions are

where

R = 2 K /N being a measure of the signal-to-noise ratio (SNR) 0 in this case. Neither of these charac­ teristic functions can be inverse transformed in closed form. It is observed that the effect of the multiple antennas, in this case, is to increase the SNR by a factor of n. This is a direct con­ sequence of the assumptions of uncorrelated noise from antenna to antenna and plane wave (spatially coherent) signal.

Other assumptions which allow an exact evaluation of the trace in (39) are possible but usually lead to characteristic functions which cannot be inverse transformed in closed form. Numerical inversion can be used in these cases but the results are necessarily valid only for the specific case postulated and cannot be con­ veniently extrapolated to other cases or used to form broad, quantative conclusions. The exact performance analysis approach was therefore dis­ continued by the authors in favor of the more productive threshold case assumption, which allows simpler interpretations of steps to take in enhancing detectabllity by specific antenna deployment.

Critique

The contribution of this paper consists of (1) an analytical development of the exact charac­ teristic functions for the test statistic computed by a time optimum data processing system employing multiple antennas for a random signal

and noise field in terms of the space-time structure of the signal and noise fields and the antenna array configuration, (2) a simple, cone ise analy­ tical demonstration that, in the threshold case the test statistic distributions for the array ca.se become asymptotically Gaussian, and (3) analy­ tical treatment in terms of physically realistic space-time correlation models of signal and noise field structures. The assumptions include Gaussian noise, stationarity, and "long" obser­ vation times.

Unfortunately the characteristic function form derived herein cannot be inverse transformed in closed form in the general case. Numerica'l solutions may be of value in specific instances but cannot be expected to provide broad quantita­ tive information. Thus the only route leading to quantitative results appears to be the threshold case for which closed form inversion is possible.

In the threshold case the test statistic is Gaussian distributed and hence its p.d.f. is uniquely determined by its first two moments. Even in non-threshold situations the accuracy to which the signal and noise fields can be statis­ tically characterized may preclude a more accurate representation of the test statistic p.d.f. Thus the threshold assumption will probably yield results which are as meaningful as can be obtained even in non-threshold situations unless an unusually accurate statistical formulation is possible.

It is of interest to note the general effects of the array upon reception capability for specific classes of random noise fields and fading signal fields in terms of their space-time correlation structure and array deployment. These aspects are currently being pursued through a detailed investigation of a number of special situations by the authors and will be presented in a later report. A thorough discussion of anticipated results is at this time premature but a number of speculations can be projected:

• (l) Performance capacity can increase with the number of antennas, n, but the extent of any improvement, as well as incremental gain, is greatly dependent upon the space-time correlation of the signal and noise fields and the antenna geometry employed.
• Deploying antennas so as to increase the noise cross-correlation between antennas increases the performance capability, assuming that the deployment does not render a similar effect upon signal cross-correlation between antennas. For example, for fully correlated (spatially coherent) noise between antennas, the interesting case of singular detection is obtained in which theoretically the noise can be completely

eliminated if that noise is the only source of interference and the antenna's front end can be ignored.

1. The effect of signal cross-correlation between antennas (correlated fading) is such that increasing this correlation, by appropriate antenna deployment, will increase the combined signal to noise level, but will also increase the "outage rate" due to spatially correlated fading. Thus deploying antennas to increase signal cross-cor­ relation under these conditions increases perfor­ mance capacity if SNR gain is the criteria and decreases performance capacity if signal loss or outage rate is the criteria.

It should be emphasized that physically realistic space time correlation models are used in the preceding analysis. The set of cross­ power spectra cited in chis paper should rea lis ti­ cally model a number of situations of engineering interest.