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Chapter 4Features of ASSP under Different Levelsof A-Priori UncertaintyAbstract. is devoted to analysis of peculiarities of adaptive spatial signal process-ing under different levels of a-priori uncertainty. First of all, it is given thoroughcharacteristics of a-priory data needed for implementation of systems with adaptivespatial signal processing. Next, the characteristic of a-priory uncertainty about theproperties of both signal and noise is given. The reasons for noise appearance andlevels of noise are shown. The third part of the chapter deals with the spatial signalprocessing under generalized parametric uncertainty about the noise properties. Thelast part is devoted to signal processing under a-priory parametric uncertainty aboutthe noise properties.4.1 Analysis of Peculiarities of ASSP with Different Levels ofA-Priori UncertaintyAbovementioned ASSP algorithms are oriented on output signal optimization forspace (space-frequency) ﬁlter under the conditions when both the amount of inter-ferences and their spatial and temporal characteristics are unknown a priori. In thecommon case, the spatial properties of the noises are unknown too. In the analysisproblems is assumed that a noise is spatially white, but this hypothesis is true onlyfor noises with the “thermal” origin [6, 13]. These conditions can be treated as aclassical example of parametric a priori uncertainty about the properties of interfer-ences and noise [34]. It should be underlined that a priori uncertainty is not total,but only parametric. Really, all optimal VWC are obtained using the correlationapproximation. It means they are true only for situations when there is no cross-correlation between a signal and interference. Thus, it should be known a priori thatinterferences are not correlative with the useful signal.In the constraints of the adaptive Bayes approach the overcoming of parametric apriori uncertainty about the properties of interferences and noise can be worked outby replacement of unknown correlation matrix Rxx, as well as vector of correlationL. Titarenko et al.: Methods of Signal Processing for Adaptive Antenna Arrays, SCT, pp. 35–50.springerlink.com c Springer-Verlag Berlin Heidelberg 2013
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36 4 Features of ASSP under Different Levels of A-Priori Uncertaintyfor input and reference signals→Rxr by corresponding consistent estimates→Rxx and→Rxr in optimal vectors of weight coefﬁcients. It means that such a replacement ismade for optimization tasks, too.In the same time, the ASSP algorithms do not use a priori information aboutthe distribution laws for signals and interferences. From this point of view theycan be treated as representatives of the class of nonparametric procedures [7, 8,34]. Such an “nonparametric nature” of ASSP procedures cannot be viewed as afactor restricting their efﬁciency, because both space and space-frequency ﬁlters arepurely linear. It is known, for example, that despite the type of distribution the ﬁlterof Kalman-Busy is the best linear ﬁlter it terms of minimum error of dispersion[14]. The mentioned conclusions are true as well for algorithms based on MRSI,MPOSR, and MPSOS criteria. Here the optimality of the space (or space-frequency)linear ﬁlter is treated according with corresponding criterion. Because of it, let usdo not take into account “nonparametric” nature of ASSP algorithms (in the senseof traditional mathematical statistic) in the further discussion. Let us discuss morethoroughly a priori data about the SIE, directly used in ASSP algorithms.All ASSP algorithms use, in a varying degree, some a priori data about eitherspatial or temporal, or even about spatial-temporal structure of the useful signal. So,in the case of MSD criterion it is assumed that the reference signal is determined,which is uncorrelated with interferences copy of useful signal s(t). In the case MRSIcriterion it is assumed that either the vector→V y= β→Vs or matrix Ryy = βRss isknown. In the case of MPOSR criterion it is assumed that the direction of inputsignal and characteristics of antenna array are known. This criterion uses such char-acteristics of AA as unit nonzero elements of the matrix C, described by 3.18, whichcorresponds to the case of AA consisting from identical antenna elements.In the case of MPSOS criterion, there is no use for a priori data about eitherspatial or temporal signal structure. But in the same time it is indirectly assumed thatthe power of interferences is signiﬁcantly greater, than the power of useful signal onthe input of AA.There are two classes of ASSP algorithms, namely parametric and nonparamet-ric. This division is made on the base of attribute of using the a priori data abouta signal structure (spatial or/and temporal). In this sense the MPSOS algorithmsbelong to the class of nonparametric algorithms, whereas other algorithms (MSD,MRSI, and MPOSR) are parametric. In turn, let us divide the parametric algorithmsby the structure of a priori data about a signal, namely scalar, vector, or matrix.Therefore, the procedures based on MSD criterion belong to the class of scalar al-gorithms. They are the algorithms 3.28, 3.37, 3.42 - 3.45, 3.46. The only conditionof their application is existence of the reference signal r(t).In turn, the class of parametric algorithms with vector organisation of a prioridata includes both MRSI and MPOSR algorithms synthesized using the hypothe-sis of distributiveness for spatial and temporal structures of a signal, in particular,algorithms 3.30, 3.38, 3.47. To apply these algorithms, it is necessary to know thecharacteristics for both signal and AA, permitting to construct the vector→V y= β→Vs.It means that such characteristics should be known as direction of signal arriving,
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4.2 Nature of a Priori Uncertainty about Properties of Signal and Noise 37the carrier frequency, partial characteristics of antenna array’s directivity, as well asinter-element distances. In the common case, it is necessary to know polarizationcharacteristics of AE and their proper and reciprocal impedances. But in contrastwith algorithms using scalar structure of a priori data, these procedures do not usea priori information about the temporal structure of a signal. Moreover, these al-gorithms can be treated as invariant to the temporal signal structure with accuracyenough for practical applications.At last, the MRSI algorithms 3.29, 3.37, 3.52 belong to the class of parametricalgorithms with the matrix structure. This class includes procedures using MPOSRcriterion to AA implemented as the space-frequency ﬁlter 3.31, 3.39, 3.48. To applythese algorithms (setting of the matrix Ryy for MRSI criterion and setting of matrixC, vector→F and “aligning” delays for MPOSR criterion), it is necessary to knowsome data about correlation (spectrum) properties of a signal in addition to a priorydata necessary to construct the vector→Vy. Therefore, in contrast to algorithms withvector structure of a priori data, the parametric procedures with matrix organizationof a priori data are not invariant to the temporal signal structure. But these algorithmscan be treated as invariant to the classes of signals with identical (or similar enough)correlation (spectrum) properties.In the boundaries of accepted division, the parametric algorithms with scalarstructure of a priori data can be interpreted as procedures of optimal linear spa-tial ﬁltration of scalar signals. In turn, the parametric algorithms with either vectoror matrix structure of a priori data should be treated as algorithms of optimal lin-ear spatial ﬁltration of vector signals (with distributed or no distributed spatial andtemporal structures correspondingly). The practical application of that or this typeof parametric ASSP algorithms depends on existence or possibility for obtaining(estimation) of necessary a priori data about a signal.4.2 Nature of a Priori Uncertainty about Properties of Signaland NoiseThe actual SIE can be different from model assumptions used for synthesis of para-metric ASSP algorithms. Particularly, it is quite possible the following situations,when there are interferences on the input of AA, which do not obey to conditionE s(t)p∗l (t) = 0. In this case, the expression 3.2 is the following one:vecX(t) =→S (t)+L∑l=1→Pl (t)+L∑l=1→Pl (t)+→N (t). (4.1)In 4.1→Pl (t) = pl→Vl;E s(t)pl∗(t) E {s2(t)}E pl2(t) = ρl; ρl ∈ ]0 ÷ 1]; Lis the number of interferences that are correlative with a signal.Such a situation takes place in the case of multiradiate high-frequency prop-agation (HFP), when signals arriving along the neighbour rays are treated as
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38 4 Features of ASSP under Different Levels of A-Priori Uncertaintyinterferences. Besides, the deliberately re-reﬂected useful signal can be treated asinterference. Thereupon, it is convenient to introduce conceptions of complete andgeneralized parametric uncertainty about the interferences’ properties to character-ize conditions, upon with the ASSP is conducted. The complete parametric a prioriuncertainty assumes the lack of any a priori data about interferences, including thedata about existence of interferences correlative with the useful signal on the inputof AA. The generalized parametric a priori uncertainty, in turn, spreads the concep-tion of parametric a priori uncertainty on the case, when it is known in advance thata model of input signals can be represented as 4.1 with unknown values of coefﬁ-cients ρl.In the case of complete a priori uncertainty about properties of interferences, theabovementioned ASSP algorithms cannot be applied. But in the case of generalizedparametric a priori uncertainty they can be applied after some modiﬁcation. In thesame time, the ASSP algorithms with matrix, scalar, and vector structures of a prioridata assume precise knowledge of corresponding parameters of a signal. It means,they allow obtaining optimal solutions only for conditions a priori certainty about asignal. But in practice, a priori known parameters of a signal can differ (sometimesvery signiﬁcantly) from their corresponding actual quantities [4, 15, 17, 36].In the cases of algorithms with vector and matrix structures of a priori data (thealgorithms of optimal linear ﬁltration of vector signals), it is necessary to have apriori information about the vector→S (t). It means these algorithms need data aboutboth the scalar signal s(t) and characteristics of AA. In this case imprecise knowl-edge about→S (t) leads to violation of equalities Ryy = βRss and→Vy= β→Vs. Reasonsof origins for a priori uncertainty about properties of→S (t) are different. In partic-ular, they include: lack of precise information about spatial (angular) position of asignal source (for example, the direction on the signal source is known with accuratewithin some angular domain); spatial evolutions of either signal source or antennaarray (that is the object carrying the AA); imprecise knowledge about the carrier fre-quency of a signal (Doppler effect): inﬂuence of both surrounding and mechanismof high-frequency propagation (ﬂuctuation for direction of signal arriving, objectmultipath propagation); distinctions due to inter-inﬂuence of antenna elements andreﬂection from local objects, characteristics of AA and corresponding model as-sumptions; ﬂuctuations of elements of AA, random alterations for amplitudes andphases of currents on the outputs of antenna elements and so on.It is possible situations when a priori data needed for construction of the matrixRyy are absent completely. For example, there is no data either about the directionof signal arriving or partial characteristics of antenna elements’ orientation. Thelast situation is typical for the case when AA is situated on an airplane. Besides, insome cases the errors in a priori data can be too big, for example, if Ryy − Rss B>Ryy − Rkk B( Ryy B= Rss B = Rkk B, . B is some matrix norm).A solution for these problems is obtaining (reﬁnement) of necessary data about→S (t) during the ASSP process directly. But in the common case a precise (oreven, approximate) assessment of required data is impossible on principle eitherdue to noise and ﬁnite size of sample or incorrect model assumptions. For example,
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4.2 Nature of a Priori Uncertainty about Properties of Signal and Noise 39during assessment of direction for signal arriving there is a postulate that it is preciseknowledge about characteristics of AA, correlation noise matrix and the numberof sources [31]. Therefore, when unknown parameters are estimated, it is possi-ble only some decrease for the errors in initial data (or decrease for the quantityδ = Ryy − Rss B). It means that it has sense to introduce the conceptions of com-plete, parametric and generalized parametric a priori uncertainty about the vector ofa signal→S (t) in connection with ASSP algorithms with vector and matrix structuresa priori data. The complete a priori uncertainty determines conditions when the dataneeded for construction the matrix Ryyare absent a priori and cannot be estimatedin a real time mode. Let us point out that the matrix Ryy =→Vy→VHy in the case of nar-rowbanded input signals. In turn, the a priori uncertainty about a signal is a prioriparametric if it is known a priori the matrix Ryy, which satisﬁes to the followingcondition:Ryy − Rss B< minkRyy − Rkk B. (4.2)At last, if there is a priori knowledge about the matrix Ryy, which can be sufﬁcientlyclose to matrix Rss (when δ = Ryy − Rss Bis a small quantity), but there is noinformation whether condition 4.2 is true, then such a situation is treated as thegeneralized parametric uncertainty.The introduced conceptions of complete, parametric and generalized parametricuncertainty can be generalized for the case of algorithms with scalar structure ofa priori data. In this case, the complete a priori uncertainty takes place when ei-ther there are some problems with generation of the reference signal or the valuer(t) has no correlation neither signal nor interference. If the following conditionsE {r(t)s∗(t)} = ρrs = 0, E r(t)pl∗(t) = ρrp = 0 take place, then there is eitherparametric (when there is a priori data about values of ρrs, ρrp) or generalized para-metric (when there is no such information) uncertainty.The abovementioned situations arise due to the fact that the reference signal isconstructed, as a rule, on the base of output signal of AA. In these conditions, thesolutions for the task of synthesis for reference signal are known only for somespeciﬁc types of temporal structures of useful signal (namely, ﬁr signals which aresimulated by pseudorandom sequences) [30]. If strong interferences present on thestage of synchronisation (the devices forming reference signal, as a rule, include“rigid” restrictors, then it is impossible to form a reference signals even for theseparticular conditions. The reference signal can be correlative with either interfer-ence, which is re-reﬂection of useful signal, or signal from the neighbour wavepath,and so on. It could be shown that in the case of algorithms with scalar structure ofa priori data about a signal the a priori uncertainty about a signal should be treatedas imprecise knowledge of the vector→S (t). Let us deal with the narrowbanded caseand let it be L = 1, L = 0 in 4.1. It gives the following equation:→Rxr= E→X (t)r∗(t) = a1→V s +a2→V1=→Vy, (4.3)
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40 4 Features of ASSP under Different Levels of A-Priori Uncertaintywhere a1 = E {s(t)r∗(t)}; a2 = E {p(t)r∗(t)}.The vector→Vy can be replaced by corresponding matrix Ryy =→Vy→V yH. In thiscase the only peculiarity of parametric algorithms with scalar structure of a pri-ori data (MSD-algorithms) is reduced to existence of dependence for the degree ofnearness of vector→Vy to vector→Vs (matrix Ryy to matrix Rss). It depends on inter-correlative properties for r(t), s(t), pk(t), as well as on ratio of powers for signaland interference.Thus, in the cases of parametric ASSP algorithms, the a priori uncertainty abouta signal should be treated as imprecise knowledge of vector→S (t) (or vector→Vy, ormatrix Ryy).The problems of ASSP theory connected with different levels of uncertaintyabout parameters of signals and interferences are discussed, for example, in [33, 36].It is shown that under the condition of complete uncertainty about a signal the para-metric algorithms cannot be applied. Under such conditions, it is recommended toapply nonparametric (MPSOS) algorithms. The other way is to consider the task ofASSP as a task of distribution for signals [2, 24, 26]. Let us restrict ourselves onlythe most important results obtained under attempts to generalize of ASSP theory onthe case of parametric (including generalized) a priori uncertainty about a signal andinterferences. Let us discuss only the N - dimensional space ﬁlter (the main results,as a rule, are true as well for the N × (M + 1) - dimensional space-frequency ﬁlter).Following the classical tradition, let us show the main analytical dependences forsignals and interferences with distributed spatial and temporal structures. It is worthto point out that these “classical traditions” have perfectly objective foundation, be-cause the “readable” analytical expressions can be obtained only for N ×N matriceswith a small rank [5, 27, 29].4.3 Methods of SSP under Generalized Parametric Uncertaintyabout the Noise PropertiesLet us assume that a priori data about a signal are known exactly, whereas the gener-alized parametric uncertainty takes place with respect to interferences. Let us pointout that the case of parametric a priori uncertainty about the properties of interfer-ences is taken into account under the synthesis of corresponding algorithms. Let usassume that the vector of input signals→X (t) is represented by expression 4.1, wherethe carrier frequencies are identical for signals and interferences. Let in this casethe AA includes isotropic and noninteracting antenna elements and let the noiseis spatial-white (Rnn = σ2n I). Let us estimate the behaviour of optimal solutions3.7, 3.15, and 3.19 if interferences present on the input of AA, which are correla-tive with a signal. Because the optimal vectors of weight coefﬁcients→WSD,→WRSI,and→WPOSR coincide up to a constant coefﬁcient, let us discuss only the expres-sion 3.19. Let us analyze the optimal VWC 3.19 under the following conditions: let
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4.3 Methods of SSP under Generalized Parametric Uncertainty 41→W1= lim(Ps σ2n )→∞→WRSI,→W2= lim(Pl σ2n )→∞→WRSI, Ps.out(→W) = β→WHRss→W. It means thatif (L+ L ) < N, then there is Ps.out(→W1) = Ps.out(→W2) = 0∀ρl > 0.To prove it, let us assume that L = 1, E s(t)p∗l (t) = ρl in the expression 4.1and let us represent the correlation matrix Rxx as the following one:Rxx = Rss + RΣ + A+ AH+ Rnn, (4.4)where there are Rss = Ps→V s→VHs ; RΣ = ∑L+1j=1 Rj j = ∑L+1j=1 Pj→V j→VHj ; A = ρl→Vl→VHs ; and,at last, Rnn = σ2n I.Using 3.15, we can get that→WRSI= βR−1xx→V s= βRin−1 →V s,Rin = RΣ + A+ AH+ Rnn. (4.5)In 4.5 Rj j, A are matrices of the unit rank, whereas Rnn is a diagonal matrix.Therefore, if inequality (L+ L ) < N takes place, then λmin(Rin) = σ2n , rank(Rin −λmin(Rin)I) = rank(RΣ + A + AH) = L + 2[33,36]. The optimal VWC 4.5 is deter-mined up to some rating coefﬁcient β, where the matrix Rin is a normalized matrix,too. Assuming that Rin is normalized by division on maximal in modulus element,we can get λmin(Rin) = σ2n ∑L+1j=1 Pj + 2Re{ρ} . Using this equality, we can cometo the limit ratio [9]:lim(Ps σ2n )→∞Rin−1= lim(Pl σ2n )→∞Rin−1= B λmin Rin , (4.6)where B(λmin (Rin)) is an adjoint matrix for the matrix Rin.On the base of 4.5 - 4.6, in turn, it can be got that→W1=→W2= B λmin Rin→V s . (4.7)It is known that the columns B(λmin (.)) are the proper vectors Rin, corresponding tothe same λmin (Rin). These vectors are orthogonal to vectors→V j, j = 1,L+ 1, as wellas to signal vector→Vs. In the same time, any linear combination of proper vectors,for example 4.7, represents a proper vector too [16]. Therefore, the vectors→W1,→W2are the proper vectors of matrix Rin, corresponding to its minimum proper number.It means they are orthogonal to the signal vector→Vs. Thus, it is possible to write that→WH1 Rss→W1=→WH2 Rss→W2= 0.As follows from abovementioned, in the case of generalized parametric uncer-tainty about properties of interferences, the VWC 3.7, 3.15, and 3.19do not provideobtaining optimal solutions, moreover, they are orthogonal asymptotically to thevector of useful signal. In the common case, which is not asymptotical, these vec-tors are not orthogonal to the signal and the ASSP quality (RSIN on the output ofthe linear space ﬁlter) depends on the values Ps σ2n , Pl σ2n , ρl, and even, on Pj σ2n .
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42 4 Features of ASSP under Different Levels of A-Priori UncertaintyThe examples for dependences of RSIN η→W from the values Ps σ2n , Pl σ2nand P1 σ2n are shown in Fig. 4.1 - Fig. 4.3 The ﬁlter is optimized by the MRSIcriterion. The symbol Pl stands for the power of correlated interference. These de-pendences (on the output of the space ﬁlter) are obtained using analytical simu-lation. It is assumed here that there is a linear equidistance antenna array, N = 5,L = 1 = L = 1, the direction of signal arriving Θs = 400(it is determined relativelyto the line of disposition of antenna elements), direction of uncorrelated interfer-ence arriving Θ1 = 600, direction of correlated interference arriving Θl = 300, in-put ratio 10lgPl σ2n = 10dB (Fig. 4.1, 4.3), signal/noise ratio 10lgPs σ2n = 20dB(Fig. 4.2, 4.3), (uncorrelated interference)/noise 10lgP1 σ2n = 20dB (Fig. 4.1),10lgP1 σ2n = 10dB (Fig. 4.2).RSIN, dBPs/σ2n, dBρl=0ρl=0,125ρl=0,25ρl=0,5ρl=0,75ρl=1Fig. 4.1 Dependence of output RSIN from input ratio signal/noiseRSIN, dBP´1/σ2n, dBρl=0ρl=0,125ρl=0,25ρl=0,5ρl=0,75ρl=1Fig. 4.2 Dependence of output RSIN from input ratio (correlated interference)/noise
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4.3 Methods of SSP under Generalized Parametric Uncertainty 43RSIN, dBP1/σ2n, dBρl=0ρl=0,125ρl=0,25ρl=0,5ρl=0,75ρl=1Fig. 4.3 Dependence of output RSIN from input ratio interference/noiseAs follows from Fig. 4.1 - Fig. 4.2), if there are interferences correlated with asignal, then RISP depends in the most degree from input ration (correlated interfer-ence)/noise. In this case, if there is ρl ≥ 0,2, then AAA cannot operate practicallyin all signal-interference situations (the value of RISP decreases more than 10 dB inrelation to potentially reachable quantity.In the abovementioned reasoning, the coefﬁcients ρl characterize only inter-correlative properties of signal envelopes, whereas their carrier frequencies are as-sumed to be identical. If in the case of identical carrier frequencies there is ρl → 1,then such a signal and interference are called spatial coherent. This case is the worstfor ASSP; it is investigates perfectly well [22, 25]. It is known a lot of solutions(approaches) providing optimization of SF under MRSI, MPOSR, and MSD cri-teria under existence of correlated (spatial coherent with a signal0 interferences[3, 11, 12, 33, 36].The variants of such solutions are very different, but there is the same basic idea,namely transformation of the vector of input signals using some operator F{.} to theform→Y (t) = F→X (in the common case, the dimension of the vector→Y (t) does notcoincide with the dimension of the vector→X (t)). The operator F{.}keeps in→Y thespatial and temporal structures of useful signal→S (t) and permits the decorrelationof interferences→Pcl (t). Then the vector→Y (t) is ﬁltrated using mentioned aboveparametric ASSP algorithms. The decorrelation of interferences (implementation ofF{.}) is provided either by the random “shift” of AA aperture supplying interfer-ences by additional phase modulation with preservation of unchangeable temporalstructure of useful signal, or by partition of→X (t) on the partially overlapped sub-vectors, and so on. All known approaches to construct the operator F{.} assumeexact knowledge of spatial structure of a signal. Besides, as a rule, it is required aredundant value of degrees of freedom for the antenna array [12].So, despite the fact that main results of classical ASSP theory are extended onthe case of generalized parametric uncertainty about the properties of interferences,
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44 4 Features of ASSP under Different Levels of A-Priori Uncertaintythe essential limitation of this theory is its orientation on exact a priori data aboutthe spatial structure of a signal. But in the real conditions these data, as a rule, haveapproximate nature. It means that the problem of applicability of ASSP in condi-tions of the generalized parametric uncertainty about the interference properties isconnected inseparably with the problem about applicability of ASSP when there isno exact a priori information about a signal.4.4 Methods of SP under a Priory Parametric Uncertaintyabout Properties of Useful SignalThe hypothesis about existence of exact a priori data used under the synthesis ofASSP parametric algorithms is not true for the majority of practical cases. In thereal conditions we should say not about the existence of errors, but rather abouttheir values and possibility to neglect their inﬂuence.Let us estimate the potential possibilities of optimal VWC 3.7, 3.15, and 3.19under conditions, when required a priori data about the vector→S (t) differ from cor-responding model assumptions that Ryy = βRss. Let us restrict ourselves by the caseof parametric a priori uncertainty about the signal properties. Let us make no differ-ence among VWC→WRSI,→WSD and→WPOSR; it means we can restrict our analysis onlyby the expression 3.19. To take into account the assumption about distributivenessof spatial and temporal structures of a signal, let us use instead of general expres-sion δ = Ryy − Rss B( Ryy B= Rss B) some equivalent conditions 4.8 and 4.9to characterize the quantity of the error in a priori data. These conditions are thefollowing ones:→V y −→Vs = δ1,→Vy =→Vs , (4.8)γsy =→Vs,→Vy = γ0. (4.9)In these conditions there are→V y =→Vs ; . stands for Euclidean vector norm inthe N - dimensional complex space; δ1 ∈ R+ ∪0,γsy = arccos→VHs→Vy→V y→Vs is a generalized angle between→Vy è→V s;γ0 ∈ R+. The condition 4.8 is more general than the condition 4.9 and it s moreinvariant to normalization of vectors→V yand→Vs. Sometimes, the component cosγsyis named the coefﬁcient of spatial correlation [22, 25].Let us represent the vector of weight coefﬁcient optimal by the MRSI criterionin two equivalent forms:→W1RSI= βR−1in→Vy, (4.10)
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4.4 Methods of SP under a Priory Parametric Uncertainty 45→W2RSI= βR−1xx→Vy . (4.11)Let us confront expressions 4.10 and 4.11 in different signal-interference environ-ments. If there is δ1 = 0, γ0 = 0 in 4.8, and 4.9, then the VWC 4.10 and 4.11 alwaysprovide accomplishment of the following unstrict inequalityηout(→W) ≥ ηin. (4.12)In 4.12 symbols ηin, ηout(→W) stand for RSIN on the input and output of a spaceﬁlter respectively. Let us point out that the relation 4.12 turns into equality iff thereare L = 1, L = 0,→Vs −→V1 = 0, (γs1 =→V s,→V1 = 0) in expression 4.1.If there are no interferences (L = 0), then the proof is a trivial one. Really, inthis case there is→W1RSI= β→W2RSI= β→Vs (here and further, the value β is used as anormalizing factor; it replaces the following phrase “with accuracy up to a constantcoefﬁcient”). Thus, there is the following equation for RSIN on the output of SF:ηout(→W) =→WHRss→W→WHRin→W= N Psσ2n= Nηin. (4.13)If there is a single interference (L = 1 in 4.1), using 4.10 and 4.11 we can get→W1RSI= β→W2RSI= α σ2H + P1→VH1→V1 I − P1→V 1→VH1→V y, (4.14)where there is α = 1 σ2H σ2H + P1→VH1→V 1 .Assuming that→Vy −→Vs =→Vy −→V1 = 0 in 4.14, the following equation canbe obtained:→W1RSI= β→W2RSI= β→Vy. It means that VWC→W1RSI,→W2RSI providein-phase summation as for a signal and for interferences and there is îηout(→W) =NηinN = ηin. In the case of→Vs −→V1 > 0, the vector 4.14 executes only in-phasesummation for a signal, providing ηout(→W) > ηin.At last, let us discuss the case with L ≥ 2. Let us presume that the followingcondition→Vk −→Vs = 0 (γks =→V k,→Vs = 0) takes place for the interferencenumber k. Using the lemma about the matrix inversion [35], there is:R−1in =⎧⎪⎪⎨⎪⎪⎩R−1gg −11+→VHy Ryy→VyR−1yy→Vy→VHy R−1yy⎫⎪⎪⎬⎪⎪⎭, (4.15)where Rgg = ∑Lj=1,j=k Pj→V j→VHj +σ2n I;→Vk=→Vs=→Vy.Substituting 4.15 into 4.10 leads to the expression
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46 4 Features of ASSP under Different Levels of A-Priori Uncertainty→W1RSI= β→W2RSI= βR−1gg→V y . (4.16)It follows from 4.16 that→W1RSI (→W2RSI) provides maximization for relationηout(→W) = Ps.outPΣout, where Ps.out is the signal power, PΣout is the total power for (K −1)interferences and the noise on the output of SF. Obviously, if ηout(→W) is maximumand Ps.out Pk.out = Ps Pk (Pk.out is a power of interference number kon the output ofSF), then there is ηout(→W) > ηin.The expression 4.12, in fact, is a background for reasonability of application forASSP algorithms implementing the optimal VWC similar to 4.10 and 4.11. Actu-ally, excluding the strictly special case γks =→Vk,→Vs = 0, they always provideexecution of inequality ηout(→W) > ηin, moreover, the value ηout(→W) reaches its po-tentially possible quantity. In the same time, there is no obvious necessity for jointconsideration vectors 4.10 and 4.11. it was found previously that these VWC areequal up to some constant coefﬁcient. It means that γ12 =→W1RSI,→W2RSI = 0.But the last equality is true iff there is δ1 = 0 in the expression 4.8. To be sure in va-lidity of this thesis, it is enough to compare both→W1RSI,→W2RSI taking into accountexpressions 4.8 and 4.9. In particular, if there are L = 0, L = 0 in the expression 4.1we can obtain the following equalities:Rin = σ2n I,Rxx = Ps→V s→VHs +σ2n I. (4.17)Due to substituting 4.17 into expressions 4.10 and 4.11, it can be obtained aftersome necessary transformations that:→W1RSI= β→Vy, (4.18)→W2RSI= β→VHs→Vs +σ2H Ps→Vy −ρsy→V s ,ρsy =→VHs→Vy . (4.19)Using 4.18 and 4.19 and assuming that→V y =→V s =√N, β = 1, we can get that:ξ→W1RSI = N2cos2γsy, (4.20)ηout→W1RSI = N cos2γsy Ps σ2n , (4.21)ξ→W2RSI = N2cos2γsy σ2n Ps2, (4.22)ηout→W2RSI = Ps σ2nN2 cos2 γsy σ2n Ps2B+ N2 cos2 γsy − 2NBcosγsy, (4.23)
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4.4 Methods of SP under a Priory Parametric Uncertainty 47where ξ (.) = Ps.out Ps; γsy =→V s,→Vy ; B = N + σ2n Ps.As follows from 4.20 - 4.23, for both vectors 4.10 and 4.11, both the power ofa signal and RSIN on the output of SF depend on the value of coefﬁcient of spacecorrelation (CSC) cosγsy.The value of CSC characterizes the degree of closeness for vectors→Vs and→Vy. Inthis case we have:limCSC→0ξ(→W1RSI) = limCSC→0ξ(→W2RSI) = 0, (4.24)limCSC→0ηout(→W1RSI) = limCSC→0ηout(→W2RSI) = 0. (4.25)Besides, it follows from 4.21 and 4.22 that the following equality takes place in thecase of the vector 4.11:lim(σ2n Ps)→0ξ(→W2RSI) = lim(σ2n Ps)→0ηout(→W2RSI) = 0 ∀ γsy > 0. (4.26)Thus, under conditions of parametric a priori uncertainty, the vectors of weight co-efﬁcients 4.10 and 4.11 are different; they can completely reject the useful signaleven if there are no interferences. Such a signal rejection can take place either dueto decrease of CSC (in the case of VWC (3.10)), or because of increase for the inputratio Signal/noise (in the case of VWC (3.11)). Moreover, in the case of VWC 4.11the useful signal can be rejected asymptotically even under the utmost small valuesfor cosγsy, δ1 in expressions 4.8 and 4.9.If there are interferences (for example, if L = 1), then the correlation matrix Rinis the following one:Rin = P1→V1→VH1 +σ2n I. (4.27)Substituting 4.27 into 4.10, we can get the following:→W1RSI= β→VH1→V 1 +σ2n P1→V y −ρ1y→V 1 ,ρ1y =→VH1→Vy . (4.28)Using 4.28 and assuming that→Vy =→Vs =→V1 =√N, we can get that:ξ→W2RSI = N + σ2n Ps N2 cos2 γsy + N4 cos2 γsy−−2 N + σ2n Ps × Re ρsyρ1yρ1s ,(4.29)ηout→W1! =Ps σ2n ξ→W1RSIB+ N2 cos2 γ1y − 2BN2 cos2 γ1y, (4.30)where ρ1s =→VHs→V 1; cosγ1s = |ρ1s|→Vs→V1 ; B = N + σ2n Ps.It follows from 4.29 and 4.30 that if there are interferences, then provided VWC4.10 depends on closeness of vectors→V y and→Vs (γsy), as well as on corresponding
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48 4 Features of ASSP under Different Levels of A-Priori Uncertaintyangles γ1y, γ1s and reciprocal combinations of coefﬁcients ρsy, ρ1y, ρ1s. In this case,limiting values 4.24 and 4.25 do not take place. It means that existence of interfer-ence leads to decrease for the depth of signal rejection.If we take into account that the matrix Rxx is a low-rank modiﬁcation of thematrix Rin (Rxx = Rin + Ps→Vs→VHs ) and use the lemma about matrix inversion, thenVWC 4.11 can be represented as the following expression:→W2RSI=⎡⎣R−1in −PsR−1in→Vs→VHs R−1in1 + Ps→VHs R−1in→Vs⎤⎦→V y, (4.31)where R−1in = α σ2n + P1→VH1→V1 I − P1→V1→VH1 ; α=1 σ2n σ2n + P1→V1→VH1 .After transformations we can use the fact that→Vy =→V s =→V1 =√N. Itpermits to transform the expression 4.31 into the following one:→W2RSI= A→Vy −P1ρ1y→V1 −αNA− P1 ρ1y2A→V y −P1ρ1y→V1NA− P1 |ρ1s|2, (4.32)where A = σ2n + P1→V1→VH1 .Analysis of VWC 4.31 shows that the limiting relation 4.26 keeps its truthfulness;in the same time the values ξ→W2RSI , η→W2RSI depend signiﬁcantly on absolutevalues of quantities cosγsy, cosγ1y, cosγ1s, as well as on mutual products ρsy, ρ1y,ρ1s. The general nature of these dependences is the same even in the case of increasefor the number of interferences. Some results of comparison for potential efﬁciencyof VWC→W1RSI,→W2RSI under the conditions of parametric a priori uncertainty aboutthe signal properties L ≥ 0 are given in [23]. The diagrams from this work showthat optimal VWC 3.19 cannot operate under conditions of parametric a priori un-certainty, even in very simple signal-interference situations. This conclusion is validfor the case of generalized parametric uncertainty, too.The problem of “inoperativeness” (or very low efﬁciency) for parametric ASSPalgorithms was many times discussed in corresponding special literature. As a rule,only the speciﬁc sources for errors’ occurrence have been researched, such as theimprecise knowledge either of arriving direction or of signal frequency, and so on. Itthe same time, the vector nature of a priori uncertainty was not taking into account inexplicit form. Nowadays, a lot of approaches is known connected with modiﬁcationof optimal vectors 3.7 and 3.19 (the vector (2.21) in the wideband case) and corre-sponding algorithms permitting providing of operability for ASSP in the conditionsof parametric a priori uncertainty. These approaches tend to decrease the sensitivityof ASSP algorithms to imprecision of a priori data about a signal [1, 3, 11, 18–21, 32]. Such algorithms are synthesized or made using some heuristics; as a rule,they are called robust algorithms of ASSP. We think that the term “robust” is not
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References 49perfectly suit to the nature of these algorithms, because in the theory of mathe-matical statistics such a deﬁnition is used for procedures, which are insensitive (upto some degree) to parameters of distribution law of observed (estimated) variable[10, 28] (in this sense, the ASSP algorithms can be treated as nonparametric). But letus the term “robustness” to determine the corresponding class of ASSP procedures,to avoid some terminological alternative versions. Of course we should specify itsnumerical content.The development of methods for synthesis of robust algorithms is one of the mostimportant directions for evolution of ASSP theory. This direction is not “closed” andnow there are very intensive researches conducted in this area.References1. Cantoni, A., Guo, L.X., Teo, K.L.: A new approach to the optimization of robust antennaarray processors. IEEE Trans. Antennas and Propag. 41(4), 403–411 (1993)2. Morozov, A.K., Licarev, N.A.: Adaptive antenna system for distribution of signals, ar-riving from different directions. Radiotechnika (9), 66–69 (1985) (in Russian)3. Alireza, M.: Analysis of the spatial ﬁltering approach to the decorrelation of coherentsources. IEEE Trans. Signal Proces. 40(3), 692–694 (1992)4. Fridlander, B.: Sensitivity analysis of the maximum likelihood direction ﬁnding algo-rithm. In: 23rd Asilomar Conf. Signals, Syst. and Comput., Paciﬁc Grove, Calif. - SanJose (1989)5. Parlet, B.: The symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs (1980)6. Gardiner, C.B.: Stochastic methods: A Handbook for the Natural and Social Sciences.Springer, Berlin (2009)7. Pitman, E.: Some basic theory of statistical inference. Springer, Berlin (1979)8. Churakov, E.P.: Optimal and adaptive systems. Energpizdat, M. (1987) (in Russian)9. Guntmacher, F.R.: Theory of matrices. Nauka, M. (1988) (in Russian)10. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics. TheApproach Based on Inﬂuence Functions. John Wiley and Sons, New York (1986)11. Loskutova, G.V.: About inﬂuence of spatial smoothing on angular resolution of signalsin aaa. Radiotechnika i Elektronika 35(12), 2557–2562 (1990) (in Russian)12. Krim, H., Viberg, M.: Two decades of array signal processing research. the parametricapproach. IEEE Signal Processing Magazine 13(4), 67–94 (1996)13. Van Trees, H.: Detection, estimation and modulation theory (Part 1). John Wiley andSons, N.J. (2001)14. Medich, J.: Statistically optimal linear estimates and control. Energia, M. (1973) (inRussian)15. Riba, J., Goldberg, J., Vazquez, G.: Robust beamforming for interference rejection inmobile communications. IEEE Trans. Signal Processing. 45(1), 148–161 (1997)16. Wilkinson, J.H.: The algebraic eigenvalue problem. Clarendon Press, Oxford (1965)17. Kim, J.W., Un, C.K.: An adaptive array robust to beam pointing error. IEEE Trans.,Signal Process. 40(6), 1582–1584 (1992)18. Kim, J.W., Un, C.K.: An adaptive array robust to beam pointing error. IEEE Trans.,Signal Process. 40(6), 1582–1584 (1992)19. Kim, J.W., Un, C.K.: A robust adaptive array based on signal subspace approach. IEEETrans. Signal Process. 41(1), 3166–3171 (1993)
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50 4 Features of ASSP under Different Levels of A-Priori Uncertainty20. Lo, K.W.: Reducing the effect of pointing error on the performance of an adaptive array.Electron Letters 26(2), 1646–1647 (1990)21. Lo, K.W.: Improving performance of adaptive array in presence of pointing errors usingnew zero-correlation method. Electron Letters 27(5), 443–445 (1991)22. Marchuk, L.A.: Spatial-temporal processing of signals in radio links. VAS, L. (1991) (inRussian)23. Marchuk, L.A.: Wiener solution and potential efﬁciency of adaptive spatial signal pro-cessing. Radiotechnika (5):75–79 (1996) (in Russian)24. Marchuk, L.A., Giniatulin, N.F., Kolinko, A.V.: Analysis of algorithms for minimizingpower of output signal in adaptive antenna arrays. Radiotechnika i Elektronika 42(6),1–6 (1997) (in Russian)25. Lin, H.-C.: Spatial correlation in adaptive arrays. IEEE Trans. Antennas and Propaga-tion 30(2), 212–223 (1982)26. Hackett, C.M.: Adaptive array can be used to separate communication signals. IEEETrans. Aerospace and Electronic Systems 17(2), 234–245 (1981)27. Lancaster, P., Tismentski, M.: The theory of matrices. Academic Press, Orlando (1985)28. Huber, P.J.: Robust Statistics. John Wiley and Sons, New York (1981)29. Bellman, R.: Introduction to matrix analysis. McGraw Hill, N.Y. (1970)30. Compton, R.T.: Adaptive Antennas. Concept and Performance. Prentice Hall, Engle-wood (1988)31. Marpl, S.: Spectral analysis and its application. Radio i Swjaz, M. (1989) (in Russian)32. Kassam, S.A., Pure, G.V.: Robust methods for signal processing. survey. Transactions ofIEEE 73(3), 324–341 (1985)33. Shan, T.-J., Kailats, T.: Adaptive beamforming for coherent signals and interference.IEEE Trans. Acoust., Speech and Signal Processing 33(3), 527–536 (1985)34. Repin, V.G., Tartakovskij, G.P.: Statistical synthesis under a priori uncertainty and adap-tation of information systems. Sovietskoje Radio, M. (1977) (in Russian)35. Wojevodin, V.V., Kuznecov, J.A.: Matrices and calculations. Nauka, M. (1984) (in Rus-sian)36. Ogawa, Y., Ohmiy, M., Itoh, K.: An lms adaptive array for multipath fading reduction.IEEE Trans. Antennas and Propag. 34(3), 17–23 (1986)
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