1. Chapter 4
Features of ASSP under Different Levels
of A-Priori Uncertainty
Abstract. 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 thorough
characteristics of a-priory data needed for implementation of systems with adaptive
spatial signal processing. Next, the characteristic of a-priory uncertainty about the
properties of both signal and noise is given. The reasons for noise appearance and
levels of noise are shown. The third part of the chapter deals with the spatial signal
processing under generalized parametric uncertainty about the noise properties. The
last part is devoted to signal processing under a-priory parametric uncertainty about
the noise properties.
4.1 Analysis of Peculiarities of ASSP with Different Levels of
A-Priori Uncertainty
Abovementioned ASSP algorithms are oriented on output signal optimization for
space (space-frequency) filter under the conditions when both the amount of inter-
ferences and their spatial and temporal characteristics are unknown a priori. In the
common case, the spatial properties of the noises are unknown too. In the analysis
problems is assumed that a noise is spatially white, but this hypothesis is true only
for noises with the “thermal” origin [6, 13]. These conditions can be treated as a
classical 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 correlation
approximation. 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 that
interferences are not correlative with the useful signal.
In the constraints of the adaptive Bayes approach the overcoming of parametric a
priori uncertainty about the properties of interferences and noise can be worked out
by replacement of unknown correlation matrix Rxx, as well as vector of correlation
L. Titarenko et al.: Methods of Signal Processing for Adaptive Antenna Arrays, SCT, pp. 35–50.
springerlink.com c Springer-Verlag Berlin Heidelberg 2013

2. 36 4 Features of ASSP under Different Levels of A-Priori Uncertainty
for input and reference signals
→
Rxr by corresponding consistent estimates
→
Rxx and
→
Rxr in optimal vectors of weight coefficients. It means that such a replacement is
made for optimization tasks, too.
In the same time, the ASSP algorithms do not use a priori information about
the distribution laws for signals and interferences. From this point of view they
can be treated as representatives of the class of nonparametric procedures [7, 8,
34]. Such an “nonparametric nature” of ASSP procedures cannot be viewed as a
factor restricting their efficiency, because both space and space-frequency filters are
purely linear. It is known, for example, that despite the type of distribution the filter
of 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 us
do not take into account “nonparametric” nature of ASSP algorithms (in the sense
of traditional mathematical statistic) in the further discussion. Let us discuss more
thoroughly a priori data about the SIE, directly used in ASSP algorithms.
All ASSP algorithms use, in a varying degree, some a priori data about either
spatial 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 MRSI
criterion it is assumed that either the vector
→
V y= β
→
Vs or matrix Ryy = βRss is
known. In the case of MPOSR criterion it is assumed that the direction of input
signal 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, which
corresponds 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 either
spatial or temporal signal structure. But in the same time it is indirectly assumed that
the power of interferences is significantly greater, than the power of useful signal on
the 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 about
a signal structure (spatial or/and temporal). In this sense the MPSOS algorithms
belong to the class of nonparametric algorithms, whereas other algorithms (MSD,
MRSI, and MPOSR) are parametric. In turn, let us divide the parametric algorithms
by 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 condition
of their application is existence of the reference signal r(t).
In turn, the class of parametric algorithms with vector organisation of a priori
data 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 the
characteristics 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,

3. 4.2 Nature of a Priori Uncertainty about Properties of Signal and Noise 37
the carrier frequency, partial characteristics of antenna array’s directivity, as well as
inter-element distances. In the common case, it is necessary to know polarization
characteristics of AE and their proper and reciprocal impedances. But in contrast
with algorithms using scalar structure of a priori data, these procedures do not use
a priori information about the temporal structure of a signal. Moreover, these al-
gorithms can be treated as invariant to the temporal signal structure with accuracy
enough for practical applications.
At last, the MRSI algorithms 3.29, 3.37, 3.52 belong to the class of parametric
algorithms with the matrix structure. This class includes procedures using MPOSR
criterion to AA implemented as the space-frequency filter 3.31, 3.39, 3.48. To apply
these algorithms (setting of the matrix Ryy for MRSI criterion and setting of matrix
C, vector
→
F and “aligning” delays for MPOSR criterion), it is necessary to know
some data about correlation (spectrum) properties of a signal in addition to a priory
data necessary to construct the vector
→
Vy. Therefore, in contrast to algorithms with
vector structure of a priori data, the parametric procedures with matrix organization
of a priori data are not invariant to the temporal signal structure. But these algorithms
can 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 scalar
structure 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 vector
or 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 and
temporal structures correspondingly). The practical application of that or this type
of 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 Signal
and Noise
The 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 condition
E 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 pl
2(t) = ρl; ρl ∈ ]0 ÷ 1]; L
is 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

4. 38 4 Features of ASSP under Different Levels of A-Priori Uncertainty
interferences. Besides, the deliberately re-reflected useful signal can be treated as
interference. Thereupon, it is convenient to introduce conceptions of complete and
generalized parametric uncertainty about the interferences’ properties to character-
ize conditions, upon with the ASSP is conducted. The complete parametric a priori
uncertainty assumes the lack of any a priori data about interferences, including the
data about existence of interferences correlative with the useful signal on the input
of 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 that
a 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, the
abovementioned ASSP algorithms cannot be applied. But in the case of generalized
parametric a priori uncertainty they can be applied after some modification. In the
same time, the ASSP algorithms with matrix, scalar, and vector structures of a priori
data assume precise knowledge of corresponding parameters of a signal. It means,
they allow obtaining optimal solutions only for conditions a priori certainty about a
signal. But in practice, a priori known parameters of a signal can differ (sometimes
very significantly) from their corresponding actual quantities [4, 15, 17, 36].
In the cases of algorithms with vector and matrix structures of a priori data (the
algorithms of optimal linear filtration of vector signals), it is necessary to have a
priori information about the vector
→
S (t). It means these algorithms need data about
both 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. Reasons
of 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 a
signal source (for example, the direction on the signal source is known with accurate
within some angular domain); spatial evolutions of either signal source or antenna
array (that is the object carrying the AA); imprecise knowledge about the carrier fre-
quency of a signal (Doppler effect): influence of both surrounding and mechanism
of high-frequency propagation (fluctuation for direction of signal arriving, object
multipath propagation); distinctions due to inter-influence of antenna elements and
reflection from local objects, characteristics of AA and corresponding model as-
sumptions; fluctuations of elements of AA, random alterations for amplitudes and
phases of currents on the outputs of antenna elements and so on.
It is possible situations when a priori data needed for construction of the matrix
Ryy are absent completely. For example, there is no data either about the direction
of signal arriving or partial characteristics of antenna elements’ orientation. The
last situation is typical for the case when AA is situated on an airplane. Besides, in
some 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 (or
even, approximate) assessment of required data is impossible on principle either
due to noise and finite size of sample or incorrect model assumptions. For example,

5. 4.2 Nature of a Priori Uncertainty about Properties of Signal and Noise 39
during assessment of direction for signal arriving there is a postulate that it is precise
knowledge about characteristics of AA, correlation noise matrix and the number
of 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 quantity
δ = 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 of
a signal
→
S (t) in connection with ASSP algorithms with vector and matrix structures
a priori data. The complete a priori uncertainty determines conditions when the data
needed for construction the matrix Ryyare absent a priori and cannot be estimated
in a real time mode. Let us point out that the matrix Ryy =
→
Vy
→
V
H
y in the case of nar-
rowbanded input signals. In turn, the a priori uncertainty about a signal is a priori
parametric if it is known a priori the matrix Ryy, which satisfies to the following
condition:
Ryy − Rss B
< min
k
Ryy − Rkk B
. (4.2)
At last, if there is a priori knowledge about the matrix Ryy, which can be sufficiently
close to matrix Rss (when δ = Ryy − Rss B
is a small quantity), but there is no
information whether condition 4.2 is true, then such a situation is treated as the
generalized parametric uncertainty.
The introduced conceptions of complete, parametric and generalized parametric
uncertainty can be generalized for the case of algorithms with scalar structure of
a 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 value
r(t) has no correlation neither signal nor interference. If the following conditions
E {r(t)s∗(t)} = ρrs = 0, E r(t)pl
∗(t) = ρrp = 0 take place, then there is either
parametric (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 is
constructed, as a rule, on the base of output signal of AA. In these conditions, the
solutions for the task of synthesis for reference signal are known only for some
specific types of temporal structures of useful signal (namely, fir signals which are
simulated by pseudorandom sequences) [30]. If strong interferences present on the
stage of synchronisation (the devices forming reference signal, as a rule, include
“rigid” restrictors, then it is impossible to form a reference signals even for these
particular 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 of
a priori data about a signal the a priori uncertainty about a signal should be treated
as imprecise knowledge of the vector
→
S (t). Let us deal with the narrowbanded case
and 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)

6. 40 4 Features of ASSP under Different Levels of A-Priori Uncertainty
where a1 = E {s(t)r∗(t)}; a2 = E {p(t)r∗(t)}.
The vector
→
Vy can be replaced by corresponding matrix Ryy =
→
Vy
→
V y
H
. In this
case 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 of
nearness 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 signal
and interference.
Thus, in the cases of parametric ASSP algorithms, the a priori uncertainty about
a signal should be treated as imprecise knowledge of vector
→
S (t) (or vector
→
Vy, or
matrix Ryy).
The problems of ASSP theory connected with different levels of uncertainty
about 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 to
apply nonparametric (MPSOS) algorithms. The other way is to consider the task of
ASSP as a task of distribution for signals [2, 24, 26]. Let us restrict ourselves only
the most important results obtained under attempts to generalize of ASSP theory on
the case of parametric (including generalized) a priori uncertainty about a signal and
interferences. 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 for
signals and interferences with distributed spatial and temporal structures. It is worth
to point out that these “classical traditions” have perfectly objective foundation, be-
cause the “readable” analytical expressions can be obtained only for N ×N matrices
with a small rank [5, 27, 29].
4.3 Methods of SSP under Generalized Parametric Uncertainty
about the Noise Properties
Let 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 point
out 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 us
assume that the vector of input signals
→
X (t) is represented by expression 4.1, where
the carrier frequencies are identical for signals and interferences. Let in this case
the AA includes isotropic and noninteracting antenna elements and let the noise
is spatial-white (Rnn = σ2
n I). Let us estimate the behaviour of optimal solutions
3.7, 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

7. 4.3 Methods of SSP under Generalized Parametric Uncertainty 41
→
W1= lim
(Ps σ2
n )→∞
→
WRSI,
→
W2= lim
(Pl σ2
n )→∞
→
WRSI, Ps.out(
→
W) = β
→
W
H
Rss
→
W. It means that
if (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 4.1
and 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
→
V
H
s ; RΣ = ∑L+1
j=1 Rj j = ∑L+1
j=1 Pj
→
V j
→
V
H
j ; A = ρl
→
Vl
→
V
H
s ; and,
at last, Rnn = σ2
n I.
Using 3.15, we can get that
→
WRSI= βR−1
xx
→
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) = σ2
n , 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) = σ2
n ∑L+1
j=1 Pj + 2Re{ρ} . Using this equality, we can come
to the limit ratio [9]:
lim
(Ps σ2
n )→∞
Rin
−1
= lim
(Pl σ2
n )→∞
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 to
the same λmin (Rin). These vectors are orthogonal to vectors
→
V j, j = 1,L+ 1, as well
as 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,
→
W2
are 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
→
W
H
1 Rss
→
W1=
→
W
H
2 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 provide
obtaining optimal solutions, moreover, they are orthogonal asymptotically to the
vector 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 of
the linear space filter) depends on the values Ps σ2
n , Pl σ2
n , ρl, and even, on Pj σ2
n .

8. 42 4 Features of ASSP under Different Levels of A-Priori Uncertainty
The examples for dependences of RSIN η
→
W from the values Ps σ2
n , Pl σ2
n
and P1 σ2
n are shown in Fig. 4.1 - Fig. 4.3 The filter is optimized by the MRSI
criterion. 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 relatively
to 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 σ2
n = 10dB (Fig. 4.1, 4.3), signal/noise ratio 10lgPs σ2
n = 20dB
(Fig. 4.2, 4.3), (uncorrelated interference)/noise 10lgP1 σ2
n = 20dB (Fig. 4.1),
10lgP1 σ2
n = 10dB (Fig. 4.2).
RSIN, dB
Ps/σ2
n, dB
ρl=0
ρl=0,125
ρl=0,25
ρl=0,5
ρl=0,75
ρl=1
Fig. 4.1 Dependence of output RSIN from input ratio signal/noise
RSIN, dB
P´1/σ2
n, dB
ρl=0
ρl=0,125
ρl=0,25
ρl=0,5
ρl=0,75
ρl=1
Fig. 4.2 Dependence of output RSIN from input ratio (correlated interference)/noise

9. 4.3 Methods of SSP under Generalized Parametric Uncertainty 43
RSIN, dB
P1/σ2
n, dB
ρl=0
ρl=0,125
ρl=0,25
ρl=0,5
ρl=0,75
ρl=1
Fig. 4.3 Dependence of output RSIN from input ratio interference/noise
As follows from Fig. 4.1 - Fig. 4.2), if there are interferences correlated with a
signal, 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 practically
in all signal-interference situations (the value of RISP decreases more than 10 dB in
relation 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 worst
for 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 the
form
→
Y (t) = F
→
X (in the common case, the dimension of the vector
→
Y (t) does not
coincide with the dimension of the vector
→
X (t)). The operator F{.}keeps in
→
Y the
spatial and temporal structures of useful signal
→
S (t) and permits the decorrelation
of interferences
→
P
c
l (t). Then the vector
→
Y (t) is filtrated using mentioned above
parametric ASSP algorithms. The decorrelation of interferences (implementation of
F{.}) is provided either by the random “shift” of AA aperture supplying interfer-
ences by additional phase modulation with preservation of unchangeable temporal
structure 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{.} assume
exact knowledge of spatial structure of a signal. Besides, as a rule, it is required a
redundant value of degrees of freedom for the antenna array [12].
So, despite the fact that main results of classical ASSP theory are extended on
the case of generalized parametric uncertainty about the properties of interferences,

10. 44 4 Features of ASSP under Different Levels of A-Priori Uncertainty
the essential limitation of this theory is its orientation on exact a priori data about
the spatial structure of a signal. But in the real conditions these data, as a rule, have
approximate nature. It means that the problem of applicability of ASSP in condi-
tions of the generalized parametric uncertainty about the interference properties is
connected inseparably with the problem about applicability of ASSP when there is
no exact a priori information about a signal.
4.4 Methods of SP under a Priory Parametric Uncertainty
about Properties of Useful Signal
The hypothesis about existence of exact a priori data used under the synthesis of
ASSP parametric algorithms is not true for the majority of practical cases. In the
real conditions we should say not about the existence of errors, but rather about
their values and possibility to neglect their influence.
Let us estimate the potential possibilities of optimal VWC 3.7, 3.15, and 3.19
under 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 case
of 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 only
by the expression 3.19. To take into account the assumption about distributiveness
of 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 4.9
to characterize the quantity of the error in a priori data. These conditions are the
following 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 in
the N - dimensional complex space; δ1 ∈ R+ ∪0,
γsy = arccos
→
V
H
s
→
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 more
invariant to normalization of vectors
→
V yand
→
Vs. Sometimes, the component cosγsy
is named the coefficient of spatial correlation [22, 25].
Let us represent the vector of weight coefficient optimal by the MRSI criterion
in two equivalent forms:
→
W1RSI= βR−1
in
→
Vy, (4.10)

11. 4.4 Methods of SP under a Priory Parametric Uncertainty 45
→
W2RSI= βR−1
xx
→
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 always
provide accomplishment of the following unstrict inequality
ηout(
→
W) ≥ ηin. (4.12)
In 4.12 symbols ηin, ηout(
→
W) stand for RSIN on the input and output of a space
filter respectively. Let us point out that the relation 4.12 turns into equality iff there
are 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, in
this case there is
→
W1RSI= β
→
W2RSI= β
→
Vs (here and further, the value β is used as a
normalizing factor; it replaces the following phrase “with accuracy up to a constant
coefficient”). Thus, there is the following equation for RSIN on the output of SF:
ηout(
→
W) =
→
W
H
Rss
→
W
→
W
H
Rin
→
W
= N Ps
σ2
n
= 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 σ2
H + P1
→
V
H
1
→
V1 I − P1
→
V 1
→
V
H
1
→
V y, (4.14)
where there is α = 1 σ2
H σ2
H + P1
→
V
H
1
→
V 1 .
Assuming that
→
Vy −
→
Vs =
→
Vy −
→
V1 = 0 in 4.14, the following equation can
be obtained:
→
W1RSI= β
→
W2RSI= β
→
Vy. It means that VWC
→
W1RSI,
→
W2RSI provide
in-phase summation as for a signal and for interferences and there is îηout(
→
W) =
Nηin
N = ηin. In the case of
→
Vs −
→
V1 > 0, the vector 4.14 executes only in-phase
summation for a signal, providing ηout(
→
W) > ηin.
At last, let us discuss the case with L ≥ 2. Let us presume that the following
condition
→
Vk −
→
Vs = 0 (γks =
→
V k,
→
Vs = 0) takes place for the interference
number k. Using the lemma about the matrix inversion [35], there is:
R−1
in =
⎧
⎪⎪⎨
⎪⎪⎩
R−1
gg −
1
1+
→
V
H
y Ryy
→
Vy
R−1
yy
→
Vy
→
V
H
y R−1
yy
⎫
⎪⎪⎬
⎪⎪⎭
, (4.15)
where Rgg = ∑L
j=1,j=k Pj
→
V j
→
V
H
j +σ2
n I;
→
Vk=
→
Vs=
→
Vy.
Substituting 4.15 into 4.10 leads to the expression

12. 46 4 Features of ASSP under Different Levels of A-Priori Uncertainty
→
W1RSI= β
→
W2RSI= βR−1
gg
→
V y . (4.16)
It follows from 4.16 that
→
W1RSI (
→
W2RSI) provides maximization for relation
ηout(
→
W) = Ps.out
PΣout
, 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 maximum
and Ps.out Pk.out = Ps Pk (Pk.out is a power of interference number kon the output of
SF), then there is ηout(
→
W) > ηin.
The expression 4.12, in fact, is a background for reasonability of application for
ASSP 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 provide
execution 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 joint
consideration vectors 4.10 and 4.11. it was found previously that these VWC are
equal 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 account
expressions 4.8 and 4.9. In particular, if there are L = 0, L = 0 in the expression 4.1
we can obtain the following equalities:
Rin = σ2
n I,Rxx = Ps
→
V s
→
V
H
s +σ2
n I. (4.17)
Due to substituting 4.17 into expressions 4.10 and 4.11, it can be obtained after
some necessary transformations that:
→
W1RSI= β
→
Vy, (4.18)
→
W2RSI= β
→
V
H
s
→
Vs +σ2
H Ps
→
Vy −ρsy
→
V s ,ρsy =
→
V
H
s
→
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 = N2
cos2
γsy, (4.20)
ηout
→
W1RSI = N cos2
γsy Ps σ2
n , (4.21)
ξ
→
W2RSI = N2
cos2
γsy σ2
n Ps
2
, (4.22)
ηout
→
W2RSI = Ps σ2
n
N2 cos2 γsy σ2
n Ps
2
B+ N2 cos2 γsy − 2NBcosγsy
, (4.23)

13. 4.4 Methods of SP under a Priory Parametric Uncertainty 47
where ξ (.) = Ps.out Ps; γsy =
→
V s,
→
Vy ; B = N + σ2
n Ps.
As follows from 4.20 - 4.23, for both vectors 4.10 and 4.11, both the power of
a signal and RSIN on the output of SF depend on the value of coefficient of space
correlation (CSC) cosγsy.
The value of CSC characterizes the degree of closeness for vectors
→
Vs and
→
Vy. In
this case we have:
lim
CSC→0
ξ(
→
W1RSI) = lim
CSC→0
ξ(
→
W2RSI) = 0, (4.24)
lim
CSC→0
ηout(
→
W1RSI) = lim
CSC→0
ηout(
→
W2RSI) = 0. (4.25)
Besides, it follows from 4.21 and 4.22 that the following equality takes place in the
case of the vector 4.11:
lim
(σ2
n Ps)→0
ξ(
→
W2RSI) = lim
(σ2
n Ps)→0
ηout(
→
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 signal
even if there are no interferences. Such a signal rejection can take place either due
to decrease of CSC (in the case of VWC (3.10)), or because of increase for the input
ratio Signal/noise (in the case of VWC (3.11)). Moreover, in the case of VWC 4.11
the useful signal can be rejected asymptotically even under the utmost small values
for cosγsy, δ1 in expressions 4.8 and 4.9.
If there are interferences (for example, if L = 1), then the correlation matrix Rin
is the following one:
Rin = P1
→
V1
→
V
H
1 +σ2
n I. (4.27)
Substituting 4.27 into 4.10, we can get the following:
→
W1RSI= β
→
V
H
1
→
V 1 +σ2
n P1
→
V y −ρ1y
→
V 1 ,ρ1y =
→
V
H
1
→
Vy . (4.28)
Using 4.28 and assuming that
→
Vy =
→
Vs =
→
V1 =
√
N, we can get that:
ξ
→
W2RSI = N + σ2
n Ps N2 cos2 γsy + N4 cos2 γsy−
−2 N + σ2
n Ps × Re ρsyρ1yρ1s ,
(4.29)
ηout
→
W1! =
Ps σ2
n ξ
→
W1RSI
B+ N2 cos2 γ1y − 2BN2 cos2 γ1y
, (4.30)
where ρ1s =
→
V
H
s
→
V 1; cosγ1s = |ρ1s|
→
Vs
→
V1 ; B = N + σ2
n Ps.
It follows from 4.29 and 4.30 that if there are interferences, then provided VWC
4.10 depends on closeness of vectors
→
V y and
→
Vs (γsy), as well as on corresponding

14. 48 4 Features of ASSP under Different Levels of A-Priori Uncertainty
angles 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 the
matrix Rin (Rxx = Rin + Ps
→
Vs
→
V
H
s ) and use the lemma about matrix inversion, then
VWC 4.11 can be represented as the following expression:
→
W2RSI=
⎡
⎣R−1
in −
PsR−1
in
→
Vs
→
V
H
s R−1
in
1 + Ps
→
V
H
s R−1
in
→
Vs
⎤
⎦
→
V y, (4.31)
where R−1
in = α σ2
n + P1
→
V
H
1
→
V1 I − P1
→
V1
→
V
H
1 ; α=1 σ2
n σ2
n + P1
→
V1
→
V
H
1 .
After transformations we can use the fact that
→
Vy =
→
V s =
→
V1 =
√
N. It
permits to transform the expression 4.31 into the following one:
→
W2RSI= A
→
Vy −P1ρ1y
→
V1 −α
NA− P1 ρ1y
2
A
→
V y −P1ρ1y
→
V1
NA− P1 |ρ1s|2
, (4.32)
where A = σ2
n + P1
→
V1
→
V
H
1 .
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 absolute
values 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 increase
for the number of interferences. Some results of comparison for potential efficiency
of VWC
→
W1RSI,
→
W2RSI under the conditions of parametric a priori uncertainty about
the signal properties L ≥ 0 are given in [23]. The diagrams from this work show
that optimal VWC 3.19 cannot operate under conditions of parametric a priori un-
certainty, even in very simple signal-interference situations. This conclusion is valid
for the case of generalized parametric uncertainty, too.
The problem of “inoperativeness” (or very low efficiency) for parametric ASSP
algorithms was many times discussed in corresponding special literature. As a rule,
only the specific sources for errors’ occurrence have been researched, such as the
imprecise knowledge either of arriving direction or of signal frequency, and so on. It
the same time, the vector nature of a priori uncertainty was not taking into account in
explicit form. Nowadays, a lot of approaches is known connected with modification
of 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 conditions
of parametric a priori uncertainty. These approaches tend to decrease the sensitivity
of 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

15. References 49
perfectly 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 (up
to 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 let
us the term “robustness” to determine the corresponding class of ASSP procedures,
to avoid some terminological alternative versions. Of course we should specify its
numerical content.
The development of methods for synthesis of robust algorithms is one of the most
important directions for evolution of ASSP theory. This direction is not “closed” and
now there are very intensive researches conducted in this area.
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