Adaptive Antenna Array Receivers for
Spread-Spectrum Signals in
Non-Gaussian Noise
1
We consider the problem of recovering
a spread-spectrum (SS) signal in
the presence of unknown highly
correlated spread-spectrum interference
and impulsive noise. In terms of basic
system and signal model considerations,
we assume availability of a narrowband
adaptive antenna array that
experiences additive white Gaussian
noise in time and across elements, as
well as impulsive disturbance with
direction-of-arrival (DOA) that changes
at the chip rate. The space–time
receiver design developed in this work
is characterized by the following
attributes: (i) Adaptive interference suppression
is pursued in the joint space–time
domain. (ii) An adaptive
parametric nonlinear front end offers
effective suppression of impulsive
disturbances at low computational
cost. (iii) An adaptive auxiliary-vector
linear filter postprocessor offers
effective, low-complexity suppression of
SS interferers and leads to superior
bit-error-rate (BER) performance under
rapid, short-data-record system
adaptation. Numerical and simulation
comparisons with plain and outlier
resistant space–time minimumvariance-
distortionless-response (MVDR)
filtering procedures are included
to illustrate and support the
theoretical developments. Ó 2001 Academic Press
Key Words: antenna arrays;
auxiliary-vector filters; CDMA; impulsive
noise; interference suppression;
space–time adaptive processing; spreadspectrum
communications
1. INTRODUCTION
Signal detection in the presence of
impulsive channel noise has been
considered extensively in the past
(for example [1–5] and references therein),
while detection of a direct-sequence
spread-spectrum (DS-SS) signal under
similar channel conditions has been
studied in [6, 7] andmore recently in [8–10].
Receiver proposals in [6] involve the
use of either a conventional signature
matched filter or a majority-vote
receiver (hard-limiter nonlinearity per chip
followed by signature matched
filtering). In [6] it is reported that neither
one of the above proposals is
universally effective against the combination of
SS interference and non-Gaussian
impulsive noise. The work in [8] follows
the principles of minimax M-estimator
designs [11] and derives a robust
version of the familiar decorrelating
DS-SS detector under the assumption
that the signatures of all SS
interferers are known. To handle unknown SS
interference, a minimax subspace
approach that relies on eigendecomposition
of the impulsive-noise corrupted input
autocorrelation matrix is also proposed
in [8]. In [9, 10] adaptive receivers
are developed that are composed of a
vector of adaptive chip-based
nonlinearities followed by an adaptive linear
tapweight filter. The structures
proposed in [9, 10] tap the relative merits of
both nonlinear and linear signal
processing and exhibit superior bit-error-rate
(BER) performance in the presence of
combined impulsive and unknown SS
interference. In particular, the
nonlinear receiver front end adapts itself to the
unknown prevailing impulsive noise
environment, while the adaptive linear
tapweight filter that follows the
nonlinearly processed chip samples combats
effectively the unknown SS
interference. This article enhances our previous
work in [9, 10] in the following
aspects. A narrowband adaptive antenna
array is employed for joint space–time
(S–T) disturbance suppression under a
generalized multipath signal model
that includes chip-rate random impulsive
disturbance with direction of arrival
changing at the chip rate. The overall S–T
receiver design objective is shifted
to superior BER performance under rapid
short-data-record adaptation. A
Hampel-type nonlinear preprocessor [10] that
encompasses the preprocessors
considered in [9] as special cases is considered.
2. SYSTEM MODEL
The baseband received signal at each
antenna element m, m D 1; : : : ;M,
is viewed as the aggregate of the
multipath received SS signal of interest
with signature code s
0
of length L (if T is the symbol
period and Tc
is
the chip period then L D
T=Tc),
K 1 multipath received DS-SS interferers
with unknown signatures s
k
, k D 1; : : : ;K 1, and
non-Gaussian (impulsive)
interference. For notational
simplicity and without loss of generality, we choose
a chip-synchronous signal setup. We
assume that the multipath spread is
of the order of a few chip intervals,
P, and the low-pass channel can be
represented as a tapped delay line
with P C 1 taps spaced at chip intervals
Tc.
After conventional chip-matched
filtering and sampling at the chip rate over a
multipath extended symbol interval of
LCP chips, the LCP data samples from
the mth antenna element, m
D 1; : : : ;M, are organized in the form of a vector r
m
given by
r
m
D
K
X1
kD0
X
P
p
D0
c
k;p
p
E
k.bksk;pCb
k
s
k;p
Cb
C
k
sC
k;p
/ak;pTmUCn;
mD 1; : : : ;M; (1)
where, with respect to the kth
SS signal,
Ek
is the transmitted energy,
bk,
b
k
, and b
C
k
are the present, the previous, and the following
transmitted
bit, respectively, and f
ck;pg
are the coefficients of the frequency-selective
slowly fading channel modeled as
independent zero-mean complex Gaussian
random variables that are assumed to
remain constant over several symbol
intervals. s
k;p
represents the p-cyclic shift of
the 0-padded by P version of
the signature of the kth SS
signal s
k
, s
k;p
is the 0-filled .L p/-left-shifted
version of s
k;0,
and sC
k;p
is the 0-filled .L p/-right-shifted
version of sk;0.
Finally, n represents additive
complex non-Gaussian (impulsive) noise and
a
k;p
is the mth coordinate of the kth
SS signal, pth path, array response
vector,
a
k;pTmU
D ej2.m1/sin
k;pd=;
mD 1; : : : ;M; (2)
where
k;p
identifies the angle of arrival of the pth
path of the kth SS
signal, is the carrier
wavelength, and d is the element spacing (usually
d D =2).
To avoid in the sequel cumbersome 2-D
data notation and filtering operations,
we decide at this point to “vectorize”
the .L C P/ M space–time data matrix
Tr
1
r2
: : : rMU
by sequencing all matrix columns in the form of a single .LCP/Mlong
column vector:
r
.LCP/M1
D VecfTr1
r2
: : : rMU.LCP/Mg:
(3)
From now on, r denotes the
joint space–time data in the
C.LCP/M
complex vector
domain.
For conceptual and notational
simplicity we may rewrite the vectorized space–
time data equation as
r D
p
E
0b0wR-MF
C iC n; (4)
where w
R-MF
, Eb0
frb
0g
D VecfTPP
p
D0
c0;ps0;pa0;pT1U
: : :
P
P
p
D0
c0;ps0;pa0;pTMUUg
is the effective space–time signature
of the SS signal of interest (signal-0) and
i identifies comprehensively both
the Inter-Symbol and the SS interference
present in (1) (
Eb0
fg denotes statistical expectation
with respect to b
0).
We
use the subscript R-MF in our
effective S–T signature notation to make a
direct association with the RAKE
Matched-Filter time-domain receiver that
is known to correlate the signature
s
0
with P C 1 size-L shifted windows
of the received signal (that
correspond to the P C 1 paths of the channel),
appropriately weighted by the
conjugated channel coefficients c
0;p,
p D 0; : : : ;P.
In our notation, the generalized S–T
RAKE operation corresponds to linear
filtering of the form w
H
R-MFr,
where H denotes the Hermitian operation.
3. RECEIVER ARCHITECTURE
For the space–time signal model of the
previous section, the general receiver
structure under consideration is given
in Fig. 1a. The receiver consists of a nonlinear
front-end in the form of a vector of
parametrized nonlinearities g.r I /:
C
.LCP/M
!
C.LCP/M,
followed by linear filter post-processing by an .L C P/M
complex tap-weight filter w.
The nonlinear preprocessor considered
in this present work employs Hampeltype
nonlinearities [10–12]
g.rI
1;
2;
3/
D Tg.rT1UI
1;
2;
3/;
: : : ;g.rT.LC P/MUI
1;
2;
3/UT
; (5)
where T denotes the transpose
operation and
g.xI
1;
2;
3/
,
8>>>>>>><
>>>>>>>:
x; if jxj<
1;
0<
1
1
x
jxj ; if
1 jxj<
2;
0<
1
2
3 jxj
3
2
1
x
jxj ; if
2 jxj
3;
0<
1
2
3
0; otherwise:
(6)
In (6) x is a complex number
and jxj denotes the magnitude of x. The linear region
of the Hampel nonlinearity has the
effect of passing the observations undistorted.
The nonlinear regions either
completely reject (remove) or “correct” the
observations. The latter is considered
as an adjustment of the magnitude while
FIG. 1. (a) General space–time
receiver structure. (b) Hampel-type nonlinearity.
148 Digital Signal Processing
Vol. 11, No. 2, April 2001
maintaining the phase. The parameters
1,
2, and
3 are
positive cut-off parameters
to be determined adaptively. The
real-valued version of (6) is shown in
Fig. 1b. We note that if we choose 0<
1 D
2 D
3 the
Hampel operation degenerates
to the familiar “puncher” and if
2!1the
Hampel operation converges
to the “clipper.” Therefore, the
Hampel preprocessor is a generalization of the
puncher and clipper preprocessors
considered in [9].
The linear filter postprocessor may be
chosen to be the space–time minimumvariance-
distortionless-response (MVDR)
solution for the nonlinearly processed
data vectors. If, without loss of
generality, we assume or we enforce that
g.w
R-MF/
DwR-MF
by selecting a sufficiently large
1
cut-off value in (6), then
w
MVDR
D kR1
g
wR-MF;
kD
w
H
R-MFR1
g
wR-MF
1
; (7)
where, w
R-MF
is the S–T RAKE matched filter in (4) and
Rg
, Efg.r/g.r/Hg
is
the S–T autocorrelation matrix of the
Hampel processed data vectors. Adaptive
SS interference suppression with
postprocessing of the form of (7) has an important
shortcoming that we attempt to improve
upon in the following section. The
data estimated sample-matrix-inversion
(SMI) “open-loop” adaptive implementation
of the S–T MVDR filter O w
MVDR.N/
D TwHR-MF
OR
1
g
.N/wR-MFU1
TOR
1
g
.N/
w
R-MFU,
where OR
g
.N/
D 1
N
P
N
n
D1
g.rn/g.rn/H
is the sample average estimate
of the S–T autocorrelation matrix over
a data record of N Hampel-processed
S–T input vectors, exhibits
disappointing short data record performance.
Constrained-LMS/RLS “closed-loop”
adaptive implementations of w
MVDR
behave
similarly [13]. Data records of size
many times the space–time product
.L C P/M are necessary to
approach satisfactorily the BER performance of the
ideal w
MVDR
filter in (7) with perfectly known R1
g
. In the following section we
address the issue of superior
small-sample-support performance in the context
of what we call auxiliary-vector (AV)
S–T processing.
4. ALGORITHMIC DEVELOPMENTS
Given a normalized constraint vector
x 2
C.LCP/M,
any linear filter w 2
C
.LCP/M
constrained to be distortionless in the
x-direction (i.e., wHx
D 1) can
be decomposed as w D x C
y, where y 2
C.LCP/M
and yH
x D 0. The S–T MVDR
filter presented in the previous
section is equivalent to the linear filter wD xCy
where x D w
kR-MFk
, wR-MF=kwR-MFk
is the normalized S–T RAKE matched
filter for the SS signal of interest
and y is the orthogonal to x vector .y
H
x D 0/
that minimizes the variance at the
output of w.
Historically, algorithmic designs that
focus on the MVDR filter part y that
is orthogonal to the constraint vector
direction x have been widely pursued in
the array processing literature and
have been known as Applebaum/Howells
arrays [14, 15], beam-space partially
adaptive processors [16], or generalized
sidelobe cancelers (GSC) [17]. Recent
developments have been influenced by
principal component analysis
reduced-rank processing principles [18, 19]. In
this context, the objective is to
approximate the orthogonal data processing
Pados, Medley, and Batalama: Adaptive
Antenna Arrays 149
branch y by an arbitrary
“blocking matrix” operator B
.LCP/M.LCP/M
(that
satisfies B
Hx
D 0) followed by processing by a weighted sum of s < .LCP/M1
“dominant” eigenvectors of the
blocked-data autocorrelation matrix. While
the eigenvector weights are usually
MS-optimum designed, there have been
many proposals for the choice of a
“dominance” criterion. Representative
examples include the s maximum
eigenvalue eigenvectors of the disturbanceonly
autocorrelation matrix [20], the s
maximum eigenvalue eigenvectors of
the blocked data autocorrelation
matrix [21, 22], or the s minimum output
variance eigenvectors [23] (termed
maximum “cross-spectral metric” design
in [24]). Reduced-rank filtering based
on filter decomposition using canonical
correlations is considered in [25] and
modular designs through factorization of
the orthogonal projection operator are
developed in [26]. A different approach
for the design of y from a
different point of view is considered in [27–32]. The
general method is termed
auxiliary-vector (AV) filtering. Instead of optimizing
the orthogonal component y as a
whole, the problem is decomposed and
optimization is pursued with respect
to a “line” direction, q, and a scalar,
, separately. Thus, the overall
filter w is approximated by the filter w
AV
D
x C q. The concept
of AV filtering pertains to a particular choice of a “line”
direction q and an MS-optimized
scalar . In [27, 28] q is chosen as a
normalized vector on the line obtained
by averaging projected line-subspaces
(vector spaces of dimension one)
generated by the received data vectors, onto
the subspace orthogonal to the
constraint-vector. In [29] a maximum crosscorrelation
criterion is proposed and q is
selected as the vector that maximizes
the magnitude of the cross-correlation
(MCC) between the output of the
constraint-vector .x/
processed data and the auxiliary-vector .q/ processed
data.
In [30–32] the AV method is
generalized to processing with multiple auxiliary
vectors that, together with the
corresponding scalars, are obtained through
conditional statistical optimization.
The overall filter w is now approximated by
w
.d/
AV
D x CP
d
i
D1
iqi,
where qi
, i D 1; : : : ;d, are orthonormal to
each other and
to the constraint-vector x.
Unconditional vector optimization of the AV weights
1;2;
: : : ;d
requires an explicit or implicit matrix inversion
operation and is
also investigated in [30–32]. In fact,
the filter produced by unconditional vector
optimization of the AV weights can be
shown theoretically to be identical to the
orthogonal multistage decomposition
filter in [33]. In this paper, we maintain
our conditional statistical
optimization approach and in addition we relax the
orthogonality condition among the
auxiliary vectors; i.e., q
i
are restricted to be
orthogonal to the constraint direction
x only. This way, an infinite sequence of
filters is produced .d!1/
as opposed to a finite sequence .d .L C P/M 1/
when orthogonality between the
auxiliary vectors is imposed. The detailed steps
of the algorithm are presented below.
The AV filter sequence fw
.d/
AV
g, d D 0; 1; 2;
: : : ; is initialized at the S–T vector
direction of interest i.e., w
.0/
AV
,wkR-MFk.
Then
w
.1/
AV
Dw
kR-MFk
1q1;
(8)
FIG. 2. Space–time
auxiliary-vector receiver structure with one auxiliary vector.
where
1
is a complex scalar and q1
is a vector in the .L C P/M
complex space
that is orthonormal with respect to
w
kR-MFk:
q
H1
w
kR-MFk
D0 and kq1kD1:
(9)
The receiver architecture that
incorporates post-filtering by w
.1/
AV
in (8) is shown
in Fig. 2. In contrast to direct
minimum output variance optimization that leads
to the optimum w
MVDR
S–T filter in (7) [9], we choose an auxiliary
vector q1
that satisfies the orthonormality
constraint in (9) and maximizes the magnitude
of the cross-correlation between
points (a) and (b) of the receiver structure in
Fig. 2 [29–32]. This way, q
1
is the vector that can capture the most (in
the
maximum MCC sense) of the disturbance
present at the output of the S–T
RAKE matched filter:
q
1
D argmax
q
E
n
w
Hk
R
-MFkg.r/
q
H
g.r/
o
D argmax
q
w
HkR-MFkRgq
; (10)
subject to q
HwkR-MFk
D 0 and qHq
D 1:
The magnitude cross-correlation
criterion function jw
Hk
R
-MFkRgqj
and the orthonormality
constraint are both phase invariant.
Hence, to avoid unnecessary
ambiguities in our presentation and
without loss of generality, we can identify
the unique auxiliary vector that is a
solution to our constraint optimization
problem and makes w
Hk
R
-MFkRgq
real nonnegative .wHk
R
-MFkRgq
0/. Standard
Lagrange multipliers derivation shows
that this vector is
q
1
D
R
gwkR-MFk
.wHk
R
-MFkRgwkR-MFk/wkR-MFk
k
RgwkR-MFk
.wHk
R
-MFkRgwkR-MFk/wkR-MFkk
: (11)
Then the complex scalar weight
1
in the receiver structure of Fig. 2 is chosen
to be the value that minimizes the
mean-square (MS) error between points (a)
and (c). Direct application of the
Yule–Walker theorem shows that this MSoptimum
value of
1
is [27–32]
1
D
q
H1
R
gwkR-MFk
q
H1
R
gq1
: (12)
This filter design approach can be
generalized to cover processing with
multiple auxiliary vectors. The AV
filter that utilizes d, d D 1; 2; 3; : : : ,
auxiliary
Pados, Medley, and Batalama: Adaptive
Antenna Arrays 151
vectors is of the form
w
.d/
AV
Dw
kR-MFk
X
d
i
D1
iqi
; (13)
where q
i
, i D 1; : : : ;d, are
orthonormal to wkR-MFk.
The weighted auxiliary
vectors are conditionally optimized in
a sequential fashion as follows. Inductively,
given q
1,
1,
: : : , qd1,
d1,
d 1, qd
is set to be the orthonormal to
w
kR-MFk
vector that maximizes the magnitude of the
cross-correlation between
w
.d1/H
AV
g.r/ and qH
d
g.r/:
q
d
D
R
gw.d1/
AV
Tw
Hk
R
-MFkRgw.d1/
AV
Uw
kR-MFk
k
Rgw.d1/
AV
Tw
Hk
R
-MFkRgw.d1/
AV
Uw
kR-MFkk
: (14)
Then the weight value
d
that minimizes the MS error between w.d1/
AV
H
g.r/
and
d
qH
d
g.r/ is
d
D
q
H
d
Rgw.d1/
AV
q
H
d
Rgqd
:
(15)
By inspection, we observe that for the
MS-optimum value of
d,
the product
dqd
is independent of the norm of qd
. Therefore, we may drop the unnecessary
normalization operation in (14).We can
also factorize the AV numerator tomake
the w
kR-MFk
orthogonal projection operation apparent. The
whole algorithm is
summarized below in the following
simple recursive form:
w
.d/
AV
Dw
.d1/
AV
dqd;
dD 1; 2; : : : ; w.0/
AV
Dw
kR-MFk;
(16)
q
d
D
.I wkR-MFkwHk
R
-MFk/Rgw.d1/
AV
; (17)
d
D
q
H
d
Rgw.d1/
AV
q
H
d
Rgqd
:
(18)
The auxiliary vector generation
proceduremay stop when q
dC1
D 0. Inthat case,
w
.d/
AV
is exactly equal to wMVDR.
Formal theoretical analysis of the sequence of
auxiliary-vector filters w
.0/
AV
,
w.1/
AV
,
w.2/
AV
;
: : : was pursued in [13] and led to the
results summarized below in the form
of a proposition.
P
ROPOSITION
4.1. Let
Rg
be a Hermitian positive definite matrix. Consider
the auxiliary-vector filter sequence
w.0/
AV
,
w.1/
AV
,
w.2/
AV
;
: : : defined by (16)–(18).
.
i/
Successive auxiliary vectors are orthogonal:
qH
d
qdC1
D 0, d
D 1; 2; 3;
: : : .
.
ii/
The generated sequence of auxiliary-vector weights
fd
g,
d D 1; 2; 3;
: : : ,
is real-valued, positive, and bounded,
0 <
1
max
d
1
min
; dD 1; 2; : : : ;
(19)
where
max
and min
are the maximum and minimum,
correspondingly,
eigenvalues of
Rg.
152 Digital Signal Processing
Vol. 11, No. 2, April 2001
.iii
/
The sequence of auxiliary vectors fqd
g,
d D 1; 2; 3;
: : : , converges to the
0 vector:
lim
d!1
q
d
D 0: (20)
.iv
/
The sequence of AV-filters w.0/
AV
;w.1/
AV
;w.2/
AV
;
: : : converges to the MVDR
filter:
lim
d!1
w
.d/
AV
Dw
MVDR
D
R
1
g
wkR-MFk
w
Hk
R
-MFkR1
g
wkR-MFk
: (21)
2
The S–T auxiliary vector filter w
.d/
AV
in (16), defined inductively through (17)
and (18), has the following advantages
in comparison with the S–T w
MVDR
filter
in [9]. First, while both filters are
a function of the S–T RAKE matched filter
w
R-MF
and the Hampel preprocessed S–T input data
autocorrelation matrix Rg,
no matrix inversion operation (neither
explicit nor implicit) is required for
the auxiliary-vector filter. The
second and most important advantage has
to do with the short data record
behavior of the filter estimators wO
.0/
AV
.N/,
wO
.1/
AV
.N/,
wO .2/
AV
.N/;
: : : , and wO MVDR.N/
2 that
are based on an N-point estimate
of the S–T autocorrelation matrix O
R
g
.N/
D 1
N
P
N
n
D1
g.rn/g.rn/H
. As illustrated
in [13], for a fixed finite
data-record-size N, the sequence fwO
.d/
AV
.N/gd
provides
an infinite number of filter
estimators with varying bias versus (co)variance
characteristics. For short data
records N, the early, nonasymptotic elements of
the generated sequence of AV
estimators offer favorable bias/covariance balance
and are seen to outperform
significantly in mean-square estimation error the
wO
MVDR.N/ D
wO .1/
AV
.N/ estimator. The latter translates to superior short data
record BER performance as illustrated
in the following section. Data record
based criteria for automatic selection
of the best filter estimator OW
.d/
AV
in the
sequence (d D 1; 2;
3; : : :) can be found in [34].
To complete the algorithmic
developments for a fully adaptive implementation
of the S–T DS-SS receiver in Fig. 1a,
we turn our attention to the Hampel
cut-off parameters
1,
2,
3.
Adaptive cutoff parameter optimization can be
pursued exactly as in [9, 10] in the
form of a decision driven minimum bit-errorrate
(MBER) stochastic approximation
recursion. For completeness purposes we
reproduce a simplified version of the
recursive algorithm in [9],
i;nC1
D
i;n C
c
n
b
0n
Ob
0
.rn;
i;n C
dn/
Ob
0
.rn;
i;n
dn/
2
dn
;
i
D 1; 2; 3; nD 1; 2; :
: : ; (22)
where c >0,
dn
D d n
1=4
,
d > 0. This recursion together with either the wO
MVDR
filter estimate through (7) or the
wO
.d/
AV
filter estimate through (16)–(18) forms a
coupled, joint optimization procedure.
2 By Proposition IV.1, wO MVDR.N/
D wO .1/
AV .N/ if O
R
g
.N/
is positive definite.
5. NUMERICAL AND SIMULATION STUDIES
We examine DS-SS signal transmissions
with speading gain L D 15 in
the presence of five SS interfering
signals .K D 6/ and impulsive noise.
The normalized synchronous signature
cross-correlations of the interfering
signals with the signal of interest
are chosen in the 20% to 30% range. The
communication channel is modeled as a
multipath Rayleigh fading channel
with three paths .P D 2/
and zero mean complex Gaussian fading coefficients
of variance one (i.e., Efj
ck;pj2g
D 1) for all paths p D 0; 1;2 and all SS
signals k D 0; : : : ;
5. The average total received interfering signal energies
E
k
P
2
pD0
Efjck;pj2g
are set equal to 7; 8; 9; 10:5; and 12 dB
for k D 1; 2; : : : ; 5,
respectively. For signal reception we
assume availability of a narrowband
adaptive antenna array withM D
5 elements. Therefore, the space–time product
for our system is .L C P/M
D .15 C 2/5 D 85. The additive impulsive channel
noise is modeled according to the
familiar -mixture disturbance model
f
.x/D
.1 /f0.x/C
f1.x/;
(23)
where 2 T0; 1U
accounts for the probability under which the noise is f
1./
distributed. The nominal pdf f
0./
is taken to be 0-mean complexGaussianwith
variance
2
0
D 1. The “contaminating” pdf f
1./
is 0-mean complex Gaussian
with variance
2
1
D
22
0
,
2 D 1000.
If the sources of impulsive disturbance are
directional with unknown but fixed
directions-of-arrival over the whole input
data record, then the impulsive
disturbance can be effectively suppressed by the
S–T linear filter processor alone. If
nondirectional “impulses” cause identical
complex valued disturbance in time at
all antenna sensors, then, again, the
disturbance can be suppressed
effectively by the S–T linear processor. Instead,
in this paper we consider the severe
form of impulsive interference, where
directional impulsive disturbances
change direction at the chip rate.
As a reference study, we begin with
the plain AWGN case . D 0/. In Fig. 3,
we plot the induced bit-error-rate as
a function of the total received energy
for the SS signal of interest for the
S–T receivers of the general form of
Fig. 1 wO
MVDR.N/,
wO .1/
AV
.N/,
and wO .6/
AV
.N/,
for a given data record size N D 400.
Multipath fading is assumed to remain
effectively constant during receiver
adaptation (N D 400 symbol
periods for this study). The BER of the ideal S–T
RAKE receiver w
R-MF
is included as a reference. We note that wO
.6/
AV
.N/
offers
the best bias/variance tradeoff
(lowest MS estimation error) for N D 400. All
BERs for all filters are analytically
evaluated over all bit combinations of the
interfering SS signals and the results
that we present are averages over 80
independent channel realizations and,
when relevant, 10 independent filter
estimates per channel.We note that for
this plain AWGNcase ( D 0 in (23)), the
adaptive nonlinear preprocessor in
Fig. 1 controlled by recursion (22) drives the
cutoff parameter
1 to
sufficiently large values so that, effectively, g.rI /D
r for
every encountered S–T data vector r.
At a glance, in terms of BER performance
comparisons, the superiority of the
wO
.6/
AV
.N/
filter is apparent.
154 Digital Signal Processing
Vol. 11, No. 2, April 2001
FIG. 3. BER versus total received
energy for the AWGN space–time channel . D 0, 2
0
D 1/. The
data record size is N D 400.
In Fig. 4 we repeat the studies of
Fig. 3 for the D 0:2 impulsive noise case.
First, the Hampel nonlinear
preprocessor is disabled; that is we let g.rI /
D r
for every S–T input vector r.
We immediately notice that all S–T receivers
collapse in the presence of impulsive
noise and exhibit BERs higher than 10
1.
Next, we reactivate the adaptive
Hampel nonlinear front-end controlled by
FIG. 4. A repetition of the
studies in Fig. 3 for impulsive noise . D 0:2, 2
0
D 1, 2
1
D 1000/ with
the Hampel preprocessor either enabled
or disabled.
Pados, Medley, and Batalama: Adaptive
Antenna Arrays 155
FIG. 5. BER versus
data-record-size for impulsive noise . D 0:2, 2
0
D 1, 2
1
D 1000/ with the
Hampel preprocessor activated.
recursion (22). As a result, the S–T
receivers recover and their relative BER
behavior parallels their behavior in
Fig. 3.
Finally, in Fig. 5 we plot the BER as
a function of the data record size N for
the D 0:2 impulsive
noise case and the Hampel nonlinearity activated. The
total received energy for the SS
signal of interest is fixed at 8 dB. The plot
illustrates the performance of the
wO
.1/
AV
.N/,
wO .2/
AV
.N/,
and wO .4/
AV
.N/
as well as the
wO
MVDR.N/
and ideal S–T RAKE receivers. As the data record size N increases,
the best bias/variance tradeoff
(lowest MS estimation error) AV filter estimator
shifts toward a higher number of AVs
d (as the sample support increases we can
“afford” a higher number of AVs).
6. CONCLUSIONS
Adaptive DS-SS receiver designs that
are resistant to unknown correlated
SS interference and impulsive noise
were considered in [9] in the form of a
vector of parametric chip-based
adaptive nonlinearities (hard-limiter, clipper, or
puncher) followed by MVDR linear
filter post-processing. In this work, (i) we
generalized the channel model to
account for multipath fading and impulsive
disturbance that changes
direction-of-arrival at the chip rate, (ii) we generalized
the parametric front end to a Hampel
processor that covers the previously
considered nonlinearities as special
cases, and (iii) we employed an adaptive
antenna array for joint space–time
adaptive interference suppression. Most
important, (iv) instead of joint S–T
MVDR postprocessing, we proposed joint
S–T auxiliary-vector (AV) linear
filtering. The benefits are (i) low computational
optimization complexity and (ii)
superior short-data-record BER performance,
156 Digital Signal Processing
Vol. 11, No. 2, April 2001
since AV filter estimates exhibit
significantly lower MS estimation error
than their conventional MVDR
counterparts (SMI/RLS/LMS). Processing with
multiple, conditionally optimized
auxiliary vectors and AV weights offers the
system designer effective control over
the bias/variance tradeoff for a given data
record size.