2.
3326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
single-antenna receiver, but, in contrast to [22], it is applicable
to the case of an arbitrary transmitted symbol constellation. In
our technique, it is only assumed that the unknown interference
is wide-sense stationary. Note that this assumption has been
frequently used in various interference rejection schemes for
CDMA communication systems [18], [22], [23]–[25] and
includes several more particular interference models such as
multitonal and autoregressive (AR) interference models [20],
[26]–[32]. Both these models are widely used in the literature,
in particular, to represent overlaid narrow-band interference
whose current value can be effectively estimated from its past
values [19], [20]. Exploiting the idea presented in [37] in the
context of array processing, we obtain a subspace which is
orthogonal to the subspace spanned by the user signals. We then
use the so-obtained subspace along with the known spreading
sequence of the user-of-interest to identify the channel vector,
and, subsequently, the signature vector of this user. We also
derive the necessary and sufﬁcient conditions which warrant
the identiﬁability of the proposed technique. Similar to other
subspace-based signature estimation techniques [4], [10]–[14],
the proposed algorithm poses some restriction on the maximum
admissible number of active users as well as the length of the
channel vector. We show that using the temporal oversampling
technique, our algorithm can also be effectively applied to the
overloaded scenarios with lengthy channels.
Note that the exact data covariance matrix can precisely
identify the channel vector. However, this matrix is not exactly
known in practice and is estimated from the data samples.
For this case, we derive a closed-form analytical expression
for the mean-squared error (MSE) of the estimated channel
vector using the ﬁrst-order perturbation theory [12], [38]–[40].
Assuming that the interference is Gaussian distributed and
using additional mild and physically justiﬁable assumptions, a
simpliﬁed version of this expression is also presented for the
high signal-to-interference (SIR) regime. From the latter ex-
pression, an impact of the key parameters (such as the number
of data samples, the received power of the user of interest, and
the received interference power) on the performance of the
proposed algorithm is studied.
The rest of our paper is organized as follows. In Section II,
we present the signal model. Our technique is presented in
Section III, where the necessary and sufﬁcient identiﬁability
conditions are also derived. The ﬁnite-sample performance of
the proposed algorithm is analyzed in Section IV. In Section V,
the temporally oversampled version of the proposed technique
is presented. Section VI contains computer simulation results.
Conclusions are drawn in Section VII.
II. SIGNAL MODEL
Consider a -user synchronous DS-CDMA system1 oper-
ating in the presence of an external interference. The received
continuous-time baseband signal can be modeled as [12]
(1)
1The synchronous case is mainly considered for the sake of notational brevity.
Extending our analysis to the asynchronous systems is direct [21].
where is the symbol period, , , and denote
the received amplitude, the th data symbol, and the signature
waveform of the th user, respectively, and is the interfer-
ence term that also includes the white Gaussian ambient noise.
We make use of the following common assumptions:
A1) The chip sequence period is the same as the symbol pe-
riod. This corresponds to the so-called short spreading
code case [36].
A2) The user channels are quasi-static, i.e., the corre-
sponding impulse channel responses are ﬁxed during
the observation period [12].
A3) For each user, the duration of the channel impulse re-
sponse is much shorter than the symbol period , so
that the effect of intersymbol-interference (ISI) can be
neglected [12], [36].
A4) The transmitted symbols are zero-mean i.i.d. random
variables with variance that are statistically indepen-
dent with the interference [12]. The original radio-fre-
quency (bandpass) interference is a zero-mean, wide-
sense stationary process with an unknown arbitrary cor-
relation function [18], [22] and, therefore, its baseband
equivalent is a zero-mean, circular, and wide-sense
stationary random process2 [19], [22], [41].
Let be the spreading factor and
denote the spreading sequence associ-
ated with the th user where stands for the transpose.
According to assumptions A1 and A2, the signature waveform
of this user is given by [12]
(2)
where is the channel impulse response of the th user, and
is the chip period.
Let us assume that has a ﬁnite support of ,
where and is a positive integer. From as-
sumption A3, we have that . Sampling (1) in the in-
terval corresponding to the th transmitted symbol of each user
and neglecting the ﬁrst ISI-contaminated samples, the
ISI-free received sampled data vector can be written as [12]
(3)
where
, ,
and
. Using (2), the signature vector can be expressed
as [12]
...
...
...
(4)
2Note that the algorithm proposed in Section III requires only that i(t) is
wide-sense stationary, while the circularity of i(t) is only used to simplify the
analytical MSE expression of the estimated channel vector that is derived in
Section IV.
3.
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3327
where . According
to (4), if the spreading code of the user-of-interest is known at
the receiver, then, provided that the channel vector is es-
timated, obtaining the signature vector is straightforward
[12]. Hence, throughout this paper we consider the problem of
channel vector estimation rather than the signature vector es-
timation. For the sake of consistency, we also assume without
any loss of generality that is a unit Euclidean norm vector
[12], that is, the normalization factor is absorbed in
. One can rewrite (3) in a more compact form as [12]
(5)
where , and
. From assumption A4 and (5), it
follows that:
(6)
where and stands for the Hermitian
transpose. Since is wide-sense stationary, the entries of
depend only on the difference between the observation times.
Hence, is a Hermitian Toeplitz matrix, and, therefore, it is
centro-Hermitian [37], [42], that is,
(7)
where is the permutation matrix with ones on the main
anti-diagonal and zeros elsewhere, and denotes complex
conjugate.
III. THE PROPOSED TECHNIQUE
Exploiting the idea presented in [37] for direction-of-arrival
(DOA) estimation, let us use (7) to facilitate estimation of the
user signature waveforms without knowing the correlation ma-
trix . Let us form the covariance difference matrix [37] as
follows:
(8)
From (6) and (7), it follows that
(9)
Note that depends on the user transmitted signals while it is
independent from the unknown interference covariance matrix
. We can rewrite (9) as [37]
(10)
where is the identity matrix. Since is an
matrix, we have that if
(11)
then is rank-deﬁcient. Considering hereafter such a case, we
have that the matrix can be eigendecomposed as
(12)
where is the diagonal matrix whose diagonal el-
ements are the nonzero eigenvalues of and is the
matrix whose columns are the eigenvectors asso-
ciated with these eigenvalues. In turn, is the
matrix whose columns are the eigenvectors
associated with the zero eigenvalues of .
The following lemma is essential for our later analysis.
Lemma 1: Assume that is an eigenvalue of and
is its associated eigenvector. Then, is also an eigenvalue
of this matrix and its associated eigenvector is . Moreover,
there exists a unitary matrix such that
(13)
Proof: The ﬁrst part of this lemma has been proved in [37].
See Appendix A for the proof of (13).
From (10) and (12) it follows that:
(14)
Since all columns of are orthogonal to all vectors in
, we have
(15)
Let us assume without any loss of generality that the ﬁrst user
is the user-of-interest. From (15), it follows that
(16)
(17)
It can be shown that (16) and (17) are equivalent, i.e., from
either of them the other can be obtained. For instance, from
(16) we have that . Substituting (13) into the
latter equation and applying conjugate operation, (17) follows.
Using this fact, either (16) or (17) can be exploited to identify
the channel vector . From (16) along with (4), it follows that
, and, therefore, is a nontrivial solution to
(18)
where . It is easy to verify that (18) is a linear
system with equations and unknowns. To
have a unique nontrivial solution for (18), it is necessary that the
number of equations is greater than or equal to the number of
unknowns, that is
(19)
Obviously, (19) also implies (11) but not vice versa.
Inequality (19) represents only the necessary condition for
uniqueness of the solution of (18). The necessary and sufﬁ-
cient conditions for unique identiﬁability of can be obtained
as follows. As is a nontrivial solution to (18), it follows
that lies in both and
. Therefore, is in the intersection between
the two latter subspaces and can be uniquely identiﬁed from (18)
if and only if
(20)
4.
3328 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
where stands for the dimension of a subspace. If we
further assume that is a full column-rank matrix, then
corresponds to a unique channel vector . Hence, can be
uniquely identiﬁed from (18) if and only if (20) holds true and
is full column-rank.
In practice, is estimated using
(21)
and the proposed method can be formulated as follows.
1) Compute the eigendecomposition of
(22)
as
(23)
where the matrices , , and are the ﬁnite-sample
estimates of the matrices , , and , respectively, and
is the diagonal matrix whose diagonal elements are the
eigenvalues of with the least absolute
values.
2) Compute and ﬁnd the least square (LS) esti-
mate of the channel vector as
(24)
where stands for the normalized eigenvector associ-
ated with the smallest (minor) eigenvalue.
IV. PERFORMANCE ANALYSIS
Using the ﬁrst-order perturbation analysis, let us derive an
approximate expression for the MSE of the channel vector esti-
mate . The following theorem holds.
Theorem 1: Assume that is estimated using (24). Then, the
MSE of the estimation error is approximately
given by
(25)
where is the pseudoinverse of and
(26)
(27)
Note that for the sake of simplicity, the time index of
has been omitted in (27). The proof of Theorem 1 is given in
Appendix B.
It should be noted that, as the interference power goes to zero,
both and converge to zero matrices. In such a case, it can
be observed from (25) that even for the ﬁnite number of data
samples, the approximate MSE of the channel vector estimate
tends to zero. It is also noteworthy that, for any arbitrary wide-
sense stationary interference, the MSE in (25) converges to zero
with the rate .
To further simplify the analysis of properties of the MSE ex-
pression (25), let us assume for the remainder of this section
that has a circular Gaussian distribution, and, therefore, the
fourth-order moments of are representable in terms of the
second-order moments [42], [43]. Note that circular Gaussian
interference has been frequently considered in the literature on
interference rejection for CDMA systems [20], [22], [30]–[32].
Before proceeding to obtain the MSE of the channel estimation
error for a circular Gaussian interference, we need the following
lemma.
Lemma 2: Assume that is a zero-mean circular Gaussian
random vector with correlation matrix and is a square ma-
trix of a conformable dimension. Then
(28)
(29)
Proof: See Appendix C.
In the following theorem, we use Lemma 2 along with (25)
to obtain an approximate expression for the MSE of the channel
vector estimate in the presence of unknown correlated Gaussian
interference. A high SIR approximation of the so-obtained MSE
expression will also be derived.
Theorem 2: Assume that is a circular Gaussian random
vector and is estimated using (24). Then, the MSE of the
estimation error is approximately given by
(30)
Moreover, if the following three conditions hold:
(31)
(32)
(33)
then (30) is simpliﬁed to
(34)
where stands for the maximum eigenvalue of a matrix,
and is the Kronecker delta.
Proof: See Appendix D.
Note that although (31) and (32) do not perfectly hold, they
appear to be reasonable practical approximations. In practice,
CDMA spreading codes are deliberately designed so that
even after passing through a frequency selective channel, the
resulting spread-spectrum signature waveforms occupy a wide
frequency band and behave as almost white pseudorandom
signals (see, e.g., [19], [28], [36], [44], [45]). Hence, in most
practical scenarios (31) and (32) should hold approximately
because the signature vectors are the sampled versions of al-
most white pseudorandom signature waveforms. The accuracy
5.
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3329
of these approximations will be validated in the simulation
section.
It should also be noted that the average received power of
the user of interest is equal to , while the average
interference power is lower bounded by the left-hand side of
(33) because
(35)
Hence, if SIR is reasonably high, it is guaranteed that (33) holds.
Based on this observation, one can consider (34) as a simple
approximation of (30) in the high SIR regime that explicitly
clariﬁes the MSE of the estimated channel vector in terms of the
number of data samples, variance of the transmitted symbols,
the received amplitude of the user of interest, and . Note
that, according to (35), the latter quantity can be viewed as a
weighted average interference power where the weighting factor
depends on the matrix and the principal angles between
and [12].
Assuming that the average interference power is equal to ,
i.e., , and is the unique nontrivial solution to
(18), one can also obtain an upper-bound for the MSE in (34)
as follows. As (18) has a unique nontrivial solution, it directly
follows that . Let us denote the positive sin-
gular values of as . Note that
the positive eigenvalues of and those of are equal
to . Since and are positive
(semi-)deﬁnite matrices, we have [46]
(36)
It should be noted that if the largest eigenvalue of is unique,
i.e., , then the ﬁrst line of (36) holds with equality
if and only if
(37)
where is the eigenvector of associated with . It follows
from (36) that the MSE of the estimated channel vector can be-
come very large if goes to zero. This is an expected result
since if , then , and, therefore, (18)
has a nontrivial solution other than .
V. TEMPORALLY OVERSAMPLED VERSION OF THE
PROPOSED ALGORITHM
Condition (19) restricts both the maximum admissible
number of active users and the channel length. However, if (19)
does not hold, one can resort to the temporal oversampling tech-
nique to facilitate identiﬁcation of lengthier channels in more
heavily loaded environments. Note that the latter technique
requires higher computational complexity and more expensive
hardware.
Let us sample the continuous-time received signal (1) with the
rate in the interval corresponding to the th transmitted
symbol where the oversampling factor is an integer. The
ISI-free part of the oversampled data vector is given by [13]
(38)
where
(39)
(40)
(41)
with
(42)
(43)
(44)
Using (2) and (40), it follows that [13]
(45)
where
(46)
(47)
and stands for the Kronecker product. Let us write (38) in the
following form
(48)
where . From as-
sumption A4 and (48) it follows that
(49)
where . Note that is a
matrix and .
Introducing
(50)
6.
3330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
from (49) and the centro-Hermitian property of we have
(51)
We can eigendecompose as
(52)
where is a
matrix which contains the eigenvectors associated with the zero
eigenvalue of . Similar to our discussion in Section III, it
can be easily shown that is a nontrivial solution to
(53)
where is a ma-
trix. To have a unique nontrivial solution for (53), it is required
that
(54)
Comparing (54) with (19), it follows that using the oversam-
pling factor of , the maximum admissible number of users is
increased by times. It can also be observed from (54) that for
a ﬁxed number of active users, the oversampling technique con-
siderably increases the maximum admissible channel length.
VI. SIMULATIONS
Computer simulations have been conducted to evaluate the
performance of the proposed algorithm and validate the ob-
tained theoretical results. In all numerical examples,
and the spreading sequence associated with each user has been
randomly drawn from the binary set of and then ﬁxed
throughout all examples. The entries of the channel vectors of
the length have been randomly and independently drawn
from a zero-mean complex Gaussian process and then have
been normalized so that and ﬁxed
throughout all examples. In all but the last example, the trans-
mitted symbols have been drawn from the QPSK constellation
with the variance . Three different models have been
used to simulate the interference vector:
• Model 1: The interference vector is a circular
Gaussian random vector such that the th entry of the
correlation matrix is [25].
• Model 2: Interference is a second-order Gaussian autore-
gressive process [22], [30] with the poles at and
, that is
(55)
where is the th entry of a vector and is the vector
of white complex Gaussian noise.
• Model 3: Interference is tonal (harmonic) with
(56)
where describes the normalized interference fre-
quency offset from the carrier frequency, and is a
random phase uniformly distributed in the interval
[18], [29].
Fig. 1. MSEs of the estimated channel versus the number of data samples N
for the ﬁrst interference model.
Fig. 2. MSEs of the estimated channel versus SIR for the ﬁrst interference
model.
In Figs. 1 through 3, the ﬁrst interference model has been
used, while in Figs. 4 and 5 the second interference model has
been applied. In Fig. 6, the third interference model is employed.
Throughout our simulations, we assume that all the users have
identical average powers and is taken in all the ﬁgures
but Fig. 3. Each point of the simulation curves is the result of
averaging over 1000 Monte Carlo realizations of interference,
noise, and the transmitted data sequences.
Fig. 1 shows the experimental MSE of the proposed algo-
rithm as well as the theoretical MSEs (30) and (34) versus
for . For the sake of comparison, the MSE curve of
the conventional (i.e., white noise assumption-based) Liu and
Xu (LX) algorithm [12] is also drawn. It can be observed from
Fig. 1 that the analytical MSE curves obtained from (30) and
(34) follow the experimental MSE curve with quite a good accu-
racy. As predicted in Section IV, these curves converge to zero
with the rate . From Fig. 1, it also follows that the MSE
of the estimated channel using the LX algorithm is constantly
7.
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3331
Fig. 3. MSEs of the estimated channel versus the number of users for the ﬁrst
interference model.
Fig. 4. MSEs of the estimated channel versus the number of data samples N
for the second interference model.
high and does not converge to zero. Note that the LX algorithm
is based on the mismatched assumption that the signal subspace
is spanned by the eigenvectors associated with the largest
eigenvalues of . Hence, even if becomes arbitrarily large
and converges to , the LX algorithm does not offer better
estimation performance.
Fig. 2 shows the experimental and analytical MSEs versus
SIR for . A substantial performance improvement can
be observed from this ﬁgure with respect to the LX algorithm.
Note that the effect of interference is negligible in high SIRs
where, as Fig. 2 demonstrates, the conventional LX algorithm
can also be used to obtain a reliable channel vector estimate.
Note also that the MSE expressions (30) and (34) are derived
using the ﬁrst-order perturbation theory, and, hence, their va-
lidity requires a small perturbation assumption. Therefore, as
shown in Fig. 2, (30) and (34) cannot accurately predict the MSE
values at very low SIRs, where the MSE of the channel vector
estimate is quite large.
Fig. 5. MSEs of the estimated channel versus SIR for the second interference
model.
Fig. 6. MSEs of the estimated channel versus SIR for the third interference
model.
Fig. 3 displays the MSE of the proposed technique versus
for and in the case of oversampling.
Three different oversampling factors of (no oversam-
pling), , and are considered. If no oversampling
is used, then, as follows from (19), the proposed algorithm fails
to estimate the channel vector for . This theoretical re-
sult is validated by Fig. 3. At the same time, it can be observed
from Fig. 3 that using the oversampled version of the proposed
algorithm with the oversampling factors of and ,
the channel vector can be reliably estimated in the presence of
up to and users, respectively.
Figs. 4 and 5 show the MSE curves versus and SIR, re-
spectively, where (55) is used to model interference. Except the
interference model, the simulation setup in Figs. 4 and 5 is iden-
tical to that of Figs. 1 and 2, respectively. As Figs. 4 and 5 show,
the proposed technique outperforms the conventional LX algo-
rithm, and the analytical MSE curves coincide with the experi-
mental MSE curves with a good precision.
8.
3332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
Fig. 6 shows the MSE curves for the proposed algorithm and
for the Buzzi and Poor (BP) algorithm [22] versus the SIR for
both the cases of BPSK and QPSK transmitted symbols. The
analytical MSE curve of the proposed algorithm is also shown
in this ﬁgure. Note that, as interference is not Gaussian, the
latter curve is obtained from (25). As it can be observed from
Fig. 6, the proposed algorithm performs equally well for both
the BPSK and QPSK transmission schemes. Moreover, both the
experimental MSE curves are quite close to the analytical curve
(25). It can also be seen from Fig. 6 that, although the perfor-
mance of the BP algorithm for the BPSK transmission scheme is
comparable to that of the proposed algorithm, the BP algorithm
becomes completely unreliable in the case of QPSK symbols.
This is, however, an expected result since the BP algorithm is
exclusively designed for the BPSK symbol case [22].
VII. CONCLUSION
In this paper, we have proposed a new blind subspace-based
signature waveform estimation technique for DS-CDMA com-
munication systems operating in the presence of unknown wide-
sense stationary interference. Using the centro-Hermitian prop-
erty of the interference covariance matrix along with the idea
of covariance differencing [37], we derive a new algorithm for
blind identiﬁcation of user signatures. In contrast to the ex-
isting algorithms [21], [22], the proposed technique can be im-
plemented using a single receiving antenna and is applicable to
arbitrary constellations of transmitted symbols. Necessary and
sufﬁcient conditions for identiﬁability of the proposed technique
have also been derived. Using the ﬁrst-order perturbation theory,
closed-form expressions for the mean-squared error of the esti-
mated channel vector have been obtained and the effects of dif-
ferent parameters on the performance of the proposed algorithm
have been studied. It has been shown that the temporally over-
sampled version of the proposed technique can be applied to
overloaded communication systems with lengthy channels. Nu-
merical examples have veriﬁed the advantages of the proposed
technique and validate its error analysis.
APPENDIX A
It follows from (8) that
(57)
To prove (13), note that
(58)
Left-multiplying (58) with , using (57), and applying conju-
gate operation, we obtain
(59)
As columns of span the null-space of , all of the columns
of are in , or, equivalently, there exists a matrix
such that
(60)
Applying Hermitian operation to both sides of (60), we have
(61)
Left-multiplying each side of (61) with the corresponding side
of (60) and using the fact that , we obtain
(62)
Since is a square matrix, from (62) it follows that is unitary.
This completes the proof.
APPENDIX B
First, note that from (15) and (26), we have
(63)
It is also obvious that
(64)
Let us denote
(65)
(66)
Using the perturbation theory, the ﬁrst-order approximation of
can be written as [12], [38]–[40]
(67)
where stands for the “equality up to the ﬁrst order,” and
(68)
As , it follows that:
(69)
Inserting (67) in (69) and applying the expectation operation to
the squared norm of the resulting expression, we obtain
(70)
Hence, to ﬁnd the MSE, we need to obtain . Note
that
(71)
where the last line of (71) follows from (64). Using (22) in the
right-hand side of (71), we have
(72)
9.
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3333
where
(73)
Assuming that the received samples corresponding to two dif-
ferent sampling times are independent from each other, the ex-
pression inside the expectation operation in the last line of (72)
is equal to , which, according to (64), is identically zero.
Hence, we have
(74)
where
(75)
(76)
and for the sake of simplicity, the time index in has been
omitted. Inserting (5) in (75) and (76), and using (63) along with
the fact that and are zero-mean independent random vectors,
we have
(77)
(78)
Computing the ﬁrst term of , we have
(79)
Note also that, as the original bandpass interference is a wide-
sense stationary random process, is a circular random vector
[41], and, therefore, the ﬁrst term in the right-hand side of (78)
is equal to zero. Using the so-obtained values of and
in (74), after straightforward manipulations it can be shown that
(80)
where is deﬁned in (27). Substituting (80) in (70), (25) di-
rectly follows. This completes our proof of the Theorem.
APPENDIX C
Assume that is a vector with the length . To prove (28),
we note that
(81)
where is the th entry of a matrix. Hence, we have
(82)
where the second line of (82) holds due to the fact that is a
circular Gaussian random vector (for example, see [42, p. 68]).
As is the covariance matrix of , we have
which directly proves (28). To prove (29), using derivations sim-
ilar to (81) and (82), we obtain
which proves (29).
APPENDIX D
Using (28) and (29) in (27), it can be readily shown that for
any circular Gaussian interference vector , we have
(83)
Substituting (83) into (25) and using (6), the approximation
(30) follows. To prove (34), note that if (31) and (32) hold true,
then, according to (14), is an orthogonal matrix
whose columns span , and, hence, the columns of
are normalized versions of the corresponding columns of
. Therefore, we have
(84)
(85)
10.
3334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
Using (84) and (85) in (68) yields
(86)
Equations (31) and (32) can be used along with (86) to obtain
the following equalities:
(87)
(88)
(89)
Using (87)–(89) in (30), it follows that:
(90)
where
(91)
Note that for any vector and positive (semi-)deﬁnite matrix
, we have
(92)
Moreover
(93)
where the third line of (93) follows from the facts that
and is Hermitian. From (92) and (93) it follows that
(94)
where stands for the absolute value. Assuming that (33)
holds, (34) immediately follows from inserting (94) into (90).
This completes the proof.
REFERENCES
[1] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ.
Press, 1998.
[2] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effective
interference, effective bandwidth and user capacity,” IEEE Trans. Inf.
Theory, vol. 45, pp. 641–657, Mar. 1999.
[3] M. Honig, U. Madhow, and S. Verdú, “Blind adaptive multiuser detec-
tion,” IEEE Trans. Inf. Theory, vol. 41, pp. 944–960, Jul. 1995.
[4] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace ap-
proach,” IEEE Trans. Inf. Theory, vol. 44, pp. 677–690, Mar. 1998.
[5] S. L. Miller, M. L. Honig, and L. B. Milstein, “Performance analysis
of MMSE receivers for DS-CDMA in frequency-selective fading chan-
nels,” IEEE Trans. Commun., vol. 48, pp. 1919–1929, Nov. 2000.
[6] K. Zariﬁ, S. Shahbazpanahi, A. B. Gershman, and Z.-Q. Luo, “Robust
blind multiuser detection based on the worst-case performance opti-
mization of the MMSE receiver,” IEEE Trans. Signal Process., vol. 53,
pp. 295–305, Jan. 2005.
[7] S. Kondo and B. Milstein, “Performance of multicarrier DS CDMA
systems,” IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996.
[8] E. A. Sourour and M. Nakagawa, “Performance of orthogonal multi-
carrier CDMA in a multipath fading channel,” IEEE Trans. Commun.,
vol. 44, pp. 356–367, Mar. 1996.
[9] U. Madhow and M. L. Honig, “MMSE interference suppression for
direct-sequence spread spectrum CDMA,” IEEE Trans. Commun., vol.
42, pp. 3178–3188, Dec. 1994.
[10] S. E. Bensley and B. Aazhang, “Subspace-based channel estimation for
code-division multiple-access communication systems,” IEEE Trans.
Commun., vol. 44, pp. 1009–1020, Aug. 1996.
[11] M. Torlak and G. Xu, “Blind multiuser channel estimation in asyn-
chronous CDMA systems,” IEEE Trans. Signal Process., vol. 45, pp.
137–147, Jan. 1997.
[12] H. Liu and G. Xu, “A subspace method for signature waveform esti-
mation in synchronous CDMA systems,” IEEE Trans. Commun., vol.
44, pp. 1346–1354, Oct. 1996.
[13] X. Wang and H. V. Poor, “Blind equalization and multiuser detection
in dispersive CDMA channels,” IEEE Trans. Commun., vol. 46, pp.
91–103, Jan. 1998.
[14] Z. Xu, P. Liu, and X. Wang, “Blind multiuser detection: From MOE to
subspace methods,” IEEE Trans. Signal Process., vol. 52, pp. 510–524,
Feb. 2004.
[15] M. K. Tsatsanis and Z. Xu, “Performance analysis of minimum
variance CDMA receivers,” IEEE Trans. Signal Process., vol. 46, pp.
3014–3022, Nov. 1998.
[16] Z. Xu and M. K. Tsatsanis, “Blind adaptive algorithms for minimum
variance CDMA receivers,” IEEE Trans. Commun., vol. 49, pp.
180–194, Jan. 2001.
[17] Q. Li, C. N. Georghiades, and X. Wang, “Blind multiuser detection
in uplink CDMA with multipath fading: A sequential EM approach,”
IEEE Trans. Commun., vol. 52, pp. 71–81, Jan. 2004.
[18] H. V. Poor and X. Wang, “Code-aided interference suppression for
DS/CDMA communications—Part I: Interference suppression capa-
bility,” IEEE Trans. Commun., vol. 45, pp. 1101–1111, Sep. 1997.
[19] S. Buzzi, M. Lops, and H. V. Poor, “Code-aided interference sup-
pression for DS/CDMA overlay systems,” Proc. IEEE, vol. 90, pp.
394–435, Mar. 2002.
[20] J.-T. Yuan and J.-N. Lee, “Narrow-band interference rejection in
DS/CDMA systems using adaptive (QRD-LSL)-based nonlinear
ACM interpolators,” IEEE Trans. Veh. Technol., vol. 52, pp. 374–379,
Mar. 2003.
[21] X. Wang and H. V. Poor, “Blind joint equalization and multiuser detec-
tion for DS-CDMA in unknown correlated noise,” IEEE Trans. Circuits
Syst. II, Analog Digit. Signal Process., vol. 46, pp. 886–895, Jul. 1999.
[22] S. Buzzi and H. V. Poor, “A single-antenna blind receiver for multiuser
detection in unknown correlated noise,” IEEE Trans. Veh. Technol., vol.
51, pp. 209–215, Jan. 2002.
[23] E. Masry, “Closed-form analytical results for the rejection of
narrow-band interference in PN spread-spectrum systems—Part I:
Linear prediction ﬁlters,” IEEE Trans. Commun., vol. 32, pp. 888–896,
Aug. 1984.
11.
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3335
[24] K.-J. Wang, Z.-C. Zhou, and Y. Yao, “Performance analysis of
nonlinear interpolation ﬁlters in DS spread spectrum systems under
narrowband interference condition,” Electron. Lett., vol. 34, pp.
1464–1465, Jul. 1998.
[25] H. Li, X. Lu, and G. B. Giannakis, “Capon multiuser receiver for
CDMA systems with space-time coding,” IEEE Trans. Signal Process.,
vol. 50, pp. 1193–1204, May 2002.
[26] L.-M. Li and L. B. Milstein, “Rejection of narrow-band interference
in PN spread spectrum systems using transversal ﬁlters,” IEEE Trans.
Commun., vol. 30, pp. 925–928, May 1982.
[27] S. Theodoridis, N. Kalouptsidis, J. Proakis, and G. Koyas, “Interfer-
ence rejection in PN spread-spectrum systems with LS linear phase
FIR ﬁlters,” IEEE Trans. Commun., vol. 37, pp. 991–994, Sep. 1989.
[28] M. G. Amin, C. Wang, and A. R. Lindsey, “Optimum interference exci-
sion in spread spectrum communications using open-loop adaptive ﬁl-
ters,” IEEE Trans. Signal Process., vol. 47, pp. 1966–1976, Jul. 1999.
[29] C. Li, G. Hu, and M. Liu, “Narrow-band interference excision in
spread-spectrum systems using self-orthogonalizing transform-do-
main adaptive ﬁlters,” IEEE J. Sel. Areas Commun., vol. 18, pp.
403–406, Mar. 2000.
[30] L. M. Garth and H. V. Poor, “Narrowband interference suppression in
impulsive channels,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, pp.
15–34, Jan. 1992.
[31] L. A. Rusch and H. V. Poor, “Narrowband interference suppression
in CDMA spread spectrum communications,” IEEE Trans. Commun.,
vol. 42, pp. 1969–1979, Feb.–Apr. 1994.
[32] Y. Fu and H. Leung, “Narrow-band interference cancellation in spread-
spectrum communication systems using chaos,” IEEE Trans. Circuits
Syst. I, Fundam. Theory Appl., vol. 48, pp. 847–858, Jul. 2001.
[33] A. M. Tulino and S. Verdú, “Asymptotic analysis of improved linear
receivers for BPSK-CDMA subject to fading,” IEEE J. Sel. Areas
Commun., vol. 19, pp. 1544–1555, Aug. 2001.
[34] S. Buzzi, M. Lops, and A. M. Tulino, “A generalized minimum-mean-
output-energy strategy for CDMA signals with improper MAI,” IEEE
Trans. Inf. Theory, vol. 48, pp. 761–767, Mar. 2002.
[35] K. Zariﬁ and A. B. Gershman, “Subspace-based blind channel estima-
tion in DS-CDMA systems with unknown wide-sense stationary inter-
ference,” in Proc. ICASSP’06, Tolouse, France, May 2006.
[36] H. V. Poor, “Active interference suppression in CDMA overlay sys-
tems,” IEEE J. Sel. Areas Commun., vol. 19, pp. 4–20, Jan. 2001.
[37] S. Prasad, R. T. Williams, A. K. Mahalanabis, and L. H. Sibul, “A
transform-based covariance matrix differentiating approach for some
classes of parameter estimation problems,” IEEE Trans. Acoust.,
Speech, Signal Process., vol. 36, pp. 631–641, May 1988.
[38] F. Li, H. Liu, and R. J. Vaccaro, “Performance analysis for DOA
estimation algorithms: Uniﬁcation, simpliﬁcation, and observations,”
IEEE Trans. Aerosp. Electron. Syst., vol. 29, pp. 1170–1184, Oct.
1993.
[39] Z. Xu, “Perturbation analysis for subspace decomposition with appli-
cations in subspace based algorithms,” IEEE Trans. Signal Process.,
vol. 50, no. 11, pp. 2820–2830, Nov. 2002.
[40] ——, “On the second-order statistics of the weighted sample covari-
ance matrix,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 527–534,
Feb. 2003.
[41] F. D. Neeser and J. L. Massey, “Proper complex random processes with
applications to information theory,” IEEE Trans. Inf. Theory, vol. 39,
pp. 1293–1302, Jul. 1993.
[42] S. Haykin, Adaptive Filter Theory, 4th ed. Upper Saddle River, NJ:
Prentice-Hall, 2002.
[43] T. W. Anderson, An Introduction to Multivariate Statistical Analysis,
2nd ed. New York: Wiley , 1984.
[44] M. Lops, G. Ricci, and A. M. Tulino, “Narrow-band-interference sup-
pression in multiuser CDMA systems,” IEEE Trans. Commun., vol. 46,
pp. 1163–1175, Sep. 1998.
[45] O. C. Ugweje and S. A. Matin, “Performance of CDMA overlay sys-
tems using ﬁltering and diversity,” IEEE Trans. Veh. Technol., vol. 52,
pp. 456–462, Mar. 2003.
[46] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:
Cambridge Univ. Press, 1999.
Keyvan Zariﬁ (S’04) received the B.Sc. and M.Sc.
degrees, both in electrical engineering, from the Uni-
versity of Tehran, Tehran, Iran, in 1997 and 2000,
respectively.
He is currently pursuing the Ph.D. degree at the
Department of Communication Systems, Darmstadt
University of Technology, Darmstadt, Germany.
From January 2002 until March 2005, he was
with the Department of Communication Systems,
University of Duisburg-Essen, Duisburg, Germany.
Since April 2005, he has been with the Darmstadt
University of Technology. From September 2002 until March 2003, he was a
visiting scholar with the Department of Electrical and Computer Engineering,
McMaster University, Hamilton, ON, Canada. His research interests include
statistical signal processing, multiuser detection, blind estimation techniques,
and MIMO communications.
Mr. Zariﬁ was student contest ﬁnalist in the IEEE International Conference
on Acoustics, Speech, and Signal Processing (ICASSP), Toulouse, France, May
2006.
Alex B. Gershman (M’97–SM’98–F’06) received
the Diploma and Ph.D. degrees in radiophysics
and electronics from the Nizhny Novgorod State
University, Russia, in 1984 and 1990, respectively.
From 1984 to 1999, he held several full-time
and visiting research appointments in Russia,
Switzerland, and Germany. In 1999, he joined the
Department of Electrical and Computer Engineering,
McMaster University, Hamilton, ON, Canada, where
he became a Professor in 2002. He held visiting pro-
fessorships with the University of Duisburg-Essen,
Duisburg, Germany (2003–2005) and the Munich University of Technology,
Munich, Germany (2004). Since April 2005, he has been with the Darmstadt
University of Technology, Darmstadt, Germany, as a professor of Communi-
cation Systems. His research interests are in the area of signal processing and
communications with the primary emphasis on array processing, beamforming,
multiantenna and multiuser communications, and estimation and detection
theory. He has coedited two books and (co-)authored several book chapters and
more than 100 journal and 150 conference papers on these subjects.
Dr. Gershman is the recipient of several awards including the 2004 IEEE
Signal Processing Society Best Paper Award, the IEEE Aerospace and Elec-
tronic Systems Society Barry Carlton Award for the best paper published in
2004, the 2002 Young Explorers Prize from the Canadian Institute for Ad-
vanced Research (CIAR), the 2001 Wolfgang Paul Award from the Alexander
von Humboldt Foundation, Germany, and the 2000 Premier’s Research Excel-
lence Award, Ontario, Canada. He also coauthored the paper that received the
2005 IEEE Signal Processing Society Young Author Best Paper Award. He is
currently Editor-in-Chief of the IEEE SIGNAL PROCESSING LETTERS and is on
the Editorial Boards of the EURASIP Journal on Wireless Communications and
Networking and the EURASIP Signal Processing Journal. He was Associate
Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 1999–2005.
He is a member of both the Sensor Array and Multichannel (SAM) and the
Signal Processing Theory and Methods (SPTM) Technical Committees of the
IEEE Signal Processing Society and is the current Chair of the SAM Technical
Committee. He was Technical Co-Chair of the IEEE International Symposium
on Signal Processing and Information Technology (ISSPIT), Darmstadt,
December 2003; Co-Chair of the IEEE/ITG International Workshop on Smart
Antennas, Duisburg, April 2005; General Co-Chair of the First IEEE Workshop
on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP),
Puerto Vallarta, Mexico, December 2005; Technical Co-Chair of the Fourth
IEEE Sensor Array and Multichannel Signal Processing Workshop, Waltham,
MA, June 2006; and Tutorial Chair of EUSIPCO, Florence, Italy, September
2006.
Be the first to comment