In this paper, we consider the problem of blind estimation of the directions of arrival (DOA's) of users' paths in a multipath DS/CDMA channel. Making use of the signal that is sampled at multiple antenna elements and using a subspace based MUSIC-like technique, we show the possibility of DOA estimation using two search methods. The first provides path delays and DOAs simultaneously, and the second provides only DOAs. Knowledge of the chip waveform is used in the first method. It is seen that the two methods exhibit good estimation accuracy, besides being extremely near-far resistant

Al-Bayati, A. K. S.

Prakriya, Shankar

Prasad, Surendra

English

Publications of the IAS Fellows

Blind Subspace DOA Estimation in Multipath  
DS/CDMA Channels 
 
A.K.S. Al-Bayati, Shankar Prakriya and Surendra Prasad* 
Dept. of Electrical Engineering 
Indian Institute of Technology, Delhi 
New Delhi – 110016, India 
*Email: sprasad@ee.iitd.ernet.in 
 
ABSTRACT 
 
In this paper, we consider the problem of 
blind estimation of the directions of arrival 
(DOA’s) of users’ paths in a multipath 
DS/CDMA channel. Making use of the signal 
that is sampled at multiple antenna elements and 
using a subspace based MUSIC-like technique, 
we show the possibility of DOA estimation 
using two search methods. The first provides 
path delays and DOAs simultaneously, and the 
second provides only DOAs. Knowledge of the 
chip waveform is used in the first method. It is 
seen that the two methods exhibit good 
estimation accuracy, besides being extremely 
near –far resistant. 
 
1. INTRODUCTION 
 
Channel parameter estimation in DS/CDMA 
systems, generally, is an important issue for 
multiuser detection. When antenna arrays are 
used at the receiver, for capacity and 
performance improvement, the directions of 
arrivals (DOA’s) of various signals become an 
important parameter needed for beamforming 
toward the desired signals and implementing 
space-time multiuser receivers [1,2,3]. 
However, most of the available techniques for 
DOA estimation do not exploit the structure of 
the DS/CDMA signal and the multiuser 
interference (MUI) [4] and therefore require an 
impractically large number of antennas. 
Recently, a few techniques have been proposed 
for DOA estimation in DS/CDMA systems such 
as [2,5,6]. 
In [2], the DOA response vector of the 
desired signal was approximated as the 
dominant eigenvector of the difference between 
the pre- and the post-correlation spatial 
matrices. A problem with this approach is that it 
relies on the large process gain of the system, 
and does not exploit the code periodicity 
(assuming use of short codes); hence its 
performance degrades for strong MUI. 
However, it should be mentioned that this 
method has the advantage of handling aperiodic 
codes. In [5], a procedure is presented for 
approximate maximum likelihood delay/DOA 
estimation. However, the method relies on the 
use of a known training sequence for the desired 
user, causing a reduction in the data throughput. 
The overhead would be specially large in fast 
time varying channels where this training would 
be needed more often. In [6], a DOA estimation 
method was proposed making use of the pre- 
and post- correlation matrices of the input signal 
and making use of the code periodicity for 
suppressing MUI, but has the disadvantage of 
requiring knowledge of the path delays and 
attenuations for all the users. 
In this paper, we propose a blind subspace 
procedure for DOA estimation of DS/CDMA 
signals in a multipath scenario. Using an 
antenna array with a certain number of 
elements, we develop two search methods for 
DOA estimation: the first provides joint 
delay/DOA estimation through a delay search 
that makes use of the knowledge of the chip 
waveform, and the second provides only the 
DOA estimates without using this knowledge. 
The system considered is a quasi-synchronous 
system for simplicity of presentation. 
 
2. THE SIGNAL MODEL 
 
We consider here a scenario of K-users in a 
quasi-synchronous DS/CDMA system using 
binary phase shift keying (BPSK) modulation, 
and a uniform linear antenna array of M-
elements with 2/λ  spacing for signal 
reception. A short spreading code is used, i.e., 
the code period P′  is taken to be one bit or 
symbol period. Without loss of generality, it is 
assumed that each user signal arrives from L 
paths. The assumption of quasi-stationarity is 
  
meant here to imply that all the multipath 
signals arrive within a few chip durations (i.e. a 
maximum of q-chips with ). Thus all the 
user signals are synchronized to within the 
tolerance of a few chips only. Hence, it will be 
possible (with negligible loss in performance) to 
carry out parameter estimation using only those 
code chips that are free of intersymbol 
interference (ISI). The number of ISI-free chips 
shall be denoted by P. After down conversion, 
the received signal is chip matched filtered and 
sampled at the chip rate to give (in the absence 
of noise) 
Pq ′<
kmkla ,,, ρ
l ,θ
)
x
x
mLL
L
m




)(
)1(
Ns
s
K
K
M
h Km,L
k ,cLL
∑ ∑
= =
=
K
k
L
l
lklkm pcnspnx
1 1
, )()(),(     (1) 
where 
al,k: is the complex gain of the lth-path of the kth- 
user channel at the first antenna. 
lkm ,,ρ : is the array response of the lth-path of the 
kth-user at the mth- antenna, expressed as 
exp( ) )sin()1( ,klmj θ−π−   where : is the DOA 
of this path. c
k
k,l(p)denotes the pth-chip of the kth-
user’s code arriving through the lth-path, taking 
into cosideration the code delay in that path and 
the pulse shape used. sk(n) denotes the kth user 
symbol at time instant n. P denotes the process 
gain considering ISI-free chips only. 
We start by constructing a signal matrix Xm 
for the samples received at the mth- antenna in 
an N-symbol block interval as 








=
),()1,(
),1(2,1()1,1(
PNNx
Pxx
m
mmm
L
MMMmX (2) 
It is easy to see that this matrix can be expressed 
in terms of the symbol and code matrices as 
follows 
                                              (3a) 
where 
m HSX =
       S                (3b) 




=
)(
)1()1(
1
21
Ns
ss
LLLL
MM
L
              (3c) [ Tm hhH mm 2,1, L= ]
                                            (4) kmkkm ,, gCh =[ ]Lkkk ,, ccC 21=             (5) 
being the code- matrix of the kth user as seen 
from the L-paths of its arrival, with 
     (6) [ Tc )()2()1( ,,,, Pccc lklklklk LL= ]
]The vector g  , given by                      m,k[ Tg LkmkLkmkkmkkm aaa ,,,2,,,21,,,1, ρρρ L=  (7) 
is the multipath parameter vector for user-k at 
the mth- antenna taking into account the path 
attenuation as well as the array response to 
signals from various paths.  Further we define 
the matrix Ym as 
  [ ]1mmm XXY +=    for m=1 to M-1.       (8) 
It is easy to see that in the absence of noise we 
can write the following matrix relation 
                                                     (9)     
where                                                                                              
′= mm HSY
                      [ ]ThhH Kmmm ,1, ′′=′ LL            (10) 
                         kmkkm ,, gCh ′=′                      (11) 



=′
LkLkkk
Lkk
k
,,,21,1,,2
,1,
cc
cc
C ρρ L
L [ ]Lkk ,1, cc ′′=∆ LL  
                                                                      (12)                           
This matrix definition of the input signal 
samples is central to the formulation in the 
coming section. 
 
3. THE DOA ESTIMATION 
 
The approach adopted here for DOA 
estimation is a subspace approach that is based 
on resolving the desired user effective code 
km,h′   in terms of path codes c . The advantage 
of this approach is that if the effective code 
( h
lk ,′
km,′ ), over two or more antennas, is properly 
expressed in terms of path codes then this can 
easily lead to DOA estimation of all paths, 
simply from estimates of ρ - values in (12). The 
proposed method will make use of the channel 
estimation procedure of [7], besides an antenna 
array at the receiver for achieving this goal. 
In the MUSIC search, the vector space of 
the received signal is decomposed into a signal 
subspace (spanning all users’ signals) and its 
orthogonal complement, which is called the 
noise subspace. In case of a single path channel, 
the signal subspace is comprised of the span of 
all the user path codes. However, in a multipath 
channel, the signal space is comprised of the 
span of the effective codes of the various users. 
Consequently, such a subspace decomposition 
in the multipath situation can not help to resolve 
the signal (through the MUSIC search) into 
individual path codes.  
To resolve the problem, we propose here, 
finding a subspace that is comprised of the span 
of the codes of individual paths for a specific 
(desired) user. Once this subspace is found, we 
can make use of its orthogonal complement in a 
  
MUSIC-like search procedure to find 
delays/DOAs or only DOAs. Fortunately, as 
shown in this paper, this can be easily done by 
using an antenna array with number of elements 
more than the number of paths /user, and by 
making use of the properties of a certain matrix 
A which is constructed for the specific desired 
user (say the kth-user) as [ ]kMkk ,1,2,1 −′′′= hhhA LL         (13) 
A useful property for this matrix is given in 
the following proposition: 
Proposition: 
     If M-antennas are used at the receiver, and 
M> the number of paths of the desired (kth) user 
and if the DOA’s of these  paths are distinct, 
then: 
1. The left null space of A is orthogonal to all 
path codes of the kth-user, i.e., to   
. { }Lkkk ,2,1, ,, ccc ′′′ LL
2 Rank(A)= number of paths for the kth –user.   
Proof: 
   From (11) and (12) we get  
                                                     (14) kk GCA ′=
Where                                             [ ]kMkkk ,1,2,1 −= gggG LL                          (15)  
   If a vector v lies in the left null space of A, 
this implies  
                                   (16) 0GCv H =′ kk
   Let  
                               (17) [ Lk bbb LL21=′Cv H ]
   Then    
                    G                  (18) [ ] 0TH =**1 Lk bb LL
   But   is a vandermonde matrix with 
distinct entries, hence it is full rank, and there is 
no solution for (18) except the trivial solution 
viz.,  b
HG k
l=0, for l=1 to L. Hence, any vector in the 
left null space of A should be orthogonal to all 
user’s path codes. It is also obvious that A has a 
rank of L , since both , and GkC′ k ,in (14), are 
full rank matrices with rank=L.  
Hence proved. 
In the presence of noise, A can be estimated 
using the subspace approach of [7], applied to 
outputs from a pair of antenna elements as given 
in (Ym), to estimate ,  for m=1   to   M-1. It 
is easy to argue that the effective codes are those 
linear combinations of the path codes which are 
orthogonal to the noise subspace
km,h′
1 of TYm . 
Taking a clue from (4) and (11) and using this 
argument, we can construct estimates for the 
effective code of the kth user at the mth antenna 
as 
TYm
                          h                       (19) mwB kkm ′=′ ,ˆ
where          
{ })(minarg wBQQBww HHH
w
m kmmk ′′= ,       (20) 
                 subject to   wHw=1 
                                               (21)                  

=′
k
k
k B0
0B
B
and                             
          (22) 








−+
=
)1()(
)()2()1(
qPbPb
qbbb
kk
kkk
k
LLLL
MMM
LL
B
where  is the binary Gold-code assigned to 
user-k and  Q  is the noise subspace of . 
Hence  
kb
m
               [ ]kMkk ,1,2,1 ˆˆˆˆ −′′′= hhhA LL        (23) 
Now, if we perform singular value 
decomposition for  as ( )then  the 
number of paths  can be estimated as the 
number of significant singular values of , and 
the left null space of A can be estimated as the 
span of the left singular vectors of  that are 
associated with the (2P- ) least significant 
singular values, i.e.   
Aˆ
Lˆ
VUA ∑=ˆ
Aˆ
Lˆ
Aˆ
                  [ ]PLn 21ˆ uu LL+=U                 (24) 
Once Un has been estimated, we can perform a 
search to find the path DOAs using one of the 
following two methods which are both based on 
the test of orthogonality of the candidate 
code/DOA to the subspace Un: 
3.1  Joint delay/DOA estimation: 
This search uses knowledge of the pulse 
shaped code waveform2 of the desired user, and 
hence, knowledge of )(τkc , a (Px1) vector 
comprised of the chip-rate samples of this 
waveform starting at a specific delay τ . This 
vector is used in a search over all the possible 
                                                          
1 The noise subspace of  here is defined as the 
subspace spanned by the left singular vectors of Y   
that are associated with the least significant (2P-K) 
singular values. For details see [7]. 
TYm
T
m
2 Obviously, this waveform is the convolution of the 
desired user binary code with the pulse shape.  
  
τ - values (typically within a period of few 
chips in quasi-synchronous systems) making use 
of the fact that any path code c (i.e. ove
antennas) should be of the form 
lk ,′
fZ , where 


τ
)fZHn
0fZ
H
n
τ
(( lf
θ
lk ,′
π
r two B.B.2T     3.2.2- The estimated path DOA’s of the desired 
user are found as the θ -values 
corresponding to the significant peaks of 
the plot of  (they should be less than 
or equal to ) . 
                                   (25) 
=
)(
)(
τ
τ
k
k
c0
0c
Z
and f is the array response vector. Hence, this 
-search will simultaneously provide DOA’s 
with the delays, as follows: 
3.1.1- For  max0 ττ to= , in steps (step<< 
chip period), perform the following two steps: 
(i)Form the matrix Z as in (25). 
(ii) Find f , a two-element vector, as  0
   f             (26) (minarg UUZf HH
f
0 n=
         subject to  1=ff H                            
                 (27) )(/1)( 0 UUZf H
H
nr =τ
3.1.2- Plot )(τr  as a function of τ , and take 
the  significant peaks of the plot, where the Lˆ
’s corresponding to these peaks represent the 
estimated path delays. If we denote the f -
vector corresponding to the l
0
th significant peak 
of )(τr  by , then the DOA’s of the desired 
user’s paths can be found as 
lf
           
for  l = 1   to    L.                              (28) 
))]1(/)2*)/1[(sin)( 1 langlelDOA fπ−=
 
3.2   DOA only estimation: 
This search is done over  without using 
the pulse shape information, by making use of 
the fact that any path code c  should be 
composed of two identical halves (upper and 
lower) except for the array response factor ρ  
multiplying the lower part. Hence any desired 
user path code should have the form of  
                                              (29) w
B
B



k
k
ρ
Hence, using this representation for a 
candidate path code, we can do a search over θ  
as follows:    
3.2.1-   For   3/3/πθ to−=    (the view 
angle)   do the following two steps: 
(i) form the candidate code matrix Cc 
given by (29), ))sin(exp( θπρ j−= . 
(ii) Test the candidate code ( C ) 
orthogonality to U
wc
n, using the measure 
)(θw
H
c U
given by 1/[min. eigen value of  
( C ) ] .     c
HCU nn
)(θw
Lˆ
 
4. SIMULATIONS 
 
To test the performance of the proposed 
DOA estimation methods, we carried out the 
following experiments via computer 
simulations. 
Exp.1:  In this experiment, we assume a quasi-
synchronous system with a process gain of 22 
employing Gold codes. We consider the 
performance as a function of SNR. We assume 
here a raised cosine chip waveform (roll off 
factor of .5), and consider DOA estimation with 
6 antennas. A scenario of 8-users is simulated 
with 3-paths /user with relative powers of (-6, -3 
and 0) dB, and an SIR=-16 dB. DOA’s of the 
desired user were (-17.2o, 4.6o, 11.4o) with 
delays of (.19,1.48, 2.13) chips.  The size of the 
block used for estimation is 200-symbols. Fig.1 
shows the rms error in DOA estimation versus 
SNR of the strongest path of the desired user. 
The figure generally shows a fair accuracy with 
the method 1 being more accurate than method 
2. It also shows that accuracy of estimation is 
better for more spatially separated paths 
(estimation of the –17.2o is better than 4.6o and 
11.4o ), and also better for higher SNRs.  
Exp.2: To show the near-far resistance of 
the proposed methods, we carried out an 
experiment similar to experiment 1 but with the 
following differences in the parameters of the 
desired user:(DOAs=-8.6o,8.5o, 17.1o.),(Delays= 
.23, .51, 2.13  chips),.Path relative powers (0,0,-
3) dB. SNR of the strongest path of the desired 
user is fixed at (6 dB). Again, 200-symbols were 
used for estimation. Fig.2 shows the rms error in 
the DOA estimation versus SIR. The figure 
clearly shows that both of the proposed search 
methods are extremely near-far resistant. 
For lack of space, we are not presenting 
here a comparison with the performance 
achieved via other methods. However, our 
experiments clearly show that the method 
proposed here has significantly better 
performance than [6]. 
 
  
5. CONCLUSIONS 
 
In this paper, a new approach for blind 
multiuser DOA estimation in multipath 
DS/CDMA channels, is presented. The approach 
employs a subspace MUSIC-like technique, and 
develops two search methods; the first is a 
delay/DOA search, and the second is a DOA-
only search. The pulse shape information is used 
only in the first method. The first method, 
generally, has higher accuracy than the second 
method,providing delays/DOAs simultaneously.  
Simulation results show that the two methods 
can provide DOA estimation of quite reasonable 
accuracy. Results also show that the method is 
very near-far resistant and hence can be 
employed in loosely power controlled scenarios. 
Hence, the proposed method can be very useful 
to base stations employing smart antennas and 
performing multiuser detection. 
 
6.  REFERENCES 
 
[1] B. Suard, A. Naguib, G. Xu, and T. Kailath, 
“ Performance analysis of CDMA mobile 
communication systems using antenna 
arrays,” in Proc. Int. Conf. Acoust. Speech, 
Signal Process., vol. VI, Apr. 1993, pp. 
153-156. 
[2] A. Naguib, A. Paulraj, and T. Kailath, “ 
Capacity improvement with base station 
antenna array in cellular CDMA,” IEEE 
Trans. Veh. Technol., vol. 43, pp.691-698, 
Aug. 1994. 
[3] X. Wang, and H. Vincent Poor, “ Space-
time multiuser detection in multipath 
CDMA channels,” IEEE Trans. on signal 
processing, vol. 47, no. 9, pp. 2356-2374, 
September 1999. 
[4] R. O. Schmidt, “ Multiple emitter location 
and signal parameter estimation,” in PROC 
RADC, Spectrum Estimation Workshop, 
NY. 243-258, 1979. 
[5] K. Wang, and H. Ge, “ On estimating 
temporal and spatial channel parameters for 
DS-CDMA system over multipath rayleigh 
fading channels,” ICASSP-2001. 
[6] Z. Lei and T. J. Lim, “ Estimation of 
directions of arrival of multipath signals in 
CDMA systems,” IEEE Trans. 
Communications, vol. 48, No. 6, pp. 1022-
1028, June 2000. 
[7] H. Liu and G. Xu, “ A subspace method 
for signature waveform estimation in 
synchronous CDMA systems,” IEEE 
Trans. Communications, vol.44, pp. 
1346-1354, Oct. 1996. 
 
 
 
 
                                     Fig.2a                                                        1.
gre
e
)
-3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
4
1.6
1.8
2
SNR (dB)
rm
s 
er
ro
r i
n 
D
O
A
 (d
e 
 
 
 
 
 
 
 
                                   Fig.1a 
 
 
 
 
 
                                                                              1
1.
O
A
 (d
 
 
 
 
 
        
-3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
2
1.4
1.6
1.8
2
SNR (dB)
rm
s 
er
ro
r i
n 
D
e
gre
e
)
                                   Fig.1b    
Fig.1  RMS error in DOA estimates 
versus SNR(strongest path) of the 
desired user,(++: method-1,  oo:method-
2). 1a: DOA=-17.2o,  1b: DOA=4.6o, 
                                                           
 
-45 -40 -35 -30 -25 -20 -15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
SIR (dB)
rm
s 
er
ro
r i
n 
D
O
A
 (d
e
gre
e
)
 
 
 
 
 
 
 
 
 
 
 
Fig.2-  RMS error in DOA estimate, for 
the desired user, versus SIR, for exp.2. 
(method 1, ++:DOA=-8.6o , 
oo:DOA=8.5o) (method 2, **:DOA=-
8.6o, xx: DOA=8.5o). 
  


Blind subspace DOA estimation in multipath DS/CDMA channels

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Blind subspace DOA estimation in multipath DS/CDMA channels

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