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Direction of Arrival Estimation Algorithm: Direction Lock
Loop
Seunghyun Min, Kwang Bok Lee, and Yong-Hwan LeeSchool of
Electrical Engineering, Seoul National University
Shinlim-dong, Kwanak-gu, Seoul 151-742, Korea
TEL: +82-2-880-8415 FAX: +82-2-880-8215
E-mail: shmin@mobile .snu.ac.kr, [email protected],
[email protected]
Abstract- In this paper, a new direction of arrival
(DOA)estimation algorithm, direction lock loop (DiLL), is
proposed.It has a similar concept to the delay lock loop (DLL) that
isused for synchronization. It estimates the DOA of a signal
byiterations, and can track the DOA of a moving source. TheDiLL
scheme is found to track better than the DOAestimation scheme based
on the PASTd, and its performanceis less sensitive to the DOA of a
signal than that of the DOAestimation scheme based on the PASTd.
The DOA estimationaccuracy and the tracking capability are
demonstrated byanalysis and computer simulations.
I. INTRODUCTION
The demand for wireless communication services isgrowing at an
explosive rate. To enhance the capacity ofwireless communication
systems, space division multipleaccess (SDMA) systems are of
considerable interest. TheSDMA systems are implemented by using
smart antennasystems. In recent years, smart antenna systems based
ondirection of arrival (DOA) estimation methods have beendeveloped.
The problem of estimating the DOA of signalsfrom array data is well
documented in [1]. Eigen structuremethods such as MUSIC and ESPRIT
have been foundwidespread use in providing estimates of the DOA
ofsignals [2], [3]. However, the computational burden of theeigen
analysis increases significantly with the number of
antenna elements.
To reduce the complexity of the eigen methods,projection
approximation subspace tracking with deflation(PASTd) was proposed
by Yang [4], [5]. This algorithmtracks the signal subspace
recursively. Using this algorithm,the eigenvectors may be tracked
easily. However, thePASTd algorithm is not for DOA estimation but
for signalsubspace estimation. Thus, an additional DOA
estimationalgorithm such as MUSIC or ESPRIT, is required toestimate
the DOA of signals.
In this paper, a new DOA estimation algorithm isproposed. Since
this algorithm is similar to the delay lockloop (DLL) used for
synchronization [6], it is referred to as
the direction lock loop (DiLL) scheme in this paper. Anerror
signal for DOA estimation is generated from thecorrelation of an
input signal and the array response vectors
whose directions are shifted from the DOA estimate.This error
signal is used to update DOA estimatesiteratively. The DiLL does
not require a separate DOAestimation. Thus, the DiLL scheme is
conceptually simple,and tracks a moving source by iterations.
This paper is organized as follows. The system model isshown in
Section II. In Section III, the new DOAestimation scheme, DiLL is
presented and thecharacteristics of the DiLL scheme are explained.
The
performance analysis of the DiLL scheme is shown inSection IV.
In Section V, numerical results are given.
Conclusions are drawn in Section VI.
II. SYSTEM MODEL
Consider a uniform linear array with Nhalf-wavelengthspacing
antenna elements. For simplicity of explanation,the single user
case is investigated in this paper. Thus, thereceived signal may be
represented as
( ) )()()( ttdt var += (1)where
l [ ]TN trtrt )(,),()( 1 L=r is an N1 vector of
received signals at time t;l ( )a is the array response vector
for a signal with a
DOA . The value of is constant when the source isfixed, or
varies when the source is moving. The vector
( )a is defined as( )[ ]Tsin1sin0)( = Njjj eee La ;
l )(td is a source signal at time t, and it is assumed
1)(2 =td for simplicity of explanation;
l v(t) is anN1 additive noise vector, which is assumedto be
spatially and temporally white, having Gaussiandistribution with
covariance matrix 2 I, where 2 isthe noise variance.
III. DIRECTION LOCK LOOP (DILL)
A. Principles of the DiLL Scheme
In this subsection, the principle of the DiLL scheme
isexplained. Fig. 1 shows a block diagram of the DiLLscheme. Note
that its structure is similar to the DLL fortime-domain
synchronization. The received signal vector iscorrelated
simultaneously with right-shifted and left-shifted
array response vectors ( ) + ia and ( )( ) ia toproduce
correlator outputs )(izR and )(izL , when the DOA
estimate at the ith time is ( )i and the shift angle is which is
set to a constant value regardless of , for
simplicity of explanation. The correlator outputs )(izR and)(izL
may be represented as
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )iviRidiiN
iz
iviRidiiN
iz
L
H
L
R
H
R
+==
++=+=
),()()(1
)(
),()()(1
)(
ra
ra (2)
where ( )21,R is a normalized spatial correlation function,which
is defined as
=
==N
n
njH eNN
R1
)sin)(sin1(
122121
1)()(
1),( aa , (3)
( ) ( )( ) ( )iiN
ivH
R va += 1 , ( ) ( )( ) ( )ii
N
ivH
L va = 1 , and
the superscriptHdenotes the Hermitian transpose.
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The difference between the amplitude-squares of twocorrelator
outputs is used to produce an error signal )(ie ,
which is defined as
( )( ) ( )iviGizizie eLR +== )()()(22
(4)
where ( )( ) iG is a direction discriminator characteristicand
is defined as
( )( ) ( ) ( )22
),(),( += iRiRiG (5)
and ( )ive
is defined as
( ) ( ) ( )
( ) ( )( ) ( ) ( )( ) ( )( ){ }iviRiviRidiviviv
LR
LRe
++
=
,,Re2
22
. (6)
The characteristics of ( )( ) iG determine the DOAestimation
performance of the DiLL and the plot of
( )( ) iG is referred to the S-curve. The error signal, e(i)
isfiltered and fed to the numerically controlled oscillator(NCO) to
update DOA estimates iteratively. Hence, theDOA estimate at the
(i+1)th time may be expressed as
( ) ( )( )ifieKii +=+ 0)()1( (7)where f(i ) is the impulse
response of the loop filter,
0K is
the NCO gain, and denotes convolution.
Fig. 2 shows the plot of G as a function of , using(5), when o0=
, N=4, and =12.24. When the DOAestimate at the ith time ( )i is
less than the actual DOA ,the error signal is larger than zero, and
thus NCO increases
the DOA estimate value at the (i+1)th time. When ( )i islarger
than , the error signal is less than zero, and thus
NCO decreases ( )1 + i . Repeating this procedure, the DOA
estimate
( )i converges to the DOA of the signal
. When
the DOA of the signal is time varying, the DiLL algorithmtracks
the DOA of a moving source through iterations.
Note that the negative S-curve slope value ensures that
the DOA estimate ( )i converges to an actual DOA, ,when the
initial DOA estimate is in a certain range which is
called a locking range. As shown in Fig. 2,+ ,zc is the
smallest positive value of , at which the S-curve has azero
value. Similarly,
,zc is the smallest negative value of
, at which the S-curve has a zero value. The lockingrange for
the DiLL is defined as ( )+ zczc , in this paper.
Since it is difficult to exactly determine+ ,zc and ,zc ,
+ ,zc may be approximated as + , which is the smallest
positive value of , at which ( )2
, R has a zero value.
Similarly, ,zc may be approximated as , which is the
smallest negative value of , at which ( )2
, +R has a
zero value. From (5), the S-curve is always zero when is90. From
these results,
+ ,zc and ,zc are approximatedas [7]
( )
( )
+
+
+
o
o
90,2sinsinmax
90,2
sinsinmin
1
,
1
,
N
N
zc
zc. (8)
From (8), it is shown that the locking range is
inverselyproportional toN, and increases as increases.
The slope of the S-curve at = plays an importantrole in the DiLL
scheme like the DLL scheme [6]. The
slope of the S-curves() at = is calculated as
( )
( )( )( )
( )
=
=i
i
iG
s
. (9)
Fig. 3 shows the slope of the S-curve as a function of using
(9), when =0and N=8. Note that the slope valueoscillates between
positive and negative values. It isrequired for the DiLL scheme to
be operated properly thatthe slope of the S-curve should be
negative and the value of
should be chosen as the less one than the first zerocrossing
point. The values in Table 1 are whatmaximize the magnitude of the
slope of the S-curve, when
=0forN=2,4 and 8.
B. Characteristics of the modification factor
In the DiLL scheme, to estimate the DOA of a signalcorrectly,
the value of the S-curve should be zero when
( ) = i . When =0,N=4 and =12.24, the value of theS-curve is
zero. This ensures that converges to 0through iterations. However,
( )( )= iG is not alwayszero. Fig. 4 shows the S-curve for =60, N=4
and=12.24, and indicates that ( )i does not approach to 60
but to 62.4. This means that the DiLL scheme is a
biasedestimator. When = , ( )[ ] ( )=GieE . It is found from
(5)
that ( )G may be represented as
( ) ( ) ( )
( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
=
+
+
=
+=
1
12
22
sinsin2
1sin
sinsin
sinsin
2
1sin
4
,,
N
nn
n
nNN
RRG
. (10)
The term in (10), ( )( ) ( )( )+ sinsinsinsin ,is not generally
zero due to the nonlinearity of the sine
function. This is why ( )G is not generally zero. From(10), ( )G
is found to increase as increases.
To remove the bias, the error signal should be modified
as follows. For simplification of representation, ( )G is
referred to as ( )m , the modification factor. The modifiederror
signal ( )iem is represented as
( ) ( ) ( ) ( ) ( )== mizizmieie LRm22
)( . (11)
In (11), the actual DOA is needed to estimate amodification
factor. However, is not available in theDOA estimation algorithm.
In the DiLL scheme, the DOA
estimate is used for the modification factor estimation.Thus,
the equation (11) may be changed to
( ) ( ) ( ) ( ) ( )ivGmizizie emLRm +== )(22
(12)
where ( ) ( ) ( )=
mGGm
, and its characteristic plot is
called the modified S-curve. Fig. 4 shows the modified S-
curve for o60= . Note that it has a zero value ato60 =
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and has a negative slope value. The DOA estimate at the(i+1)th
time may be expressed as
( ) ( )( )ifieKii m +=+ )()1( 0 . (13)
IV. PERFORMANCE ANALYSIS
In this section, the DOA estimation error variance for the
DiLL scheme is analyzed. The DOA estimation error at the
ith time ( )i may be defined as ( ) ( )ii =
. From (12)
and (13), the DOA estimation error at the (i+1)th time maybe
expressed as
( ) ( ) ( ) ( ) ( )ifiviGKiiem
+=+ 10
. (14)
If the DOA estimation error is small enough, ( ) iGm
may be approximated as ( ) ( )ism , where
( ) ( )( )
( )( ) =
=
i
m
mi
iGs
and ( )ms is a negative value. Thus,
the DOA estimation error may be approximated as
( ) ( ) ( ) ( ) ( )( )
( )ifs
ivisKiim
em
+=+ 01 . (15)
Thus, the DOA estimation error may be represented inthe z-domain
as
( ) ( ) ( )
( ) ( )( )
( )
( )
+
=
m
e
m
m
s
zV
zzFsK
zzFsKz
1
0
1
0
11 (16)
where ( )zF is the z-transform of the loop filter ( )if
.Therefore, using (16), the DOA estimation error variance
in the steady state, 2s
, may be expressed as [6]
( )( )
( )= 2
2
m
Le
s
BivVars
(17)
where LB is the two-sided noise bandwidth of the closed
loop transfer function, ( ) ( ) ( )
( ) ( )( ) 10
10
11
+
=zzFsK
zzFsKzH
m
m . By
using (6), the variance of ( )ive
may be expressed as
( )( ) ( ) ( ) ( ) ( )( )( )++= ,,,Re222
RRRqN
ivVarse
(18)
where ( )s
q is defined as
( ) ( ) ( )22
,, ++= sss RRq . (19)
When and are defined as ( )
0
2
2
!2
1
=
=
ss
sq and
( ) 0== ssq, ( )
sq may be represented as [6]
( ) += 2ssq . (20)
Using (17), (18) and (20), 2s
may be represented as
( ) ( ) ( ) ( )( )( )+++
= ,,,Re22 2
2
22
RRRsN
Bss
m
L
(21)
or solving for 2s
( )
( )
( )
= 2
2
2
2 2
21
2
m
L
m
Lm
L
sN
B
sN
BsN
Bs
(22)
where2
1
=
and
( ) ( ) ( )( )++=
,,,Re2 RRR . When the
BL/ is small, 2s
of the DiLL can be linearly
approximated as in (15), thus( )
12
2
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ESPRIT scheme, when the DOA of a signal start to changeat the
100th symbol time, the DOA tracking error increasessignificantly,
and then, the DOA tracking error decreases asthe rate of the DOA
angle change decreases. The DOAtracking error again increases, as
the rate of DOA anglechange increases. Unlike the PASTd with
ESPRIT, theDiLL scheme is relatively insensitive to the variation
in the
rate of DOA angle change. The reason is that ifBLis largerthan
the rate of DOA angle change, the tracking error isonly related
with the noise variance. Thus, the trackingerror is insensitive to
the rate of DOA angle change.
From these results, it is found that the DiLL scheme hasthe
better tracking capability than the PASTd with ESPRITwhen the two
schemes have the same DOA estimationerror variance.
VI. CONCLUSIONS
In this paper, a DOA estimation algorithm, DiLL isproposed and
analyzed. It estimates the DOA of a signaliteratively by using the
difference of the correlation of an
input signal and the array response vectors whosedirections are
shifted from the DOA estimate.
The DiLL scheme is found to track better than thePASTd based DOA
estimation scheme, and its
performance is less sensitive to the DOA of a signal thanthat of
the PASTd based DOA estimation scheme when thetwo scheme have the
same DOA estimation error variance.
REFERENCES
[1] L. C. Godara, Applications of Antenna Arrays to
Mobile Communications, Part II: Beam-Forming and
Direction-of-Arrival Considerations,Proc. IEEE, vol.85, no. 8,
pp. 1195-1245, Aug. 1997.
[2] R. O. Schmidt, Multiple Emitter Location and Signal
Parameter Estimation, IEEE Trans. Antennas
Propagat., vol. 34, no. 3, pp. 276-280, Mar. 1986.
[3] R. Roy and T. Kailath, ESPRIT-Estimation of Signal
Parameters via Rotational Invariance Techniques,
IEEE Trans. Acoust., Speech, Signal Processing, vol.
37, no. 7, pp. 984-995, Jul. 1989.
[4] B. Yang, Projection Approximation Subspace
Tracking, IEEE Trans. Signal Processing, vol. 43, no.
1, pp. 95-107, Jan. 1995.
[5] B. Yang, An Extension of the PASTd Algorithm to
Both Rank and Subspace Tracking, IEEE Trans.
Signal Processing, vol. 43, no. 9, pp. 179-182, Sep.
1995.
[6] M. K. Simon, Noncoherent pseudonoise code tracking
performance of spread spectrum receivers, IEEE
Trans. Commun., vol. 25, no. 3, pp. 327-345, Mar.
1977
[7] R. A. Monzingo and T. W. Miller, Introduction to
Adaptive Arrays, 1980.
[8] B. D. Rao and K. V. S. Hari, Performance Analysis of
ESPRIT and TAM in Determining the Direction of
Arrival of Plane Waves in Noise,IEEE Trans. Acoust.,
Speech, Signal Processing, vol. 37, no. 12, pp. 1990-
1995, Dec. 1989.
Table 1. (degrees)
N=2 26.28N=4 12.24N=8 6.12
Downconverting
and
sampling.
.
.
1
2
N
| |2
| |2
)(ir
-
Loop
filterNCO
)( iem
-
+)(ie
)( a
)(izL
)(izRspatial
correlator
array response
vector generator
spatialcorrelator
)( +a
( )mmodificationfactor generator
Fig.1. Block diagram of the DiLL scheme
-90 -60 -30 0 30 60 90
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
S-curve
+
zc,-
-
zc,+
DOA estimate, (degrees)
S-curve
Fig. 2. S-curve (=0,N=4, =12.24)
0 10 20 30 40 50 60
-0.3
-0.2
-0.1
0.0
0.1
slope(/de
grees)
(degrees)
Fig. 3. The slope of the S -curve (=0,N=8)
( )
( )2
2
,
,
+
R
R
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-90 -60 -30 0 30 60 90
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
^
62.4O
S-curve
DOA estimate, (degrees)
unmodified S-curve
modified S-curve
Fig. 4. Modified and Unmodified S-curves
(=60,N=4, =12.24)
-60 -30 0 30 60
10-1
100
101
N=2
N=4
N=8
s
2
(degrees
2)
DOA of signal, (degrees)
DiLL-analysis
DiLL-simulation
unmodified DiLL(N=8)
PASTd with ESPRIT(N=8)
Fig. 5. DOA estimation error variance
0 1000 2000 3000 4000
-60
-40
-20
0
20
40
60
80
DOAofasignal(d
egrees)
number of symbols
real trajectory
DiLL
PASTd with ESPRIT
(a)
0 1000 2000 3000 400010
-1
100
101
DiLL
PASTd with ESPRIT
Ensemble-averagedsquareerror
number of symbols
(b)
Fig. 6. DOA tracking of the time-varying (i)
