Cramer-Rao bounds of DOA estimates for BPSK and QPSK Modulated signals

Abstract

This paper focuses on the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive circular complex Gaussian noise. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform are given. These CRBs are compared, on the one hand, with those obtained with different a priori knowledge and, on the other hand, with CRBs under the noncircular and circular complex Gaussian distribution and with different deterministic CRBs. It is shown in particular that the CRBs under the noncircular [respectively, circular] complex Gaussian distribution are tight upper bounds on the CRBs under the BPSK [respectively, QPSK] distribution at very low and very high signal-to-noise ratios (SNRs) only. Finally, these results and comparisons are extended to the case of two independent BPSK or QPSK distributed sources where an explicit expression of the CRB for the DOA parameters alone is given for large SNR.

Full text

Page 1

STOCHASTIC CRAMER-RAO BOUNDS OF DOA ESTIMATES FOR BPSK AND QPSK
MODULATED SIGNALS
Jean-Pierre Delmas, Habti Abeida
GET/INT, D´ epartement CITI, UMR-CNRS 5157, Evry, France
ABSTRACT
This paper focuses on the stochastic Cramer-Rao bound
(CRB) of direction of arrival (DOA) estimates for binary
phase-shift keying (BPSK) and quaternary phase-shift key-
ing (QPSK) modulated signals corrupted by additive circu-
lar complex Gaussian noise. Explicit expressions of the
CRB for the DOA parameter alone in the case of a sin-
gle signal waveform are given. Finally, these results are
extended to the case of two independent BPSK distributed
sources whereanexplicitexpressionoftheDOAparameters
alone is given for large SNR.
1. INTRODUCTION
Stochastic CRB’s play an important role in DOA estima-
tionbecause they serve as a benchmark for the performance
of actual estimators and because the stochastic CRB can be
achieved asymptotically (in the number of measurements)
by the stochastic maximum likelihood (ML) method. Un-
fortunately, stochastic CRB’s appear to be prohibitive to
compute for non-Gaussian processes includingdiscrete sig-
nal waveforms. Andtothebest ofourknowledge, nocontri-
bution has yet dealt with stochastic CRB for non-Gaussian
signal waveforms.
To cope with this difficulty, a method sometimes used
is to assume that the signals are arbitrary deterministic
sequences while the noise is circular complex Gaussian,
so that the distribution is still Gaussian and the associ-
ated deterministic CRB is easily deduced (see e.g., [1, rel.
(B.3.11)]). But the corresponding deterministic (or condi-
tional)ML method does not achieve this deterministicCRB
because the deterministic likelihoodfunctiondoes not meet
the required regularity conditions. Consequently, this de-
terministic CRB is only a non-attainable lower bound on
the variance of any unbiased DOA estimator. To deal with
non-Gaussian processes, another solutionis to suppose that
the signals are Gaussian but not necessarily complex circu-
lar. In that case, the associated CRB is under rather general
conditions(see e.g., [2, p. 293]) the largest CRB among the
class of arbitrary distributionswith given covariance matri-
ces. This approach was used in [3] for non-circular com-
plex signal waveforms such as discrete signals. But the as-
sociated CRB is only an upper bound on the true stochas-
tic CRB. Faced with the drawbacks of the two aforemen-
tioned approximations, we need an explicit expression of
the stochastic CRB under non-Gaussian distributions.
In this paper, we derive explicit expressions of the
stochastic CRB for the DOA parameter alone in the case
of BPSK and QPSK signal waveforms observed in additive
circular complex Gaussian noise.
2. DATA MODEL
Consider a BPSK or QPSK modulated signal impingingon
an arbitrary array of
sensors. The received signals are
bandpass filtered and after down-shifting the sensor signal
to baseband, the in-phase and quadrature components are
paired to obtain complex signals. We assume Nyquist shap-
ing and ideal sample timing so that the inter-symbol inter-
ference at each symbol spaced sampling instance can be ig-
nored. In the absence of frequency offset but with possible
phase offset, the signals at the output of the matched filter
can be represented as:
?
????
???????
????????
where
DOA parameter
?
?is the steering vector parametrized by the scalar
We suppose
whereare independent, identically,
distributed (IID) random symbols taking values
?
?.
?
?
?
??
?
?
.
?
?
?
?
??!#%'?
*
'?+?,?/000/2
3
?[resp.
346763:4676 ] with equal probabilitiesfor BPSK [resp.
QPSK] modulations, where
unknown parameters.
mean complex circular Gaussian with
Consequentlyare IID
variableswhoseprobabilitydensityfunction(PDF)ismixed
Gaussian:
=
?
and
?
?
are considered as
are IID -variate zero
*
??
+
?,?/000/2
?
?*
???B
?
+
??
?
DEG .
*
?
?
+
?,?/000/2
?
-dimensional random
H
*
?
?IK+
?
?
LM
G
?
?
G
D
N
OP
,?
?QSUWXZ
%[\]
%^_`
%
Sb
Z
bc
with
L
?
6
and
'
P
?
3
?
[resp.
L
?d
and
'
P
?
3
4
6763:
4
676 ] for BPSK [resp. QPSK] modulated sig-
nals and where
Khjk
?
*
?
D
??
?
?
=
?
?
?
?
+
2
.
II - 730-7803-8484-9/04/$20.00 ©2004 IEEEICASSP 2004
«¬

Page 2

3. STOCHASTIC CRB FOR BPSK AND QPSK
SIGNALS
Because the distribution of these models are simple mixed
Gaussian, an explicit expression of the Fisher information
matrix (FIM) is proved in [6] using well known properties
of the Gaussian distribution.
Theorem 1 The FIM associated with the stochastic BPSK
and QPSK modulatedsignals are given by
E
????
?
??
?
E
?
?
?
E
?
?
with
E
?
?
??
G
?
b
c
*
?
?
?
?
b
%
?
b
c
?
?
*
?
++
?
G
?
%
??
c
?
?
*
?
+
?
G
?
%
?
?
c
?
?
*
?
+
?
G
?
b
c
*
?
?
?
?
*
?
++
?
and
E
?
?
?
?
G
?
b
%
?
b
c
*
?
?
?
?
*
?
++
?
?
b
%
!#$
%
#'
%
!
?
b
c
*
?
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?
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*
?
++
?
?
b
%
!#
$
%
#
'
%
!
?
b
c
*
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*
?
++
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b
%
*#
'
%
*
b
?
b
c
*
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?
?
?
*
?
++
-/
?
E
0
???
?
??
?
E
2
?
?
?
E
2
?
?
with
E
2
?
?
??
G
?
b
c
*
?
?
?
?
b
%
?
b
c
?
?
*
3
?
++
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G
?
%
?
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c
?
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*
3
?
+
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G
?
%
?
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c
?
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*
3
?
+
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G
?
b
c
*
?
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?
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*
3
?
++
?
?
*
E
2
?
+?/?
?
6
?
?
?
?
?
?
D
7
?
?
*
?
?
?
+
?
?
*
?
6
+
8
*
E
2
?
+
?/
?
?
*
E
2
?
+
?
/?
?
6
?
?
?
:
?B
?
?
'
?
:
?
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D
7
?
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*
?
?
?
+
?
?
*
?
6
+
8
*
E
2
?
+
?
/
?
?
6
?
?
?
?
?
'
?
??
?
?
D
?
?
?
*
?
?
?
?
:
?B
?
?
'
?
:
?
?
?
'
?
?
?
+
?
?
*
?
6
+
?
with
followingdecreasing function of
?
hjk
?
G
?
b
%
?
b
c
and
?
'
?
hjk
?
<
#
%
<>
% and where
?
?and
?
?are the
? :
?
?
*
?
+
hjk
?
?
Q
3
46
M
?AC
Q
CE
?
?
Q
F
b
b
HIJL
*
E
4
6
?
+
N
E
?
?
?
*
?
+
hjk
?
?
Q
3
4
6
M
?
AC
Q
C
?
Q
F
b
b
HIJL
*
E
4
6
?
+
N
E
?
Consequently the following explicit expressions for the
CRB for the parameter DOA alone are easily derived:
OPR
????
*?
?+
?
?
?
S
?
U
?
?
?
D
?
?
?
V
S
?
?
?
?
?
*
?
+
V
(1)
OPR
0
???
*?
?+
?
?
?
S
?
U
?
?
?
D
?
?
?
V
S
?
?
*
?
?
?
+
?
?
*
3
?
+
?
?
*6
?
?
+
?
?
*
3
?
+
?
*
?
?
?
+
?
?
?
*
3
?
+
V
(2)
where
U
?is the purely geometrical factor
In the absence of phase offset or after correcting it (i.e.
parameter
6
?
'
?
B
WX
#
%
?
'
?.
=
?known), these CRB’s for
?
?become
OPRZ\
????
*?
?
+
?
?
?
S
?
6?
?
'
?
?
?
?
?
D
?
?
?
V
S
?
?
?
?
?
*
?
+
V
OPRZ\
0
???
*?
?+
?
?
?
S
?
6?
?
'
?
?
?
?
?
D
?
?
?
V
?a

?
?
?
*
?
?
3
G
!#
$
%
#
'
%
!
b
*#
'
%
*
b
+
?
?
*
3
?
+
-b
/
?
Because the BPSK [resp. QPSK] modulation is non-
circular [resp.circular] complex to the second-order, it
makes sense to compare the stochastic CRB’s (1) and (2)
to the CRB’s associated with respectively non-circular1
(NCG) [3] or circular (CG) complex Gaussian distribution
that can be considered as upper bounds on the true stochas-
ticCRB’s(see e.g., [2, p. 293]). More precisely, afterrecall-
ingthese CRB’sunderGaussian distributionsforthe conve-
nience of the reader
OPRc
Zd
*?
?+
?
?
?
S
?
U
?
?
?
?
D
?
?
?
?
?
6
?
?
?
D
?
?
?
?
V
OPR
Zd
*?
?+
?
?
?
S
?
U
?
?
?
?
D
?
?
?
?
?
?
?
?
D
?
?
?
?
V
?
we have
OPR
????
*?
?
+
OPR
c
Zd
*?
?
+
?
?
*
?
?
?
?
*
?
++
*
?
?
?
?
3
+
OPR
0
???
*?
?+
OPR
Zd
*?
?
+
?
?
?
*
?
?
?
+
?
?
*
3
?
+
*
?
?
*6
?
?
+
?
?
*
3
?
+
?
*
?
?
?
+
?
?
?
*
3
?
++
*
?
?
?
3
+
?
Wenotethat theseratiosdependon
to 1 when
not monotone as it is numerically shown in the next section.
?
hjk
?
G
?
b
%
?
b
c
onlyandtend
? tends to
e
. However this dependence in
? is
1Because
the non-circular complex Gaussian distribution associated with
fgiklmnfoiklo for the BPSK modulation, we consider
fgi
k
l
mn
foi
k
lons .
II - 74
¬¬

Page 3

4. NUMERICAL EXAMPLES
The purpose of this section is to illustratethe results of sec-
tion 3 and to extend them to the case of two independent
BPSK distributedsources. We consider throughoutthissec-
tion, one or two independent sources impinging on a uni-
form linear array of
sensors spaced a half-wavelength
apart for which
?
?
?
?
*
??
?!
>
?
?????
?!
?
G
Q
?
?
>
?
+
2
.
4.1. Single source case
The first experiment illustrates the results of section 3.
Fig.1 shows the ratios
hjk
Z
?
?
????
?
>
%
?
Z
?
?
???
?
>
%
?
and
Z
?
?
?
???
?
>
%
?
Z
?
?
??
?
>
%
?
as
a function of
ure that the CRB’s under the non-circular [resp.
cular] complex Gaussian distribution are tight upper
bounds on the CRB’s under the BPSK [resp.
distribution at very low and very large SNR’s only.
?
?
G
?
b
%
?
b
c
.We see from that fig-
cir-
QPSK]
−10−5051015202530
0.85
0.9
0.95
1
ρ (dB)
r1(θ1)
QPSK
BPSK
Fig.1 Ratios
??
g
??
m
???
n
? "
????$%
%
'
? "
???
$%
%
'
and
??
g
??
m
???
n
? "
?
???$%
%
'
? "
??
$%
%
'
as a function of
/
???
n
12
b
%
2
b
c
.
4.2. Two sources case
We consider now two independent BPSK distributed
sources. Because the PDF of
sian PDF’s, the associated stochastic CRB appears to be
prohibitive to compute. Consequently we use a numerical
approximation derived from the strong low of large num-
bers, i.e.
?
?is a mixture of 4 Gaus-
OPR
????
*?
?
?
?
?
+
?
4
E
Q
?
67
?
?
9
?
/?
9
?
?
with
?
?
*
E
6
+
?
/
P
?
:;=
2
'
>
C
?
?
2
2
'
O
?,?
S
?
:@
H
*
?
?IK+
?
?
?
V
S
?
:@
H
*
?
?IK+
?
?
P
V
(3)
where
H
*
?
?IK+
?
?
d
M
G
?
?
G
D
?
O
A
,?
?Q
SUWX
BDE
S
b
Z
bc
with
F
A
hjk
?
*
?
?
G
A
/?
?
!#
%
??
?
G
A
/
?
?
!#
b
+
2
where
*
G
?/?
?
G
?/
?
+
?
*
?
??
?
?
+,
*
G
?
/?
?
G
?
/
?
+
?
*
?
??
?
?
+,
*
GH
/?
?
GH
/
?
+
?
*
?
??
?
?
+,
*
G
?
/?
?
G
?
/
?
+
?
*
?
??
?
?
+and where
K
hjk
?
*
?
D
??
?
?
=
?
?
?
?
??
?
?
=
?
?
?
?
+
2
and
J
hjk
?
*
?
?
?
?
?
+. At
high SNR’s (more precisely for
an explicit expression of the FIM can be derived by condi-
tioning the derivative
different couples
bols. The followingexpression is proved in [6]:
?
?
b
%
?
b
c
K
?and
?
?
b
b
?
b
c
K
?)
LMNO
?Q
W
RS
?
L
>
?
,
T
???????
U
w.r.t. the
of sym-
*
'?/?
?
'?/
?
+
?
*
G
P
/?
?
G
P
/
?
+
P
,?/000/
?
E
????
?
??
VW
?
G
?
b
c
X
2
X
2
X
E
?
?
X
?
E
?
YZ
(4)
with
E
?
?
VW
?
G
?
b
?
?
b
c
?
?
b
?
!#
$
?
#'
?
!
?
b
c
?
?
b
?
!#
$
?
#
'
?
!
?
b
c
?
?
b
?
*#
'
?
*
b
?
b
c
YZ
,
T
???
6 .
We clearly see that the entries correponding to sources 1
and 2 are decoupled. Consequently, for large SNR’s and
independent sources, the CRB for the DOA of one source is
independent of the parameters of the other source and
OPR
????
*?
?
?
?
?
+
?
?
?
VW
?
[
%
?
b
c
?
b
%
\
\
?
[
b
?
b
c
?
b
b
YZ
?
Furthermore,theCRB’s
for each DOA are those of the single source case. We note
that this property is quite different from the behavior of the
CRB under the Gaussian distributionand the deterministic
CRB, forwhich the CRB for the DOA of one source depends
on the DOA of the other source. More precisely, it is proved
[1, result R9]that these twoCRB’stendtothesame limitas
all SNR’s increase. For independent sources, they are given
by
OPR
????
*?
?
+and
OPRZ\
????
*?
?
+
OPR
Zd
*?
?
?
?
?
+
?
?
?
VW
?
]
%
?
b
c
?
b
%
\
\
?
]
b
?
b
c
?
b
b
YZ
with
^
?
hjk
?
6
7
?
?
'
?
??
?
_
?
*?
?
?
?
?
+
8,
T
???
6 , where
_
?
*?
?
?
?
?
+
hjk
?
?
*
:
?
'
?
B
?
?
:
?
?
:
?
'
?
B
?
H
Q
?
:
?
+
?
6
a
*
?
B
?
?
'
?
?
B
H
Q
?
?
?
?
'
?
B
?
H
Q
?
+
?
?
?
:
?B
?
?
?
:
?
?
The second experiment considers two independent and
equipowered BPSK distributed sources. Fig.2 compares
OPR
????
*?
?
+given by (3) with the CRB under the non-
circular complex Gaussian distribution. And to be fair,
this comparison must be done under the same a priori that
II - 75
¬¬

Page 4

the two sources are independent. For that reason, we use
the expression of the CRB obtained in [4] which can take
this a priori information into account. Fig.2 exhibits the
ratio
as a function of the DOA separation
for two values of the circularity phase separation
?
Z
?
?
????
?
>
%
?
Z
?
?
???
?
>
%
?
?
?
?
?
?
?
=
hjk
=
?
?
=
?. This figure shows, that contrary to the sin-
gle source case, the CRB under the non-circular complex
Gaussian distribution is a very loose upper bound on the
CRB under the BPSK distribution except for large values
of the DOA and phase separation. Consequently, maximum
likelihood (ML) solutions such as the EM approaches [5]
outperform the traditional ML estimator under the Gaus-
sian distributionspecifically for small DOA and phase sep-
aration.
00.10.2 0.30.4 0.50.6
10−4
10−3
10−2
10−1
100
DOA separation (rd)
r2(θ1)
∆φ=0.1 rd
∆φ=0.02 rd
∆φ=0.3 rd
Fig.2 Ratio
tion for different values of the circularity phase separation
and
Fig.3exhibits
the DOA separation for two SNR’s.
contraryto
notincreasesignificantly
DOA separation.
??
?
k
g
??
m
???
n
? "
????$%
%
'
? "
???
$%
%
'
as a functionof the DOA separa-
?? for
?
n
?
???
n
???? .
OPR
????
*?
?
+
asafunction of
We see that
OPR
c
Zd
*?
?
+,
OPR
????
*?
?
+
does
thewhendecreasing
This explains the behavior of
for low DOA separations.
?
*?
?
+
hjk
Z
?
?
????
?
>
%
?
Z
?
?
???
?
>
%
?
0.050.10.15 0.2 0.250.3 0.35 0.4
10−3
10−2
10−1
DOA separation (rd)
CRB(θ1)
5dB
12dB
Fig.3
and
???
"
???
g
??
m as a function of the DOA separation for
?
n
?
??
n
??
s
?
? .
Finally, Fig.4 exhibits the domain of validity of the
high SNR approximation (4). We see from this figure that
this domain depends not only on
?
and SNR, but also
on the DOA separation, e.g.
old is about 4dB for
for
?
?!, the thresh-
?
?
?
\
???
N
and 8dB for
?
?
?
\
?
\
$
?
N. The larger the DOA separation is or the larger
is, the larger the domain of validityof the approximationis.
?
051015
10−3
10−2
10−1
100
101
SNR (dB)
CRB(θ1)
exact (∆θ=0.05rd)
exact (∆θ=0.1rd)
approximate (SNR>>1)
x
x
x
x
M=3
M=6
Fig.4
the SNR for different values of the DOA separation.
Approximate and exact value of
???
"
???
g
??
m as a function of
5. CONCLUSION
In this paper, we have proved that for a single source,
the CRB’s under the non-circular [resp. circular] complex
Gaussian distribution are tight upper bounds on the CRB’s
under the BPSK [resp. QPSK] distributionat very low and
very large SNR’s only. For two independent BPSK sources,
the CRB under the non-circular Gaussian distribution is a
very loose upper bound on the CRB under the BPSK dis-
tribution. And the difference between these CRB’s is more
prominent for small DOA and phase separation. Furthe-
more for high SNR’s, the CRB for the DOA of one source
is independent of the parameters of the other source.
6. REFERENCES
[1] P. Stoica, A. Nehorai, “Performance study of conditional and
unconditionaldirection of arrival estimation,” IEEE Trans. on
Acoustics Speech and Signal Processing, vol. 38, no. 10, pp.
1783-1795,Oct. 1990.
[2] P. Stoica, R. Moses, “Introduction to spectral analysis,”
Prentice-Hall, Upper Saddle River, NJ, 1997.
[3] H. Abeida, J.P. Delmas, “Stochastic Cramer-Rao bound for
non-circular signals with application to DOA estimation,”
submitted to IEEE Trans. on Signal Processing,2003.
[4] J.P. Delmas, “Asymptotically optimal estimation of DOA
for non-circular sources from second-order moments,” Proc.
ICASSP, Hong Kong, pp. 185-188, April 2003.
[5] M. Lavielle, E. MoulinesandJ.F. Cardoso,“Amaximum like-
lihoodsolutionto DOAestimationfor discretesources,” Proc.
Seventh IEEE Workshopon SP, pp. 349-353, 1994.
[6] H. Abeida, J.P. Delmas, “Cramer-Rao bounds of DOA esti-
mates for BPSK and QPSK modulated signals,” submited to
IEEE Trans. on Signal Processing,2003.
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