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Noise analysis for a new digital beamformer onreceiveonly
Sheng Hong*, Dongkai Yang**, Kefei Liu, Shuliang Gao, Huagang Xiong, Qishan Zhang
School of Electronic and Information Engineering
Beihang University
Beijing, China
*fengqiao1981@gmail.com
** edkyang@buaa.edu.cn
Abstract—This paper provides noise analysis for a new kind of
digital beamformer onreceiveonly which uses pseudorandom
spreading sequences as spreading sequence weights. It
investigates the noise influence on the magnitude and phase of
the array correlation output. Aiming at the corruption of white
additive noise, we respectively adapt Minimal Error
Probability (MEP) detection to evaluate the array correlation
output magnitude and combine Maximal likelihood estimation
(MLE) with Minimum Distance Decision (MDD) to evaluate
the array correlation output phase. From the simulation, we
succeed in detecting the magnitude of the array correlation
output and estimating its phase within noise environment.
Keywords Digital beamformer, Noise analysis, Detection
and Estimation
I.
INTRODUCTION
The digital beamforming has overwhelming advantages
over analog one such that it saves many attenuators and
phase shifters. But it has deficiencies like complicated and
timeconsuming adaptive digital beamforming algorithm.
Those algorithms are of no effect in case it can not be
evaluated by the hardware faster than the users’ motion.
Aiming at the defects of previous smart antennas, Frank
B.Gross and Carl M. Elam[1] propose a new digital
beamformer onreceiveonly adapting spreading sequences
as weights. It can process multiple angles of arrival
simultaneously without adaptive algorithm, and it is not
limited by acquisition or tracking speeds. It also doesn’t need
analog attenuators and phase shifters. In Frank’s paper, it
doesn’t account for noise influence for this new beamformer,
nor does it tell how to extract array correlation output
information among the noise background. In this paper we
make research on the influence of noise and deduct the
formula of the array correlation output. And we adapt
Minimal Error Probability (MEP) detection to evaluate the
array correlation output magnitude and combine Maximal
likelihood estimation (MLE) with Minimum Distance
Decision (MDD) to evaluate the array correlation output
phase.
The remainder is provided as following parts. II shows
the principle for digital beamformer onreceiveonly. III
demonstrates the noise influence to array correlation output.
IV shows detection and estimation of array correlation
output. V is the conclusion.
II.
PRINCIPLE FOR BEAMFORMER ONRECEIVEONLY
The generic beamformer onreceiveonly [1] is
described as Fig.1, wherein the array weights
orthogonal spreading sequences.
Without loss of generality and for the proof of concept,
( )
nt
β
are
the array used in this discussion will be an Nelement linear
array. The incoming signals arrive at angles
2 . . . L. The total spread array output is called the received
signal vector and is given by
( )( )( )y t tt
= βx
(1)
( ) [( ),,( )]
N
ttt
ββ=
β
?
Corresponding to the actual Nelement antenna array is a
second virtual array modeled in memory. The memory array
is based on the calibrated array and has N virtual outputs for
each expected direction
(
θ
= ?
memory steering vectors are created based upon K expected
anglesofarrival
=
Aaa
?
(2)
where
A is the matrix of steering vectors for expected
direction
direction
lθ where l = 1,
rTr
where
1
T
,
1
( ) t[ ( ),x t,( )]t
rT
N
x
=
x
?
.
1,,)
kkK
. The array signal
k θ as
]
1
[,,
eee
K
e
k θ and
k θ . The memory has K complex output waveforms,
e
ka is the steering vector for expected
Fig. 1 Beamformer onreceiveonly with spreading sequences as array
weights.
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one for each expected direction
given by
( ) ( )
kk
tt
=
yβa (3)
According to reference [1], the best correlations occur
when the actual angle of arrival matches the expected angle
of arrival. The correlation can be used as a discriminate for
detection. The general complex correlation output, for the k
th expected direction, is given as
t T
re
kkk
t
∫
where
k θ . Each memory output is
eTe
*
( )( ) t dt
k
j
φ
R y t yR e
+
=⋅=
(4)
k R is the complex correlation at expected angle
kφ the average correlation phase at expected angle
k θ ，each chip is of length
length
TMτ=
(5)
k θ
and
cτ and the entire sequence is of
c
The correlation magnitude 
discriminate to determine if a signal is present at the
expected angle
. If the discriminate exceeds a
predetermined threshold then a signal is deemed present and
the phase is calculated. Since it is assumed that the emitter
phase modulation (PM) is nearly constant over the code
length ()
c
Mτ
, the correlator output phase angle is
approximately the average of the emitter PM. Thus,
arg()
kkk
Rm
φ =≈ ? (6)
where
1/( )
kk
t
∫
modulation at angle

k R
is used as the
k θ
t T
+
m
?
T m t dt
=
, the average of emitter’s
k θ .
III. NOISE INFLUENCE TO ARRAY CORRELATION OUPUT
A. Magnitud and Phase Distortion of Array Correlation
Output
We investigate the equivalent baseband noise influence
to the array correlation output. When there is a signal
coming at direction
among L different incoming
()
L
θθ
…
, we choose a virtual array vector at
direction
vector and the realistic array output,
t T
re
kk
t
e
Men
e
M Nen
=⋅+
−
++⋅
−
k θ
directions
1
k θ and want to see the correlation function of that
(sin sin)
( )
i
0
(sinsin)
1
(
( )
i
t
0
2 /
π λ
(sin sin)
( )
i
2 /
π λ
(sinsin)
1,
( ) ( )t dt
1
()
1
)
1
()
1
lk
l
lk
k
lk
l
lk
j k d
⋅ ⋅ ⋅
N
L
jm t
l
j k d
⋅ ⋅ ⋅
l
jm
k
jdN
L
jm t
l
jd
l
l k
≠
Ry ty
e
en
e
θθ
θθ
θθ
θθ
+
∗
−⋅
−
=
⋅⋅ ⋅−⋅
⋅ ⋅ ⋅−
=
=⋅
−
=⋅+⋅
−
∫
∑
∑
(7)
wherein
function of a spreading sequence, N is the number of array
elements and
lclsl
nNjN
=+
is the complex Gaussian noise
with inphase component
N
component
sl
N . In the case above, it can be easily deducted
that the fist term of the equation is far larger than the second
term. Hence, the formula can be described as follows,
( )
0
()
kk
R M Nen
≈⋅+
(8)
Assuming there are two signals coming at 450 and 00,
when they are received in the receiver, they are added with
Gaussian noise and suffer output distortion. As seen from
Fig. 2 and Fig. 3, the array correlation output for signals at
450 and 00 suffer a magnitude and phase distortion, when there
is Gaussian noise with variance 1.
0
M
is the absolute value of autocorrelation
cl
and Quadrature phase
ki
jmt
Fig. 2 Magnitude of array correlation output without/with noise
Fig. 3 Phase of array correlation output at 00without/with noise, where the
solid line means original signal and the dashed line means the received
signal.
The magnitude of array correlation output is polluted
by the noise and phase of array correlation output is also
distorted, either. The way should be found to extract the
useful information from the noise environment.
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IV. DETECTION AND ESTIMATION OF ARRAY
CORRELATION OUTPUT
A. Minimal Error Probability Detection
Aiming at the white Gaussian noise influence to the
array, we use classical Bayes detection Minimal Error
Probability (MEP) to judge the array correlation output
magnitude. Assume that under
e± with complex noise and under
zero with complex noise. It can be seen that the observations
under the two hypotheses are
:(
:()
kcs
HR M N NjN
=+
where
c
N and
s
N are respectively inphase and quadrature
components of white Gaussian noise with zeromean and
variance
σ .
2
k R
S
M N σ
probability density functions are respectively list as follows
for hypotheses
1
H and
0
H .
(
1
1
2
exp
22
σ
⎝
(
0
0
22
⎝⎠
where
I ⋅ is modified Bessel function of first kind with
order zero.
1
H the
k R is a constant
H the
voltage
j
0
k R is
10
00
)
j
kcs
HRM N eNjN
±
=++
(9)
2
Choose
222
0
=
, then according to [3],
)
2
0
2
11
()
s H
ss
p S HI
σ
σ
⎛
⎜
⎞
⎟
⎠
+
=−
(10)
)
1exp
s H
s
p S H
⎛
⎜
⎞
⎟
=−
(11)
0()
Set priori probabilities
10
1
2
pp
==
, and we assume
throughout this paper that the cost of a wrong decision is 1
and the cost of a correct decision is 0. According to [4], it
has
(
(
0
0
s H
pS H
When
( ) 1S
Λ> ,
1
H is true. Otherwise, when
H is true. When ( ) 1S
Λ= , we can say either is true of
and
0
H .
Apparently, the
( )S
Λ
function with variant S . It can be easily got the function
relationship between
( )S
Λ
and S as follows Fig. 4,
wherein the noise variance is 1.
Firstly, assume
asset
1
H is true when
0
SS
>
SS
<
.
When the same signal is transported in the AWGN
channels with different noise variance
)
)
1
1
( )
S
s H
pS H
Λ=
(12)
( ) 1S
Λ< ,
H
01
is a monotone increasing
0 S satisfying
0
()1
H is true when
S
Λ= . Then we can
. Otherwise
0
0
2
σ , the decision
threshold
conveniently, we calculate the function relationship between
0 S and SNR (Eb/N0) as follows. From Fig. 5, we can
choose different thresholds according to different SNRs.
0 S
changes with the
2
σ
. To illustrate
Fig. 4 The function relationship between
( )S
Λ
and S
Fig. 5 The function relationship between
With the effort of the MEP detection, we can get the
Error Probability curve when we adapt different SNR for
this detection. As seen from Fig. 6, the errordetection
probability decreases with the increasing SNR.
0 S and SNR (Eb/N0)
Fig. 6 The errordetection probability Pe versus SNR.
B. Maximal Likelihood Estimation
The phase information of array correlation output
reflects the average of emitter’s modulation
However, the phase is also corrupted by the noise. We adapt
Maxim Likelihood Estimation (MLE) method and Minimal
)(tml
.
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Distance Decision (MDD) to extract the phase information
from the noise background.
Suppose that
)(tml
is a sequence with +1 and 1 chips.
We make hypotheses that the
e+ under
1
H and the
Before observation the voltage is corrupted by an additive
noise. It can be seen that the observations under the two
hypotheses are
:(
kc
HRM N eN jN
HR M N eN jN
=++
where
c
N and
s
N are respectively inphase and quadrature
components of white Gaussian noise with zeromean and
variance
σ .
According to [5], we estimate the phase information by
using MLE sufficient statistic as follows.
Im()
arctan
Re( )
k
R
⎣⎦
Hence, the two hypotheses are represented as
sin1
: arctan()
cos1
sin1
: arctan()
cos1
c
N
+
Easily known, the probability functions of tan(
under hypotheses
1
H and
0
H are symmetric axially with
0
k
m =
. The probability functions of tan(
hypotheses
1
H is centralized on the side of
However, The probability functions of tan(
hypotheses
0
H is centralized on the side of
we use the Minimal Distance Decision to discriminate
H by comparing
k
m and 0. We can get that
true when
0
k
m >
. Otherwise, when
When
0
k
m =
, we can say either is true of
k R source output is a constant
j
k R source output is
j
e− under
0
H .
10
00
)
:()
j
s
j
kcs
+
−
=++
(13)
2
k
k
R
m
Λ
⎡
⎢
⎤
⎥
=
(14)
1
0
s
k
c
N
s
k
N
N
+
Hm
Hm
+
+
=
−
=
(15)
)
k
m
)
m >
)
under
0
. Hence,
k
m
under
0
k
.
k
m
k
m <
1
H
and
01
H is
0
k
m <
,
0
H .
H is true.
1
H and
0
Set priori probabilities
10
1
2
pp
==
, and we can obtain
the error probability of decision by theoretical method as
follows.
()
1[
(sin10)(cos1
2
(sin1 0)(cos1
1[
( sin10) (cos1
2
( sin1 0) (cos1
sin1 cos1sin1cos1
( ) ()() (QQQQ
σσσ
101110
()
0)
0)]
0)
0)]
)
err
rSrc
rSrc
rSrc
rSrc
Pp p say HH is true p p say HH is true
pNpN
pNpN
pNpN
pNpN
σ
=+
=+>+<
++<+>
+−+>+>
+−+<+<
=−+−
(16)
wherein
2
2
( )
α
1
2
y
Q
edy
α
π
+∞−
=∫
.
We utilize MonteCarlo method to verify our joint
method with estimation and decision. From Fig. 7, we can
see the simulation result is quite fit for the theoretical curve,
which verifies our deduction.
Fig. 7 Comparison between simulation result and theoretical curve on bit
error rate versus SNR.
V.
CONCLUSION
This paper provides noise analysis for a new kind of
digital beamformer onreceiveonly. We investigate the
noise influence on the magnitude and phase of the array
correlation output and abstract the issue to some specific
detection, estimation and decision problems. Pointing to the
Gaussian noise influence, we adapt Minimal Error
Probability (MEP) detection to evaluate the array correlation
output magnitude. And we combine Maximal likelihood
estimation (MLE) with Minimum Distance Decision (MDD)
to evaluate the array correlation output phase. From the
simulation, it certifies our detection, estimation and decision
method correct and feasible.
ACKNOWLEDGMENT
This paper is supported by China National Science Fund
with No.60872062 and 863 National Hightech Research and
Development Program with No.2007AA12Z340.
REFERENCES
[1] Frank B. Gross, Carl M. Elam, “A new digital beamforming approach
for SDMA using spreading sequence array weights,” Signal
Processing, v 88, n 10, Oct. 2008, pp. 24252430
[2] Haykin, Simon, Communication Systems, second ed., Wiley, New
York, 1983, pp.580.
[3] Yinqing Zhou, Random process theory 2ed edition[M], Beijing:
Publishing House of Electronics Industry 2006.10,pp.175p178
[4] Van Tree,Harry L, Detection, estimation, and modulation theory[M],
USA, John Wiley & Sons, Inc, 2001, pp.4252
[5] Kay S M, Fundamentals of Statistical Signal Processing: Estimation
Theory［M］. Beijing: Publishing House of Electronics Industry
2006,pp.427428.
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