Pulse or digital communications – Spread spectrum – Direct sequence
Reexamination Certificate
2002-04-22
2004-04-06
Vo, Don N (Department: 2631)
Pulse or digital communications
Spread spectrum
Direct sequence
Reexamination Certificate
active
06717979
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention concerns a method for estimating the direction of arrival of a reference signal emitted by a signal source. In particular, the present invention can be used for estimating the direction of arrival of a signal emitted by a mobile terminal.
2. Discussion of the Background
Before reviewing the state of the art in the field of DOA (direction of arrival) estimation, the technique of passive beamforming will be shortly introduced.
An adaptive antenna generally comprises an antenna array and a beamformer as shown on FIG.
1
. The antenna
100
may have an arbitrary geometry and the elementary sensors
100
1
, . . . ,
100
L
may be of an arbitrary type. We consider an array of L omnidirectional elements immersed in the far field of a sinusoidal source S of frequency ƒ
0
. According to the far-field condition we may consider a plane wave arriving from the source in direction (&PHgr;
0
,&thgr;
0
). The first antenna is arbitrarily taken as the time origin. The travelling time difference between the l th element and the origin is given by:
τ
l
(
ϕ
0
,
θ
0
)
=
<
r
i
→
,
u
→
(
ϕ
0
,
θ
0
)
>
c
(
1
)
where {right arrow over (r)}
l
is the position vector of the l th element, {right arrow over (u)}(&PHgr;
0
,&thgr;
0
) is the unit vector in the direction (&PHgr;
0
,&thgr;
0
), c is the speed of propagation, and <, > represents the inner product. For a uniform linear array (ULA) i.e. a linear array of equispaced elements, with element spacing d aligned along the X-axis and the first element located at the origin (1) may be expressed as:
τ
l
(
θ
0
)
=
d
c
(
l
-
1
)
cos
θ
0
(
2
)
The signal received from the first element can be expressed in complex notation as:
m(t)exp(j2&pgr;f
0
t) (3)
where m(t) denotes the complex modulating function.
Assuming that the wavefront on the l th element arrives &tgr;
l
(&PHgr;
0
,&thgr;
0
) later on the first element, the signal received by the l th element can be expressed as:
m(t)exp(j2&pgr;ƒ
0
(t+&tgr;
l
(&PHgr;
0
,&thgr;
0
))) (4)
The expression is based upon the narrow-band assumption for array signal processing, which assumes that the bandwidth of the signal is narrow enough and the array dimensions are small enough for the modulating function to stay almost constant over &tgr;
l
(&PHgr;
0
,&thgr;
0
), i.e. the approximation m(t)≅m(t+&tgr;
l
(&PHgr;
0
,&thgr;
0
)) is valid. Then the signal received at the l th element is given by:
x
l
(
t
)=
m
(
t
)exp(
j
2
&pgr;ƒ
0
(
t+&tgr;
l
(&PHgr;
0
,&thgr;
0
)))+
n
l
(
t
) (5)
where n
l
(t) is a random noise component, which includes background noise and electronic noise generated in the lth channel. The resulting noise is assumed temporally white Gaussian with zero mean and variance equal to &sgr;
2
.
Passive beamforming consists in weighting the signals received by the different elements with complex coefficients &ohgr;
l
and summing the weighted signals to form an array output signal. By choosing the complex coeffcients, it is possible to create a receiving pattern exhibiting a maximum gain in the direction &thgr;
0
of the source. The array output signal can be expressed as the product of m(t) and what is commonly referred to as the array factor F:
F
(
θ
)
=
∑
l
=
1
L
ω
l
exp
(
j
(
l
-
1
)
κ
d
cos
θ
)
(
6
)
where &kgr;=2&pgr;/&lgr; is the magnitude of the so-called wave-vector and &lgr; is the wavelength of the emitted signal. If we denote the complex weight &ohgr;
l
=&rgr;
l
exp(j&phgr;
l
), the array factor can be written:
F
(
θ
)
=
∑
l
=
1
L
ρ
l
exp
(
j
(
φ
l
+
(
l
-
1
)
κ
d
cos
θ
)
(
7
)
If we choose &phgr;
l
=−(l−1)&kgr;d cos &thgr;
0
, the maximum response of F(&thgr;) will be obtained for angle &thgr;
0
, i.e. when the beam is steered towards the wave source.
We suppose now that the array factor is normalized and that &rgr;
l
=1/L. If we consider an arbitrary direction of arrival &thgr;.
F
(
θ
)
=
1
L
∑
l
=
1
L
exp
(
j
κ
d
(
l
-
1
)
(
cos
θ
-
cos
θ
0
)
)
(
8
)
that is, by denoting &psgr;=&kgr;d(cos &thgr;−cos &thgr;
0
):
F
(
ψ
)
=
1
sin
(
L
ψ
/
2
)
L
sin
(
ψ
/
2
)
exp
(
j
(
L
-
1
)
ψ
/
2
)
(
9
)
If we denote w=(&ohgr;
1
, . . . , &ohgr;
L
)
T
the vector of the weighting coefficients used in the beamforming, the output of the beamformer can be simply expressed as w
H
x. The vector w for steering the beam in the look direction &thgr;
0
is w=a(&thgr;
0
) where a(&thgr;
0
)=[1,exp(j &kgr;d cos &thgr;
0
), . . . , exp(j(L−1)&kgr;d cos &thgr;
0
)]
T
. If R
N
=&sgr;
2
I is the covariance matrix of the uncorrelated noise, the power of the noise component at the array output may be written:
P
N
=
w
H
R
N
w
=
σ
2
L
(
10
)
In other words, the noise power at the array output is 2/Lth the noise power present at each element. Thus, beamforming with unity gain in the signal direction has reduced the uncorrelated noise by a factor L and thereby increased the output signal to noise ratio (SNR).
Turning now to a more general case where M point sources are present in the far field, the signal received by an element l can be written:
x
l
(
t
)
=
∑
m
=
1
M
s
m
(
t
)
exp
(
j
(
l
-
1
)
π
cos
θ
m
)
+
n
l
(
t
)
(
11
)
If we consider the sampled signals at sampling times nT, n={1, . . . , N}, denoted n for sake of simplification (11) can be rewritten:
x
(
n
)
=
∑
m
=
1
M
a
(
θ
m
)
s
m
(
n
)
+
n
(
n
)
(
12
)
where a(&thgr;
m
)=[1,exp(j&pgr; cos &thgr;
m
), . . . , exp(j(L−1)&pgr; cos &thgr;
m
)]
T
is a vector called the array response, x(n) is the vector of the received signals at time n and n(n)=[n
1
(n),n
2
(n), . . . , n
L
(n)]
T
is the noise vector. The sampled array output x(n) can be expressed as a matrix product:
x
(
n
)=
As
(
n
)+
n
(
n
) (13)
where A=[a(&thgr;
1
),a(&thgr;
2
), . . . , a(&thgr;
M
)] is the L×M matrix the columns of which are the vectors a(&thgr;
m
) and s(n)=[s
1
(n),s
2
(n), . . . , s
M
(n)]
T
is the signal vector.
We suppose that the signals and noise samples are stationary and ergodic complex-valued random processes with zero mean, uncorrelated with the signals and uncorrelated each other. They are modeled by temporally white Gaussian processes and have identical variance &sgr;
2
.
Most of the DOA estimation techniques are based upon the calculation of an estimate of the spatial covariance matrix R:
R
=
E
[
x
(
n
)
x
H
(
n
)
]
=
lim
N
→
∞
1
N
∑
n
=
1
N
x
(
n
)
x
H
(
n
)
(
14
)
which can be rewritten according to the matrix notation of (13):
R=APA
H
+&sgr;
2
I
(15)
where P is the source covariance matrix and I is the identity matrix.
Not surprisingly, since R reflects the (spatial) spectrum of the received signal and the directions of arrival are obviously linked with the peaks of the spectrum, most of the DOA estimation techniques make use of the spectral information contained in R.
The simplest; DOA estimation technique (also called conventional DOA estimation) merely amounts to finding the peaks of the spatial spectrum:
P
B
F
=
a
H
(
θ
)
R
a
(
θ
)
L
2
(
16
)
Brunel Loïc
Ribeiro Dias Alexandre
Mitsubishi Electric Information Technology Centre Europe B.V.
Oblon, Spivack, McClelland, Maier & Neustadt, P.C.
Vo Don N
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