Tài liệu Digital Signal Processing Handbook P69 ppt

Barroso, V.A.M. & Moura, J.M.F. “Beamforming with Correlated Arrivals in Mobile Communications”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
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69
Beamforming with Correlated
Arrivals in Mobile
Communications
1
Victor A.N. Barroso
Instituto Superior T
´
ecnico,
Instituto de Sistemas e Rob
´
otica
Jos
´
e M.F. Moura
Carnegie Mellon University
69.1 Introduction
69.2 Beamforming
Minimum Output Noise Power Beamforming (MNP)
69.3 MMSE Beamformer: Correlated Arrivals
69.4 MMSE Beamformer for Mobile Communications
Modelof the ArrayOutput
•
MaximumLikelihoodEstimation

of
H
69.5 Experiments
69.6 Conclusions
Acknowledgments
References
69.1 Introduction
The classical definition of a beamformer basically specifies its goal: to estimate the signal waveform
arriving at the array from a given direction. Beamformers are spatial processors that combine the
signals impinging on an array of captors. Combining the outputs of the captors forms a narrow beam
pointing towards the direction of the source (look direction). This narrow beam can discriminate
between sources spatially located at distinct sites. This important property of beamformers is used
to design techniques that localize active or passive sources particularly in RADAR/ SONAR systems.
In the last two decades, beamforming methods have had significant theoretical and practical
advances. This, together with other technological advances, has broadened the application of so-
phisticated beamforming techniques to a diversity of areas, including imaging, geophysical and
oceanographic exploration, astrophysical exploration, and biomedical. See [19, 20] for an excellent
overview of modern beamforming techniques and applications.
Communications is another attractive application area for beamforming. In fact, beamforming
has been widely used for directional transmission and reception as well as for sector broadcasting
in satellite communications systems. More recently, due to the drastic increase of users in cellular
1
Initial date of submission of this article September 28, 1995.
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radio systems [10, 15, 18], including indoors and outdoors mobile systems, it is increasingly being
recognized that the design of base station and mobile antennas based on beamforming methods
improves significantly the system’s spectrum efficiency [1, 11]. In turn, this enables accommodating
larger numbers of users [3, 16]. The most striking argument in favor of using advanced beamforming

techniques suchasadaptive or blind beamforming for mobile communicationsisbasedonthe idea of
Space Division Multiple Access(SDMA) schemes. With SDMA, several mobiles share simultaneously
the same frequency channel by creating virtual channels in the spatial domain. Another important
argument in favor of using beamforming in cellular radio is that beamforming yields flexible signal
processingschemesthat properlyhandlemultipatheffects whicharetypicalin radio communications.
Multipath is the term given when the same signal arrives at the destination through different paths.
This may arise when signals bounce off obstacles in their path of propagation. At the receiver,
these arrivals are correlated. Their recombination causes severe signal distortions and fading. In
limiting cases, the power of the received signal can become so small that the reliability of the data
communications link is completely lost.
In this section we design a multichannel beamformer to combat multipath effects. The receiver
uses a base station antenna array which handles several radio links operating simultaneously at the
same carrier frequency, while preserving the reliability of the communications. The approach relies
on statistical signal processingmethods, yielding a solution that operates in a blind mode with respect
to the parameters that specify the propagation channel. This means that, except for a few quantities
related to system specifications, e.g., link budget and array geometry, the receiver that we describe
here does not assume any prior knowledge about the locations of the sources and of the structures
of the ray arrivals, including directions of arrival and correlations. The simulation results show the
excellent performance of this multichannel beamformer in SDMA schemes.
The chapter is organized as follows. In Section 69.2 we introduce the beamforming problem (see
also [20]), and classical beamformers such as the delay-and-sum beamformer, the minimum output
noise power beamformer, and the minimum variance beamformer. We show that these beamformers
presentseveredrawbacks when operating inmultipathenvironments. Section69.3 presentsasolution
to the beamforming problem for the case of correlated arrivals. This solution is based on a minimum
mean square error (MMSE) approach. We compare the performance of this beamformer with the
performance of the beamformers introduced in Section 69.2. We emphasize, in particular, the case
of multipath propagation. In this section, we also discuss issues regarding the implementation of the
minimum mean square error beamformer. In Section 69.4, we describe a method to implement the
minimum mean square error beamformer in the context of a digital mobile communications system.
The method operates in a blind mode and strongly exploits the structure of the received multipath

data. Since the propagation channel parameters, e.g., angles of arrivals of the multiple paths, are
not known, we estimate them with a maximum likelihood approach supported on a finite mixture
distribution model of the array data. We maximize the likelihood function with an iterative scheme.
We describe an efficient procedure to initialize the iterative algorithm. In general, this procedure
convergesrapidly to the global maximum of the likelihoodfunction. Section 69.5 presents simulation
results obtained with data synthesized by a simple mobile communications simulator. These results
confirm the excellent performance of the MMSE beamformer described in the paper.
69.2 Beamforming
Beamforming is an array processing technique for estimating a desired signal waveform impinging on
an array of sensors from a given direction. This technique applies to both narrowband and wideband
signals. Here, we will consider only the narrowband case.
Let s(t) be the complex envelope of the source radiated signature. Under the farfield assumption
the signal at the receiving array is a planar wavefront, see Fig. 69.1. In this case, and according to the
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FIGURE 69.1: Source/receiver geometry.
model derived in [20], the complex envelope of the signal received at each sensor of a uniform and
linear array of N omnidirectional sensors is
s
n
(t) = s(t − τ
n
)e
−jω
0
τ
n
,
(69.1)

where ω
0
is the carrier frequency and τ
n
is the intersensor propagation delay. Let d, c, and θ
0
be, respectively, the distance between sensors, the propagation velocity, and the direction of arrival
(DOA). The intersensor delays are then
τ
n
=
(n − 1)d
c
sin θ
0
,n= 1, 2,...,N .
(69.2)
Because of the narrowband assumption, we can make the simplification
s(t − τ
n
)  s(t)
in Eq. (69.1). This means that, for the values of τ
n
of interest, the source complex envelope s(t) is
slowly varying when compared with the carrier e
jω
0
t
.
We model each array sensor by a quadrature receiver, its output being given by

z
n
(t) = s
n
(t) + n
n
(t),
(69.3)
where s
n
(t) is the complex envelope of the signal component and n
n
(t) is a complex additive dis-
turbance, such as sensor noise, ambient noise, or another signal interfering with the desired one.
Collecting in a vector z(t) all the responses of the N sensors of the array to a narrowband source
coming at the array from the DOA= θ
0
,wegettheN-dimensional complex vector
z(t) = a(θ
0
)s(t) + n(t ) .
(69.4)
The vector a(θ
0
) is referred to as the steering vector for the DOA θ
0
.
The elements of the steering vector a(θ
0
) are given by a

n
(θ
0
) = e
−jω
0
τ
n
,n = 1, 2,...,N.The
noise vectorn(t ) isan N-dimensional complexvector collecting the N sensor noises n
n
(t). In general,
it includes components correlated with the desired signal as in multipath propagation environments.
With multipath, several replicas of the same signal, each one propagatingalong a different path, arrive
at the array with distinct DOAs.
In beamforming, the goal is to estimate the source signal s(t) given a(θ
0
). The narrowband
beamformer is illustrated in Fig. 69.2. The output of the beamformer is
y(t) = w
H
z(t) ,
(69.5)
where w =[w
1
,w
2
,w
3
,...,w

N
]
T
is a vector of complex weights. We use the notation {·}
T
to
denote vector and matrix transposition, and {·}
H
for transposition followed by complex conjugation.
The beamformer is completely specified by the vector of weights w.
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FIGURE 69.2: Narrowband beamformer.
In the absence of the noise term n(t ) in (69.4), it is readily seen that choosing
w = (1/N )a(θ
0
),
the beamformer output is y(t) = s(t). This corresponds to the simplest implementation of the
narrowband beamformer, known as the delay–and–sum (DS) beamformer: it combines coherently
the signal replicas received at each sensor after compensating for their corresponding relative delays.
The interpretation of the DS beamformer operation is rather intuitive. However, we may ask
ourselves the following question: is DS the best we can do to estimate the desired signal when the
disturbance n(t ) is present? To answer the question satisfactorily, we begin by noting that, in the
presence of noise, the output of the DS beamformer is
y(t) = s(t) + (1/N)a
H
(θ
0
)n(t) .

The influence of the error term on the estimate y(t) of s(t) depends basically on the structure of
n(t). The optimal design of beamformers depends now on the choice of an adequate optimization
criterion that takes into account the disturbance vector, with the goal of improving in some sense the
quality of the desired estimate. In the sequel, we will consider several cases of practical interest.
69.2.1 Minimum Output Noise Power Beamforming (MNP)
To reduce the effect of the error term at the beamformer output, we formulate the beamforming
problem as follows:
find the weight vector w such that the noise output power
E



w
H
n(t))


2

is minimized subject to the constraint w
H
a(θ
0
) = 1 ,
where E{·} denotes the statistical average. The cost function is
E





w
H
n(t))



2

= w
H
R
n
w
(69.6)
with R
n
the covariancematrix of the disturbance vector n(t ), i.e., R
n
= E{n(t)n
H
(t)}. The constraint
guarantees that the signal along the look direction θ
0
is not distorted.
The solution to this constrained optimization problem is obtained by Lagrange multipliers tech-
niques. It is given by
w = (a
H
(θ
0

)R
−1
n
a(θ
0
))
−1
R
−1
n
a(θ
0
).
(69.7)
The vector w in Eq. (69.7) is the gain of the MNP beamformer [20].
When the source signal is uncorrelated with the disturbance,
E{s(t)n
H
(t)}=0 ,
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it can be shown that the weight vector (69.7) of the MNP beamformer takes the form
w = (a
H
(θ
0
)R
−1
a(θ

0
))
−1
R
−1
a(θ
0
),
(69.8)
where
R = E

z(t)z
H
(t)

(69.9)
is the covariance matrix of the array data vector z.Thevectorw in Eq. (69.8) is the gain of the
minimum variance (MV) beamformer [20]. The MV beamformer minimizes the total output power
E




w
H
z(t))




2

= w
H
Rw
subject to w
H
a
H
(θ
0
) = 1. The MV beamformer presents an important advantage over the MNP
beamformer. While to implement the MNP beamformer we need to know the covariance matrix R
n
of the disturbance vector n, in general, to implement the MV beamformer it is sufficient to estimate
the array covariance matrix R using the available data z.
We discuss how to estimate R.LetT time samples (snapshots) of the array response vector z(t)
be available. An estimate of R is the data sample covariance matrix R
s
:
R
s
=
1
T
T

t=1
z(t)z
H

(t) .
(69.10)
Under technical conditions that we will not discuss here, the sample covariance matrix, R
s
, converges
(in the appropriate sense) to the array covariance matrix R when T approaches infinity. This means
that, for a large enough number T of snapshots, we can replace R in Eq. (69.8)byR
s
, without a
significant performance degradation.
We provide an alternative interpretation to the MNP beamformer. Using Eq. (69.7)inEq.(69.5),
and taking into account Eq. (69.4), we see that the output of the MNP beamformer has a signal
component s(t) and an error term w
H
n(t) with average power
P
o
= w
H
E

n(t)n
H
(t)

w =

a
H
(θ

0
)R
−1
n
a(θ
0
)

−1
.
(69.11)
Since the power of the signal is preserved and the MNP beamformer minimizes the power of the
noise at its output, the MNP beamformer maximizes the output signal-to-noise ratio (SNR).
FIGURE 69.3: (a) Single source in white noise; (b) uncorrelated interference; (c) correlated interfer-
ence.
We will not discuss in detail the behavior of the MNP and MV beamformers. The reader is referred
to the work in [2]. We list some of the properties of the MNP and MV beamformers in two scenarios
of practical interest.
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Case 1: Single Source in White Noise
Here (see Fig. 69.3(a)), we assume that the noise n(t ) is sensor noise. We model it as n(t ) = u(t )
where the components of u(t ) are jointly independent and identically distributed samples of zero
mean white noise sequences with variance σ
2
, i.e.,
R
n
= R

u
= σ
2
I ,
where I is the identity matrix. The sensor noise models the thermal noise generated at each receiver
and is assumed independent of (thus, uncorrelated with) the source signal. If S is the power of the
desired signal, the SNR at each sensor is SNR
i
= S/σ
2
. Also, from Eq. (69.7), we conclude that when
the additive noise is white, as in this case, the MNP beamformer reduces to the DS beamformer.
Moreover, computingthe powerattheoutputof the beamformer with Eq.(69.11)forthisparticular
situation yields P
o
= σ
2
/N. This means that, at the output of the beamformer, the signal-to-noise
ratioisSNR
o
= N SNR
i
.
We conclude, for the case of a single source in white noise, that the DS beamformer is optimum
in the sense of maximizing the output signal-to-noise ratio SNR
o
. Further, SNR
o
increases linearly
with the number N of array sensors.

Case 2: Directional Interferences and White Noise
Now, we assume that the disturbance n(t ) is the superposition of possibly several directional
interferences and white noise. Without loss of generality, we consider the case of a single interferer:
n(t) = a(θ
i
)i(t) + u(t ) ,
(69.12)
where i(t) is the signal radiated by the interferer, θ
i
is the DOA of the interference signal, and u(t ) is
the white noise vector. In general, we assume that u(t ) is uncorrelated with i(t).
Case 2.1: Uncorrelated Arrivals
This is the case where the desired signal and the interference are generated by distinct sources,
see Fig. 69.3(b). It is clear that under this assumption, s(t) and n(t) are uncorrelated. As we
emphasized before, this is the situation where the MNP beamformer (69.7) is equivalent to the MV
beamformer (69.8).
The covariance of the noise n(t ) is now
R
n
= a(θ
i
)S
i
a
H
(θ
i
) + σ
2
I ,

where S
i
is the average power of the interference i(t). At the DOA=θ
i
, the beamformer has an
amplitude response
|w
H
a(θ
i
)|=
|β|
1 + (1 −|β|
2
)INR
,
(69.13)
where
INR = S
i
/(σ
2
/N)
is the interference-to-noise ratio (INR), and β = (1/N)a
H
(θ
i
)a(θ
0
) measures the spatial coherence

between the desired source and the interference.
Well Separated Arrivals
When the signal and interference are well separated, their spatial coherence is small, i.e., |β|1.
In Eq. (69.13), the denominator is approximately given by 1 + INR. The net effect is that the
beamformer output along the interference direction decreases when INR increases. In other words,
the MNP and the MV beamformers direct a beam with gain 1 towards the DOA of the desired signal
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and null the interference. The interference canceling property is reflected on the average power of
the beamformer output error which is evaluated to
P
o
=
σ
2
N
1
1 −|β
2
|
INR
1+INR
.
(69.14)
For large INR and well-separated DOAs, P
o
 (σ
2
/N). This means that the interference contributes

little to the estimation error at the output of the beamformer.
Close Arrivals
When the source and the interferer are spatially close, their spatial coherence is large, |β|1,
and the output at the interference DOA is  1. This means that the MNP and MV beamformers no
longer have the ability to discriminate the two sources.
We conclude from the simple analysis of these two cases that the DOA discrimination capability is
strongly related to the spatial resolution of the array geometry through the parameter β. In practice,
to improve upon the resolution of a linear and uniform array, we increase, when feasible, the number
N of sensors. This results in narrower beamwidths which can resolve closer arrivals.
Case 2.2: Correlated Arrivals
This is the case where the interference i(t) is correlated with the desired signal s(t), i.e.,
E{s(t)i
∗
(t)}=ρ =|ρ|e
jφ
ρ
= 0 .
We denote complex conjugate by (·)
∗
.
With reference to Fig. 69.3(c), we discuss a simple example where the interference results from a
secondary path generated by a reflector (multipath propagation)
i(t) = γs(t).
(69.15)
The complex parameter, γ , accounts for the relative attenuation and delay of the reflected path. The
correlation factor, ρ, between i(t) and s(t) is, in this case, given by
ρ = γ/|γ | .
The desired signal and the disturbance vector n(t) are now correlated and the MV beamformer is no
longer equivalent to the MNP beamformer. Recall that the MV beamformer attempts to minimize
the total output power under the constraint of a unitary gain at the DOA of the desired source. As the

array output vector has a correlated signal component at a different DOA, to minimize the output
powermaycause the desired signal itself to be strongly attenuated. This is the signal cancellation effect,
typical of MV beamforming when operating in multipath environments like the one just considered.
On the contrary, the behavior of the MNP beamformer is independent of the correlation degree
between the desired signal and the disturbance: the MNP beamformer filters out correlated arrivals
just as if they were uncorrelated interferences.
To implement the MNP beamformer, besides the DOA of the desired signal, we also need to know
the covariance matrix R
n
of the disturbance vector. In general, this covariance is not known a priori.
It has to be estimated using the available data, and this can be a rather complicated task, not discussed
here.
In this section, we discussed the MNP solution to the beamforming problem. The MNP beam-
former is optimum in the sense of maximizing the output SNR. When the noise is white, the DS
beamformer is recovered as the optimum solution for the single source case. It points a beam towards
the source DOA and reduces the sensor noise power by a factor of N, see Fig. 69.4(a). We also saw
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