# Sparse Array Transceiver Design for Enhanced Adaptive Beamforming in MIMO
Radar

Syed A. Hamza1, Weitong Zhai2, Xiangrong Wang2 and Moeness G. Amin3 The work
by W Zhai and X Wang is supported by National Natural Science Foundation of
China under Grant No. 62071021 and No. 61827901. 1School of Engineering,
Widener University, Chester, PA 19013, USA
2School of Electronic and Information Engineering, Beihang University,
Beijing, 100191, China
3Center for Advanced Communications, Villanova University, Villanova, PA
19085, USA
Emails<EMAIL_ADDRESS>{wtzhai<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Sparse array design aided by emerging fast sensor switching technologies can
lower the overall system overhead by reducing the number of expensive
transceiver chains. In this paper, we examine the active sparse array design
enabling the maximum signal to interference plus noise ratio (MaxSINR)
beamforming at the MIMO radar receiver. The proposed approach entails an
entwined design, i.e., jointly selecting the optimum transmit and receive
sensor locations for accomplishing MaxSINR receive beamforming. Specifically,
we consider a co-located multiple-input multiple-output (MIMO) radar platform
with orthogonal transmitted waveforms, and examine antenna selections at the
transmit and receive arrays. The optimum active sparse array transceiver
design problem is formulated as successive convex approximation (SCA)
alongside the two-dimensional group sparsity promoting regularization. Several
examples are provided to demonstrate the effectiveness of the proposed
approach in utilizing the given transmit/receive array aperture and degrees of
freedom for achieving MaxSINR beamforming.

###### Index Terms:

MIMO radar, sparse array transceiver, adaptive beamforming, group sparsity,
successive convex approximation

##  I Introduction

Non-uniform sparse arrays have emerged as an effective technology which can be
deployed in various active and passive sensing modalities, such as acoustics
and radio frequency (RF) applications, including GPS[1]. Fundamentally, sparse
array design seeks optimum system performance, under both noise and
interference, while being cognizant of the limitations on cost and aperture.
Within the RF beamforming paradigm, maximizing the signal to interference plus
noise ratio (SINR) amounts to concurrently optimizing the array configuration
and beamforming weights. Equivalently, the problem can be cast as continuously
selecting the best antenna locations to deliver MaxSINR under time-varying
environment. In lieu of constantly moving antennas to new positions, a more
practical and feasible approach is to have a uniform full array and then
switch among antennas with a fixed number of front-end chains, yielding the
MaxSINR sparse beamformer for a given environment.

The above environment-dependent approach differs from the environment-
independent sparse array design that seeks to increasing the spatial
autocorrelation lags and maximizing the contiguous portion of the coarray
aperture for a limited number of sensors. The main task, therein, is to enable
DOA estimation involving more sources than the physical sensors [2, 3, 4, 5,
6]. For beamforming applications, the environment-independent design criteria
typically strives to achieve desirable beampattern characteristics such as
broad main lobe and minimum sidelobe levels and frequency invariant
beampattern for wideband design [7, 8, 9, 10, 11, 12].

Environment-dependent sparse receive beamformer striving to achieve MaxSINR
can potentially provide sparse configurations that improve target detection
and estimation accuracy [13, 14, 15, 16, 17, 18, 19, 20, 21]. In this paper,
we consider MaxSINR receive beamformer design for MIMO radar operation. The
beamformer design, in this case, jointly selects the optimum transmit and
receive sensor locations for implementing an efficient receive beamformer. We
assume that transmit sensors emit orthogonal waveforms. The MIMO radar
implementing sensor-based orthogonal transmit waveforms does not benefit from
coherent transmit processing gain achieved when using directional beamforming.
In this context, the proposed approach is fundamentally different from the
parallel sparse array MIMO beamforming designs, which primarily rely on
optimizing the transmit sensor locations and the corresponding correlation
matrix of the transmit waveform sequence [22, 23, 24, 25, 26]. The proposed
approaches, therein, maximize the transmitted signal power towards the
perspective target locations while minimizing the cross-correlation of the
target returns to achieve efficient receive beamforming characteristics.
Moreover, existing design schemes essentially pursue an uncoupled
transmit/receive design which is in contrast to the proposed approach that
incorporates sparse receiver design in conjunction with transmit array
optimization.

The proposed transceiver beamforming design problem, in essence, translates to
configuring the transmit/receive array with the corresponding optimum receiver
beamforming weights. To this end, we pursue a data dependent approach that
jointly optimizes the beamforming sensor positions and weights. The
optimization only requires the knowledge of the perspective desired source
locations and does not assume any knowledge of the interfering sources. We
optimally select $M_{t}$ from $M$ transmitters and $N_{r}$ from $N$ receivers.
Such selection is a binary optimization problem, and is NP-hard. In order to
avoid extensive computations associated with enumeration of all possible array
configurations, we apply convex relaxation. The design problem is posed as SCA
with two-dimensional reweighted mixed $l_{1,2}$-norm penalization to jointly
invoke sparsity in the transmit and receive dimensions.

The rest of the paper is organized as follows: In the next section, we state
the problem formulation for the MIMO beamformer design in the case of a pre-
specified array configuration. Section III elaborates on the sparse array
design by semidefinite relaxation to jointly find the optimum sparse
transmit/receive array geometry. In the subsequent section, simulations are
presented to demonstrate the offerings of the proposed sparse array transmit
design. The paper ends with concluding remarks.

##  II Problem Formulation

We consider a MIMO radar with $M$ transmitters and $N$ receivers illuminating
the scene through omni-directional beampattern. This is typically achieved
through emissions of orthogonal waveforms across the transmit array.
Specifically, we pursue a co-located MIMO arrangement with the transmit and
receive arrays in close vicinity. As such, a far-field source is presented by
equal angles of departure and arrival. Consider $K$ target sources arriving
from $\\{\theta_{s,k}\\}$. The received baseband data
$\mathbf{x}(n)\in\mathbb{C}^{MN\times 1}$ after matched filtering at the $N$
element uniformly spaced receiver array at time instant $n$ is given by,

$\mathbf{x}(n)=\sum_{k=1}^{K}s_{k}(n)\mathbf{b}(\theta_{s,k})+\sum_{l=1}^{L_{c}}c_{l}(n)\mathbf{b}(\theta_{j,l})+\sum_{q=1}^{Q}\mathbf{i}_{q}(n)+\mathbf{v}(n),$
(1)

where, $s_{k}(n)\in\mathbb{C}$ is the $k$th reflected target signal. The
extended steering vector $\mathbf{b}(\theta)$ of the virtual array is
$\mathbf{b}(\theta)=\mathbf{a}_{t}(\theta)\otimes\mathbf{a}_{r}(\theta)$. In
the case of uniform transmit and receive linear arrays with a respective
inter-element spacing of $d_{t}$ and $d_{r}$, the transmit and receive
steering vectors are given by,

$\mathbf{a}_{t}(\theta)={}[1\,\,\,e^{j2\pi(d_{t}/\lambda)cos\theta}{}\\\
\,.\,.\,.\,\,\,\,\,\,\,e^{j2\pi(M-1)(d_{t}/\lambda)cos\theta}]^{T},$ (2)

and

$\mathbf{a}_{r}(\theta)={}[1\,\,\,e^{j2\pi(d_{r}/\lambda)cos\theta}{}\\\
\,.\,.\,.\,\,\,\,\,\,\,e^{j2\pi(N-1)(d_{r}/\lambda)cos\theta}]^{T}.$ (3)

The variance of additive Gaussian noise $\mathbf{v}(n)\in\mathbb{C}^{MN\times
1}$ is $\sigma_{v}^{2}$ at the receiver output. There are $L_{c}$
interferences $c_{l}(n)$ mimicking target reflected signal and $Q$ narrowband
interferences $\mathbf{i}_{q}(n)$ $\in\mathbb{C}^{MN\times 1}$. The latter is
defined as the Kronecker product of the receiver steering vector
$\mathbf{a}_{r}(\theta_{i,q})$ and the matched filtering output of the
interference $\mathbf{j}_{q}(n)$ such that
$\mathbf{i}_{q}(n)=\mathbf{j}_{q}(n)\otimes\mathbf{a}_{r}(\theta_{i,q})$. The
received data vector $\mathbf{x}(n)$ is then linearly combined to maximize the
output SINR. The output signal $y(n)$ of the optimum beamformer for MaxSINR is
given by [27],

$y(n)=\mathbf{w}^{H}\mathbf{x}(n)$ (4)

where $\mathbf{w}$ is the beamformer weight at the receiver. The optimal
solution $\mathbf{w}_{o}$ can be obtained by solving the optimization problem
that seeks to minimize the interference power at the receiver output while
preserving the desired signal. The constraint minimization problem can be cast
as,

$\displaystyle\underset{\mathbf{w}\in\mathbb{C}^{MN}}{\text{minimize}}$
$\displaystyle\quad\mathbf{w}^{H}\mathbf{R_{x}}\mathbf{w},$ (5) s.t.
$\displaystyle\quad\mathbf{w}^{H}\mathbf{R}_{s}\mathbf{w}=1,$

where the source correlation matrix is $\mathbf{R}_{s}$ $=$
$\sum_{k=1}^{K}\sigma_{s,k}^{2}$
$\mathbf{b}(\theta_{s,k})\mathbf{b}^{H}(\theta_{s,k})$, with
$\sigma^{2}_{s,k}=E\\{s_{k}(n)s_{k}^{H}(n)\\}$ denoting the average received
power from the $k$th target return. The data correlation matrix,
$\mathbf{R}_{\mathbf{x}}\approx(1/T)\mathbf{x}(n)\mathbf{x}(n)^{H}$, is
directly estimated from the $T$ received data snapshots. The solution to the
optimum weights in (5) is given by
$\mathbf{w}_{o}=\mathscr{P}\\{\mathbf{R}_{\mathbf{x}}^{-1}\mathbf{R}_{s}\\}$,
with the operator $\mathscr{P}\\{.\\}$ representing the principal eigenvector
of the input matrix. This optimum solution yields the MaxSINR, SINRo, given by
[27],

$\text{SINR}_{o}=\Lambda_{max}\\{\mathbf{R}^{-1}_{n}\mathbf{R}_{s}\\},$ (6)

which is the MaxSINR is the maximum eigenvalue ($\Lambda_{max}$) of the
product of the two matrices, the inverse of interference plus noise
correlation matrix and the desired source correlation matrix. It is clear that
the resulting solution holds irrespective of the array configuration, whether
the array is uniform or sparse. For the latter, the performance of MaxSINR
beamformer is intrinsically tied to the array configuration. The sparse
optimization of the above formulation is explained in the next section.

##  III Sparse array design

The separate sensor selection problem for a joint transmit and receive design
is a combinatorial optimization problem and can’t be solved in polynomial
time. We formulate the sparse array design problem by exploiting the structure
of the received signal model and solve it by applying sequential convex
approximation. To exploit the sparse structure of the joint transmit and
receive sensor selection, we introduce a two dimensional $l_{1,2}$-mixed norm
regularization to recover group sparse solutions. One dimension pertains to
the transmitter sparsity, whereas the other sparsity pattern is associated
with the receiver side. Moreover, this two-dimensional sparsity pattern is
entwined and coupled, meaning that when one transmitter is discarded, all $N$
receiving data pertaining to this transmitter is zero. Similarly, the receiver
is discarded only when its received data corresponding to all the $M$
transmitters is zero. The structure of the optimal sparse beamforming weight
vector is elucidated in (7), where $\checkmark$ denotes a sensor location
activated, or selected, and $\times$ denotes a sensor not activated.

$\displaystyle\overbrace{\underbrace{\left(\begin{array}[]{@{}rrrr@{}}\checkmark\\\
\times\\\ \times\\\ \checkmark\\\ \vdots\\\
\times\end{array}\right)}_{\text{(Tx 1
active)}}}^{\mathbf{w}(1,N)}\quad\overbrace{\underbrace{\left(\begin{array}[]{@{}rrrr@{}}\checkmark\\\
\times\\\ \times\\\ \checkmark\\\ \vdots\\\
\times\end{array}\right)}_{(\text{Tx 2
active)}}}^{\mathbf{w}(N+1,2N)}\quad\overbrace{\underbrace{\left(\begin{array}[]{@{}rrrr@{}}\times\\\
\times\\\ \times\\\ \times\\\ \vdots\\\ \times\end{array}\right)}_{(\text{Tx 3
inactive)}}}^{\mathbf{w}(2N+1,3N)}\quad\ldots\overbrace{\underbrace{\left(\begin{array}[]{@{}rrrr@{}}\checkmark\\\
\times\\\ \times\\\ \checkmark\\\ \vdots\\\
\times\end{array}\right)}_{(\text{Tx $M$ active)}}}^{\mathbf{w}(N(M-1)+1,MN)}$
(31)

In (7), each column vector denotes the receive beamformer weights for a fixed
transmit location. It is noted that the optimal sparse beamformer,
corresponding to the active transmit and receive locations, follows a group
sparse structure along the transmit and receive dimensions (the consecutive
$N$ sensors are discarded vertically or the consecutive $M$ sensors are
discarded horizontally. It is evident that the missing transmit sensor at
position $3$, for example, translates to the sparsity along all the
corresponding $N$ entries of $\mathbf{w}$ ($N$ consecutive $\times$ vertically
in (7)). Similarly, the group sparsity is also invoked across the received
signals corresponding to all transmitters ($M$ consecutive $\times$
horizontally in (7)).

###  III-A Group Sparse solutions through SCA

The problem in (5) can equivalently be rewritten by swapping the objective and
constraint functions as follows,

$\displaystyle\underset{\mathbf{w\in\mathbb{C}}^{MN}}{\text{maximize}}$
$\displaystyle\quad\mathbf{w}^{H}\bar{\mathbf{R}}_{s}\mathbf{w}$ (32) s.t.
$\displaystyle\quad\mathbf{w}^{H}\mathbf{R_{x}}\mathbf{w}\leq 1$

where $\bar{\mathbf{R}}_{s}=-\mathbf{R}_{s}$. The sparse MIMO configuration of
uniformly spaced receivers and transmitters with a respective inter-element
spacing of $d_{r}$ and $d_{t}=Nd_{r}$ is employed for data collection. The
covariance matrix $\mathbf{R_{x}}$ of the full receive virtual array can be
then obtained by the matrix completion method proposed in [17].

The beamforming weight vectors are generally complex valued, whereas the
quadratic functions are real. This observation allows expressing the problem
with only real variables which is typically accomplished by replacing the
correlation matrix $\bar{\mathbf{R}}_{s}$ by $\tilde{\mathbf{R}}_{s}$ and
concatenating the beamforming weight vector accordingly [28],

$\tilde{\mathbf{R}}_{s}=\begin{bmatrix}\text{real}(\bar{\mathbf{R}}_{s})&-\text{imag}(\bar{\mathbf{R}}_{s})\\\
\text{imag}(\bar{\mathbf{R}}_{s})&\text{real}(\bar{\mathbf{R}}_{s})\\\
\end{bmatrix},\tilde{\mathbf{w}}=\begin{bmatrix}\text{real}({\mathbf{w}})\\\
\text{imag}({\mathbf{w}})\\\ \end{bmatrix}$ (33)

Similarly, the received data correlation matrix $\mathbf{R_{x}}$ is replaced
by $\mathbf{\tilde{R}_{x}}$. The problem in (32) then becomes,

$\displaystyle\underset{\mathbf{\tilde{w}\in\mathbb{R}}^{2MN}}{\text{minimize}}$
$\displaystyle\quad\mathbf{\tilde{w}}^{{}^{\prime}}\tilde{\mathbf{R}}_{s}\mathbf{\tilde{w}},$
(34) s.t.
$\displaystyle\quad\mathbf{\tilde{w}}^{{}^{\prime}}\mathbf{\tilde{R}_{x}}\mathbf{\tilde{w}}\leq
1,$

where ′ denotes transpose operation. After expressing the constraint in terms
of real variables, we convexify the objective function by utilizing the first
order approximation iteratively,

$\displaystyle\underset{\mathbf{\tilde{w}\in\mathbb{R}}^{2MN}}{\text{minimize}}$
$\displaystyle\quad\mathbf{m}^{(k)}{{}^{\prime}}\mathbf{\tilde{w}}+b^{(k)},$
(35) s.t.
$\displaystyle\quad\mathbf{\tilde{w}}^{{}^{\prime}}\mathbf{\tilde{R}_{x}}\mathbf{\tilde{w}}\leq
1,$

where, $\mathbf{m}^{(k)}$ and ${b^{(k)}}$, updated at the $k+1$ iteration, are
given by
$\mathbf{m}^{(k+1)}=2\tilde{\mathbf{R}}_{s}\mathbf{\tilde{w}}^{(k)},b^{(k+1)}=-\mathbf{\tilde{w}}^{(k)}{{}^{\prime}}\tilde{\mathbf{R}}_{s}\mathbf{\tilde{w}}^{(k)}$,
respectively. Finally, to invoke sparsity in the beamforming weight vector,
the re-weighted mixed $l_{1,2}$ norm is adopted primarily for promoting group
sparsity,

$\displaystyle\underset{\mathbf{\tilde{w}},\mathbf{c},\mathbf{r}}{\text{minimize}}\quad$
$\displaystyle\quad\mathbf{m^{(k)}{{}^{\prime}}}\mathbf{\tilde{w}}+b^{(k)}+\alpha(\mathbf{p}^{{}^{\prime}}\mathbf{c})+\beta(\mathbf{q}^{{}^{\prime}}\mathbf{r})$
(36) s.t.
$\displaystyle\quad\mathbf{\tilde{w}}^{{}^{\prime}}\mathbf{\tilde{R}_{x}}\mathbf{\tilde{w}}\leq
1,$ (12a) $\displaystyle\quad||\mathbf{P}_{i}\odot\mathbf{\tilde{w}}||_{2}\leq
c_{i},$ (12b) $\displaystyle\quad 0\leq c_{i}\leq 1,\quad i=1,...,M$ (12c)
$\displaystyle\quad||\mathbf{Q}_{j}\odot\mathbf{\tilde{w}}||_{2}\leq r_{j},$
(12d) $\displaystyle\quad 0\leq r_{j}\leq 1,\quad j=1,...,N$ (12e)
$\displaystyle\quad\mathbf{1}_{M}^{{}^{\prime}}\mathbf{c}=M_{t},$ (12f)
$\displaystyle\quad\mathbf{1}_{N}^{{}^{\prime}}\mathbf{r}=N_{r},$ (12g)

and

$\displaystyle\mathbf{P}_{i}=[\overbrace{0...0...0}^{\mbox{\footnotesize$\begin{array}[]{c}N\
elements\\\ 1st\
group\end{array}$}}...\overbrace{1...1...1}^{\mbox{\footnotesize$\begin{array}[]{c}N\
elements\\\ ith\
group\end{array}$}}...\overbrace{1...1...1}^{\mbox{\footnotesize$\begin{array}[]{c}N\
elements\\\ (M+i)th\
group\end{array}$}}...\overbrace{0...0...0}^{\mbox{\footnotesize$\begin{array}[]{c}N\
elements\\\ 2Mth\ group\end{array}$}}]^{{}^{\prime}},$ (45)

$\displaystyle\mathbf{Q}_{j}=[\ \overbrace{\underbrace{0\ ...\ 0}_{(j-1)\ 0s}\
1\ 0\ ...\ 0}^{\mbox{\footnotesize$\begin{array}[]{c}N\ elements\ of\\\ the\
1st\ group\end{array}$}}\ .\ .\ .\ \overbrace{\underbrace{0\ ...\ 0}_{(j-1)\
0s}\ 1\ 0\ ...\ 0}^{\mbox{\footnotesize$\begin{array}[]{c}N\ elements\ of\\\
the\ 2Mth\ group\end{array}$}}]^{{}^{\prime}}.$ (50)

Here, $\odot$ means the element-wise product, $\mathbf{c}$ and $\mathbf{r}$
are two auxiliary binary selection vectors, $\mathbf{P}_{i}\in\\{0,1\\}^{2MN}$
in (12b) is the transmission selection matrix, which is used to select the
real and imaginary parts of the $N$ weights corresponding to the $i$th
transmitter, as shown in (13). Eqs. (12c) and (12f) indicate that we can
select $M_{t}$ transmitters at most. Similarly, $\mathbf{Q}_{j}$ in (12d) is
the receiver selection matrix, as shown in (14). Eqs. (12e) and (12g) indicate
that we can select $N_{r}$ receivers at most. The two parameters $\alpha$ and
$\beta$ are used to control the sparsity of the transmitters and the
receivers, $\mathbf{p}$ and $\mathbf{q}$ are the reweighting coefficient
vectors for the transmitters and receivers respectively and the detailed
update method is given below.

Algorithm 1 SCA-Sparse Transmit/Receive Beamformer Design

0: $M$, $N$, $M_{t}$, $N_{r}$, $\alpha=\beta=0.5$, $\alpha_{0}=\beta_{0}=1$,
target direction $\theta_{s}$, auto-correlation matrix $\mathbf{R}_{s}$ and
$\mathbf{R}_{x}$

0: Optimal weight $\mathbf{w}$, location and number of receivers and
transmitters. Initialization: Initialize $\mathbf{w}$, $\mathbf{p}$,
$\mathbf{q}$, $\mathbf{m}$ as all ones matrix. Set $\alpha_{0}$ = 1 and
$\beta_{0}$ = 1. Initialize the tradeoff parameters $\alpha$ and $\beta$
according to the sparsity requirement.

while $||\mathbf{\tilde{w}}^{(k+1)}-\mathbf{\tilde{w}}^{(k)}||_{2}\geq
10^{-5}$ do

a) Convert $\mathbf{w}$, $\mathbf{R}_{s}$ and $\mathbf{R}_{x}$ to the real
domain to get $\mathbf{\tilde{w}}$, $\mathbf{\tilde{R}}_{s}$ and
$\mathbf{\tilde{R}}_{x}$ according to Eq.(9).

b) Update $\mathbf{p}^{(k+1)}$ and $\mathbf{q}^{(k+1)}$ according to Eq.(13)

c) Update $\mathbf{\tilde{w}}^{(k+1)}$ according to Eq.(12).

d) Convert $\mathbf{\tilde{w}}^{(k+1)}$ back into the complex solution
$\mathbf{w}^{(k+1)}$ by
$\mathbf{w}^{(k+1)}(i)=\mathbf{\tilde{w}}^{(k+1)}(i)+j\mathbf{\tilde{w}}^{(k+1)}(i+MN)$.

end while

return $\mathbf{w}$

###  III-B Reweighting Update

A common method of updating the reweighting coefficient is to take the
reciprocal of $|\mathbf{w}|$ [29]. However, this can’t control the number of
elements to be selected. Thus, similar to [30], in order to make the number of
selected antennas controllable, we update the weights using the following
formula,

${p}_{i}^{(k+1)}=\frac{1-c_{i}^{(k)}}{1-e^{-\beta_{0}c_{i}^{(k)}}+\epsilon}-(\frac{1}{\epsilon})(c_{i}^{(k)})^{\alpha_{0}}$
(51)

Here, $\alpha_{0}$ and $\beta_{0}$ are two parameters that control the shape
of the curve, and the parameter $\epsilon$ avoids the unwanted explosive case.
The essence of this re-weight update method is to take a large positive
penalty for a entry close to zero and a small negative reward for the entry
close to 1. As a result, the entries of the two selection vectors $\mathbf{c}$
and $\mathbf{r}$ tend to be either 0 or 1. The update of the re-weight vector
$\mathbf{q}$ follows the same rule. Through iterative regression, we can find
$M_{t}$ transmitters and $N_{r}$ receivers. In addition, by controlling the
values of $\alpha$ and $\beta$, we can obtain different sparsities of
transmitters and receivers [31]. The proposed algorithm for joint
transmit/receive beamformer design is elaborated further in Algorithm 1.

##  IV Simulations

In this section, we demonstrate the effectiveness of our proposed sparse MIMO
radar from the perspective of output SINR. We compare the performance of the
optimal MIMO array transceiver configured according to the proposed algorithm
with randomly configured MIMO arrays. In practice, the interference may be
caused by co-existence in the same bandwidth or being deliberately positioned
at some angles transmitting the same waveform as targets of interest. We
consider both kinds of interferences.

###  IV-A Example 1

In this example, we fixed the direction of the interferences, and change the
arrival angle of the target from $0^{\circ}$ to $90^{\circ}$. We consider a
full uniform linear MIMO array consisting of 8 transmitters (M = 8) and 8
receivers (N = 8). We select $M_{t}=4$ transmitters and $N_{r}=4$ receivers.
For the receiver, we set the minimum spacing of the sensors to $\lambda/2$,
while for the transmitter, we set the minimum spacing of the sensors to
$N\lambda/2$. Suppose there are two interferences with an interference to
noise-ratio (INR) of 13dB and arrival angles of
$\theta_{q}=[40^{\circ},70^{\circ}]$. The SNR of the desired signal is fixed
at 20dB. We simulate co-existing interferences and deliberate interferences.
The output SINR versus target angle is shown in Fig.1. In this figure, case 1
corresponds to two deliberate interferences, whereas case 2 represents two co-
existing interferences. It can be seen that, in both cases, the optimal sparse
MIMO arrays exhibit better performance than randomly selected sparse arrays.
Fig.2, depicts the configuration of the optimal MIMO array for a target at
$80^{\circ}$.

###  IV-B Example 2

In this example, we consider the scenario where the interferences are
spatially close to the target which causes a great adverse impact on the array
performance. We change the angle of the target from $0^{\circ}$ to
$90^{\circ}$. For each target angle, two interferences are generated from the
proximity of $\pm 5^{\circ}$ away from the target. We consider a full uniform
linear MIMO array consisting of 20 transmitters (M = 20) and 20 receivers (N =
20). We select 5 transmitters and 5 receivers to compose the sparse MIMO
array. The other simulation parameters remain the same as those in example 1.
Again, we plot the output SINR versus target angle, as shown in Fig. 3. In
case 1, the two interferences are deliberate, whereas in case 2, both
interferences are co-existing. It can be observed that when the interferences
are spatially close to the target, the proposed optimal MIMO sparse array
exhibits more superiority than randomly configured sparse MIMO arrays.

Figure 1: Relationship between output SINR and target angle with the
directions of interferences being fixed. Figure 2: Optimal MIMO transceiver
configuration when the target is from $80^{\circ}$. Figure 3: Relationship
between the output SINR and the target angle when the interferences are closed
to the target.

##  V Conclusion

In this paper, the problem of sparse transceiver design for MIMO radar was
considered. For a given number of transmitting and receiving sensors, a sparse
MIMO array structure in terms of MaxSINR was jointly designed. A mixed-norm
reweighted regularization was utilized to promote two-dimensional sparsity. It
was shown by simulation examples that the proposed sparse MIMO transceiver
selection algorithm provides superior interference suppression performance to
randomly designed array.

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# Hybrid Fusion for 802.11ax Wi-Fi-based Passive Radars Exploiting Beamforming
Feedbacks

Martin Willame12, Hasan Can Yildirim2, Laurent Storrer 2, François Horlin2,
Jérôme Louveaux1,  2ULB - OPERA: Université libre de Bruxelles, Brussels,
Belgium  1UCLouvain - ICTEAM: Université catholique de Louvain, Louvain-la-
Neuve, Belgium  {martin.willame,
jerome.louveaux}@uclouvain.be,{hasan.can.yildirim, laurent.storrer,
<EMAIL_ADDRESS>
###### Abstract

Passive Wi-Fi-based radars (PWRs) are devices that enable the localization of
targets using Wi-Fi signals of opportunity transmitted by an access point.
Unlike active radars that optimize their transmitted waveform for
localization, PWRs align with the 802.11 amendments. Specifically, during the
channel sounding session preceding a multi-user multiple-input multiple-output
downlink transmission, an access point isotropically transmits a null data
packet (NDP) with a known preamble. From these known symbols, client user
equipments derive their channel state information and transmit an unencrypted
beamforming feedback (BFF) back to the access point. The BFF comprises the
right singular matrix of the channel and the corresponding stream gain for
each subcarrier, which allows the computation of a beamforming matrix at the
access point. In a classical PWR processing, only the preamble symbols from
the NDP are exploited during the channel sounding session. In this study, we
investigate multiple target localization by a PWR exploiting hybrid
information sources. On one hand, the joint angle-of-departure and angle-of-
arrival evaluated from the NDP. On another hand, the line-of-sight angle-of-
departures inferred from the BFFs. The processing steps at the PWR are defined
and an optimal hybrid fusion rule is derived in the maximum likelihood
framework. Monte-Carlo simulations assess the enhanced accuracy of the
proposed combination method compared to classical PWR processing based solely
on the NDP, and compare the localisation performance between client and non-
client targets.

###### Index Terms:

Passive Wi-Fi Radar, Beamforming Feedback, MU-MIMO, Multitarget Localization,
Maximum Likelihood

##  I Introduction

Passive Wi-Fi-based Radars (PWRs) are devices employed for target localization
utilizing Wi-Fi signals transmitted by an Access Point (AP) [1]. The
communication signal emitted by the AP to its clients within the environment
is reflected off the targets and received by the PWR. The PWR captures and
demodulates known preamble orthogonal frequency division multiplexing (OFDM)
modulated symbols to estimate the localization parameters of the targets.
Unlike active radars which optimize their transmitted waveform for
localization, the design of PWRs is intrinsically linked to the evolution of
the 802.11 standard [2]. Consequently, any wireless communication technology
specified within the Wi-Fi standard presents an opportunity to enhance PWR
localization capabilities.

In particular, the initiation phase of a multi-user multiple-input multiple-
output (MU-MIMO) downlink transmission between an AP and multiple client user
equipments (UEs) could be leveraged. To determine its beamforming coefficient,
an 802.11ax AP initiates a channel sounding session during which null data
packets (NDP) containing known OFDM symbols are transmitted to the client UEs.
Based on the NDP, the client UEs can evaluate their channel state information
(CSI) and transmit back unencrypted beamforming feedback (BFF) to the AP.
These MU-MIMO BFFs comprise a subcarrier-averaged stream gain and, for each
subcarrier, the right singular matrix of the channel and the quantized delta
streams, enabling the AP to compute its beamforming matrix [3]. However,
traditional processing methods employed by PWRs during the channel sounding
session solely rely on the known preambles provided by the NDP.

Previous works in the literature explore the utilization of the BFF for
various purposes. For instance, an experimental investigation conducted by [4]
focuses on BFF-based angle-of-departure (AoD) estimation. The study
demonstrates that the estimation error of AoD using BFF is comparable to that
achieved through CSI-based estimation methods. In [5], the right singular
vectors contained within the BFF are sniffed at a PWR to reconstruct the
precoded MIMO preamble signals. This approach enables the localization of
targets using the beamformed Wi-Fi signals. However, to the best of the
author’s knowledge, there has been no investigation into the hybrid
combination of the CSI evaluated at a PWR during the channel sounding session
with the information provided by the BFF to enhance the localization accuracy
of the radar system. Additionally, there is no previous work leveraging the
delta signal-to-noise ratio (SNR) transmitted in the MU-MIMO BFF for this
purpose.

In this study, our focus lies on the localization of $K$ targets by a PWR
exploiting a hybrid source of information: the joint AoDs and angles-of-
arrival (AoAs) extracted from the NDP transmitted by an AP, and the line-of-
sight (LoS) AoD inferred from the BFF transmitted by the client UEs.
Consequently, the PWR must implement a combination rule for the optimal
exploitation of both sources of information.

The maximum likelihood (ML) framework enables the derivation of an optimal
joint AoD/AoA-based fusion rule. However, in scenarios involving the detection
of multiple targets, the complexity of the ML algorithm increases, as its
brute force implementation requires a $2K$-dimensional search across the
positions of all targets [6]. In our previous analysis [7], we introduced a
fusing methodology for a multistatic OFDM radar localization based on known
preambles only, effectively decoupling the $2K$-dimensional ML estimator into
$K$ per-target two-dimensional searches. The proposed alternating summation
method relies on a pre-estimation of the targets’ parameters acquired via the
multiple signal classification (MUSIC) algorithm. This method takes into
account the varying ability of the different radar pairs constituting the
multistatic configuration to localize different targets.

The aim of this paper is to explore the advantages of utilizing the BFFs at a
PWR for localizing multiple targets. Our contributions are outlined as
follows:

  * •

We detail the processing steps at a PWR to extract both the CSI from the NDP
transmitted by the AP and the BFFs transmitted by the client UEs during the
MU-MIMO channel sounding session.

  * •

We formulate the ML estimator for localizing multiple targets through the
hybrid fusion of the CSI and the BFFs. We demonstrate that the alternating
summation method proposed in [7] for a multistatic radar configuration can be
adapted by computing an approximated sample covariance matrix of the channel
using the right singular vectors and the quantized subcarrier stream gain
contained in the BFF.

  * •

Numerical simulations demonstrate the benefits of exploiting the BFFs compared
to localization based solely on the CSI. Furthermore, the localization
improvement is compared for client and non-client targets.

The structure of the paper is organized as follows: Section II presents the
sensing steps at the PWR during the channel sounding session. Section III
describes the system model. Section IV details the mathematical formulation of
the hybrid radar processing at the PWR. In Section V, numerical results are
provided to evaluate the benefit of the proposed fusion.

###  I-A Notations

The vectors and matrices are defined as $\mathbf{a}$ and $\mathbf{A}$,
respectively. The trace, the transpose and the Hermitian transpose are denoted
by $\operatorname{Tr}\left\\{\mathbf{A}\right\\},\mathbf{A}^{\mathrm{T}}$ and
$\mathbf{A}^{\mathrm{H}}$, respectively. The Moore-Penrose inverse is defined
as
$\mathbf{A}^{+}=\left(\mathbf{A}^{\mathrm{H}}\mathbf{A}\right)^{-1}\mathbf{A}^{\mathrm{H}}$.
The vector and the Frobenius matrix norm are written $\lVert\mathbf{a}\rVert$
and $\lVert\mathbf{A}\rVert_{\mathrm{F}}$. The identity matrix is denoted by
$\mathbf{I}$ and the Kronecker product is denoted by $\otimes$.

##  II Channel Sounding Session Sensing

In this study, we investigate the sensing capability of a PWR during the
channel sounding session defined by the 802.11ax amendment. The scenario is
illustrated in Figure1.

Figure 1: Illustration of the scenario. The hybrid configuration comprises an
AP, a PWR, multiple client and non-client targets. In the illustration, solid
lines represent the incident waveforms, while dashed lines denote the
reflected waveforms. Blue lines represent the NDP, and orange lines depict the
BFF.

From a communication standpoint, the AP initiates a downlink MU-MIMO
transmission with the client UEs. To compute or update its beamforming
steering matrix, the AP initiates an explicit channel sounding session with
the following steps:

  1. C1)

The AP isotropically transmits an NDP to the client UEs.

  2. C2)

The client UEs evaluate their CSI for each transmitting-receiving antenna
pairing from known high efficiency long training fields (HE-LTF) in the NDP.

  3. C3)

Each client UE performs a singular value decomposition (SVD) of the obtained
channel matrix to extract the right singular matrix of the channel and the
quantized delta stream for each subcarrier, along with a subcarrier-averaged
stream gain. These parameters are subsequently transmitted back to the AP in
an unencrypted and compressed BFF.

  4. C4)

The AP decompresses the BFFs and computes its precoding matrix for beamformed
data transmission.

From a sensing perspective, the PWR aims to localize the clients and other
potential targets within its coverage area. To achieve this, the PWR operates
as follows:

  1. S1)

It evaluates its CSI from the NDP transmitted isotropically by the AP.

  2. S2)

It sniffes the BFFs transmitted by the client UE and uncompresses their
content.

  3. S3)

It combines the estimated CSI with the information contained in the BFFs to
enhance the localization of the targets.

##  III System Model

In this section, we provide the system model for the configuration depicted in
Figure 1. On one hand, the AP initiates a channel sounding session to conduct
a downlink MU-MIMO transmission to the client UEs. As described in Section II,
the AP isotropically transmits an NDP to obtain feedback from the client UEs,
from which the beamforming steering matrix can be evaluated. If the AP has
$N_{A}$ antennas, then the NDP comprises $N_{A}$ known HE-LTF OFDM symbols on
$Q$ subcarriers with a subcarrier spacing of $\Delta_{f}$.

On the other hand, the PWR intends to exploit this NDP to localize $C$ clients
and $K\scalebox{0.75}[1.0]{$-$}C$ non-client targets within its coverage area.
The positions of these $K$ targets are defined by the vectors
$\mathbf{x}=[x_{1}\ \dots\ x_{K}]^{\mathrm{T}}$ and $\mathbf{y}=[y_{1}\ \dots\
y_{K}]^{\mathrm{T}}$, where indices $k=1\dots C$ correspond to the positions
of the client UEs. The AP, the PWR and the client UEs are all equipped with
uniform linear arrays of respectively $N_{A}$, $N_{P}$ and $N_{u}$ antennas,
where $u=1\dots C$. For simplicity, we assume that the arrays of the AP and
the PWR are oriented towards the coverage area and have half-wavelength
spacing. To express the channel model, we assume that only multipath signals
featuring a single reflection on a target significantly impact the observed
channel model. In this single-bounce model, signals with multiple reflections
are thus omitted [8].

In the remainder of this section, we describe the radar channel model between
the AP and the PWR, as well as the communication channel model between the AP
and the client UEs.

###  III-A Passive Radar Channel Model

To describe the radar channel between the AP and the PWR, we assume that the
direct LoS signal, along with clutter contributions, are effectively
suppressed from the estimated channel. This can be obtained by exploiting
previously transmitted known preamble symbols. The resulting radar channel is
thus defined by the $K$ multipath components resulting from reflections on the
client and non-client targets. The AoD and the AoA corresponding to the
$k\textsuperscript{th}$ target are denoted by $\varphi_{k}$ and
$\vartheta_{k}$, respectively. All angles are defined between the wavefront
and the normal vector of their corresponding antenna array. The corresponding
complex steering vectors are thus given by $\mathbf{a}(\varphi_{k})=[1\
e^{j\pi\sin(\varphi_{k})}\dots\
e^{j\pi(N_{A}-1)\sin(\varphi_{k})}]^{\mathrm{T}},\in\mathbb{C}^{N_{A}\times
1}$ and $\mathbf{a}(\vartheta_{k})=[1\ e^{j\pi\sin(\vartheta_{k})}\dots\
e^{j\pi(N_{P}-1)\sin(\vartheta_{k})}]^{\mathrm{T}},\in\mathbb{C}^{N_{P}\times
1}$. The set of all AoDs and AoAs are denoted by $\boldsymbol{\Phi}$ and
$\boldsymbol{\Theta}$, respectively. Throughout this paper, for the sake of
notation simplicity, we do not explicitly denote the dependence of the angle
on $(\mathbf{x},\mathbf{y})$. The baseband equivalent channel matrix for the
radar channel is defined for each subcarrier $q$ as follows

$\mathbf{H}_{q}^{r}=\mathbf{A}(\boldsymbol{\Theta})\ \mathbf{B}_{q}^{r}\
\mathbf{A}^{\mathrm{H}}(\boldsymbol{\Phi}),\ \in\mathbb{C}^{N_{P}\times
N_{A}},$ (1)

where $\mathbf{A}(\boldsymbol{\Phi})=[\mathbf{a}(\varphi_{1})\ \dots\
\mathbf{a}(\varphi_{K})]\in\mathbb{C}^{N_{A}\times K}$ is the AoD steering
matrix, $\mathbf{A}(\boldsymbol{\Theta})=[\ \mathbf{a}(\vartheta_{1})\ \dots\
\mathbf{a}(\vartheta_{K})]\in\mathbb{C}^{N_{P}\times K}$ is the AoA steering
matrix, and $\mathbf{B}_{q}^{r}\in\mathbb{C}^{K\times K}$ is the radar channel
coefficient diagonal matrix. The diagonal entries of the matrix are defined by
the channel vector $\boldsymbol{\beta}_{q}^{r}=[\beta_{q,1}^{r}\ \dots\
\beta_{q,K}^{r}]^{\mathrm{T}},\in\mathbb{C}^{K\times 1}$. The received signal
model at the PWR is thus given by

$\mathbf{Z}_{q}^{r}=\mathbf{H}_{q}^{r}\mathbf{S}_{q}+\mathbf{N}_{q}^{r},$ (2)

where $\mathbf{N}_{q}^{r}\in\mathbb{C}^{N_{P}\times N_{A}}$ denotes the radar
Additive White Gaussian Noise (AWGN) matrix, and
$\mathbf{S}_{q}\in\mathbb{C}^{N_{A}\times N_{A}}$ is the transmitted signal
per subcarrier. The columns of $\mathbf{S}_{q}$ represent the different HE-LTF
OFDM symbols, and the rows represent the different AP antennas. Both the
client UE and the PWR know the structure of the transmitted $\mathbf{S}_{q}$
matrix from the information contained in the preamble of the NDP, as it is
well defined in the 802.11ax amendment [3].

###  III-B Communication Channel Model

The communication channel between the AP and the $u\textsuperscript{th}$
client UE consists of the LoS link and $K_{u}\scalebox{0.75}[1.0]{$-$}1$
multipath components resulting from reflections on other targets and the
clutter. In this scenario, the clutter contributions cannot be effectively
removed from the communication channel, as the client UE does not perform any
suppression before computing its BFF. However, it is assumed that the LoS link
is the strongest. The LoS AoD is denoted by $\varphi_{u}$ and the set
$\boldsymbol{\Phi}_{u}$ comprises all multipath AoDs including the reflection
on both the other $K-1$ targets and the clutter. The baseband equivalent
channel matrix for the communication channel of the $u\textsuperscript{th}$
client UE is defined for each subcarrier $q$ as follows

$\mathbf{H}_{u,q}^{c}=\boldsymbol{\beta}_{u,q}^{c}\mathbf{a}^{\mathrm{H}}(\varphi_{u})+\mathbf{B}_{u,q}^{c}\
\mathbf{A}^{\mathrm{H}}(\boldsymbol{\Phi}_{u}),\in\mathbb{C}^{N_{u}\times
N_{A}},$ (3)

where $\boldsymbol{\beta}_{u,q}^{c}\in\mathbb{C}^{N_{u}\times 1}$ and
$\mathbf{B}_{u,q}^{c}\in\mathbb{C}^{N_{u}\times(K_{u}-1)}$ represent the LoS
and multipath communication channel coefficients, respectively. Each column
represents a different path, while the rows represent the different antennas
of the UE. These coefficients encompass the linearly increasing phases across
subcarriers and across receiving antennas, attributed to the range and AoA of
the paths, respectively, as well as the attenuation defined by the radar range
equation. The received signal model at the client UE is given by

$\mathbf{Z}_{u,q}^{c}=\mathbf{H}_{u,q}^{c}\mathbf{S}_{q}+\mathbf{N}_{u,q}^{c},$
(4)

where $\mathbf{N}_{u,q}^{c}\in\mathbb{C}^{N_{u}\times N_{A}}$ denotes the AWGN
matrix.

##  IV Hybrid Radar Processing

We formulate the ML combination rule at the PWR for determining the positions
of the $K$ targets based on the sniffed BFFs and the processing of the NDP. In
multitarget localization scenarios, we seek the set of $(x,y)$ positions of
the $K$ targets that maximizes the sum of the log-likelihood function of each
independent source of information. Therefore, the brute-force maximization
implies solving a $2K$-dimensional problem. In our previous work [7], we
demonstrated that in a multistatic scenario comprising multiple AP/PWR radar
pairs, the $2K$-dimensional log-likelihood functions can be approximately
decoupled into $K$ per-target two-dimensional functions. This method relies on
a pre-estimate of the target angles
$(\widehat{\boldsymbol{\Phi}},\widehat{\boldsymbol{\Theta}})$ at each radar
pair using the MUSIC algorithm. Furthermore, this so-called alternating
summation method relies only on the sample covariance matrices.

In this section, we develop the ML estimator based on the received signals for
the radar channel in (2) and the communication channels in (4) to show how the
alternating summation method presented in [7] for a multistatic scenario can
be adapted to the hybrid fusion considered in this paper.

###  IV-A Maximum Likelihood Fusion

As described in Section III, the PWR and the client UEs exploit the HE-LTF
symbols transmitted isotropically by the AP to evaluate the radar and
communication channel matrices, respectively. The CSI matrices estimated by
the PWR and the client UEs are thus obtained by multiplying the received
signals in (2) and (4) by $\mathbf{S}_{q}^{\mathrm{H}}$ since
$\mathbf{S}_{q}\mathbf{S}_{q}^{\mathrm{H}}=N_{A}\mathbf{I}$.

To localize the $K$ targets in the scene, the PWR exploits a hybrid source of
information: the joint AoD/AoA extracted from the NDP and the LoS AoD from the
BFFs. This is highlighted in the ML formulation by expressing the estimated
CSI matrices as

$\displaystyle\widetilde{\mathbf{h}}_{q}^{r}$
$\displaystyle=\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})\
\boldsymbol{\beta}_{q}^{r}+\mathbf{n}_{q}^{r\prime},$ (5)
$\displaystyle\widetilde{\mathbf{H}}_{u,q}^{c}$
$\displaystyle=\boldsymbol{\beta}_{u,q}^{c}\mathbf{a}^{\mathrm{H}}(\varphi_{u})+\mathbf{N}_{u,q}^{c\prime},$
(6)

where $\widetilde{\mathbf{h}}_{q}^{r}\in\mathbb{C}^{N_{A}N_{P}\times 1}$ is
the vector obtained by vectorizing the radar CSI matrix,
$\mathbf{n}_{q}^{r\prime}$ and $\mathbf{N}_{u,q}^{c\prime}$ are the noise
vector and matrix of variances $\sigma_{r}^{2}$ and $\sigma_{u}^{2}$ resulting
from the symbol equalization, and
$\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})=[\mathbf{a}^{\prime}(\varphi_{1},\vartheta_{1})\
\dots\
\mathbf{a}^{\prime}(\varphi_{K},\vartheta_{K})]\in\mathbb{C}^{N_{A}N_{P}\times
K}$ represents the joint AoD/AoA matrix in which
$\mathbf{a}^{\prime}(\varphi_{k},\vartheta_{k})=\mathbf{a}(\varphi_{k})\otimes\mathbf{a}(\vartheta_{k}),\mathbb{C}^{N_{A}N_{P}\times
1}$. Observe that in (6), the multipath term $\mathbf{B}_{u,q}^{c}\
\mathbf{A}^{\mathrm{H}}(\boldsymbol{\Phi}_{u})$ is omitted from the model of
the observations as only the LoS AoD will be retrieved from the BFFs. This
model mismatch has thus an impact on the ML estimator but can not be avoided
in this scenario.

The parameters to be estimated are defined by the vector
$\boldsymbol{\gamma}=[\mathbf{x}^{\mathrm{T}}\mathbf{y}^{\mathrm{T}}\\{\boldsymbol{\beta}_{q}^{r}\\}_{q=1\dots
Q}\\{\boldsymbol{\beta}_{u,q}^{c}\\}_{q=1\dots Q}^{u=1\dots C}]^{\mathrm{T}}$.
It is assumed that the number of targets to be localized is known, as methods
for estimating $K$ are available in the literature [9]. Notice that our study
solely focuses on angle-based ML localization and does not utilize range
information. Therefore, each channel coefficient vector
$\boldsymbol{\beta}_{q}^{r},\boldsymbol{\beta}_{u,q}^{c}$ is independently
estimated, as the linear phase increase across subcarriers, defined by the
range of each target, is not exploited. Considering independent noise
contributions for the estimated CSIs, the combined likelihood function is
derived as the product of individual Gaussian density functions. After taking
the natural logarithm of the combined likelihood function, the following
expression must be maximized

$\widehat{\boldsymbol{\gamma}}=\operatorname*{arg\,max}_{\boldsymbol{\gamma}}\mathcal{L}(\boldsymbol{\gamma})=\operatorname*{arg\,max}_{\boldsymbol{\gamma}}\mathcal{L}^{r}(\boldsymbol{\gamma})+\sum_{u=1}^{C}\mathcal{L}_{u}^{c}(\boldsymbol{\gamma}),$
(7)

where the log-likelihood from the PWR and the clients CSIs are respectively
given by

$\displaystyle\mathcal{L}^{r}(\boldsymbol{\gamma})$
$\displaystyle=\frac{-1}{2\sigma_{r}^{2}}\sum_{q=1}^{Q}\lVert\widetilde{\mathbf{h}}_{q}^{r}-\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})\
\boldsymbol{\beta}_{q}^{r}\rVert^{2},$ (8)
$\displaystyle\mathcal{L}_{u}^{c}(\boldsymbol{\gamma})$
$\displaystyle=\frac{-1}{2\sigma_{u}^{2}}\sum_{q=1}^{Q}\lVert\widetilde{\mathbf{H}}_{u,q}^{c}-\boldsymbol{\beta}_{u,q}^{c}\mathbf{a}^{\mathrm{H}}(\varphi_{u})\rVert_{\mathrm{F}}^{2},$
(9)

First, we maximize with respect to the channel coefficients
$\boldsymbol{\beta}_{q}^{r}$ and $\boldsymbol{\beta}_{u,q}^{c}$ to obtain
closed-form expressions as function of the angles. Following the solution of
the resulting linear least squares problem, the ML estimate of the channel
coefficients for every subcarrier $q$ is expressed as
$\widehat{\boldsymbol{\beta}}_{q}^{r}(\boldsymbol{\Phi},\boldsymbol{\Theta})=\left(\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})\right)^{+}\
\widetilde{\mathbf{h}}_{q}^{r}$ and
$\widehat{\boldsymbol{\beta}}_{u,q}^{c}(\varphi_{u})=\widetilde{\mathbf{H}}_{u,q}^{c}\left(\mathbf{a}^{+}(\varphi_{u})\right)^{\mathrm{H}}$.

Substituting these estimates back into (8) and (9), and after performing some
mathematical manipulations, the two log-likelihood functions can be
reformulated as

$\displaystyle\mathcal{L}^{r}(\boldsymbol{\gamma})$
$\displaystyle=\frac{Q}{2\sigma_{r}^{2}}\operatorname{Tr}\left\\{\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})\left(\mathbf{A}^{\prime}(\boldsymbol{\Phi},\boldsymbol{\Theta})\right)^{+}\
\widetilde{\mathbf{R}}^{r}\right\\},$ (10)
$\displaystyle\mathcal{L}_{u}^{c}(\boldsymbol{\gamma})$
$\displaystyle=\frac{QN_{u}}{2\sigma_{u}^{2}}\operatorname{Tr}\left\\{\mathbf{a}(\varphi_{u})\mathbf{a}^{+}(\varphi_{u})\
\widetilde{\mathbf{R}}_{u}^{c}\right\\},$ (11)

in which the sample covariance matrices of the channel averaged over the
subcarriers are given by

$\displaystyle\widetilde{\mathbf{R}}^{r}$
$\displaystyle=\frac{1}{Q}\sum_{q=1}^{Q}\widetilde{\mathbf{h}}_{q}^{r}\
(\widetilde{\mathbf{h}}_{q}^{r})^{\mathrm{H}}.$ (12)
$\displaystyle\widetilde{\mathbf{R}}_{u}^{c}$
$\displaystyle=\frac{1}{QN_{u}}\sum_{q=1}^{Q}(\widetilde{\mathbf{H}}_{u,q}^{c})^{\mathrm{H}}\
\widetilde{\mathbf{H}}_{u,q}^{c},$ (13)

###  IV-B Channel State Information and Beamforming Feedback

As described in Section II, the PWR does not have access to the full CSI from
the client UEs, but only to the BFFs. Each client UE constructs this feedback
by evaluating the SVD of the estimated CSI matrix, yielding:

$\widetilde{\mathbf{H}}_{u,q}^{c}=\widetilde{\mathbf{U}}_{u,q}^{c}\widetilde{\boldsymbol{\Sigma}}_{u,q}^{c}(\widetilde{\mathbf{V}}_{u,q}^{c})^{\mathrm{H}},$
(14)

where $\widetilde{\mathbf{U}}_{u,q}^{c}$ and
$\widetilde{\mathbf{V}}_{u,q}^{c}$ represent the left and right unitary
singular matrices, respectively, and
$\widetilde{\boldsymbol{\Sigma}}_{u,q}^{c}$ is the singular values diagonal
matrix. As defined in the 802.11ax standard [3], the MU-MIMO BFF contains the
compressed right singular vectors $\widetilde{\mathbf{V}}_{u,q}^{c}$ for each
subcarrier, the subcarrier-averaged stream gain and for each stream and
subcarrier the delta signal-to-noise (SNR) with respect to this subcarrier-
averaged stream gain. This gives access to a quantized version of the singular
value matrix $\widetilde{\boldsymbol{\Sigma}}_{u,q}^{c}$. Details on the
compression can be found in [3]. Assuming the LoS link between the AP and the
client UE to be much stronger than the multipath components, it can be shown
that the sample covariance matrix in (13) can be approximated from the BFF as
follows:

$\widetilde{\mathbf{R}}_{u}^{c}\approx\frac{1}{QN_{u}}\sum_{q=1}^{Q}\widetilde{\mathbf{V}}_{u,q}^{c,1}\widetilde{\boldsymbol{\Sigma}}_{u,q}^{c,1}(\widetilde{\mathbf{V}}_{u,q}^{c,1})^{\mathrm{H}}.$
(15)

where $\widetilde{\mathbf{V}}_{u,q}^{c,1}$ and
$\widetilde{\boldsymbol{\Sigma}}_{u,q}^{c,1}$ are obtained by keeping only the
strongest stream of the SVD. The alternating summation method with an
association step is thus evaluated at the PWR from the sample covariance
matrices defined in (12) and (15).

##  V Simulation Results

(a)

(b)

Figure 2: The impact of the signal-to-noise ratio on the performance metrics
of the various methods is depicted. The solid and dashed lines correspond to
the localization performance for the client and non-client targets,
respectively.

This section evaluates the enhancement brought by the exploitation of the BFFs
at the PWR on the accuracy of the radar in localizing targets within its
coverage area. The joint NDP and BFF processing outlined in Section IV is
compared through $30,000$ simulations with a MUSIC algorithm based on the NDP
only. It is also compared to a sum of the MUSIC pseudo-spectrums obtained from
the NDP and the BFFs. The following performance metrics are studied:

  1. 1.

Hit Rate: A hit is acknowledged when a peak is detected within a vicinity of 2
meters from the true target position; otherwise, it is considered a miss.

  2. 2.

RMSE: The root mean square error (RMSE) is computed between the true target
position and the corresponding detected position. To ensure a fair comparison
between the methods, only the targets resulting in a hit by all methods are
considered in the evaluation of the RMSE.

For each simulation, we analyze a fixed AP and PWR setup designated to locate
two client targets and a non-client target randomly positioned. The number of
antennas are $N_{A}=N_{u}=N_{P}=4$ and the number of subcarriers is $Q=512$.
Figure 2 depicts the results obtained for the hit rate and the RMSE for the
client and non client targets as functions of the SNR. The SNR is defined here
as the mean of the quotient of the radar channel coefficients of each path by
the noise variance. Note that the same noise variance is considered at the
clients and the PWR. It is observed that for both the client and the non-
client target, the hybrid fusion of the NDP and the BFF based on the
alternating summation method enhances the achievable RMSE compared to the
exploitation of the NDP only. However, the hybrid fusion rule must be well
defined, as a simple combination of MUSIC pseudo-spectrums deteriorates the
RMSE. Furthermore, it can be observed that only the alternating projection
method achieves an improved RMSE for the client target compared to the non-
client one. This results from the improved ability of the alternating
summation method to account for the varying localization ability of the NDP
and the BFF. Regarding the hit rate, the alternating summation presents
similar performance at high SNRs as the NDP-based MUSIC. At low SNRs, we
observe that it improves the hit rate for the client targets. However, it can
decrease the hit rate of the non-client target since the association step is
not always well performed at low SNRs. Globally, these results confirm the
benefit of the hybrid combination when the fusion rule ùis appropriately
chosen.

##  VI Conclusion

In this paper, we investigate the multitarget localization capabilities of a
PWR during the MU-MIMO channel sounding session initiated by an AP. Both the
joint AoD/AoA extracted from the NDP transmitted by the AP and the LoS AoD
extracted from the BFFs transmitted by the client UEs are explored for their
potential utility by the PWR. The proposed hybrid radar fusion method is
derived from the maximum likelihood framework. It is shown to solely rely on
the computation, for all source of information, of the approximated sample
covariance matrices of the channel state information. Numerical simulations
presented in this study validate the effectiveness of the proposed approach.
The results highlight the benefits of exploiting BFF in a hybrid fusion for
target localization compared to classical PWR processing based solely on NDPs.
In future works, the exploitation of range information from the NDP would
further enhance localization accuracy.

## References

  * [1] F. Colone, P. Falcone, C. Bongioanni, and P. Lombardo, “Wifi-based passive bistatic radar: Data processing schemes and experimental results,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 48, no. 2, pp. 1061–1079, 2012.
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  * [3] “Wireless LAN medium access control (MAC) and physical layer (PHY) specifications amendment 1: Enhancements for high-efficiency WLAN,” _IEEE Std 802.11ax-2021_ , pp. 1–767, 2021.
  * [4] S. Itahara, S. Kondo, K. Yamashita, T. Nishio, K. Yamamoto, and Y. Koda, “Beamforming feedback-based model-driven angle of departure estimation toward legacy support in wifi sensing: An experimental study,” _IEEE Access_ , vol. 10, pp. 59 737–59 747, 2022.
  * [5] H. C. Yildirim, L. Storrer, M. Willame, J. Louveaux, P. De Doncker, and F. Horlin, “Passive wi-fi-based radars with 802.11ax MU-MIMO signals: AoD estimation with a single antenna,” in _2023 IEEE 3rd International Symposium on Joint Communications & Sensing (JC&S)_, 2023, pp. 1–6.
  * [6] P. Stoica and K. Sharman, “Maximum likelihood methods for direction-of-arrival estimation,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 38, no. 7, pp. 1132–1143, 1990.
  * [7] M. Willame, L. Storrer, H. C. Yildirim, F. Horlin, and J. Louveaux, “Maximum Likelihood Alternating Summation for Multistatic Angle-based Multitarget Localization,” Mar. 2024. [Online]. Available: http://arxiv.org/abs/2403.13992
  * [8] S. Bartoletti, A. Conti, and M. Z. Win, “Device-free counting via OFDM signals of opportunity,” in _2018 IEEE International Conference on Communications Workshops (ICC Workshops)_ , 2018, pp. 1–5.
  * [9] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 33, no. 2, pp. 387–392, 1985.


#  On common zeros of entire functions of exponential growth

B. Kazarnovskii  Institute for Information Transmission Problems of the
Russian Academy of Sciences
<EMAIL_ADDRESS>
###### Abstract.

For systems of equations with an infinite set of roots, one can sometimes
obtain Kushnirenko-Bernstein-Khovanskii type theorem if replace the number of
roots by their asymptotic density. We consider systems of entire functions
with exponential growth in the space $\mathbb{C}^{n}$, and calculate the
asymptotic distribution of their common zeros in terms of the geometry of
convex sets in the space $\mathbb{C}^{n}$.

###### Contents

  1. 1 Introduction
    1. 1.1 Asymptotics of Zero Distribution
    2. 1.2 Functions of Exponential Growth
    3. 1.3 Averaged Distribution of Roots
    4. 1.4 Main Result
  2. 2 Averaged Distribution on a Complex Manifold
  3. 3 Regular Subspaces
  4. 4 Proof of Theorem 1
  5. 5 Proof of Theorem 2

##  1\. Introduction

###  1.1. Asymptotics of Zero Distribution

Consider the set $X$ of common zeros of entire functions of exponential growth
$f_{1},\ldots,f_{m}$ in ${\mathbb{C}}^{n}$. Let $X_{\rm reg}$ denote the set
of points $z\in X$ such that the co-dimension of the analytic set $X$ in the
neighborhood of the point $z$ is equal to $m$. In the generic situation,
$X_{\rm reg}=X$. Further, we consider $X$ as a current
$\mathfrak{M}(f_{1},\ldots,f_{m})$, i.e., as the integration of compactly
supported differential forms of degree $2n-2m$ over the analytic set $X_{\rm
reg}$. For example, if $m=n$, then
$\mathfrak{M}(f_{1},\ldots,f_{n})=\sum_{x}\delta(x)$, where $\delta(x)$ is the
delta function with support at an isolated root $x$ of the system
$f_{1}=\ldots=f_{n}=0$. For a fixed $t>0$, let us denote the function
$f_{j}(tz)$ by $f_{t,j}(z)$. If the limit of the currents
$\frac{1}{t^{m}}\mathfrak{M}(f_{t,1},\ldots,f_{t,m})$ exists as $t\to+\infty$,
we call it the _asymptotic distribution_ of the variety of common zeros of
entire functions of exponential growth $f_{1},\ldots,f_{m}$.

######  Example 1.1.

Let $f$ be a quasi-polynomial in one variable $z$ with spectrum $K$, i.e., a
finite sum of the form

$f(z)=\sum_{\xi\in K\subset{\mathbb{C}}}p_{\xi}(z)\>{\rm e}^{\xi z},$

where $p_{\xi}$ are polynomials in the variable $z$. Then the asymptotic
distribution of zeros of the function $f$ exists. In this case, the current of
the asymptotic distribution is a distribution in ${\mathbb{C}}$. If
$K\subset{\rm Re}{\mathbb{C}}$, we denote by $[\alpha,\beta]$ the minimal
interval in ${\rm Re}{\mathbb{C}}$ containing all points of the finite set
$K$. If the polynomials $p_{\alpha},p_{\beta}$ are non-zero, then the value of
the asymptotic density current $\mathfrak{M}(f)$ on a compactly supported
function $\varphi$ is equal to $\frac{\beta-\alpha}{2\pi}\int_{{\rm
Im}{\mathbb{C}}}\varphi\>dy$.

###  1.2. Functions of Exponential Growth

Recall that a linear functional on the space of holomorphic functions on the
complex manifold $M$ is called an analytic functional on $M$; see [1]. Let
${{\mathbb{C}}^{n}}^{*}$ be the space of linear functionals in
${\mathbb{C}}^{n}$, and $\mu$ be an analytic functional in
${{\mathbb{C}}^{n}}^{*}$, i.e., $\mu$ is a linear functional on the space of
entire holomorphic functions in ${{\mathbb{C}}^{n}}^{*}$. For a fixed
$z\in{\mathbb{C}}^{n}$, consider $\>{\rm e}^{\xi(z)}$ as an entire function
with $\xi\in{{\mathbb{C}}^{n}}^{*}$ as its argument. Define
$\hat{\mu}(z)=\mu(\>{\rm e}^{\xi(z)})$. It is worth noting that the entire
function $\hat{\mu}$ on ${\mathbb{C}}^{n}$ is called the Borel-Fourier
transform (or Laplace transform; see [2]) of the analytic functional $\mu$.
The Borel-Fourier transform is a bijective map from the space of analytic
functionals in ${{\mathbb{C}}^{n}}^{*}$ to the space of all entire functions
of exponential growth in ${\mathbb{C}}^{n}$, i.e., entire functions
$g\colon{\mathbb{C}}^{n}\to{\mathbb{C}}$, such that

$\exists(C,a)\colon\>\forall z\in{\mathbb{C}}^{n}\colon|g(z)|\leq C\>{\rm
e}^{a|z|}.$

For an analytic functional $\mu$, there exists a compact set $K\subset M$ for
which the following holds. For any open neighborhood $U$ of the compact set
$K$, there exists a constant $C_{U}$ such that

$\forall f\colon|\mu(f)|\leq C_{U}\sup_{U}|f(\xi)|.$ (1.1)

The compact set $K$ is called the support111There is no guarantee of the
existence of the minimal support of an analytic functional; see [1, Chapter 8]
or [2, §4.5]. of the functional $\mu$.

Let $\mathcal{S}(K)$ be the space of analytic functionals in
${{\mathbb{C}}^{n}}^{*}$ with support $K$. Denote by $\hat{\mathcal{S}}(K)$
the space of entire functions in ${\mathbb{C}}^{n}$, consisting of Borel-
Fourier transforms of functionals $\mu\in\mathcal{S}(K)$. For $K\subset{\rm
Re}{{\mathbb{C}}^{n}}^{*}$, the exact description of functions in
$\hat{\mathcal{S}}(K)$ is given by the Paley-Wiener theorem; see, for example,
[3, Theorem 1.7.7].

######  Example 1.2.

If the compact set $K\subset{{\mathbb{C}}^{n}}^{*}$ is finite, then the space
$\hat{\mathcal{S}}(K)$ contains all quasi-polynomials with spectrum $K$, i.e.,
entire functions in ${\mathbb{C}}^{n}$ of the form

$\sum_{\xi\in K}p_{\xi}(z)\>{\rm e}^{\xi(z)},$ (1.2)

where the coefficients $p_{\xi}$ are polynomials in the variables
$z_{1},\ldots,z_{n}$.

Define the function $h_{K}\colon{\mathbb{C}}^{n}\to{\mathbb{R}}$ as
$h_{K}(z)=\max_{\xi\in K}{\rm Re}\xi(z)$. Recall that $h_{K}$ is called the
support function of the compact set $K$. From (1.1), it follows (see §3) that
for any function $f\in\hat{\mathcal{S}}(K)$

$\forall
z\in{\mathbb{C}}^{n}\colon\>\operatorname*{\overline{lim}}_{t\to+\infty}\frac{\log|f(tz)|}{t}\leq
h_{K}(z)$ (1.3)

###  1.3. Averaged Distribution of Roots

Let
$V_{1}\subset\hat{\mathcal{S}}(K_{1}),\ldots,V_{m}\subset\hat{\mathcal{S}}(K_{m})$
be finite-dimensional spaces. Recall that we consider the variety of roots of
the system

$f_{1}=\ldots=f_{m}=0;\,0\neq f_{i}\in V_{i}$ (1.4)

as the current $\mathfrak{M}(f_{1},\ldots,f_{m})$. Using arbitrary Hermitian
inner products $\langle*,*\rangle_{i}$ in the spaces $V_{i}$, we define the
_averaging $\mathfrak{M}(V_{1},\ldots,V_{m})$ of the current_
$\mathfrak{M}(f_{1},\ldots,f_{m})$ over all systems (1.4); see Definition 2.1
in §2.

######  Definiton 1.1.

For a finite-dimensional space $V\subset\hat{\mathcal{S}}(K)$, let
$V(t)=\\{f_{t}(z)=f(tz)\colon f\in V\\}$. For a Hermitian inner product
$\langle*,*\rangle$ in $V$, define $\langle f_{t},g_{t}\rangle=\langle
f,g\rangle$. If, in the topological space of currents in ${\mathbb{C}}^{n}$,
the limit

$\mathfrak{M}^{\infty}(V_{1},\ldots,V_{m})=\lim_{t\to+\infty}\frac{1}{t^{m}}\mathfrak{M}(V_{1}(t),\ldots,V_{m}(t))$

exists and does not depend on the choice of inner products
$\langle*,*\rangle_{j}$ in spaces $V_{j}$, then we call it the _averaged
asymptotic distribution_ of roots of systems (1.4).

###  1.4. Main Result

Recall that, by definition, for a function $f$ on the complex manifold $M$,
the value of the 1-form $d^{c}f$ on the tangent vector $\xi$ is equal to
$df(-i\xi)$. If continuous functions $h_{1},\ldots,h_{m}\colon
M\to{\mathbb{R}}$ are plurisubharmonic functions on the manifold $M$, then
$dd^{c}h_{1}\wedge\ldots\wedge dd^{c}h_{m}$ is a well-defined positive current
of degree $2m$; see, for example, [1].

######  Remark 1.1.

Currents of the form $dd^{c}h_{1}\wedge\ldots\wedge dd^{c}h_{n}$ are called
Monge-Ampère measures. Such measures traditionally appear in problems related
to the distribution of zeros of holomorphic systems of equations; see [1, 4,
5].

The support functions $h_{i}$ of the compacts
$K_{i}\subset{{\mathbb{C}}^{n}}^{*}$ are convex and, consequently,
plurisubharmonic. In particular, the current $dd^{c}h_{1}\wedge\ldots\wedge
dd^{c}h_{n}$ is a non-negative measure in ${\mathbb{C}}^{n}$.

Below is defined the concept of a regular finite-dimensional subspace in
$\hat{\mathcal{S}}(K)$; see Definition 3.1 in §3, as well as an example at the
end of §1.4.

######  Theorem 1.

Let
$V_{1}\subset\hat{\mathcal{S}}(K_{1}),\ldots,V_{m}\subset\hat{\mathcal{S}}(K_{m})$
be regular spaces. Then the current of the averaged asymptotic distribution of
roots of systems (1.4) exists and is given by the formula

$\mathfrak{M}^{\infty}(V_{1},\ldots,V_{m})=\frac{1}{\pi^{m}}dd^{c}h_{1}\wedge\ldots\wedge
dd^{c}h_{m},$

where $h_{i}$ is the support function of compact $K_{i}$.

######  Definiton 1.2.

Let $h$ be the support function of a convex compact
$K\subset{{\mathbb{C}}^{n}}^{*}$. Put ${\rm pvol}(K)=\int_{B}(dd^{c}h)^{n}$,
where $B$ is the ball of radius $1$ centered at the origin. The value ${\rm
pvol}(K)$ is called the _pseudo-volume of the convex body_ $K$. If
$h_{1},\ldots,h_{n}$ are the support functions of convex compacts
$A_{1},\ldots,A_{n}$, then

${\rm pvol}(A_{1},\ldots,A_{n})=\int_{B}dd^{c}h_{1}\wedge\ldots\wedge
dd^{c}h_{n}$

is called the _mixed pseudo-volume of convex bodies_ $A_{1},\ldots,A_{n}$.

######  Remark 1.2.

The pseudo-volume is a unitarily invariant valuation on convex bodies in
complex vector spaces; see [6].

From Theorem 1, the following statements follow.

######  Corollary 1.1.

Let $V_{1},\ldots,V_{n}$ be regular subspaces in the spaces
$\hat{\mathcal{S}}(K_{1}),\ldots,\hat{\mathcal{S}}(K_{n})$, respectively. Then
the asymptotic average, over all systems (1.4), of the number of roots in a
growing-radius ball centered at the origin is asymptotically given by

${\rm pvol}\left({\rm conv}(K_{1}),\ldots,{\rm conv}(K_{n})\right)\>r^{n},$

where ${\rm conv}(K)$ denotes the convex hull of the compact $K$.

If $K\subset{\rm Re}{{\mathbb{C}}^{n}}^{*}$, then the pseudo-volume of $K$ is
equal to its volume as a body of full dimension in the $n$-dimensional real
space ${\rm Re}{{\mathbb{C}}^{n}}^{*}$. Therefore we get the following
statement; see [7].

######  Corollary 1.2.

Let $K_{1},\ldots,K_{n}\subset{\rm Re}{{\mathbb{C}}^{n}}^{*}$. Then

$\mathfrak{M}^{\infty}(V_{1},\ldots,V_{n})=\frac{n!}{(2\pi)^{n}}{\rm
vol}\left({\rm conv}(K_{1}),\ldots,{\rm conv}(K_{n})\right)\>\mu_{n},$

where ${\rm vol}(*,\ldots,*)$ is the mixed volume of convex bodies in
$n$-dimensional real space ${\rm Re}{{\mathbb{C}}^{n}}^{*}$, and $\mu_{n}$ is
a measure in ${\mathbb{C}}^{n}$ with support in ${\rm Im}{\mathbb{C}}^{n}$,
defined as an integral of a function with respect to the Lebesgue measure on
the space ${\rm Im}{\mathbb{C}}^{n}$.

Finally, let us provide an example, which is the source of the concept of a
regular space. Later in §1.4, it is assumed that the compact $K$ is a finite
set. Let $\hat{\mathcal{S}}_{N}(K)$ denote the space of quasi-polynomials of
the form (1.2), such that ${\rm deg}(p_{\xi})\leq N$ for all $\xi\in K$. For
example, $\hat{\mathcal{S}}_{0}(K)$ consists of exponential sums with spectrum
$K$, i.e., functions of the form $\sum_{\xi\in
K,c_{\xi}\in{\mathbb{C}}\>}c_{\xi}\>{\rm e}^{\xi(z)}$.

######  Theorem 2.

Any finite-dimensional space $V\subset\hat{\mathcal{S}}(K)$, such that
$V\supset\hat{\mathcal{S}}_{0}(K)$, is regular. In particular, for any $N\geq
0$, the space of quasi-polynomials $\hat{\mathcal{S}}_{N}(K)$ is regular.

From here follows the old result on the asymptotic density of the variety of
roots of systems of exponential sums [4, 8].

##  2\. Averaged Distribution on a Complex Manifold

Let $V_{1},\ldots,V_{m}$ be finite-dimensional spaces of holomorphic functions
on an $n$-dimensional manifold $X$, and $\langle*,*\rangle_{i}$ be a Hermitian
scalar product in the space $V_{i}$. The roots of the system

$f_{1}=\ldots=f_{m}=0;\,0\neq f_{i}\in V_{i}$ (2.1)

depend solely on the projections $p_{i}=\pi_{i}(f_{i})$ of the points
$f_{i}\in V_{i}\setminus 0$ onto the projectivizations $\mathbb{P}_{i}$ of the
spaces $V_{i}$. We consider the variety of roots of the system (2.1) as a
current $\mathfrak{M}(f_{1},\ldots,f_{m})$. Furthermore, if necessary, we
identify the system (2.1) with a point
$(p_{1},\ldots,p_{m})\in\mathbb{P}_{1}\times\ldots\times\mathbb{P}_{m}$. Using
the Hermitian metrics $\langle*,*\rangle_{i}$, we define and calculate the
averaging $\mathfrak{M}(V_{1},\ldots,V_{m})$ of the current
$\mathfrak{M}(f_{1},\ldots,f_{m})$ over all systems (2.1). For this purpose,
we employ the corresponding products of $\langle*,*\rangle_{i}$ and the
Fubini-Study metrics in the spaces $\mathbb{P}_{i}$; see, for example, [9].
Let $\Omega_{i}$ denote the corresponding volume form in $\mathbb{P}_{i}$,
normalized as $\int_{\mathbb{P}_{i}}\Omega_{i}=1$.

######  Definiton 2.1.

Suppose $\pi_{i}(f_{i})=p_{i}$, where $\pi_{i}\colon V_{i}\setminus
0\to\mathbb{P}_{i}$ is the projection mapping. Identify the system (2.1) with
a point
$(p_{1},\ldots,p_{m})\in\mathbb{P}_{1}\times\ldots\times\mathbb{P}_{m}$. The
current on $X$, defined as

$\mathfrak{M}(V_{1},\ldots,V_{m})=\int_{\mathbb{P}_{1}\times\ldots\times\mathbb{P}_{m}}\mathfrak{M}(f_{1},\ldots,f_{m})\>\Omega_{1}\wedge\ldots\wedge\Omega_{m}$

is called the averaged distribution of roots of systems of the form (2.1).

######  Definiton 2.2.

Furthermore, we assume that

$\forall(x\in X,i\leq m)\>\exists f\in V_{i}\colon f(x)\neq 0$

Let $V^{*}_{i}$ be the space of linear functionals on $V_{i}$. Define the
mapping $\Theta_{i}\colon X\to V^{*}_{i}\setminus 0$ as $\Theta_{i}(x)\colon
f\mapsto f(x)$. Consider the Hermitian product $\langle*,*\rangle^{*}_{i}$ in
the space $V^{*}_{i}$, conjugate to the product $\langle*,*\rangle_{i}$. For
$x\in X$, let
$\|x\|_{i}=\sqrt{\langle\Theta_{i}(x),\Theta_{i}(x)\rangle^{*}_{i}}$.

######  Theorem 3.

It holds that

$\mathfrak{M}(V_{1},\ldots,V_{m})=\frac{1}{(2\pi)^{m}}dd^{c}\log\|x\|^{2}_{1}\wedge\ldots\wedge
dd^{c}\log\|x\|^{2}_{m}$

The assertion of the theorem is local. That is, for its proof, one can replace
$X$ with any small neighborhood of the point $x\in X$. Therefore, by Sard’s
lemma, the theorem is reduced to the case where the mapping

$X\xrightarrow{\Theta_{1}\times\ldots\times\Theta_{m}}(V^{*}_{1}\setminus
0)\times\ldots\times(V^{*}_{m}\setminus
0)\xrightarrow{\pi^{*}_{1}\times\ldots\times\pi^{*}_{m}}\mathbb{P}^{*}_{1}\times\ldots\times\mathbb{P}^{*}_{m},$

where $\mathbb{P}^{*}_{i}$ is the dual of $\mathbb{P}_{i}$ projective space,
and $\pi^{*}_{i}\colon V^{*}_{i}\to\mathbb{P}^{*}_{i}$ is the projection
mapping, is a closed embedding.

Let $\omega^{*}_{i}$ be a Kähler form in $\mathbb{P}^{*}_{i}$, such that its
integral over the projective line is equal to one. Then the form
$\frac{1}{2\pi}dd^{c}\log(\|f\|^{*}_{i})^{2}$ in $V^{*}_{i}\setminus 0$ is the
pullback of the Kähler form $\omega^{*}_{i}$ in $\mathbb{P}^{*}_{i}$ under the
projection mapping. Therefore, Theorem 3 can be reduced to the following
statement.

######  Theorem 4.

Let $X$ be a closed complex manifold with a boundary in
$\mathbb{P}^{*}_{1}\times\ldots\times\mathbb{P}^{*}_{m}$,
$p_{i}\in\mathbb{P}_{i}$, and $H(p_{i})$ be a projective hypersurface in
$\mathbb{P}^{*}_{i}$ defined by the equation $p_{i}(*)=0$. Consider
$H(p_{1},\ldots,p_{m})=H(p_{1})\times\ldots\times H(p_{m})$. Then, for any
differential form $\varphi$ of degree $2n-2m$ on the manifold
$\mathbb{P}^{*}_{1}\times\ldots\times\mathbb{P}^{*}_{m}$, it holds that

$\int_{(p_{1},\ldots,p_{m})\in\mathbb{P}_{1}\times\ldots\times\mathbb{P}_{m}}\left(\int_{X\cap
H(p_{1},\ldots,p_{m})}\varphi\right)\Omega_{1}\wedge\ldots\wedge\Omega_{m}=\frac{1}{(2\pi)^{m}}\int_{X}\varphi\wedge\omega^{*}_{1}\wedge\ldots\wedge\omega^{*}_{m}$

Theorem 4 can be identified as the Crofton formula for the product of
projective spaces, as announced in [8]. Here is a brief derivation of Theorem
4.

As is customary in integral geometry (see, for example, [9]), Theorem 4 can be
reformulated in the language of double fibrations. Let

$\Gamma(\mathbb{P})=\\{(q,p)\colon p\in\mathbb{P},q\in\mathbb{P}^{*},p\in
H(q)\\},$

$\gamma(\mathbb{P})\colon(p,q)\mapsto q$, and
$\delta(\mathbb{P})\colon(p,q)\mapsto p$ be the two projection mappings from
$\Gamma(\mathbb{P})$ to $\mathbb{P}^{*}$ and $\Gamma(\mathbb{P})$ to
$\mathbb{P}$, respectively. Consider the double fibration

$\mathbb{P}^{*}_{1}\times\ldots\times\mathbb{P}^{*}_{m}\xleftarrow{\gamma}\Gamma(\mathbb{P}_{1})\times\ldots\times\Gamma(\mathbb{P}_{m})\xrightarrow{\delta}\mathbb{P}_{1}\times\ldots\times\mathbb{P}_{m},$
(2.2)

where $\gamma=\gamma(\mathbb{P}_{1})\times\ldots\times\gamma(\mathbb{P}_{m})$,
$\delta=\delta(\mathbb{P}_{1})\times\ldots\times\delta(\mathbb{P}_{m})$. Let
$\delta^{*}$ and $\gamma_{*}$ be the mappings of the inverse and direct images
of differential forms, corresponding to the mappings $\delta$ and $\gamma$.
Then, Theorem 4 transforms into the equality

$\gamma_{*}\delta^{*}\>\Omega_{1}\wedge\ldots\wedge\Omega_{m}=\omega^{*}_{1}\wedge\ldots\wedge\omega^{*}_{m}.$
(2.3)

The fibration (2.2) is a direct product of $m$ double fibrations

$\mathbb{P}^{*}_{i}\xleftarrow{\gamma(\mathbb{P}_{i})}\Gamma(\mathbb{P}_{i})\xrightarrow{\delta(\mathbb{P}_{i})}\mathbb{P}_{i}.$

Therefore, (2.3) reduces to the case $m=1$, i.e., to the following statement.

######  Proposition 2.1.

Let $\Gamma=\\{(p,q)\in\mathbb{P}^{*}\times\mathbb{P}\colon\>\langle
p,q\rangle=0\\}$, $\gamma=\gamma(\mathbb{P})$, and
$\delta=\delta(\mathbb{P})$. Then it holds that
$\gamma_{*}\delta^{*}\Omega=\omega^{*}$.

The last proposition is a well-known formula of the Crofton type; see [9]. Its
proof is based on the commutativity of the action of the unitary group on
$\mathbb{P}$, $\mathbb{P}^{*}$, and $\Gamma$ with the mappings $\gamma$ and
$\delta$.

##  3\. Regular Subspaces

Let $V$ be a finite-dimensional subspace in $\hat{\mathcal{S}}(K)$ with a
fixed Hermitian scalar product $\langle*,*\rangle$. Let $B\subset V$ denote
the ball of radius $1$ centered at the origin. Below, we will define the
concept of regularity for the space $V$. In this definition, the Hermitian
product $\langle*,*\rangle$ is used. However, as it can be easily observed,
the regularity property of $V$ does not depend on the choice of
$\langle*,*\rangle$.

Consider the function $\max_{f\in B}|f(tz)|$ in ${\mathbb{C}}^{n}$ with
parameter $t>0$. According to Definition 1.1, the quantity $\max_{f\in
B}|f(tz)|$ is equal to the norm of the linear functional $f_{t}\mapsto
f_{t}(z)$ in the space $V(t)$.

######  Definiton 3.1.

Call the space $V$ regular if, as $t\to+\infty$, the parameter-dependent
function

$z\mapsto\frac{\log\max_{f\in B}|f(tz)|}{t}$

locally uniformly converges to the support function $h_{K}$ of the compact set
$K$.

######  Example 3.1.

Let $L\subset K$. Then, if ${\rm conv}(K)\neq{\rm conv}(L)$, any subspace
$\hat{\mathcal{S}}(K)$ consisting of functions belonging to
$\hat{\mathcal{S}}(L)$ is not regular. Conversely, if ${\rm conv}(K)={\rm
conv}(L)$, any regular subspace $\hat{\mathcal{S}}(L)$ is also a regular
subspace of $\hat{\mathcal{S}}(K)$.

######  Proposition 3.1.

For any arbitrarily small $\varepsilon>0$, there exists a constant $C$ such
that for any $t>0$,

$\frac{\log\max_{f\in B}|f(tz)|}{t}\leq\frac{C}{t}+h_{K}(z)+\varepsilon|z|.$

Proposition 3.1 reduces the question of regularity to a lower bound for the
function $\frac{\log\max_{f\in B}|f(tz)|}{t}$ depending on the growing
parameter $t$.

######  Corollary 3.1.

Assume that for any nonzero $z\in{\mathbb{C}}^{n}$, there exists a
neighborhood $U_{z}$ of the point $z$ and a function $F_{z}\in V$ such that

(*) as $t$ grows, the function $\frac{\log|F_{z}(tw)|}{t}$ on $U_{z}$
uniformly converges to $h_{K}(w)$.

Then the space $V$ is regular.

###### Proof.

Let $K_{z}$ denote the open cone in ${\mathbb{C}}^{n}$, consisting of points
of the form $\\{\tau w\colon w\in U_{z}\\}$. Then condition (*) is satisfied
by replacing the neighborhood $U_{z}$ with the cone $K_{z}$. Consider a finite
covering of the space ${\mathbb{C}}^{n}$ by cones of the form $K_{z}$ and
apply condition (*) to each of these cones. ∎

######  Corollary 3.2.

Assume that the space $V_{1}$ is regular, and $V_{1}\subset
V\subset\hat{\mathcal{S}}(K)$. Then the space $V$ is also regular.

###### Proof.

The functions $F_{z}$ from Corollary 3.1 can be found in the subspace $V_{1}$.
∎

Next in §3, the proof of Proposition 3.1 is provided. To do this, we use the
topology of uniform convergence on compacts in the space of entire functions
and the corresponding weak topology in the space of analytic functionals in
${{\mathbb{C}}^{n}}^{*}$.

######  Lemma 3.1.

Let $A$ be a compact set of analytic functionals with support $K$. Then for
any open neighborhood $U$ of the compact $K$, there exists a constant $C_{U}$
such that for all $\mu\in A$,

$\forall f\colon|\mu(f)|\leq C_{U}\sup_{U}|f(\xi)|$

###### Proof.

Considering the compactness of the set $A$, it follows from (1.1). ∎

######  Lemma 3.2.

Let $A$ be a compact set of analytic functionals with support $K$. Then for
any arbitrarily small $\varepsilon>0$, there exists a constant $C$ such that

$\forall\mu\in A\colon\>|\hat{\mu}(z)|\leq C\>{\rm
e}^{h_{K}(z)+\varepsilon|z|}$

###### Proof.

Let $U=K+\varepsilon B_{1}$, where $B_{1}\subset{{\mathbb{C}}^{n}}^{*}$ is a
ball of radius $1$ centered at $0$. Since 1) $\sup_{\xi\in U}|\>{\rm
e}^{\xi(z)}|=\sup_{\xi\in U}\>{\rm e}^{{\rm Re}\xi(z)}=\>{\rm e}^{h_{U}(z)}$,
2) $h_{U}(z)=h_{K}(z)+\varepsilon|z|$, and 3) by definition,
$\hat{\mu}(z)=\mu(\>{\rm e}^{\xi(z)})$, the required estimate follows from
Lemma 3.1. ∎

######  Lemma 3.3.

Let $A$ be a compact set of analytic functionals with support $K$. Then for
any arbitrarily small $\varepsilon>0$, there exists a constant $C$ such that
for any $t>0$,

$\forall\mu\in
A\colon\>\frac{\log|\hat{\mu}(tz)|}{t}\leq\frac{C}{t}+h_{K}(z)+\varepsilon|z|.$

###### Proof.

By Lemma 3.2, $|\hat{\mu}(tz)|\leq C\>{\rm e}^{h_{K}(tz)+\varepsilon|tz|}$.
Taking logarithms and using the homogeneity of the functions $|z|$ and
$h_{K}$, we obtain the desired statement. ∎

_Proof of Proposition_ 3.1. Let $A$ be the set of analytic functionals in
${{\mathbb{C}}^{n}}^{*}$ such that their image under the Borel-Fourier
transform belongs to the unit ball $B\subset V$. The Borel-Fourier transform
is continuous and bijective. Therefore, the set $A$ is compact. Hence, the
desired statement follows from Lemma 3.3.

##  4\. Proof of Theorem 1

Let us recall that, according to Theorem 3, for the flow of the averaged
distribution of roots $\mathfrak{M}(V_{1},\ldots,V_{m})$ (see Definition 2.1),
it holds that

$\mathfrak{M}(V_{1},\ldots,V_{m})=\frac{1}{(2\pi)^{m}}dd^{c}\log\|x\|^{2}_{1}\wedge\ldots\wedge
dd^{c}\log\|x\|^{2}_{m},$

where $\|x\|_{i}(z)$ is the norm of the linear functional $f\mapsto f(z)$ on
the space $V_{i}$; see Definition 2.2. Let $\|x\|_{t,i}(z)$ denote the norm of
the linear functional $f_{t}\mapsto f_{t}(z)$ on the space $V_{i}(t)$; see
Definition 1.1. Then, according to the definition of the asymptotic density
flow (see Definition 1.1),

$\mathfrak{M}^{\infty}(V_{1},\ldots,V_{m})=\lim\frac{1}{t^{m}}\frac{1}{(2\pi)^{m}}dd^{c}\log\|x\|_{t,1}^{2}\wedge\ldots\wedge
dd^{c}\log\|x\|_{t,m}^{2}.$

The regularity property of spaces $V_{i}$ means that the function
$\frac{1}{t}\log\|x\|_{t,i}^{2}(z)$ locally converges uniformly to $2\log
h_{K_{i}}(z)$ as $t$ grows; see Definition 3.1. Now the desired statement
follows from the continuity property of the complex Monge-Ampère operator

$(g_{1},\ldots,g_{m})\mapsto dd^{c}g_{1}\wedge\ldots\wedge dd^{c}g_{m}$

with respect to the topology of locally uniform convergence of continuous
plurisubharmonic arguments $g_{i}$; see, for example, [10, 7].

##  5\. Proof of Theorem 2

According to Corollary 3.2, it is sufficient to prove Theorem 2 in the case
when $V=\hat{\mathcal{S}}_{0}(K)$, i.e., when $V$ is the space of exponential
sums.

Let $0\neq z\in{\mathbb{C}}^{n}$. Corollary 3.1 reduces the proof of Theorem 2
to the construction of an exponential sum $F_{z}\in V$ such that, as $t$
grows, the function $\frac{\log|F_{z}(tw)|}{t}$ converges uniformly to
$h_{K}(w)$ in some neighborhood of the point $z\in{\mathbb{C}}$. The
construction of the function $F_{z}$ is given below.

######  Definiton 5.1.

For $0\neq z\in{\mathbb{C}}^{n}$, consider a real linear functional on
${{\mathbb{C}}^{n}}^{*}$ $\varphi_{z}(\xi)={\rm Re}\xi(z)$. Associate with the
point $0\neq z\in{\mathbb{C}}^{n}$ the face $\Delta(z)$ of the polytope ${\rm
conv}(K)$ consisting of points $\xi\in{\rm conv}(K)$ where the functional
$\varphi_{z}$ on ${\rm conv}(K)$ attains its maximum. Call $\Delta(z)$ the
supporting face of the point $z$.

######  Lemma 5.1.

If the point $x\in{\mathbb{C}}$ is sufficiently close to $z$, then the
supporting face $\Delta(x)$ is a face of the polytope $\Delta(z)$.

###### Proof.

Follows from the definition of supporting faces. ∎

######  Lemma 5.2.

Let $\Delta_{1}=\Delta(z),\Delta_{1},\ldots,\Delta_{N}$ be the set of all
faces of the polytope $\Delta(z)$. Choose a function
$F_{z}(w)=\sum_{\xi\in\Delta(z)\cap K,\>c_{\xi}\neq 0}c_{\xi}\>{\rm
e}^{\xi(w)}$ so that $\forall i\colon\sum_{\xi\in\Delta_{i}\cap
K}c_{\xi}\>{\rm e}^{\xi(z)}\neq 0$. Then, the condition (*) from Corollary 3.1
is satisfied.

###### Proof.

Let $\Delta_{i}$ be the supporting face that is close to $z$ of the point
$x\in{\mathbb{C}}^{n}$. Then, by construction, $|F_{z}(tx)|=A(t)\>{\rm
e}^{h_{K}(tx)}+B(t)$ for all $t>0$, where $A(t),B(t)$ are bounded for all
$t>0$. Hence, it follows that $\frac{\log|F_{z}(tx)|}{t}$ locally uniformly
converges to $h_{K}(x)$, i.e., the condition (*) is satisfied. ∎

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