Improving MIMO channel estimation via receive power feedback

ChaoZhang1 HangZou2 SamsonLasaulce3 LucasSaludjian4

1Central South University & 2TII & 3CRAN-Nancy & 4RTE France

Estimer l’état du canal est un problème important dans les réseaux sans fil. À
cette fin, il importe d’exploiter toutes les informations disponibles pour
améliorer autant que possible la précision de l’estimation de canal. Il
s’avère que le problème de l’exploitation de l’information associée au
feedback de puissance reçu (par exemple, l’indicateur de force du signal reçu
-RSSI-) n’a pas été identifié et résolu; dans cette configuration, on suppose
que l’émetteur reçoit des commentaires de tous les récepteurs présents. Pour
résoudre ce problème, des outils d’estimation classiques peuvent être
utilisés. L’utilisation du MMSE correspondant est toujours bénéfique, tandis
que la pertinence de l’utilisation de l’estimateur MAP dépendrait du rapport
signal sur bruit (SNR) de fonctionnement.

###### Abstract

Estimating the channel state is known to be an important problem in wireless
networks. To this end, it matters to exploit all the available information to
improve channel estimation accuracy as much as possible. It turns out that the
problem of exploiting the information associated with the receive power
feedback (e.g., the received signal strength indicator -RSSI-) has not been
identified and solved; in this setup, the transmitter is assumed to receive
feedback from all the receivers in presence. As shown in this paper, to solve
this problem, classical estimation tools can be used. Using the corresponding
MMSE is shown to be always beneficial, whereas the relevance of using the MAP
estimator would depend on the operating SNR.

##  1 Introduction

The acquisition of channel information is essential to optimize the system
performance in many wireless networks such as orthogonal frequency-division
multiplexing (OFDM) and multiple input multiple output (MIMO) (see e.g., [1,
2, 3, 4]). Generally the channel information is not perfectly known in
practical systems and needs to be estimated by sending pilot sequences [5] or
using blind channel estimation [6]. Improving the quality of channel
estimation is a well studied problem for both academic researchers as well as
engineers in the communication industry. Recent studies of channel estimation
through downlink training concentrated on massive MIMO systems. The overhead
of training sequence is huge in such systems with a large number of antennas
which makes the channel estimation even more challenging. The behavior of MMSE
estimator of low-rank channel covariance matrix is studied in [2]. It is shown
that training overhead can be substantially reduced by exploiting the low-rank
property of the channel covariance matrix in the asymptotic low-noise regime.
In [3], an open and closed-loop training framework using successive channel
prediction/estimation at the user for FDD massive MIMO systems is proposed.
Similarly, a small amount of feedback is required in low signal-to-noise ratio
(SNR) regime with the immense transmit antennas or with inaccurate prior
channel estimates setup for closed-loop systems. More interestingly, in [4],
downlink beamforming training, although increases pilot overhead and
introduces additional pilot contamination, improves significantly the
achievable downlink rate. Most of the works aims at finding efficient training
schemes to improve the channel estimate, especially in low SNR regime.
However, apart from the training phase, the channel estimation can also be
enhanced during data transmission, which is the purpose of this paper.

In this paper, by exploiting the receive power feedback [7], also referred to
as received signal strength indicator (RSSI), we propose a novel technique to
better estimate the channel in MIMO systems. Since feedback resources are
extremely limited in practical systems, it is important to efficiently use
these power domain feedback resources. For example, authors in [8] consider a
sub-optimal scheduling scheme for a multiuser MIMO (MU-MIMO) to decrease the
load on the feedback. Received signal-to-noise-plus-interference-ratio (SINR)
at each user is quantized for both fixed thresholds and adaptable ones both
achieving a remarkable reduction of feedback bits. It is worth mentioning that
the usage of such information could be highly case-depending. Authors in [9]
show that the quantization method for a single cell could highly degrade the
performance of coordinated multipoint (CoMP) system for quantization error
vector being no longer isotropic. To this end, we propose to use receive power
feedback at a much lower rate compared to the classical channel state
information (CSI) feedback rate. We will focus on two specific estimators:
minimum mean square error (MMSE) and maximum a posteriori (MAP) estimators.
Both estimators have been widely used to acquire CSI but only through pilot
training, e.g., in [5][10].

##  2 Problem statement

We consider a MIMO broadcast channel consisting of a base station (BS)
equipped with $N$ transmit antennas and $K$ user equipments (UEs) with single
antenna. The channel model can be described by

$y_{j}=\mathbf{h}_{j}^{\mathrm{H}}\mathbf{x}+z_{j}$ (1)

where $y_{j}$ represents the received signal at UE $j$,
$\mathbf{h}_{j}=(h_{1j},\dots,h_{Nj})^{\mathrm{T}}\in\mathbb{C}^{N}$ is the
vector of channel coefficients from BS to UE $j$,
$\mathbf{x}=(x_{1},\dots,x_{N})^{\mathrm{T}}$ is the vector of independent
transmitted symbols at BS and $z_{j}$ is the received Gaussian noise at UE $j$
following $z_{j}\sim\mathcal{CN}(0,N_{0})$. The channel matrix,
$\mathbf{H}=[\mathbf{h}_{1},\dots,\mathbf{h}_{K}]$, is assumed to follow a
block fading model, that is, $\mathbf{H}$ remains constant by block and more
precisely over each transmitted frame. Indeed, like modern cellular networks
standards, we assume that the transmission take places in the form of frames,
each frame being itself divided into $M$ time-slots, and each time slot
comprises $T$ symbols. In frequency division duplex (FDD) wireless systems,
one can assume that each UE $j\in\\{1,\dots,K\\}$ estimates $\mathbf{h}_{j}$
from downlink training symbols and feeds the estimated channel information
back to the BS. This is the setting we assume here. The conventional approach
to estimate the channel is to use pilot signals and training in the signal
domain only. Assume orthogonal pilot signals with power $P$. If $\beta$
symbols are used for training, the observation at Receiver $j$ that is,
$\mathbb{s}_{j}=(s_{1j},\dots,s_{Nj})^{T}$ is given by

$\mathbb{s}_{j}=\sqrt{\beta P}\mathbf{h}_{j}+\mathbb{z}_{j}$ (2)

where the estimation noise
$\mathbb{z}_{j}=(z_{1j},z_{2j},\dots,z_{Nj})^{T}\sim\mathcal{CN}(0,N_{0}\mathbb{I}_{N})$.
The signal-to-noise ratio (SNR) in the training phase is defined as
$\mathrm{SNR}=\frac{\beta P}{N_{0}}$. The observations $\mathbb{s}_{j}$ can be
used to estimate $\mathbf{h}_{j}$. The approach retained in this paper is to
de-noise the above estimates not only by using the channel statistics (as done
in previous works such as [11]) but also by using some information that can be
acquired in the power domain, namely the receive power feedback samples sent
by the UEs to the BS. Indeed, in wireless systems such as 3G and 5G systems,
for each transmitted time slot $m^{\prime}$, the BS receives a feedback sample
of the power received by UE $j$. This feedback is used e.g., for power control
or scheduling at the BS. Our key observation is that this information can also
be used to improve channel estimation, and more precisely the modulus
information accuracy. Suppose the transmitted symbols at BS are not
correlated, i.e.
$\mathbb{E}[x_{i}(m^{\prime})x_{j}(m^{\prime})^{\mathrm{H}}]=0$ for $i\neq j$,
the average receive power at UE $j$ for time slot $m^{\prime}$ can be written
as

$R_{j,m^{\prime}}=\sum_{i=1}^{N}g_{ij}P_{i}(m^{\prime})+N_{0}$ (3)

where $g_{ij}=|h_{ij}|^{2}$ represents the channel gain and
$P_{i}(t)=\mathbb{E}[|x_{i}(m^{\prime})|^{2}]$ represents the transmit power
of the $i$-th antenna at time-slot $m^{\prime}$. Our goal is to find new
estimates $\mathbb{h}_{j}^{\text{X}:m}$ by exploiting this power domain
information, where $\text{X}\in\\{\text{MMSE},\text{MAP}\\}$ indicates the
type of considered estimation criterion, and $m$ the number of time slots
exploited. Without loss of generality, we assume that
$\mathbb{h}_{j}\sim\mathcal{CN}(0,\mathbb{D}_{j})$ where
$\mathbb{D}_{j}=\mathrm{diag}\\{\sigma_{1j}^{2},\dots,\sigma_{Nj}^{2}\\}$.

##  3 Novel estimates with receive power feedback

In this section, we explain our technique for the single band scenario. The
extension to the multi-band case is straightforward since the estimation over
each band can be processed independently. Apart from the observation
$\mathbf{s}_{j}$ during the downlink training phase, we use receive power
feedback introduced by (3) as an additional information to improve the
estimation quality.

###  3.1 MMSE estimator

Denote by $\mathbf{R}_{j}^{(m)}=(R_{j,1},\dots,R_{j,m})$ the vector of random
RSSI observed at $j$ over $m$ time slots. Correspondingly,
$\mathbf{r}_{j}^{(m)}=(r_{j,1},\dots,r_{j,m})$ denotes a vector of RSSI
realizations. The optimal MMSE estimator knowing $\mathbf{r}_{j}^{(m)}$ can be
written as

$\mathbb{h}_{j}^{\mathrm{MMSE}:m}=\mathbb{A}_{j}^{\mathrm{OPT:m}}\mathbb{s}_{j}$
(4)

where $\mathbb{A}_{j}^{\mathrm{OPT:m}}$ is obtained from:

$\mathbb{A}_{j}^{\mathrm{OPT:m}}\in\underset{\mathbb{A}_{j}}{\arg\min}\quad\mathbb{E}_{\mathbb{h}_{j},\mathbb{z}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|\mathbb{A}_{j}\mathbb{s}_{j}-\mathbb{h}_{j}|^{2}]$
(5)

The corresponding optimal solution $\mathbb{A}_{j}^{\mathrm{OPT:m}}$ expressed
as:

$\mathbb{A}_{j}^{\mathrm{OPT:m}}=\mathbb{E}_{\mathbb{h}_{j},\mathbb{z}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{h}_{j}\mathbb{s}_{j}^{\mathrm{H}}]\mathbb{E}_{\mathbb{h}_{j},\mathbb{z}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{s}_{j}\mathbb{s}_{j}^{\mathrm{H}}]^{-1}$
(6)

With our model, this could be rewritten as:

$\mathbb{A}_{j}^{\mathrm{OPT:m}}={\sqrt{\beta
P}\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{h}_{j}\mathbb{h}_{j}^{\mathrm{H}}]}\mathbb{W}^{-1}$
(7)

where $W={\beta
P}\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{h}_{j}\mathbb{h}_{j}^{\mathrm{H}}]+N_{0}\mathbb{I}_{K}$.

The
$\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{h}_{j}\mathbb{h}_{j}^{\mathrm{H}}]$
term in (7) is not always easy to express. Therefore, we also provide details
on how such an estimator can be designed in practice. First, it can be
verified that the non-diagonal elements of
$\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[\mathbb{h}_{j}\mathbb{h}_{j}^{\mathrm{H}}]$
equal to zero, i.e.

$\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[h_{ij}h_{kj}^{*}]=0\quad(k\neq
i)$ (8)

From (8), it can be seen that $\mathbb{A}_{j}^{\mathrm{OPT:m}}$ is a diagonal
matrix. Thus the MMSE estimator expresses:

${h}_{ij}^{\mathrm{MMSE}:m}=\frac{\sqrt{\beta
P}\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|{h}_{ij}|^{2}]}{{\beta
P}\mathbb{E}_{\mathbb{h}_{j}|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|{h}_{ij}|^{2}]+N_{0}}s_{ij}$
(9)

The classical MMSE estimator can be obtained as a special case

${h}_{ij}^{\mathrm{MMSE}:0}=\frac{\sqrt{\beta P}\sigma_{ij}^{2}}{{\beta
P}\sigma_{ij}^{2}+N_{0}}s_{ij}.$ (10)

Knowing the optimal estimator, we can calculate the conditional MSE
(conditioned to $\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}$) as a result of
using the MMSE given by (7). This conditional MSE is denoted by
$\mathrm{D}_{j}(\mathbf{r}_{j}^{(m)})$ when $m$ RSSI observations are
available, and can be evaluated as
$\mathrm{D}_{j}(\mathbf{r}_{j}^{(m)})=\mathbb{E}_{|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|\mathbb{A}_{j}\mathbb{s}_{j}-\mathbb{h}_{j}|^{2}]$.
Finally, the real MSE of the estimator with $m$ RSSI measurements can be
expressed as
$\Delta_{j;m}=\mathbb{E}_{\mathbf{r}_{j}^{(m)}}[\mathrm{D}_{j}(\mathbf{r}_{j}^{(m)})]$.
We use $D^{*}_{j}$ and $\Delta_{j;m}^{*}$ to denote the conditional and real
MSEs when using the optimal estimator given in (7). This yields:

$\displaystyle\mathrm{D}_{j}^{*}(\mathbf{r}_{j}^{(m)})=$
$\displaystyle\sum_{i=1}^{N}\frac{N_{0}\mathbb{E}_{|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|h_{ij}|^{2}]}{\beta
P\mathbb{E}_{|\mathbf{R}_{j}^{(m)}=\mathbf{r}_{j}^{(m)}}[|h_{ij}|^{2}]+N_{0}}$
(11) $\displaystyle\Delta_{j;m}^{*}=$
$\displaystyle\mathbb{E}^{*}_{\mathbf{r}_{j}^{(m)}}[\mathrm{D}^{*}_{j}(\mathbf{r}_{j}^{(m)})]$
(12)

Next, we prove that the performance in terms of the real MSE given in (12) is
at least as good as the classical estimator for the proposed estimator. We
also prove that lesser MSE can be achieved if we use more feedback samples. We
formalize this result with the following proposition.

######  Proposition 3.1.

The MSE resulting from the proposed estimator in (7) is a decreasing function
of the number of RSSI feedbacks available at any receiver $j$, i.e.,
$\Delta^{*}_{j;m}\geq\Delta^{*}_{j;m+1}$. Additionally, the MSE is lower
bounded by a constant

$\Delta^{*}_{j;m}\geq\sum_{i=1}^{N}\mathbb{E}_{h_{ij}}\left[\frac{N_{0}|h_{ij}|^{2}}{\beta
P|h_{ij}|^{2}+N_{0}}\right]$ (13)

###  3.2 MAP estimator

As with the MMSE estimator, we can use the information provided by RSSI
measurements to propose a novel MAP estimator which reduces the MSE. Denoting
$m$ RSSI measurements available at the $j$-th receiver by
$\mathbf{r}_{j}^{(m)}$, the MAP estimate can be given by

$\begin{split}\widehat{h}_{ij}^{\text{MAP:m}}&\in\arg\max_{h_{ij}}f_{h|s,r}(h_{ij}|{s}_{ij},\mathbf{r}_{j}^{(m)})\\\
&\in\arg\max_{h_{ij}}f_{s|h,r}(s_{ij}|h_{ij},\mathbf{r}_{j}^{(m)})f_{r|h}(\mathbf{r}_{j}^{(m)}|h_{ij})f_{h}(\mathbf{r}_{j}^{(m)}|h_{ij})\\\
&\in\arg\max_{h_{ij}}f_{r|h}(\mathbf{r}_{j}^{(m)}|h_{ij})\exp(\frac{|s_{ij}-\sqrt{\beta
Ph_{ij}}|^{2}}{N_{0}})\exp(\frac{|h_{ij}|^{2}}{\sigma_{ij}^{2}})\end{split}$
(14)

When no RSSI measurements are available i.e., $m=0$, this reduces to the
classical MAP as follows

$\widehat{h}_{ij}^{\text{MAP}:0}=\frac{\sqrt{\beta
P}\sigma_{ij}^{2}}{N_{0}+\beta P\sigma_{ij}^{2}}s_{ij},$ (15)

which also coincides with the classical MMSE. However, when the RSSI estimates
are known, $f_{r|h}(\mathbf{r}_{j}^{(m)}|h_{ij})$ becomes a Dirac function.
Therfore, the optimization must be performed under the constraints that
$R_{j}^{(1)}=r_{j}^{(1)},\dots,R_{j}^{(m)}=r_{j}^{(m)}$. The optimization
problem can be rewritten as:

$\begin{split}&\underset{\mathbb{h}_{j}}{\min}\quad\frac{(\mathbb{s}_{j}-\sqrt{\beta
P}\mathbb{h}_{j})^{H}(\mathbb{s}_{j}-\sqrt{\beta
P}\mathbb{h}_{j})}{N_{0}}+{\mathbb{h}_{j}^{H}\mathbb{D}_{j}^{-1}\mathbb{h}_{j}}\\\
&\mathrm{s.t.}\quad\mathbb{h}_{j}^{\mathrm{H}}\mathbb{P}(n)\mathbb{h}_{j}+N_{0}=r_{j}^{(n)},\,\,\forall\,\,n\in\\{1,2,\dots,m\\}\end{split}$
(16)

where $\mathbb{P}(n)=\mathrm{diag}(P_{1}(n),\dots,P_{N}(n))$. This problem can
be solved by using the Lagrange multiplier method. The optimal solution
$\widehat{\mathbb{h}}^{\text{MAP:m}}_{ij}$ can be written as

$\widehat{\mathbb{h}}^{\text{MAP:m}}_{ij}=\frac{\sqrt{\beta
P}}{N_{0}(\frac{1}{\sigma^{2}_{ij}}+\frac{\beta
P}{N_{0}}+\sum_{n=1}^{m}\lambda_{n}P_{i}(n))}{s}_{ij}$ (17)

where $\lambda_{n}$ can be obtained from the constraint
$\lambda_{n}({\mathbb{h}_{j}}^{H}\mathbb{P}(n){\mathbb{h}_{j}}-r_{j}^{(n)}+N_{0})=0$
for all $n\in\\{1,2,\dots,m\\}$. If the number of timeslots $m$ is less than
the number of antennas, then there are possibly infinite solutions for the
above system of equations. To circumvent this problem, we assume that we can
perfectly know $|h_{ij}|^{2}$ or $g_{ij}$ for $i\in S$ by exploiting $m$ RSSI
measurements where $S$ is defined as follows:
$S=\\{i\in\\{1,\dots,N\\}|\quad|h_{ij}|^{2}\,\,\text{can be obtained from
RSSI}\\}$. Based on this relaxation, the RSSI measurements are treated as the
measurements of channel gains/magnitude $|h_{ij}|^{2}$, and the optimization
problem defined by (16) can be simplified as:

$\begin{split}&\underset{\mathbb{h}_{j}}{\min}\quad\frac{(\mathbb{s}_{j}-\sqrt{\beta
P}\mathbb{h}_{j})^{H}(\mathbb{s}_{j}-\sqrt{\beta
P}\mathbb{h}_{j})}{N_{0}}+{\mathbb{h}_{j}^{H}\mathbb{D}_{j}^{-1}\mathbb{h}_{j}}\\\
&\mathrm{s.t.}\quad|h_{ij}|^{2}=g_{ij},\,\,\forall\,\,i\in S\end{split}$ (18)

The solution of the simplified problem can be found by solving the following
equations:

$\widehat{{h}}^{\text{MAP:m}}_{ij}=\frac{\sqrt{\beta
P}}{N_{0}(\frac{1}{\sigma^{2}_{ij}}+\frac{\beta P}{N_{0}}+q_{i})}{s}_{ij}\\\ $
(19) $\frac{\beta P|s_{ij}|^{2}}{[(\frac{N_{0}}{\sigma_{ij}^{2}}+{\beta
P})+N_{0}\lambda_{i}]^{2}}=g_{ij},\,\,\forall\,\,i\in S$ (20)

where $q_{i}=\lambda_{i}$ when $i\in S$ and $q_{i}=0$ otherwise. With (20),
for $i\in S$, we have:

$\lambda_{i}=\frac{\displaystyle{\pm\sqrt{\frac{\beta
P|s_{ij}|^{2}}{g_{ij}}}-(\frac{N_{0}}{\sigma_{ij}^{2}}+{\beta P})}}{N_{0}}.$
(21)

Therefore, the MAP estimator can be simplified and rewritten as:

$\widehat{h}_{ij}^{\text{MAP:m}}=\left\\{\begin{array}[]{ll}\sqrt{g_{ij}}\frac{s_{ij}}{|s_{ij}|}&\hbox{
$i\in S$}\\\ \\\ \widehat{h}_{ij}^{\text{MAP}:0}&\hbox{ otherwise}\\\ \\\
\end{array}\right.$ (22)

The proposed MAP will ensure the maximum a posteriori probability. However the
minimization of the MSE is not always guaranteed. Indeed, it can be seen from
the following proposition that the proposed MAP brings more MSE in the low SNR
regime.

######  Proposition 3.2.

Assuming all the channel gains can be obtained from the RSSI measurements,
i.e., $S=\\{1,\dots,N\\}$, the following equalities can be derived

$\underset{\mathrm{SNR}\rightarrow
0}{\lim}\mathbb{E}_{h_{ij},z_{ij}}[|\widehat{h}_{ij}^{\text{MAP:m}}-h_{ij}|^{2}]=2\underset{\mathrm{SNR}\rightarrow
0}{\lim}\mathbb{E}_{h_{ij},z_{ij}}[|\widehat{h}_{ij}^{\text{MAP}}-h_{ij}|^{2}]$
(23)
$\underset{\mathrm{SNR}\rightarrow\infty}{\lim}\mathbb{E}_{h_{ij},z_{ij}}[|\widehat{h}_{ij}^{\text{MAP:m}}-h_{ij}|^{2}]=\frac{1}{2}\underset{\mathrm{SNR}\rightarrow\infty}{\lim}\mathbb{E}_{h_{ij},z_{ij}}[|\widehat{h}_{ij}^{\text{MAP}}-h_{ij}|^{2}]$
(24)

##  4 Numerical results

In the simulations, we fix number of transmit antennas as$N=4$, number of
users as $K=4$ and observe the distortion performance at different SNR. For
ease of exposition, we assume $\mathbb{D}_{j}$ is an identity matrix in the
simulations. i.e. $\mathbb{D}_{j}=\mathbb{I}_{N}$. As seen from (10) and (15),
the classical MMSE coincides with the classical MAP in our model. We define
the relative MSE reduction as

$\frac{\Delta_{j;0}^{*}-\Delta_{j;m}^{\mathrm{X}*}}{\Delta_{j;0}^{*}}\times
100\%$ (25)

with $\text{X}\in\\{\text{MMSE},\text{MAP}\\}$. We assume that from each
feedback, we can perfectly reconstruct one channel gain, i.e., with $m$
feedbacks, Receiver $j$ can acquire $g_{1j},\dots,g_{mj}$.

First, we study the influence of the prior information on the MMSE estimator.
We compare the reduced MSE in terms of number of feedback samples for
different SNR. Fig. 1 shows that the MSE can be mitigated by using the RSSI
measurements and the MSE decreases with more measurements available. In the
low SNR regime, the MMSE estimator does not perform well since
$\mathbf{s}_{j}$ is multiplied by a constant matrix $\mathbb{A}_{j}$. Hence,
even if we can improve the selection of $\mathbb{A}_{j}$ by knowing the RSSI,
the MSE can not be reduced much due to the high noise level. On the other
hand, in the high SNR regime, the classical MMSE estimator is already
accurate. This results in limited gains for the high SNR regime too. Our
technique can bring more improvements in the mid-SNR range.

We conduct a similar analysis for the new MAP estimator. The results are shown
in Fig. 2. From Fig. 2, we see that unlike in the MMSE case, the MSE does not
necessarily reduce with higher number of RSSI feedback samples. Indeed, as
proved in Prop. III.2, the MSE for the new MAP might actually increase in the
low SNR regime. It has been checked that the degradation in estimation is
close to 100% when SNR is less than -40 dB. However, when SNR increases, the
RSSI measurements can lead to a significant reduction in MSE and it can even
outperform the new MMSE.

Figure 1: The MMSE estimator brings a significant improvements in the range
[-10dB,+10dB] Figure 2: The MAP estimator performs very well for sufficiently
high SNRs

##  5 Conclusion and future works

Under some classical assumptions on the residual noise produced by least-
squares estimates, it is known that the de-noising matrices obtained by using
MMSE and MAP de-noising coincide [11]. In the presence of additional prior
coming from the power domain, this result is no longer true, which explains
that MMSE and MAP performance do not perform similarly here. In terms of MSE
performance, it is recommended to use the proposed MMSE in the moderate SNR
regime and the proposed MAP in the high SNR regime. By doing so, RSSI feedback
can be used not only for resource allocation purposes (as currently done) but
also for improving channel estimation. In future works, we intend to extend
this framework to goal-oriented communication systems, where estimation
quality is not only to recover the signal itself but also to serve the system
to better accomplish the goal [12][13][14].

## Acknowledgement

This work is partly supported by PGMO project fund from the Fondation Hadamard
des Mathématiques.

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