# Trade-Offs in Decentralized Multi-Antenna Architectures: Sparse Combining
Modules for WAX Decomposition

Juan Vidal Alegría, and Fredrik Rusek This paper is built upon previous
results presented at the 2022 IEEE ICC conference [1].

###### Abstract

With the increase in the number of antennas at base stations (BSs),
centralized multi-antenna architectures have encountered scalability problems
from excessive interconnection bandwidth to the central processing unit (CPU),
as well as increased processing complexity. Thus, research efforts have been
directed towards finding decentralized receiver architectures where a part of
the processing is performed at the antenna end (or close to it). A recent
paper put forth an information-lossless trade-off between level of
decentralization (inputs to CPU) and decentralized processing complexity
(multiplications per antenna). This trade-off was obtained by studying a newly
defined matrix decomposition–the WAX decomposition–which is directly related
to the information-lossless processing that should to be applied in a general
framework to exploit the trade-off. The general framework consists of three
stages: a set of decentralized filters, a linear combining module, and a
processing matrix applied at the CPU; these three stages are linear
transformations which can be identified with the three constituent matrices of
the WAX decomposition. The previous work was unable to provide explicit
constructions for linear combining modules which are valid for WAX
decomposition, while it remarked the importance of these modules being sparse
with 1s and 0s so they could be efficiently implemented using hardware
accelerators. In this work we present a number of constructions, as well as
possible variations of them, for effectively defining linear combining modules
which can be used in the WAX decomposition. Furthermore, we show how these
structures facilitate decentralized calculation of the WAX decomposition for
applying information-lossless processing in architectures with an arbitrary
level of decentralization.

###### Index Terms:

WAX decomposition, MIMO, Massive MIMO, LIS, decentralized processing, linear
equalization, matched filter.

##  I Introduction

Multi-antenna architectures constitute a mature technology which keeps
developing to improve wireless communication links. Their main benefits
include increased data rates and reliability due to the exploitation of space-
division multiplexing and diversity. Current research on multi-antenna
architectures is trending towards scaling up the number of antennas in order
to further increase spectral efficiency and spatial resolution. This trend can
be seen, e.g., in massive multiple-input multiple-output (MIMO) [2, 3] and
large intelligent surface (LIS) [4], where massive MIMO considers base
stations (BSs) with hundreds of antennas, while LIS goes beyond by considering
whole walls of electromagnetically active material.

Several prototypes of massive MIMO have been developed and tested [5, 6, 7].
In the prototypes from [5, 7], centralized processing results in scalability
issues due to the increased data-rates between the antennas and the central
processing unit (CPU), which scales with the number of antennas. These issues
become even more concerning in LIS, where practical deployments are expected
to include a number of antennas at least an order of magnitude greater than
massive MIMO [8].111Discrete surfaces approximate continuous ones when
sampling is dense enough [4, 9]. Cell-free massive MIMO[10, 11, 12, 13] is
also likely to suffer from scalability issues due to the large number of
access points (APs) distributed throughout large geographical areas. Our
system model will consider a general multi-antenna architecture which can be
generalized to more specific applications, e.g., the ones previously
mentioned.

Decentralized pre-processing of the received signals at the antenna end (or
nearby) allows to reduce the dimension of the data that needs to be
transmitted to a CPU[14, 15, 16]. In the recent years, there has been a trend
towards considering more decentralized architectures [17, 18, 15, 16, 19, 14,
20, 21, 22, 23, 24] in order to cope with scalability issues arising in large-
scale multi-antenna architectures. The literature on decentralized massive
MIMO includes a number of solutions, ranging from fully-decentralized
architectures [21, 20, 22, 17], where channel state information (CSI) does not
have to be available at the CPU, to partially decentralized architectures,
where some of the processing tasks are distributed, but neither full [18, 24],
nor partial CSI [14] is available at the CPU. We can also find decentralized
solutions tailored for other large-scale multi-antenna systems such as for
cell-free massive MIMO [11, 25], or for extra-large scale MIMO (XL-MIMO) [24,
26], which can be seen as a system with a number of antennas in the regime of
massive MIMO where the antenna array is deployed throughout a large surface
such that spatial non-stationarities appear [27].

In [28], an information-lossless trade-off between the number of connections
to a CPU and number of multiplications per antenna is
presented.222Information-lossless here means that the mutual information
between the post-processed and the user data is equal to the mutual
information between the received data and the user data. To this end, a
general framework is considered which can accommodate classical centralized
processing architectures, decentralized architectures such as [14], as well as
a wide range of intermediate architectures. Unlike [19], where a system-level
trade-off between different decentralized architectures, algorithms, and data
precision is studied, [28] gives a fundamental trade-off between level of
decentralization and decentralized processing complexity. The information-
lossy regime of said trade-off is considered in [8, 15], while we restrict our
work to the information-lossless regime. Hence, the results from [17, 18, 15,
19, 20, 21, 22, 23, 24, 26] lie essentially outside the scope of our work
since they rely on the usage of linear equalizers which incur information-
losses before symbol detection, and/or they focus on the symbol detection
problem, which we disregard in this work. Furthermore, most of these works
focus on the implementation of solutions as decentralized as possible, while
our aim is to understand the trade-offs arising when we can have different
levels of decentralization. Thus, we consider the general framework from [28],
which corresponds to a generic architecture useful in the analysis of the
information-lossless regime of decentralized linear equalization. Note that
BER is not a suitable metric for judging the results presented in this
work,333BER can be made arbitrarily small when operating at rates below
capacity [29] with marginal loss when considering practical channel coding
methods, e.g., LDPC [30]. while channel capacity is perfectly achievable under
this framework.

The WAX decomposition, as originally introduced in [28], is a matrix
decomposition which has direct correspondence with the information-lossless
linear processing to be applied in an architecture with an arbitrary level of
decentralization. It thus allows to characterize the information-lossless
trade-off between level of decentralization and decentralized processing
complexity. The idea is to decompose the channel matrix into the product of a
(block-diagonal) decentralized processing matrix $\boldsymbol{W}$, a linear
combining module $\boldsymbol{A}$, and a CPU processing matrix
$\boldsymbol{X}$. In [28, Theorem 1], the requirements for the existence of
the WAX decomposition are only proved for randomly chosen channel matrices and
using fixed randomly chosen combining modules $\boldsymbol{A}$ (for definition
of ”randomly chosen” see Notation). In [13], the applicability of the WAX
decomposition is generalized to sparse channel matrices, showing that channel
sparsity can degrade the trade-off given in [28]. On the other hand, [28]
remarks the importance of employing a simple sparse combining matrix
$\boldsymbol{A}$ with 1s and 0s, so that it could be efficiently implemented
through hardware modules, i.e., generalizing the trivial combining modules
from purely decentralized architectures (e.g., the sum module from [14]) or
common centralized architectures (i.e., an identity module). However, [28]
only presents necessary conditions for an $\boldsymbol{A}$ to be valid for WAX
decomposition.

The current paper is a continuation of the work presented in [28], and it
further extends the results from [1]. Thus, our aim is to fill some of the
gaps from [28] by presenting a set of constructions for $\boldsymbol{A}$ which
consist of sparse structures of 1s and 0s,444This condition is slightly
relaxed in degenerate cases as will be discussed. and which can be proved
valid for WAX decomposition under different parameter settings. The proven
existence of these constructions strengthens the practicality of the WAX
decomposition for the exploitation of the trade-off between level of
decentralization and decentralized processing complexity from [28].
Furthermore, we exploit the structure of said $\boldsymbol{A}$ matrices to
define a decentralized scheme for computing the information-lossless
decentralized filters without the need of aggregating the full CSI at any
single point. We also extend [28, Theorem 1] by proving the converse (only if)
statement for arbitrary combining modules, thus showing that the information-
lossless trade-off studied [28] is of fundamental nature and it is not
possible to operate without loss beyond it. The list of contributions are
summarized next:

  * •

We prove that there exists no combining module, $\boldsymbol{A}$, attaining a
less-restrictive information-lossless trade-off than the one obtained in [28,
Theorem 1], which was only proved for randomly chosen $\boldsymbol{A}$.

  * •

We present an equivalent formulation of the WAX decomposition which describes
the information-lossless regime without the need of taking into account any
processing at the CPU. This was already included in [1].

  * •

We present 3 sparse structures for $\boldsymbol{A}$ and prove their validity
for WAX decomposition. Only one of these structures was included in [1]. The
new structures allow for more freedom in the exploitation without loss of the
achievable trade-off, which corresponds to a novel generalization of the
trade-off from [28] with marginal loss.

  * •

We present two transformations for $\boldsymbol{A}$ that maintain its
validity. One of them was included in [1].

  * •

We present a general algorithm to construct a matrix $\boldsymbol{A}$ that
allows for the exploitation of the achievable trade-off for any set of system
parameters. Unfortunately, we were unable to formally prove the validity of
the $\boldsymbol{A}$ matrices constructed using said algorithm.

  * •

We present a decentralized scheme for computing the information-lossless
decentralized filters which generalizes the one included in [1] to the new
$\boldsymbol{A}$ structures presented in this work.

The rest of the paper is organized as follows. Section II presents the system
model and discusses the relevant background from [28]. Section III presents
the main theoretical results, including the converse of [28, Theorem 1], and
the equivalent formulation of the WAX decomposition. In Section IV, we discuss
different ways of constructing a valid combining matrix $\boldsymbol{A}$.
Section V describes the decentralized scheme for computing the decentralized
filters considering the valid $\boldsymbol{A}$ structures. In Section VI, we
present some examples as well as a discussion of the previous results.
Finally, Section VII concludes the paper.

Notation: In this paper, lowercase, bold lowercase and bold uppercase letters
stand for scalars, column vectors and matrices, respectively. When using the
mutual information operator, $I(\cdot;\cdot)$, bold uppercase sub-scripts
refer to random vectors instead of their realizations. The operations
$(\cdot)^{\text{T}}$, $(\cdot)^{*}$ and $(\cdot)^{\text{H}}$ denote transpose,
conjugate, and conjugate transpose, respectively. The operation
$(\cdot)^{\dagger}$ denotes Moore-Penrose inverse. The operation
$\mathrm{diag}(\cdot,\ldots,\cdot)$ outputs a block diagonal matrix with the
input matrices as the diagonal blocks. $\boldsymbol{A}\otimes\boldsymbol{B}$
denotes the Kronecker product between matrices $\boldsymbol{A}$ and
$\boldsymbol{B}$. $\mathbf{I}_{i}$ corresponds to the identity matrix of size
$i$, $\boldsymbol{1}_{i\times j}$ denotes the $i\times j$ all-ones matrix, and
$\boldsymbol{0}_{i\times j}$ denotes the $i\times j$ all-zeros matrix (absence
of one such index indicates that the matrix is square). The notation
$[\boldsymbol{A}]_{i:j,\ell:k}$ denotes a matrix formed by rows $i$ to $j$ and
columns $\ell$ to $k$ of $\boldsymbol{A}$ (as in Python vector notation,
absence of one or more indexes indicates that start/end of the included rows
or columns corresponds to the first/last row or column of $\boldsymbol{A}$,
respectively). In this paper, a randomly chosen matrix corresponds to a
realization of a random matrix where any submatrix of it is full-rank with
probability 1, e.g., a realization of an independent and identically
distributed (IID) Gaussian matrix.

##  II System model

Let us consider $K$ single-antenna users transmitting to an $M$-antenna BS,
with $M>K$, through a narrow-band channel. The $M\times 1$ received complex
baseband vector can be expressed as

$\boldsymbol{y}=\boldsymbol{H}\boldsymbol{s}+\boldsymbol{n},$ (1)

where $\boldsymbol{H}$ is the $M\times K$ channel matrix, $\boldsymbol{s}$ is
the $K\times 1$ vector of symbols transmitted by the users, and
$\boldsymbol{n}$ is a zero-mean complex white Gaussian noise vector with
sample variance $N_{0}$. The $M$ antennas are divided into $M_{\mathrm{P}}$
groups (or panels) with $L$ antennas ($M/L$ evaluates to an integer). Thus, we
can express the channel matrix as
$\boldsymbol{H}=[\boldsymbol{H}_{1}^{\text{T}}\,\boldsymbol{H}_{2}^{\text{T}}\,\dots\boldsymbol{H}_{M_{\mathrm{P}}}^{\text{T}}]^{\text{T}}$
where $\boldsymbol{H}_{m}$ corresponds to the $L\times K$ local channel matrix
seen by panel $p$, for $p\in\\{1,\dots,M_{\mathrm{P}}\\}$. Each panel
multiplies the received vector by an $L\times L$ matrix,
$\boldsymbol{W}_{m}^{\text{H}}$ $m\in\\{1,\dots,M_{\mathrm{P}}$}, thus
generating $L$ outputs,555From [28], the restriction of having the same number
of antennas and outputs in each panel can be relaxed through an equivalent
transformation without constraining the validity of our analysis. $L\leq K$.
The aggregated outputs are combined through a fixed $T\times M$ matrix,
$\boldsymbol{A}^{\text{H}}$, $T\leq M$. We can view
$\boldsymbol{A}^{\text{H}}$ as a hardware combining module which can be
predesigned, but is fixed once deployed. The resulting vector is forwarded to
a CPU, which can apply further processing. In order be able to relate the
resulting linear processing to common strategies, e.g., MRC, ZF, MMSE, etc, we
assume that the processing at the CPU can be given by a matrix multiplication
with a $K\times T$ matrix $\boldsymbol{X}^{\text{H}}$ (see [28] for further
details). The post-processed vector is then given by

$\boldsymbol{z}=\boldsymbol{X}^{\text{H}}\boldsymbol{A}^{\text{H}}\boldsymbol{W}^{\text{H}}\boldsymbol{y},$
(2)

where $\boldsymbol{W}$ is an $M\times M$ block diagonal matrix of the form

$\boldsymbol{W}=\mathrm{diag}\left(\boldsymbol{W}_{1},\boldsymbol{W}_{2},\dots,\boldsymbol{W}_{M_{\mathrm{P}}}\right).$
(3)

The matrices $\boldsymbol{W}$ and $\boldsymbol{X}$ can be recalculated for
every channel realization, while the matrix $\boldsymbol{A}$ remains unchanged
once the system is deployed (we can think of it as a fixed hardware combining
module). The framework under study is represented in Fig. 1. Note that, during
the whole uplink transmission, information is only flowing from the antennas
towards the CPU, unlike message passing approaches like [24, 26, 23]. This
means that there is no extra delay with respect to common centralized
architectures, apart from the delay associated to the computation of the
decentralized filters which is only done once per coherence interval.

Figure 1: Framework considered in this paper during an uplink transmission.

The main challenge of the current framework is to maximize the information
rate at which the users can transmit to the BS, i.e.,
$I_{\boldsymbol{Z},\boldsymbol{S}}(\boldsymbol{z};\boldsymbol{s})$, or,
correspondingly,666Note that $\boldsymbol{X}$ cannot possibly increase the
maximum information rate at which the users can transmit (recall data-
processing inequality [29]). The main purpose of it is to be able to consider
specific linear receiver schemes, e.g., zero-forzing (ZF), matched filter
(MF), etc.
$I_{\boldsymbol{Y},\boldsymbol{S}}(\boldsymbol{A}^{\text{H}}\boldsymbol{W}^{\text{H}}\boldsymbol{y};\boldsymbol{s})$.
In this paper we will aim at applying information lossless processing, where
$I_{\boldsymbol{Y},\boldsymbol{S}}(\boldsymbol{A}^{\text{H}}\boldsymbol{W}^{\text{H}}\boldsymbol{y};\boldsymbol{s})=I_{\boldsymbol{Y},\boldsymbol{S}}(\boldsymbol{y};\boldsymbol{s})$.
Note that the application of $\boldsymbol{X}$ is not strictly necessary since
it cannot possibly increase the information rate.

The framework under study allows for an information-lossless trade-off between
the number of multiplications per antenna, $L$, and the number of inputs to
the CPU, $T$. Said trade-off was identified in [28], where initial results are
presented. In the present work we aim at presenting new results that allow for
practical exploitation of the trade-off.

Having the number of antennas per panel equal to the multiplications per
antenna, both given by $L$ in this work, might seem like unnecessarily
restrictive. In [28], the number of antennas per panel considered was an
arbitrary number $N$, leading to $\boldsymbol{W}_{m}$ matrices of size
$N\times L$. However, the most important results in said paper consider the
case $N=L$ due to its intrinsic generality in the information-lossless
scenario. Note that, in order to achieve information-lossless processing, we
require $N\leq L$, while if $N$ divides $L$, [28, Lemma 2] shows that there is
a direct mapping to the case where $N=L$. Furthermore, from a practical
perspective, minimum interconnection bandwidth (i.e., outputs per panel) in
the information-lossless case is achieved for $N=L$. Considering all the
above, we find it reasonable to focus on the case where the number of antennas
per panel coincides with the number of outputs per panel as in the presented
framework. However, it would be straightforward to consider panels formed by
several of these groups of $L$ antennas as in [13].

The framework discussed so far shows how the system operates during the data
phase, where the users are transmitting data within one coherence block, so
the corresponding $\boldsymbol{W}$ and $\boldsymbol{X}$ matrices have already
been calculated for the current channel realization $\boldsymbol{H}$. In this
work we also focus on what is being done during the training phase.
Specifically, we want to find decentralized schemes to compute the
information-lossless decentralized filters to be applied.777By decentralized
here we mean that each panel has access to its local channel,
$\boldsymbol{H}_{m}$ and it can share some reduced data with a number of other
panels to find the processing to be applied. Since the application of
$\boldsymbol{X}$ at the CPU cannot possibly increase mutual information (as
previously discussed), we restrict our problem to proposing a decentralized
scheme that allows us to compute the equalizer that each panel has to apply,
i.e., $\boldsymbol{W}_{m}\;\forall m$, such that the overall processing is
information-lossless. In this way, the data arriving at the CPU will contain
the same amount of information from the users as in the centralized case. As
we will see, the structure of $\boldsymbol{A}$ plays a big role in how the
decentralized computation of $\boldsymbol{W}$ can be performed. Thus, we will
explore how certain structures for $\boldsymbol{A}$ allow the definition of
decentralized schemes for obtaining $\boldsymbol{W}_{m}$ at each panel.

###  II-A Background

As we mentioned earlier, the system model considered in this work was already
studied in [28], where we can find important results which will be required
for our analysis. From [28, Lemma 1], the framework under study can achieve
information lossless processing if and only if we can decompose the channel
matrix $\boldsymbol{H}$ into the so called WAX decomposition

$\boldsymbol{H}=\boldsymbol{W}\boldsymbol{A}\boldsymbol{X},$ (4)

where $\boldsymbol{W}$, $\boldsymbol{A}$ and $\boldsymbol{X}$ correspond to
the matrices from (2), i.e., $\boldsymbol{A}$ is fixed by design while
$\boldsymbol{W}$ and $\boldsymbol{X}$ can be tuned to $\boldsymbol{H}$. Note
that, according to [28, Lemma 1], selecting $\boldsymbol{W}$ and
$\boldsymbol{X}$ in (2) such that (4) is fulfilled leads to information-
lossless processing within our framework. The main result of the applicability
of WAX decomposition is given in [28, Theorem 1], which states that, for a
fixed randomly chosen $\boldsymbol{A}\in\mathbb{C}^{M\times T}$, a randomly
chosen $\boldsymbol{H}\in\mathbb{C}^{M\times K}$ admits WAX decomposition with
probability 1 if

$T>\max\left(M\frac{K-L}{K},K-1\right).$ (5)

An alternative formulation of (5), can be given by considering the restriction
on the other trade-off parameter, $L$. This results in

$L>K\frac{M-T}{M},$ (6)

where we restrict ourselves to the regime $T\geq K$ where there exists an
information-lossless trade-off between $T$ and $L$ (for $T<K$ there would be
information-loss no matter the value of $L$). Defining $T_{\mathrm{P}}=T/L$ we
have

$L>K\frac{M_{\mathrm{P}}-T_{\mathrm{P}}}{M_{\mathrm{P}}},$ (7)

which may ease comparison with the results to be presented.

In this paper, however, we will explore specific structures for
$\boldsymbol{A}$ matrices and prove their validity for WAX decomposition. We
consider the same definition as in [28, Definition 1] for the validity of
$\boldsymbol{A}$, i.e., a randomly chosen $\boldsymbol{H}$ admits WAX
decomposition with probability 1 for a valid $\boldsymbol{A}$. Note that [28]
only provides necessary conditions for valid $\boldsymbol{A}$ matrices which
are not randomly chosen, as well as a method to test if a specific
$\boldsymbol{A}$ matrix is valid for some fixed dimensions (not
generalizable). It is one of our desires to find structures for
$\boldsymbol{A}$ that allow for a trade-off between $L$ and $T$ as close as
possible to (7) (for $T\geq K$).

##  III New results on the WAX decomposition

###  III-A The necessary information-lossless trade-off

In[28, Theorem 1], the condition (5) for the existence of WAX decomposition
was only proved for a randomly chosen $\boldsymbol{A}$. However, it is unclear
if there exist any other selection of $\boldsymbol{A}$ that may attain a
better trade-off than the one defined in (5). The following theorem shows that
(5) is not only a sufficient condition for the existence of the WAX
decomposition, but also a necessary condition.

###### Theorem 1

Let $\boldsymbol{A}$ be an arbitrary $M\times T$, and $\boldsymbol{H}$ be an
$M\times K$ randomly chosen matrix. The WAX decomposition of $\boldsymbol{H}$,
given by (5), can only exists if (5) is satisfied. Furthermore,
$\boldsymbol{A}$ should be of rank $T$ to be able to attain (5).

###### Proof:

See Appendix A. ∎

Theorem 1 states that the fundamental trade-off between the number of
multiplications per antenna ($L$), and the number of inputs to the CPU ($T$),
is ultimately governed by (5) (or its alternative formulations). For the rest
of the paper we assume $M\geq T\geq K$, which is the regime where the
information-lossless trade-off between $L$ and $T$ applies.

###  III-B The equivalent formulation of the WAX decomposition

Let us divide $\boldsymbol{A}$ into two blocks

$\boldsymbol{A}=\begin{bmatrix}\boldsymbol{A}_{\mathrm{T}}\\\
\boldsymbol{A}_{\mathrm{B}}\end{bmatrix},$ (8)

where $\boldsymbol{A}_{\mathrm{T}}$ is a $T\times T$ matrix corresponding to
the top part of $\boldsymbol{A}$, and $\boldsymbol{A}_{\mathrm{B}}$ is the
$(M-T)\times T$ matrix corresponding to the bottom part of $\boldsymbol{A}$.
We next provide a theorem corresponding to an equivalent formulation of the
WAX decomposition.

###### Theorem 2

Assume that $T_{\mathrm{P}}=T/L$ evaluates to an integer value, and that
$\boldsymbol{A}_{\mathrm{T}}$ is full-rank. Then, the WAX decomposition of
some $M\times K$ matrix $\boldsymbol{H}$, given by (4), exists if and only if
we can find a full-rank $\boldsymbol{W}$ (corresponding to (3)) such that

$\boldsymbol{B}^{\text{T}}\boldsymbol{W}^{-1}\boldsymbol{H}=\boldsymbol{0}_{(M-T)\times
K},$ (9)

where the matrix $\boldsymbol{B}$ is defined as

$\boldsymbol{B}=\begin{bmatrix}\boldsymbol{A}_{\text{B}}\boldsymbol{A}_{\text{T}}^{-1}&-\mathbf{I}_{M-T}\end{bmatrix}^{\text{T}}.$

###### Proof:

Let us assume $\boldsymbol{W}$ in (4) to be full-rank; correspondingly,
$\boldsymbol{W}_{m}$ are also full-rank $\forall m$. Note that, considering
[28, Lemma 3], the WAX decomposition of a randomly chosen $\boldsymbol{H}$
exists if and only if then there exists a full-rank $\boldsymbol{W}$ that
achieves said decomposition. From (4) we can get

$\boldsymbol{X}=\boldsymbol{A}_{\mathrm{T}}^{-1}\mathrm{diag}\left(\boldsymbol{W}_{1},\boldsymbol{W}_{2},\dots,\boldsymbol{W}_{T_{\mathrm{P}}}\right)^{-1}\begin{bmatrix}\boldsymbol{H}_{1}\\\
\boldsymbol{H}_{2}\\\ \vdots\\\ \boldsymbol{H}_{T_{\mathrm{P}}}\end{bmatrix},$
(10)

where $\boldsymbol{A}_{\mathrm{T}}$ is full-rank by assumption. On the other
hand, selecting $\boldsymbol{X}$ as in (10) implies that, in order to fulfill
(4), we only need to fulfill

$\boldsymbol{W}\boldsymbol{A}_{\text{B}}\boldsymbol{X}=\begin{bmatrix}\boldsymbol{H}_{T_{\mathrm{P}}+1}\\\
\boldsymbol{H}_{T_{\mathrm{P}}+2}\\\ \vdots\\\
\boldsymbol{H}_{M_{\mathrm{P}}}\end{bmatrix}.$ (11)

If we substitute (10) in (11) and do some simple matrix manipulations we get
(9). ∎

We should note that the assumptions taken in Theorem 2 are not as restrictive
as they seem. In fact, they are fairly reasonable within our framework:

  * •

If $L$ is small with respect to $T$, restricting $T_{\mathrm{P}}=T/L$ to
integers will only have a minor effect on the achievable optimum trade-off
between $T$ and $L$ from (5). For arbitrary $T$ this restriction can translate
to an increase of $T-L\lfloor T/L\rfloor<L$ CPU inputs. Furthermore, in the
optimum trade-off regime we have $0\leq L\leq K$ and $K\leq T\leq M$, so $L$
is small with respect to $T$ throughout much of the trade-off, specially as
$M$ grows large.

  * •

Having full-rank $\boldsymbol{A}_{\text{T}}$ can be relaxed by re-indexing the
diagonal blocks of $\boldsymbol{W}$, i.e., only a submatrix formed by
$T_{\mathrm{P}}$ out of the $M_{\mathrm{P}}$ horizontal blocks of dimensions
$L\times T$ in $\boldsymbol{A}$ should be full-rank. Moreover, from Theorem 1,
$\boldsymbol{A}$ should be of rank $T$ to attain (5), so a $T\times T$
submatrix of it should be full-rank.

Therefore, we will keep these assumptions throughout the rest of the paper.

The importance of Theorem 2 resides in the fact that it provides an
alternative formulation of the WAX decomposition without any need to consider
$\boldsymbol{X}$. Since the WAX decomposition allows for information-lossless
processing within the framework under study (see [28]), the new formulation,
given in (9), will also assure information-lossless processing. Thus, we can
see (9) as the restriction on the $\boldsymbol{W}_{m}$ matrices $\forall m$ in
order to achieve information-lossless processing until the CPU is reached
(under the assumptions of Theorem 2).

Another important implication of Theorem 2 is that we can construct a valid
$\boldsymbol{A}$ matrix by selecting
$\boldsymbol{A}_{\text{B}}\boldsymbol{A}_{\text{T}}^{-1}$ such that there
exists a $\boldsymbol{W}$ that satisfies (11) for any randomly chosen
$\boldsymbol{H}$ (except those in a zero-probability set). We can thus note
that we have full freedom in selecting $\boldsymbol{A}_{\text{T}}$ (as long as
it is full-rank) since this matrix can be compensated through a full-rank
transformation on $\boldsymbol{A}_{\text{B}}$.

Throughout the rest of the paper, we will focus on the study of
$\boldsymbol{A}$ matrices formed as

$\boldsymbol{A}=\widetilde{\boldsymbol{A}}\otimes\mathbf{I}_{L},$ (12)

where $\widetilde{\boldsymbol{A}}$ is now an $M_{\mathrm{P}}\times
T_{\mathrm{P}}$ matrix. Even though it may seem like an unnecessary
restriction, (12) is in fact a desirable construction for a number of reasons:

  * •

Any $\boldsymbol{A}$ matrix resulting from (12) will be inherently sparse
since it would have a minimum of $(M-M_{\mathrm{P}})T$ zeros out of its $MT$
elements.

  * •

The combining module resulting from (12) has a simple hardware implementation
since it only requires to scale and phase-shift the aggregated output of each
panel before combining it with other panels. In fact, our goal is to eliminate
the scaling and phase-shifting so that only sum modules are required.

  * •

The equivalent formulation of the WAX decomposition (9) can simplify greatly
through (12), as will be apparent in Corollary 1. Hence, it will lead to
increased mathematical tractability, allowing to prove the validity of some
interesting $\boldsymbol{A}$ structures.

The main concern that can raise from fixing (12) is that we may sacrifice
achievability of the optimum trade-off (7), which is defined for randomly
chosen $\boldsymbol{A}$. However, if we are able to reach a bound arbitrarily
close to (7) we could conclude that there is no loss associated to (12).

Given (12), it becomes natural to extend the definition from [28, Definition
1] and talk about valid $\widetilde{\boldsymbol{A}}$ matrices for WAX
decomposition. Considering (8), we can now write

$\displaystyle\boldsymbol{A}_{\text{T}}=\widetilde{\boldsymbol{A}}_{\text{T}}\otimes\boldsymbol{I}_{L},$
(13)
$\displaystyle\boldsymbol{A}_{\text{B}}=\widetilde{\boldsymbol{A}}_{\text{B}}\otimes\boldsymbol{I}_{L},$

where $\widetilde{\boldsymbol{A}}_{\text{T}}$ and
$\widetilde{\boldsymbol{A}}_{\text{B}}$ are matrices of dimensions
$T_{\mathrm{P}}\times T_{\mathrm{P}}$ and
$(M_{\mathrm{P}}-T_{\mathrm{P}})\times T_{\mathrm{P}}$, respectively. In order
to simplify upcoming notation, let us define

$\Phi=M_{\mathrm{P}}-T_{\mathrm{P}}.$ (14)

The next corollary comes as a direct consequence of Theorem 2 whenever we have
(12).

###### Corollary 1

Assume that $\boldsymbol{A}$ is of the form (12), and that
$\widetilde{\boldsymbol{A}}_{\text{T}}$ is full rank. If we define the matrix

$\widetilde{\boldsymbol{B}}=\begin{bmatrix}\widetilde{\boldsymbol{A}}_{\text{B}}\widetilde{\boldsymbol{A}}_{\text{T}}^{-1}&-\mathbf{I}_{M_{\mathrm{P}}-T_{\mathrm{P}}}\end{bmatrix}^{\mathrm{T}},$
(15)

the WAX decomposition of some $M\times K$ matrix $\boldsymbol{H}$, given by
(4), exists if and only if we can find full-rank $\boldsymbol{W}_{m}$ matrices
such that

$\begin{bmatrix}\boldsymbol{W}_{1}^{-1}\\!&\\!\boldsymbol{W}_{2}^{-1}\\!&\\!\dots\\!&\\!\boldsymbol{W}_{M_{\mathrm{P}}}^{-1}\end{bmatrix}\\!\begin{bmatrix}\tilde{\boldsymbol{b}}_{1}^{\mathrm{T}}\otimes\boldsymbol{H}_{1}\\\
\tilde{\boldsymbol{b}}_{2}^{\mathrm{T}}\otimes\boldsymbol{H}_{2}\\\ \vdots\\\
\tilde{\boldsymbol{b}}_{M_{\mathrm{P}}}^{\mathrm{T}}\otimes\boldsymbol{H}_{M_{\mathrm{P}}}\end{bmatrix}\\!=\boldsymbol{0}_{L\times
K\Phi},$ (16)

where $\tilde{\boldsymbol{b}}_{m}^{\mathrm{T}}$, for
$m=1,\dots,M_{\mathrm{P}}$, correspond to the rows of
$\widetilde{\boldsymbol{B}}$. A more compact notation for (16) is achieved by
considering the face-splitting product, $(\cdot)\bullet(\cdot)$, which
corresponds to a special case of the Khatri-Rao product dividing the left
matrix into its rows, i.e.,

$\begin{bmatrix}\boldsymbol{W}_{1}^{-1}&\boldsymbol{W}_{2}^{-1}&\dots&\boldsymbol{W}_{M_{\mathrm{P}}}^{-1}\end{bmatrix}\left(\widetilde{\boldsymbol{B}}\bullet\boldsymbol{H}\right)=\boldsymbol{0}_{L\times
K\Phi}.$ (17)

###### Proof:

Let us take Theorem 2 and substitute (13) in (9). Simple matrix manipulation
leads to (16). ∎

Corollary 1 provides a new formulation of the WAX decomposition, now taking
into account (12). The main benefit of this new formulation is that the
diagonal blocks of $\boldsymbol{W}^{-1}$ come in the form of a block row
matrix instead of a block diagonal matrix, which will simplify the tasks of
proving valid $\widetilde{\boldsymbol{A}}$ structures. As happened for
$\boldsymbol{A}$, we can note that the validity of
$\widetilde{\boldsymbol{A}}$ for WAX decomposition depends only on
$\widetilde{\boldsymbol{B}}$, i.e., the product
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}\widetilde{\boldsymbol{A}}_{\mathrm{T}}^{-1}$
will determine the validity of $\widetilde{\boldsymbol{A}}$. Our next goal is
to come up with clever ways of constructing the product
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}\widetilde{\boldsymbol{A}}_{\mathrm{T}}^{-1}$
which can lead to valid $\widetilde{\boldsymbol{A}}$.

##  IV Constructing valid $\widetilde{\boldsymbol{A}}$ matrices

###  IV-A Transforming $\widetilde{\boldsymbol{A}}$ while maintaining validity

Taking into account the results from the previous section, we will start by
stating some transformations on $\widetilde{\boldsymbol{A}}$ that maintain its
validity for WAX decomposition. These may be useful for proving the validity
of specific constructions for $\widetilde{\boldsymbol{A}}$, or for generating
new $\widetilde{\boldsymbol{A}}$ structures from those that can be proved
valid.

###### Proposition 1

Assume a valid $\widetilde{\boldsymbol{A}}$ for WAX decomposition. If we
construct
$\widetilde{\boldsymbol{A}}^{\prime}=\widetilde{\boldsymbol{A}}\boldsymbol{\Theta}$,
where $\boldsymbol{\Theta}$ can be any $T_{\mathrm{P}}\times T_{\mathrm{P}}$
full-rank matrix, $\widetilde{\boldsymbol{A}}^{\prime}$ is also valid for WAX
decomposition.

###### Proof:

Considering (15) we have that

$\displaystyle\widetilde{\boldsymbol{B}}^{\prime}$
$\displaystyle=\begin{bmatrix}\widetilde{\boldsymbol{A}}_{\text{B}}^{\prime}\widetilde{\boldsymbol{A}}_{\text{T}}^{\prime}{}^{-1}&-\mathbf{I}_{\Phi}\end{bmatrix}^{\mathrm{T}}$
$\displaystyle=\begin{bmatrix}\widetilde{\boldsymbol{A}}_{\text{B}}\boldsymbol{\Theta}\boldsymbol{\Theta}^{-1}\widetilde{\boldsymbol{A}}_{\text{T}}^{-1}&-\mathbf{I}_{M_{\mathrm{P}}-T_{\mathrm{P}}}\end{bmatrix}^{\mathrm{T}},$
$\displaystyle=\widetilde{\boldsymbol{B}}.$

From Corollary 1 the validity of $\widetilde{\boldsymbol{A}}^{\prime}$ is only
determined by $\widetilde{\boldsymbol{B}}^{\prime}$, which leads to
Proposition 1. ∎

The previous proposition can be also trivially extended to $\boldsymbol{A}$ if
we disregard the restriction (12). This proposition also remarks that the
selection of $\widetilde{\boldsymbol{A}}_{\text{T}}$ does not affect the
validity of $\widetilde{\boldsymbol{A}}$ as long as it is full-rank, since it
can be compensated by selecting $\boldsymbol{\Theta}$.

###### Proposition 2

Assume $\widetilde{\boldsymbol{A}}$ is valid for WAX decomposition. If we
construct
$\widetilde{\boldsymbol{A}}^{\prime}=\boldsymbol{P}\widetilde{\boldsymbol{A}}$,
where $\boldsymbol{P}$ can be any $M_{\mathrm{P}}\times M_{\mathrm{P}}$
permutation matrix, $\widetilde{\boldsymbol{A}}^{\prime}$ is also valid for
WAX decomposition.

###### Proof:

It is enough to notice that applying a permutation matrix on
$\widetilde{\boldsymbol{A}}$ only corresponds to a re-indexing of the
$\boldsymbol{W}_{m}$ matrices in (3), which does not affect the solvability of
(4). ∎

The previous propositions focused on applying transformations on
$\widetilde{\boldsymbol{A}}$ that maintain its validity for WAX decomposition.
However, as we will see, one way to explore valid $\widetilde{\boldsymbol{A}}$
matrices is to explore $\widetilde{\boldsymbol{B}}$ matrices of the form (15)
that allow us to solve (17). Thus, let us define valid
$\widetilde{\boldsymbol{B}}$ for WAX decomposition as such matrices allowing
for a solution to (16), i.e., leading to a valid $\boldsymbol{A}$ through (13)
and (15).

###  IV-B Constructing $\widetilde{\boldsymbol{A}}$ from predesigned
$\widetilde{\boldsymbol{B}}$

In Section III we noted that properties of $\widetilde{\boldsymbol{B}}$, given
by (15), determine the validity of a matrix $\widetilde{\boldsymbol{A}}$. We
can thus construct an $\widetilde{\boldsymbol{A}}$ by first specifying a valid
$\widetilde{\boldsymbol{B}}$ and then extracting an underlying
$\widetilde{\boldsymbol{A}}$. More specifically, we should only define the
product
$\widetilde{\boldsymbol{A}}_{\text{B}}\widetilde{\boldsymbol{A}}_{\text{T}}^{-1}$
giving a valid $\widetilde{\boldsymbol{B}}$, and then we can extract a valid
$\widetilde{\boldsymbol{A}}$ from the possible
$\widetilde{\boldsymbol{A}}_{\text{B}}$ and
$\widetilde{\boldsymbol{A}}_{\text{T}}$.

If we consider the $\Phi\times T_{\mathrm{P}}$ upper part of
$\widetilde{\boldsymbol{B}}$, given by
$(\widetilde{\boldsymbol{A}}_{\text{B}}\widetilde{\boldsymbol{A}}_{\text{T}}^{-1})^{\text{T}}$,
we can note that we have no loss of generality if we set

$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}},$ (18)

since we can still generate any possible $\widetilde{\boldsymbol{B}}$ of the
form (15) by choosing $\widetilde{\boldsymbol{A}}_{\text{B}}$. 888Note that
with (18), $\widetilde{\boldsymbol{A}}_{\text{B}}$ would directly correspond
the top $T_{\mathrm{P}}$ rows of $\widetilde{\boldsymbol{B}}$, which are the
only ones that can be changed for Corollary 1 to apply. Any other full-rank
$\widetilde{\boldsymbol{A}}_{\text{T}}$ can be selected by considering the
transformation in Proposition 1, although said transformation would also
change $\widetilde{\boldsymbol{A}}_{\text{B}}$. On the other hand, the
physical implication of having (18) is also practically desirable, since this
would result in an $\widetilde{\boldsymbol{A}}$ with minimum number of 1s in
its first $T_{\mathrm{P}}$ rows, i.e., it corresponds to the sparsest possible
$\widetilde{\boldsymbol{A}}_{\text{T}}$. The reason is that such
$\widetilde{\boldsymbol{A}}_{\text{T}}$ leads, through (12), to an
$\boldsymbol{A}$ matrix with a single 1 per row in its first $T$ rows, thus
attaining the lower bound from [28, Lemma 6], which corresponds to a lower
bound on the number of 1s per row of $\boldsymbol{A}$ for it to be valid.
Therefore, in what follows, we consider $\widetilde{\boldsymbol{A}}$ matrices
such that (18) is fulfilled. We remark that such selection does not impact the
validity of $\widetilde{\boldsymbol{A}}$ since if we can find a valid
$\widetilde{\boldsymbol{A}}$ with a different
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}$, we can always find a valid
$\widetilde{\boldsymbol{A}}^{\prime}$ with
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}^{\prime}=\mathbf{I}_{T_{\mathrm{P}}}$
by invoking Proposition 1 with
$\boldsymbol{\Theta}=\widetilde{\boldsymbol{A}}_{\mathrm{T}}^{-1}$. Thus, (18)
should not be seen as a restriction, but a beneficial selection of
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}$ achieving maximum sparsity without
loss.

The following proposition presents a structure for $\boldsymbol{A}$, taking
into account the previous assumptions, which is proved to be valid for WAX
decomposition.

###### Proposition 3

Assume that $\boldsymbol{A}$ is given by (12), with
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$, and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ constructed as

$\widetilde{\boldsymbol{A}}_{\mathrm{B}}=\left[\begin{matrix}\boldsymbol{1}_{\Phi\times
1}&\boldsymbol{0}_{\Phi\times
J}&\underbrace{\begin{matrix}\boldsymbol{I}_{\Phi}&\cdots&\boldsymbol{I}_{\Phi}\end{matrix}}_{Q_{1}=\left\lfloor\frac{T_{\mathrm{P}}-1}{\Phi}\right\rfloor}\end{matrix}\right],$
(19)

where $J=T_{\mathrm{P}}-1-Q_{1}\Phi$, and where

$Q_{1}=\left\lfloor\frac{T_{\mathrm{P}}-1}{\Phi}\right\rfloor.$ (20)

A randomly chosen matrix $\boldsymbol{H}$ admits WAX decomposition with
probability 1 for the given $\boldsymbol{A}$ if

$L\geq\frac{K}{1+Q_{1}},$ (21)

Furthermore, $\boldsymbol{W}_{1}$ (defined in (3)) can be fixed to an
arbitrary $L\times L$ full-rank matrix without affecting the solvability of
the WAX decomposition.

###### Proof:

Selecting
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$ and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ as in (19) leads to

$\widetilde{\boldsymbol{B}}=\left[\begin{matrix}\boldsymbol{1}_{\Phi\times
1}&\boldsymbol{0}_{\Phi\times
J}&\underbrace{\begin{matrix}\boldsymbol{I}_{\Phi}&\cdots&\boldsymbol{I}_{\Phi}\end{matrix}}_{Q_{1}}&-\mathbf{I}_{\Phi}\end{matrix}\right]^{\mathrm{T}}.$

From Corollary 1, we can solve the equivalent formulation of the WAX
decomposition, given in (16), with the restriction of having full-rank
$\boldsymbol{W}_{m}\;\forall m$. If we invoke (16), we get the set of
equations

$\boldsymbol{W}_{1}^{-1}\boldsymbol{H}_{1}=\sum_{q=0}^{Q_{1}+1}\boldsymbol{W}_{J+r+q\Phi}^{-1}\boldsymbol{H}_{J+r+q\Phi},\;\;r=1,\dots,\Phi.$
(22)

Note that we have ignored the negative sign associated to the last identity
block in $\widetilde{\boldsymbol{B}}$ since this can be absorbed without loss
of generality by the corresponding $\boldsymbol{H}_{m}$ blocks. Let us
consider $\boldsymbol{W}_{1}$ to be fixed to an arbitrary $L\times L$ full-
rank matrix (e.g., $\boldsymbol{W}_{1}=\mathbf{I}_{L}$), since this is the
only $\boldsymbol{W}_{m}$ shared in all the $\Phi$ equations from (22). Note
that the selection of $\boldsymbol{W}_{1}$, as long as it is full-rank, does
not affect the solvability of (22) because this matrix can be absorbed by
$\boldsymbol{H}_{1}$ (or by the rest of the $\boldsymbol{W}_{m}$ matrices)
without changing its nature. Then, through trivial linear algebra arguments,
namely counting equations and variables in the resulting linear system, and
assuming randomly chosen $\boldsymbol{H}$ (i.e., $\boldsymbol{H}_{m}$ are also
randomly chosen $\forall m$ and their sum will reduce rank with probability
0), we can independently solve each of the $\Phi$ equations whenever (21) is
fulfilled. ∎

The trade-off between $T_{\mathrm{P}}$ and $L$ given by (21) can be linked to
the optimum trade-off for randomly chosen $\boldsymbol{A}$, given in (7), by
assuming that $\Phi=M_{\mathrm{P}}-T_{\mathrm{P}}$ divides $T_{\mathrm{P}}-1$.
In this case we would have,

$L\geq K\frac{(M_{\mathrm{P}}-T_{\mathrm{P}})}{M_{\mathrm{P}}-1},$ (23)

which for $M_{\mathrm{P}}\gg 1$ corresponds approximately to the same bound as
in (7) (for small $M_{\mathrm{P}}$, the gap can be linked to the loss of
degrees of freedom when fixing $\boldsymbol{W}_{1}^{-1}$). We can thus
conclude that there is essentially no loss in restricting (12). Note that,
unlike (23), the optimum trade-off (7) cannot be achieved with equality, which
further promotes the equivalence between (7) and (23). Furthermore, due to the
integer nature of the variables under consideration, in most cases, both
trade-offs would give the same effective parameter restrictions. Let us thus
refer to (23) as the achievable trade-off. The achievable trade-off results
from fixing one of the diagonal blocks of $\boldsymbol{W}$ to identity, as in
the proof of Proposition 3.

The main restriction of the construction for $\boldsymbol{A}$ considered in
Proposition 3 is that the only meaningful points of the achievable trade-off
(21) are those where $\Phi$ divides $T_{\mathrm{P}}-1$, since except for those
points, there would be an increase in the number of inputs to the CPU, given
by $T=LT_{\mathrm{P}}$, without a corresponding decrease in the
multiplications per antenna, given by $L$. This restriction becomes specially
concerning when we have $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$, since in this
regime Proposition 3 cannot exploit any trade-off between $T$ (or
$T_{\mathrm{P}}$) and $L$. Thus, the following proposition considers a novel
structure for $\boldsymbol{A}$ that allows for exploitation of the trade-off
between $T$ and $L$ in the regime $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$.

###### Proposition 4

Let $\boldsymbol{A}$ be given by (12), with
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$, and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ constructed as

$\widetilde{\boldsymbol{A}}_{\mathrm{B}}=\begin{bmatrix}\alpha_{1}\boldsymbol{1}_{(T_{\mathrm{P}}-1)\times
1}&\mathbf{I}_{T_{\mathrm{P}}-1}\\\ \vdots&\vdots\\\
\alpha_{Q_{2}-1}\boldsymbol{1}_{(T_{\mathrm{P}}-1)\times
1}&\mathbf{I}_{T_{\mathrm{P}}-1}\\\ \alpha_{Q_{2}}\boldsymbol{1}_{\Pi\times
1}&\left[\mathbf{I}_{T_{\mathrm{P}}-1}\right]_{1:\Pi,:}\end{bmatrix},$ (24)

where $\Pi=\Phi-(Q_{2}-1)(T_{\mathrm{P}}-1)$, i.e., the last column block is
cropped to fit the dimensions, and where

$Q_{2}=\left\lceil\frac{\Phi}{T_{\mathrm{P}}-1}\right\rceil.$ (25)

Furthermore, $\alpha_{i}\in\mathbb{C}\backslash\\{0\\}$ can be arbitrarily
selected as long as

$\alpha_{i}=\alpha_{j}\iff i=j,\;\forall i,j\in\\{1,\dots,Q_{2}\\}.$

A randomly chosen matrix $\boldsymbol{H}$ admits WAX decomposition with
probability 1 for the given $\boldsymbol{A}$ if

$L\geq\frac{K}{1+\frac{1}{Q_{2}}}.$ (26)

Moreover, $\boldsymbol{W}_{1}$ (defined in (3)) can be fixed to an arbitrary
$L\times L$ full-rank matrix without affecting the solvability of the WAX
decomposition.

###### Proof:

See Appendix B ∎

If we assume values of $M_{\mathrm{P}}$ and $T_{\mathrm{P}}$ such that (25)
gives an integer without the need of the ceiling operator, the trade-off in
(26) leads again to (23). However, with the $\boldsymbol{A}$ structure given
in Proposition 4 we can now select parameters that allow to exploit the trade-
off in the regime $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$. The following
proposition presents a structure for $\boldsymbol{A}$ which can be seen as
combination of the structures from Propositions 3 and 4, and which allows more
freedom in the exploitation of the achievable trade-off in the regime
$T_{\mathrm{P}}\geq M_{\mathrm{P}}/2+1$.

###### Proposition 5

Let $\boldsymbol{A}$ be given by (12), with
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$, and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ constructed as

$\widetilde{\boldsymbol{A}}_{\mathrm{B}}=\left[\begin{matrix}\boldsymbol{1}_{\Phi\times
1}&\left[\boldsymbol{1}_{Q_{2}\times
1}\otimes\mathbf{I}_{J}\right]_{1:\Phi,:}&\underbrace{\begin{matrix}\boldsymbol{I}_{\Phi}&\cdots&\boldsymbol{I}_{\Phi}\end{matrix}}_{Q_{1}}\end{matrix}\right],$
(27)

where $Q_{1}\geq 1$ and $J$ are defined in Proposition 3, while $Q_{2}$ is now
given by

$Q_{2}=\left\lceil\frac{\Phi}{J}\right\rceil.$ (28)

A randomly chosen matrix $\boldsymbol{H}$ admits WAX decomposition with
probability 1 for the given $\boldsymbol{A}$ if

$L\geq\frac{K}{1+Q_{1}+\frac{1}{Q_{2}}}.$ (29)

Moreover, $\boldsymbol{W}_{1}$ (defined in (3)) can be fixed to an arbitrary
$L\times L$ full-rank matrix without affecting the solvability of the WAX
decomposition.

###### Proof:

See Appendix C ∎

Note that for $Q_{1}=0$, the previous structure degenerates to the case from
Proposition 4, where some elements from the first column of
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ should be changed to fulfill the
additional $\alpha_{i}$ requirements. Furthermore, for $J=0$ (i.e., $\Phi$
divides $T_{\mathrm{P}}-1$) the previous structure leads directly to the one
presented in Proposition 3.

As happened in the previous cases, we can still reach the achievable trade-off
(23) whenever we have parameters such that $Q_{2}$ in (28) evaluates to an
integer value without the need of the ceiling operator. However, we can also
reach it if we have parameters such that $Q_{1}$ evaluates to an integer value
without the floor operation, since this would lead to $J=0$ and $Q_{2}$ would
tend to infinity, so we could remove it altogether. Thus, the structure from
Proposition 5 has a looser requirement so as to reach the achievable trade-off
in the regime $T_{\mathrm{P}}\geq M_{\mathrm{P}}/2+1$ as compared to structure
from Proposition 3, where $Q_{1}$ had to evaluate to an integer value without
the floor operation. Thus, the $\boldsymbol{A}$ structure defined in
Proposition 5 allows for a broader selection of parameters leading to the
achievable trade-off (23), hence increasing the freedom in the exploitation of
said trade-off.

###  IV-C General construction of valid $\widetilde{\boldsymbol{A}}$

A natural generalization of the structure given in Proposition 5, which
already corresponds to a generalization of the structures from Propositions 3
and 4, consists of filling the dimensions of
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ with full identity matrices,
alternating horizontal and vertical allocation until all dimensions are
exhausted. This method is presented in Algorithm 1, where the first column of
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ is given by $\alpha_{i}$ so that it
can accommodate degenerated cases as the one in Proposition 4. The following
conjecture aims at generalizing the validity of the structures defined by
Algorithm 1.

Algorithm 1 Generalized $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ for WAX
decomposition.

0: $M_{\mathrm{P}}$, $T_{\mathrm{P}}$

0: $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ Initialize:
$\left[\widetilde{\boldsymbol{A}}_{\text{B}}\right]_{1,:}\\!=\\!\begin{bmatrix}\alpha_{1},\dots,\alpha_{\Phi}\end{bmatrix}^{T}$
$R_{\text{row}}\\!=\\!T_{\mathrm{P}}\\!-\\!1$, $R_{\text{col}}\\!=\\!\Phi,$
$i=0,$ $i_{\mathrm{row}}=1,$ $i_{\mathrm{col}}=2$

while $R_{\text{col}}>0$ and $R_{\text{row}}>0$ do

$i=i+1$

if $R_{\text{col}}>R_{\text{row}}$ then

$Q_{i}=\left\lfloor R_{\text{col}}/R_{\text{row}}\right\rfloor$

$[\widetilde{\boldsymbol{A}}_{\text{B}}]_{i_{\mathrm{row}}:(i_{\mathrm{row}}+Q_{i}R_{\text{row}}),i_{\mathrm{col}}:(i_{\mathrm{col}}+Q_{i}R_{\text{row}})}\\!=\\!\boldsymbol{1}_{1\times
Q_{i}}\otimes\mathbf{I}_{R_{\text{row}}}$

$R_{\text{col}}=R_{\text{col}}-Q_{i}\cdot R_{\text{row}}$

$i_{\text{col}}=i_{\text{col}}+Q_{i}\cdot R_{\text{row}}$

else

$Q_{i}=\left\lfloor R_{\text{row}}/R_{\text{col}}\right\rfloor$

$[\widetilde{\boldsymbol{A}}_{\text{B}}]_{i_{\mathrm{row}}:(i_{\mathrm{row}}+Q_{i}R_{\text{col}}),i_{\mathrm{col}}:(i_{\mathrm{col}}+Q_{i}R_{\text{col}})}\\!=\\!\boldsymbol{1}_{Q_{i}\times
1}\otimes\mathbf{I}_{R_{\text{col}}}$

$R_{\text{row}}=R_{\text{row}}-Q_{i}\cdot R_{\text{col}}$

$i_{\text{row}}=i_{\text{row}}+Q_{i}\cdot R_{\text{col}}$

end if

end while

###### Conjecture 1

Let $\boldsymbol{A}$ be given by (12), with
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$, and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ constructed through Algorithm 1. A
randomly chosen matrix $\boldsymbol{H}$ admits WAX decomposition with
probability 1 for the given $\boldsymbol{A}$ if

$L\geq\frac{K}{1+Q_{\mathrm{tot}}},$ (30)

where, given $Q_{i}$ for $i=1,\dots,N_{\text{Q}}$, which are defined in
Algorithm 1, and $N_{\text{Q}}$, corresponding to the iteration $i$ where
dimensions are exhausted, we have

$Q_{\mathrm{tot}}=Q_{1}+\frac{1}{Q_{2}+\frac{1}{\ddots+\frac{1}{Q_{N_{\text{Q}}}}}}.$
(31)

Furthermore, in the regime $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$, the first
column of $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$, given by
$[\alpha_{1},\dots,\alpha_{\Phi}]^{\mathrm{T}}$, should fulfill the same
restrictions as in Proposition 4.

#### Supporting arguments

We first note that $N_{\text{Q}}$ is determined by $T_{\mathrm{P}}$ and
$M_{\mathrm{P}}$ (as more thoroughly discussed later), leading to integers in
the range
${N_{\text{Q}}\in\\{1,\dots,\mathrm{min}(T_{\mathrm{P}}-1,\Phi)\\}}$. Then,
for every value of $N_{\text{Q}}$ an equation similar to (47) can be obtained,
which should be proved solvable. However, after extensive work on the matter,
a formal proof for general $N_{\text{Q}}$ has not been found. We have only
been able to test this formula through thorough simulations without
encountering a single exception to it. One simulation procedure we employed to
check the conjecture was to randomly define a large number of combinations of
$K$, $M_{\mathrm{P}}$, and $T_{\mathrm{P}}$, and for each of these
combinations construct an $\boldsymbol{A}$ matrix through Algorithm 1
(together with (12) and (18)) using different values for $L$. Then,
considering [28, Theorem 2], we tried to perform WAX decomposition of a
randomly chosen $\boldsymbol{H}$ (e.g., an IID Gaussian matrix realization),
which would either be possible (i.e., $\boldsymbol{A}$ is valid) or not (i.e.,
$\boldsymbol{A}$ may not be valid). The simulation results led to valid
$\boldsymbol{A}$ matrices if and only if Conjecture 1 was satisfied. $\square$

Figure 2 illustrates with an example how Algorithm 1 is used to define the
$T_{\mathrm{P}}-1$ last columns of $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$.
We can immediately notice that its iterations are equivalent to the steps of
the Euclidean algorithm for finding the greatest common divisor (GCD) between
$T_{\mathrm{P}}-1$ and $\Phi=M_{\mathrm{P}}-T_{\mathrm{P}}$. In fact, we can
see that $Q_{\mathrm{tot}}$, given in (31), corresponds to a continued
fraction expansion [31] of $(T_{\mathrm{P}}-1)/\Phi$. Hence, the value for
$N_{\text{Q}}$, from Conjecture 1, is equal to the number of steps to
calculate $\mathrm{GCD}(T_{\mathrm{P}}-1,\Phi)$. Furthermore, since
$T_{\mathrm{P}}$ and $M_{\mathrm{P}}$ are restricted to integers,
$(T_{\mathrm{P}}-1)/\Phi$ corresponds to a rational number, so its continued
fraction expansion will always be finite [31]. Thus, we can substitute
$Q_{\mathrm{tot}}$ in (30) by $(T_{\mathrm{P}}-1)/\Phi$, which gives directly
the achievable bound (23). On the other hand, the number of 1s in the last
$T_{\mathrm{P}}-1$ columns of $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$, which
gives its sparsity, corresponds to

$\left\|\left[\widetilde{\boldsymbol{A}}_{\mathrm{B}}\right]_{:,2:T_{\mathrm{P}}}\right\|^{2}_{\mathrm{F}}=\Phi-\mathrm{GCD}(T_{\mathrm{P}}-1,\Phi).$

Figure 2: Example of how Algorithm 1 constructs the last $T_{\mathrm{P}}-1$
columns of $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ for $M_{\mathrm{P}}=12$
and $T_{\mathrm{P}}=9$.

The reader may also note here the direct relation between Conjecture 1 and
Propositions 3-5. When $N_{\mathrm{Q}}=1$, i.e., $\Phi$ divides
$T_{\mathrm{P}}-1$, Conjecture 1 directly corresponds to Proposition 3 (for
this case we can choose $\alpha_{i}=1$). Furthermore, when $N_{\mathrm{Q}}=2$,
i.e., $J=T_{\mathrm{P}}-1-Q_{1}\Phi$ divides $\Phi$, Conjecture 1 leads to
either Proposition 4 (in the $T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$ regime) or
Proposition 5 (in the $T_{\mathrm{P}}\geq M_{\mathrm{P}}/2+1$ regime, where we
can choose $\alpha_{i}=1,\,\forall i$). Thus, although we lack a formal proof
for Conjecture 1, we may use Algorithm 1 as a general strategy for
constructing $\widetilde{\boldsymbol{A}}$ since it merges the previous results
whenever they attain the achievable trade-off (23).

Another thing to remark is that, for centralized architectures, where we can
identify $T_{\mathrm{P}}=M_{\mathrm{P}}$, our structures degenerate to the
trivial $\widetilde{\boldsymbol{A}}=\mathbf{I}_{M_{\mathrm{P}}}$, i.e., the
combining module can be disregarded altogether. Moreover, for information-
lossless fully-decentralized architectures (e.g., local-MF as in [14]), where
we can identify $T_{\mathrm{P}}=1$ (for this case we have $L=K$), our
structures degenerate to
$\widetilde{\boldsymbol{A}}=\mathbf{1}_{M_{\mathrm{P}}\times 1}$ (taking the
case $\alpha_{i}=1$), i.e., the combining module would correspond to a sum
module that combines all the outputs from the decentralized filters. This
remarks the relevance of the presented work as a generalization of
architectures with an arbitrary level of decentralization, since it allows for
a wide range of architectures from centralized to fully-decentralized, and
where both extremes can be considered within the same framework.

##  V Decentralized computation of $\boldsymbol{W}$

The structures for $\boldsymbol{A}$ presented in the previous section are not
only interesting for their sparsity and validity for WAX decomposition, but we
can also use them to create decentralized schemes for computing
$\boldsymbol{W}$. As previously discussed, decentralized here means that each
panel would find the $\boldsymbol{W}_{m}$ to be applied, i.e., leading to
information-lossless processing, by exchanging reduced data with the rest of
the panels so that the full channel matrix $\boldsymbol{H}$ needs not be
collected at any single point. Using the equivalent formulation of the WAX
decomposition (9), or (16) with (12), we can find $\boldsymbol{W}_{m}$
matrices achieving information-lossless processing without having to compute
$\boldsymbol{X}$.

The $\boldsymbol{A}$ structures from Propositions 3, 4, and 5, allow the use
of a tree-based scheme, such as the one illustrated in Fig. 3 where we have
conveniently re-indexed the $\boldsymbol{W}_{m}$ and $\boldsymbol{H}_{m}$
matrices to make them general for all three cases. Specifically, we identify
now $\boldsymbol{W}_{0}$ with the original $\boldsymbol{W}_{1}$ from (3),
which is the $\boldsymbol{W}_{m}$ that can be arbitrarily selected in
Propositions 3-5. The tree scheme consists of a reference panel, which is
connected through a one-way link to $N_{1}$ processing panels, i.e., having a
local processing unit (LPU), each of which communicates with a set of $N_{2}$
passive panels. For simplicity, the reference panel makes use of the available
freedom provided by Propositions 3-5 by fixing
$\boldsymbol{W}_{0}=\mathbf{I}_{L}$. This way, $\boldsymbol{W}_{0}$ has no
effect, so the reference panel only needs to share its $L\times K$ local
channel matrix $\boldsymbol{H}_{0}$ with the $N_{1}$ processing
panels.999$\boldsymbol{W}_{0}$ can also be fixed to any other full-rank
matrix. In that case, either all processing panels have previous knowledge of
$\boldsymbol{W}_{0}$ for their computations, or the reference panel should
share the $L\times K$ matrix resulting from the multiplication
$\boldsymbol{W}_{0}^{-1}\boldsymbol{H}_{0}$ instead of $\boldsymbol{H}_{0}$
directly. Hence, any selection other than $\boldsymbol{W}_{0}=\mathbf{I}_{L}$
leads to higher computation complexity. Each group of $N_{2}$ passive panels
would share their local channels to their corresponding processing panel,
which would then use them to compute all the $\boldsymbol{W}_{m}$ matrices
that have to be applied in its group (including itself). Lastly, the
processing panels would send each $\boldsymbol{W}_{m}$ to the corresponding
passive panels in their group so that they can apply them.

In order to understand why the $\boldsymbol{A}$ structures from Propositions
3-5 can make use of the decentralized scheme from Fig. 3, we will refer to the
proofs of said propositions. For Proposition 3, we can see that the equivalent
formulation of the WAX decomposition can be solved by solving a set of
independent equations of the form (22), where the left-hand side (LHS), which
is the only part shared in all equations, is associated to the reference panel
($\boldsymbol{W}_{1}$, here re-indexed to $\boldsymbol{W}_{0}$, which is later
fixed in the proof), and the right-hand side (RHS) can be associated to a
group of panels of which one would be the processing panel and the rest the
passive panels. Each processing panel would only need the $\boldsymbol{H}_{m}$
matrices of the rest of the panels in the group, as well as the one from the
reference panel, to be able to solve its equation, corresponding to one out of
the $\Phi$ independent equations from (22). For Proposition 4, the reference
panel determines the LHS of (38). On the other hand, (38) can be divided into
a set of independent equations of the form (39), only sharing
$\boldsymbol{H}_{0}$ (or $\widetilde{\boldsymbol{H}}_{0}$ in the proof), and
each of which can be solved at one processing panel by accumulating the
involved $\boldsymbol{H}_{m}$ matrices. The same is true for Proposition 5
where instead of (39) we would have (47) solved at each processing panel.

Figure 3:  Architecture for decentralized computation of the
$\boldsymbol{W}_{m}$ matrices for the $\widetilde{\boldsymbol{A}}$ given in
Propositions 3-5. Blue arrows indicate sharing of local CSI, and red arrows
indicate sharing of decentralized filters after computation.

Table I gives the resulting parameters $N_{1}$ and $N_{2}$ of the
decentralized scheme in Fig.3 for the different structures from Propositions
3-5. Said parameters are related to the number of independent equations and
the number of involved passive panels in each equation, respectively, as
explained before. In the case of Propositions 4 and 5, we are assuming that
$Q_{2}$ evaluates to an integer without the need of the ceiling operator;
otherwise, the last group of panels would have a number of passive panels
smaller than $N_{2}$ due to the cropping of the corresponding equation. Note
that, in all cases, several independent equations can be solved at a single
processing panel by gathering the corresponding $\boldsymbol{H}_{m}$ matrices
at said panel. Thus, the values of $N_{1}$ from Table I could be trivially
reduced by a corresponding increase in $N_{2}$.

To conclude this section, we have shown that, not only can we define
architectures with an arbitrary level of decentralization in the data phase
(i.e., by employing the framework from Fig. 1), but, during the training
phase, and if $\boldsymbol{A}$ is suitably selected, these architectures can
be used for computing in a decentralized manner the decentralized processing
to be applied (i.e., by considering schemes like the one in Fig. 3).

$\boldsymbol{A}$ structure | $N_{1}$ | $N_{2}$
---|---|---
Proposition 3 | $\Phi$ | $Q_{1}$
Proposition 4 | $T_{\mathrm{P}}-1$ |  $Q_{2}$ (from (25))
Proposition 5 | $J$ |  $Q_{2}$ (from (28))
TABLE I: Decentralized scheme parameters.

##  VI Numerical results and examples

In Section IV, we presented some constructions for $\boldsymbol{A}$ that were
proved to be valid for WAX decomposition. The current section aims at
providing some discussion, as well as useful examples, to further understand
the differences of said constructions, and the circumstances under which they
reach the achievable trade-off (23).

In Section IV, we discussed the requirements for Propositions 3-5 to achieve
(23), namely that either $Q_{1}$ or $Q_{2}$ should evaluate to an integer
without requiring the floor/ceiling operator, respectively. Instead of
obtaining $Q_{1}$ or $Q_{2}$ from $T_{\mathrm{P}}$ and $M_{\mathrm{P}}$, we
can also take them as arbitrary integers, and substitute the resulting $K/L$
in (21), (26), and (29) (after restricting the inequality for the case of
equality) to get the ratios $K/L$ that achieve (23) for the structures in
Propositions 3, 4, and 5, respectively. The reason is that we can always find
a combination of integers $M_{\mathrm{P}}$ and $T_{\mathrm{P}}$ leading to the
corresponding $Q_{1}$ or $Q_{2}$ without the need of the respective
floor/ceiling operators. An alternative interpretation of the presented
structures is that they are directly defined by a (truncated) continued
fraction expansion of the ratio $K/L$, corresponding to (31). The structure
from Algorithm 1 considers the full continued fraction expansion of $K/L$, the
structure from Proposition 3 is given by a fraction expansion of $K/L$
truncated to a single term ($Q_{1}$), and the structure from Proposition 5
(which degenerates to the one from Proposition 4 for $Q_{1}=0$) is given by a
fraction expansion of $K/L$ truncated to two terms ($Q_{1}$ and $Q_{2}$). Fig.
4 shows the possible $K/L$ ratios achieving (23) for the $\boldsymbol{A}$
structures from Propositions 3-5. Proposition 4 is the only one having values
in the interval $(1,2)$, associated to the regime
$T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$, as previously mentioned. However, the
structure from Proposition 5 would also reach the points from Proposition 4 by
selecting the first column of $\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ as in
(24) with $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. Note that, for Proposition
5, the values in the interval [2,3] can be shifted to any other interval
[i,i+1] with $i\geq 2$ (which corresponds to increasing $Q_{1}$). Any other
value for $K/L$ can be obtained by using Algorithm 1, since any positive
rational number can be decomposed into a continued fraction of the form (31)
[31], while $K/L-1$ is inherently restricted to positive rational numbers
since $K$ and $L$ are restricted to integers, and we have $L\leq K$.

Figure 4: Values of $K/L$ achieving (23) in the interval $[1,3]$ with the
$\boldsymbol{A}$ structures from the different propositions.

As previously discussed, the results presented in this paper lead to
conditions that exhibit direct connection to the ratio $K/L$. However, the
main value of the original condition (5), as presented in [28], and which has
now been shown to be fundamental due to Theorem 1, is to give a trade-off
between $L$, the number of multiplications per antenna, and $T$, the required
connections to a CPU. It is thus of special interest to outline the explicit
relation between $T$ and $L$ in the conditions obtained in this work: (21),
(26) and (29), as well as (23), which ultimately governs all the previous
conditions (apart from being attained by the structure from Conjecture 1). On
the other hand, the achievable trade-off from (23) can be straightforwardly
translated into a condition between $L$ and $T$ (instead of $T_{\mathrm{P}}$)
if we multiply both the numerator and denominator of the RHS by $L$. However,
there may be points of this trade-off not attainable by the proposed
structures, as we will illustrate next.

In Fig. 5 we compare the trade-off between $T$ and $L$ considering the
different strategies for constructing $\boldsymbol{A}$. The dashed blue line
corresponds to the trade-off defined in (5), which is proved to be attained by
randomly chosen $\boldsymbol{A}$ [28, Theorem 1], while Theorem 1 shows that
it is also the optimum trade-off. The red line corresponds to the achievable
trade-off (23), which can be attained by all the structures presented in this
work under favorable parameter combinations. We can see that there is a minor
gap between the achievable trade-off and the optimum one, which is mainly
noticeable as $L$ grows. This gap can be explained by the exhaustion of
degrees of freedom when fixing one $L\times L$ matrix, which clearly grows
with $L$. The rest of the points correspond to the achievable points that can
be exploited in practice through the proposed structures. We have used [28,
Theorem 2], i.e., by performing WAX decomposition of a randomly chosen
$\boldsymbol{H}$, to check that the proposed $\boldsymbol{A}$ structures are
valid at these points, thus confirming the theoretical claims from
Propositions 3-5, as well as Conjecture 1. The purple triangles correspond to
the achievable points with randomly chosen $\boldsymbol{A}$ after considering
the integer restriction of the variables $T$, $L$, and $M_{\mathrm{P}}$.
Hence, these points correspond to the fundamental limits of multi-antenna
architectures with an arbitrary level of decentralization, while the main
motivation of the current work is to get as close as possible to these points
with structured sparse constructions for $\boldsymbol{A}$. The red circles
correspond to the $\boldsymbol{A}$ structures defined through Proposition 3,
which was already included in the conference version [1]. The remaining points
are novel contributions achieved by Propositions 4 and 5, as well as by
Conjecture 1. As we can see, the achievable points when constructing
$\boldsymbol{A}$ as in Conjecture 1 have minor gap (if any) with respect to
the achievable trade-off due to the integer nature of $T_{\mathrm{P}}$ from
the limitation $T_{\mathrm{P}}=T/L$. As for the other constructions, we see
that the achievable points for the constructions from Propositions 4 and 5 get
fairly close to the points achieved by Conjecture 1, while the main difference
is that they exploit different regimes of the trade-off. We again note that
the achievable points from Proposition 3 only allowed for reductions in the
regime $T_{\mathrm{P}}\geq M_{\mathrm{P}}/2+1$, which justifies the poor
performance in the right part of the plots. These results remark the
importance of the novel structures presented in this work for better
exploitation of the trade-off between level of decentralization and
decentralized complexity.

Figure 5: Comparison of fundamental trade-off (5) with achievable trade-off
(23) and achievable points with the presented structures. We assume $M=120$,
$K=9$ (left) and $K=15$ (right).

The following example illustrates the differences between the presented
strategies for constructing $\boldsymbol{A}$. By focusing on the regime
$T_{\mathrm{P}}\geq M_{\mathrm{P}}/2+1$, we intentionally skip Proposition 4
due to its correspondence with Proposition 5 for $Q_{1}=0$.

###### Example 1

Let $T_{\mathrm{P}}=6$, $M_{\mathrm{P}}=9$, and $K=40$. We then have $\Phi=3$.
Let $\boldsymbol{A}_{1}$, $\boldsymbol{A}_{2}$, and $\boldsymbol{A}_{3}$, be
$\boldsymbol{A}$ matrices constructed using Propositions 3, 5, and Algorithm
1, respectively. Such matrices are found in (33), where every pair of
$\mathbf{I}_{L}$ matrices in each row block of $\boldsymbol{A}$ can be seen as
a sum module which combines the outputs from the two respective
panels.101010We have horizontally flipped
$[\widetilde{\boldsymbol{A}}_{\mathrm{B}}]_{:,2:T_{\mathrm{P}}}$ from
Algorithm 1, which is possible considering Propositions 1 and 2, to remark its
similarity with the other constructions. Note that the structure from
Proposition 4 does not apply here since we are not in the regime
$T_{\mathrm{P}}<M_{\mathrm{P}}/2+1$. Using Propositions 3, 5, and Conjecture
1, we can find the possible values for $L$ to be

$\displaystyle L_{1}\geq 20,\;\;\;\;\;\;L_{2}\geq 16,\;\;\;\;\;\;L_{3}\geq
15,$

respectively. On the other hand, (23) leads to the restriction $L\geq 15$.
Thus, in this case Algorithm 1 gives the only structure reaching the
achievable trade-off. However, the structure from Proposition 5 gets
considerably closer to it than the one from Proposition 3.

We next show a practical example of how to use the theory developed in this
work for designing a BS with constrained decentralized processing complexity.

###### Example 2

Let us have a massive MIMO BS with $M=64$ antennas serving $K=10$ users. If we
choose an arbitrary number of multiplications per antenna $L$, we would like
to see which methods can be used for constructing a combining module
$\boldsymbol{A}$, and what would be their resulting minimum number of inputs
to the CPU. Recall that $L$ should be an integer dividing $M$ to be able to
group the antennas into $M_{\mathrm{P}}=M/L$ panels. The number of inputs to
the CPU is given by $T=T_{\mathrm{P}}L$, where we can find the minimum
achievable integer $T_{\mathrm{P}}$ from (23) by

$T_{\mathrm{P},\min}=\left\lceil M_{\mathrm{P}}-\frac{M-L}{K}\right\rceil,$
(32)

which, using Conjecture 1, will always be possible by constructing
$\boldsymbol{A}$ through Algorithm 1.

  * •

Let us have $L=2$, which gives $M_{\mathrm{P}}=32$ and the achievable trade-
off $T_{\mathrm{P}}\geq 25.8$ leading to $T_{\mathrm{P},\min}=26$. If we
choose to construct $\boldsymbol{A}$ through Proposition 3, we start by using
$T_{\mathrm{P}}=T_{\mathrm{P},\min}$ to get $\Phi=6$ from (14). This gives
$Q_{1}=4$ by (20), which leads to the restriction $L\geq 2$. So we get the
desired $L$ without the need to increase $T_{\mathrm{P}}$. If we use
Proposition 5 this restriction transforms to $L\geq 1.94$. Thus, for this
case, due to the integer restrictions, there is no difference in terms of
inputs to the CPU of defining $\boldsymbol{A}$ from Propositions 3, 5, or
Algorithm 1, since all give $T=52$. We would recommend Proposition 3 for its
simplicity and greater sparsity.

  * •

Let us have $L=4$, which gives $M_{\mathrm{P}}=16$ leading to
$T_{\mathrm{P},\min}=10$. If we want to construct $\boldsymbol{A}$ through
Proposition 3, we proceed as before using first
$T_{\mathrm{P}}=T_{\mathrm{P},\min}$ to get $Q_{1}=1$, which leads to $L\geq
5$. In this case the desired $L$ is not possible, so we increase
$T_{\mathrm{P}}=T_{\mathrm{P},\min}+1=25$, and calculate again $Q_{1}=2$,
which leads to $L\geq 3.33$. This means that in order to use $\boldsymbol{A}$
from Proposition 3 we need $T=100$ inputs to the CPU. Instead, if we try to
construct $\boldsymbol{A}$ from Proposition 5, we start again with
$T_{\mathrm{P}}=T_{\mathrm{P},\min}$, leading to $Q_{1}=2$ and $Q_{2}=2$. This
gives the restriction $L\geq 4$, which is already fulfilled by the desired
one. Thus, using Proposition 5 to define $\boldsymbol{A}$ would require $T=96$
inputs to the CPU, i.e., there is no loss with respect to the achievable gain,
which can also be reached by defining $\boldsymbol{A}$ through Algorithm 1.

$\begin{matrix}\scalebox{0.8}{$\boldsymbol{A}_{\mathrm{T}}$}\left\\{\begin{matrix}\\\
\\\ \\\ \\\ \\\ \\\ \end{matrix}\right.\\\ \\\ \\\ \\\
\end{matrix}\begin{matrix}\\\ \\\ \\\ \\\ \\\ \\\
\scalebox{0.8}{$\boldsymbol{A}_{\mathrm{B}}$}\left\\{\begin{matrix}\\\ \\\ \\\
\end{matrix}\right.\end{matrix}\boldsymbol{A}_{1}=\;\begin{bmatrix}\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\\\
\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\end{bmatrix},\;\;\boldsymbol{A}_{2}=\begin{bmatrix}\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\\\
\mathbf{I}_{L}&\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\end{bmatrix},\;\;\boldsymbol{A}_{3}=\begin{bmatrix}\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\\\
\mathbf{I}_{L}&\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}&\mathbf{I}_{L}&\boldsymbol{0}\\\
\mathbf{I}_{L}&\mathbf{I}_{L}&\mathbf{I}_{L}&\boldsymbol{0}&\boldsymbol{0}&\mathbf{I}_{L}\end{bmatrix}$
(33)

##  VII Conclusions

We have continued with the work on WAX decomposition by filling some gaps from
[28]. We have proved that the trade-off given in [28] is fundamental in the
sense that no decentralized system falling withing our general framework can
perform beyond it. We have defined an equivalent formulation of the WAX
decomposition without the need of considering the CPU processing matrix
$\boldsymbol{X}$. We have used said equivalent formulation to prove some
properties that allow to transform the combining matrix $\boldsymbol{A}$ while
maintaining its validity. We have also proved the validity of 3 structures for
$\boldsymbol{A}$ which lead to an achievable version of the trade-off in [28]
under different system parameter settings. An ad hoc method for constructing
$\boldsymbol{A}$ such that the achievable trade-off is reached for any system
parameter setting is also presented. We have defined a decentralized scheme
for obtaining the information-lossless decentralized filters
$\boldsymbol{W}_{m}$ to be applied at different panels without the need to
aggregate their CSIs.

Future work can include jointly considering the sparse combining modules
$\boldsymbol{A}$ in scenarios where the channel matrix $\boldsymbol{H}$ is
also sparse or has rank-deficiencies. More clever decentralized schemes, e.g.,
those which could be also employed with $\boldsymbol{A}$ matrices constructed
through the general ad hoc method from Algorithm 1, could also be explored. It
would also be desirable to come up with a formal proof for the validity of the
$\boldsymbol{A}$ matrices constructed through the ad hoc method.

## Appendix A: Proof of Theorem 1

The necessary condition $T\geq K$, in (5) stated as $T>(K-1)$ due to the
integer nature of $T$, comes trivially by the fact that
$\mathrm{rank}(\boldsymbol{W}\boldsymbol{A}\boldsymbol{X})\leq T$ and
$\mathrm{rank}(\boldsymbol{H})=K$ with probability 1 for randomly chosen
$\boldsymbol{H}$. Let us thus assume $T\geq K$. If we invoke [28, Lemma 3], we
can conclude that a randomly chosen $\boldsymbol{H}$ admits WAX decomposition
if and only if we can find full-rank $\boldsymbol{W}$ solving the linear
system

$\boldsymbol{A}\boldsymbol{X}=\boldsymbol{W}^{-1}\boldsymbol{H}.$

The previous expression can be vectorized as in [28] giving

$\left[\mathbf{I}_{K}\otimes\boldsymbol{A}\;\;\;-(\boldsymbol{H}^{\mathrm{T}}\otimes\mathbf{I}_{M})\widetilde{\mathbf{I}}_{\boldsymbol{W}}\right]\begin{bmatrix}\mathrm{vec}(\boldsymbol{X})\\\
\mathrm{vec}(\boldsymbol{W}_{1})\\\ \vdots\\\
\mathrm{vec}(\boldsymbol{W}_{M_{\mathrm{P}}})\end{bmatrix}=\boldsymbol{0}_{MK\times
1},$ (34)

where $\widetilde{\mathbf{I}}_{\boldsymbol{W}}$ corresponds to an $M^{2}\times
ML$ matrix having $\mathbf{I}_{L}$ blocks separated by blocks of zeros so as
to disregard the zeros in $\mathrm{vec}(\boldsymbol{W})$. The rank of the
block $\mathbf{I}_{K}\otimes\boldsymbol{A}$, which multiplies
$\mathrm{vec}(\boldsymbol{X})$, is given by

$\mathrm{rank}(\mathbf{I}_{K}\otimes\boldsymbol{A})=KR_{\boldsymbol{A}},$

where $R_{\boldsymbol{A}}=\mathrm{rank}(\boldsymbol{A})$. If
$\mathrm{vec}(\boldsymbol{X})$ is in the null-space of
${\mathbf{I}_{K}\otimes\boldsymbol{A}}$, then
$[\mathrm{vec}(\boldsymbol{W}_{1})^{T},\dots,\mathrm{vec}(\boldsymbol{W}_{M_{\mathrm{P}}})^{T}]^{T}$
should be in the null-space of
$-(\boldsymbol{H}^{\mathrm{T}}\otimes\mathbf{I}_{M})\widetilde{\mathbf{I}}_{\boldsymbol{W}}$
(full-rank with probability 1), leading to a more restrictive condition than
(5), $K<L$. Thus, we can remove the subspace of $\mathrm{vec}(\boldsymbol{X})$
that falls in the null-space of $\mathbf{I}_{K}\otimes\boldsymbol{A}$, which
means that can rewrite (34) as

$\left[\boldsymbol{C}\;\;-(\boldsymbol{H}^{\mathrm{T}}\otimes\mathbf{I}_{M})\widetilde{\mathbf{I}}_{\boldsymbol{W}}\right]\begin{bmatrix}\tilde{\boldsymbol{x}}\\\
\mathrm{vec}(\boldsymbol{W}_{1})\\\ \vdots\\\
\mathrm{vec}(\boldsymbol{W}_{M_{\mathrm{P}}})\end{bmatrix}=\boldsymbol{0}_{MK\times
1},$ (35)

where $\boldsymbol{C}$ is now an $MK\times KR_{\boldsymbol{A}}$. Since
$\boldsymbol{H}$ is a randomly chosen matrix, the block
$-(\boldsymbol{H}^{\mathrm{T}}\otimes\mathbf{I}_{M})\widetilde{\mathbf{I}}_{\boldsymbol{W}}$
adds full-rank to $\boldsymbol{C}$ with probability 1. Hence, the
$MK\times(KR_{\boldsymbol{A}}+ML)$ matrix
$\left[\boldsymbol{C}\;\;-(\boldsymbol{H}^{\mathrm{T}}\otimes\mathbf{I}_{M})\widetilde{\mathbf{I}}_{\boldsymbol{W}}\right]$
is full-rank with probability 1, which means that it has non-empty null-space
only if

$MK<KR_{\boldsymbol{A}}+ML.$ (36)

After simple manipulation of (36), and noting that $R_{\boldsymbol{A}}\leq T$,
where equality corresponds to $\boldsymbol{A}$ having rank T, we reach the
necessary condition (5) .

## Appendix B: Proof of Proposition 4

Selecting
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$ and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ as in (24) leads to

$\widetilde{\boldsymbol{B}}=\begin{bmatrix}\begin{matrix}\alpha_{1}\boldsymbol{1}_{(T_{\mathrm{P}}-1)\times
1}&\mathbf{I}_{T_{\mathrm{P}}-1}\\\ \vdots&\vdots\\\
\alpha_{Q_{2}-1}\boldsymbol{1}_{(T_{\mathrm{P}}-1)\times
1}&\mathbf{I}_{T_{\mathrm{P}}-1}\\\ \alpha_{Q_{2}}\boldsymbol{1}_{\Pi\times
1}&\left[\mathbf{I}_{T_{\mathrm{P}}-1}\right]_{1:\Pi,:}\end{matrix}&\scalebox{1.5}{$-\mathbf{I}_{\Phi}$}\end{bmatrix}^{\mathrm{T}}.$
(37)

If we use Corollary 1, we can substitute $\widetilde{\boldsymbol{B}}$ in the
equivalent formulation of the WAX decomposition, given in (17). We then fix
$\boldsymbol{W}_{1}^{-1}$ to some arbitrary full-rank matrix, for simplicity
let us have $\boldsymbol{W}_{1}^{-1}=\mathbf{I}_{L}$ (any other full rank-
matrix can be absorbed by $\boldsymbol{H}_{1}$ or by the remaining
$\boldsymbol{W}_{m}$’s), so (17) gives

$\displaystyle-\begin{bmatrix}\alpha_{1}&\cdots&\alpha_{Q_{2}}\end{bmatrix}\otimes\boldsymbol{H}_{1}=\begin{bmatrix}\boldsymbol{W}_{2}^{-1}&\cdots&\boldsymbol{W}_{M_{\mathrm{P}}}^{-1}\end{bmatrix}$
(38)
$\displaystyle\times\left(\begin{bmatrix}\begin{matrix}\mathbf{I}_{T_{\mathrm{P}}-1}&\cdots&\left[\mathbf{I}_{T_{\mathrm{P}}-1}\right]_{1:\Pi,:}\end{matrix}\\\
\scalebox{1.3}{$-\mathbf{I}_{\Phi}$}\end{bmatrix}\bullet\begin{bmatrix}\boldsymbol{H}_{2}\\\
\vdots\\\ \boldsymbol{H}_{M_{\mathrm{P}}}\end{bmatrix}\right)$ .

We can notice that the face-splitting product $(.)\bullet(.)$ only substitutes
in the left matrix each 1 at row $m$ by $\boldsymbol{H}_{m}$. Furthermore,
(38) corresponds to a series of $T_{\mathrm{P}}-1$ independent equations of
the form

$\displaystyle\begin{bmatrix}\alpha_{1}&\cdots&\alpha_{Q_{2}}\end{bmatrix}\otimes\widehat{\boldsymbol{H}}_{0}=\begin{bmatrix}\widehat{\boldsymbol{W}}_{1}^{-1}&\cdots&\widehat{\boldsymbol{W}}_{Q_{2}+1}^{-1}\end{bmatrix}$
(39)
$\displaystyle\times\begin{bmatrix}\widehat{\boldsymbol{H}}_{1}&\widehat{\boldsymbol{H}}_{1}&\cdots&\widehat{\boldsymbol{H}}_{1}\\\
\widehat{\boldsymbol{H}}_{2}&\boldsymbol{0}_{L\times
K}&\cdots&\boldsymbol{0}_{L\times K}\\\ \boldsymbol{0}_{L\times
K}&\widehat{\boldsymbol{H}}_{3}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\boldsymbol{0}_{L\times K}\\\ \boldsymbol{0}_{L\times
K}&\cdots&\boldsymbol{0}_{L\times
K}&\widehat{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}$ ,

where $\widehat{\boldsymbol{H}_{i}}$ corresponds to a re-indexing of the
respective $\boldsymbol{H}_{m}$, including a possible change of sign
($\widehat{\boldsymbol{H}}_{0}=-\boldsymbol{H}_{1}$ is the only matrix shared
in all equations), so we can think of them as $L\times K$ blocks from a
randomly chosen matrix. Note that we may also require the ability to solve a
sub-problem of (39) with $1$ less column block, i.e., substituting $Q_{2}$ by
$Q_{2}-1$. This is due to the possibly cropped block in (24) or (38),
$\left[\mathbf{I}_{T_{\mathrm{P}}-1}\right]_{1:\Pi,:}$, which would lead to
$(T_{\mathrm{P}}-1-\Pi)$ equations having $Q_{2}-1$ instead of $Q_{2}$ column
blocks in (39). However, this sub-problem is less restrictive than (24), as we
will see.

Let us multiply from the right both sides of (39) by the full rank matrix
$\mathrm{diag}(\widehat{\boldsymbol{V}}_{1},\dots,\widehat{\boldsymbol{V}}_{1})$,
where $\widehat{\boldsymbol{V}}_{1}$ is the $K\times K$ right unitary matrix
from the singular value decomposition of $\widehat{\boldsymbol{H}}_{1}$. If we
further use the fact that any full-rank block diagonal matrix being multiplied
between the $\widehat{\boldsymbol{W}}_{m}^{-1}$’s and
$\widehat{\boldsymbol{H}}_{m}$’s block matrices in the RHS of (39), as well as
any full-rank $L\times L$ matrix that multiplies from the left both sides of
(39), can be absorbed by the corresponding $\widehat{\boldsymbol{W}}_{m}^{-1}$
matrices, we reach

$\displaystyle([\alpha_{1}\;\,\cdots\;\,\alpha_{Q_{2}}]\otimes[\mathbf{I}_{L}\;\;\widetilde{\boldsymbol{H}}_{0}])=\begin{bmatrix}\widetilde{\boldsymbol{W}}_{1}^{-1}&\\!\cdots\\!&\widetilde{\boldsymbol{W}}_{Q_{2}+1}^{-1}\end{bmatrix}$
(40)
$\displaystyle\scalebox{0.9}{$\times\begin{bmatrix}[\mathbf{I}_{L}\;\boldsymbol{0}_{L\times(K-L)}]&[\mathbf{I}_{L}\;\boldsymbol{0}_{L\times(K-L)}]&\cdots&[\mathbf{I}_{L}\;\boldsymbol{0}_{L\times(K-L)}]\\\
[\mathbf{I}_{L}\;\widetilde{\boldsymbol{H}}_{2}]&\boldsymbol{0}_{L\times
K}&\cdots&\boldsymbol{0}_{L\times K}\\\ \boldsymbol{0}_{L\times
K}&[\mathbf{I}_{L}\;\widetilde{\boldsymbol{H}}_{3}]&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\boldsymbol{0}_{L\times K}\\\ \boldsymbol{0}_{L\times
K}&\cdots&\boldsymbol{0}_{L\times
K}&[\mathbf{I}_{L}\;\widetilde{\boldsymbol{H}}_{Q_{2}+1}]\end{bmatrix}$}$ ,

where $\widetilde{\boldsymbol{H}}_{i}$, $i\in\\{0,2,\dots,(Q_{2}+1)\\}$, are
now $L\times(K-L)$ blocks from a randomly chosen matrices.111111Note that the
multiplication by a common unitary matrix from the right to generate each
$\widetilde{\boldsymbol{H}}_{i}$ can be seen as a common rotation to their
original random unitary matrices, thus it does not affect the randomly chosen
property Equation (40) corresponds to the system of equations

$\left\\{\begin{array}[]{l}\widetilde{\boldsymbol{W}}_{1}^{-1}+\widetilde{\boldsymbol{W}}_{i+1}^{-1}=\alpha_{i}\mathbf{I}_{L}\\\
\widetilde{\boldsymbol{W}}_{i+1}^{-1}\widetilde{\boldsymbol{H}}_{i+1}=\alpha_{i}\widetilde{\boldsymbol{H}}_{0}\end{array}\right.,\;\;\;i=1,\dots,Q_{2}.$
(41)

We can now isolate
$\widetilde{\boldsymbol{W}}_{i+1}^{-1}=\alpha_{i}\mathbf{I}_{L}-\widetilde{\boldsymbol{W}}_{1}^{-1}$
from the first line of (41), and then substitute it in the second line to
reach
$\alpha_{i}\widetilde{\boldsymbol{H}}_{i+1}-\widetilde{\boldsymbol{W}}_{1}^{-1}\widetilde{\boldsymbol{H}}_{i+1}=\alpha_{i}\widetilde{\boldsymbol{H}}_{0}$.
After reordering terms and merging the inequalities for $i=1,\dots,Q_{2}$,
into matrix notation, we can write

$\widetilde{\boldsymbol{W}}_{1}^{-1}\widetilde{\boldsymbol{\mathcal{H}}}=\begin{bmatrix}\alpha_{1}(\widetilde{\boldsymbol{H}}_{2}-\widetilde{\boldsymbol{H}}_{0})&\cdots&\alpha_{Q_{2}}(\widetilde{\boldsymbol{H}}_{Q_{2}+1}-\widetilde{\boldsymbol{H}}_{0})\end{bmatrix},$
(42)

where
$\widetilde{\boldsymbol{\mathcal{H}}}=\begin{bmatrix}\widetilde{\boldsymbol{H}}_{2}&\cdots&\widetilde{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}$
gives an $L\times Q_{2}(K-L)$ randomly chosen matrix, which is thus full-rank
with probability 1. The block matrix on the RHS of (42) is also a randomly
chosen matrix, and thus full-rank with probability 1, as long as
$\alpha_{i}\neq 0$ $\forall i$. We now note that (42) corresponds to a linear
equation solvable for $L\geq Q_{2}(K-L)$, which directly gives us the
condition (26).

It remains to prove that we have a full-rank solution for each
$\widetilde{\boldsymbol{W}}_{i}^{-1}$ (then each corresponding
$\boldsymbol{W}_{m}^{-1}$ would also be full-rank). Solving for
$\widetilde{\boldsymbol{W}}_{1}^{-1}$ in (41) gives the set of solutions

$\widetilde{\boldsymbol{W}}_{1}^{-1}\\!=\\!\begin{bmatrix}\alpha_{1}(\widetilde{\boldsymbol{H}}_{2}-\widetilde{\boldsymbol{H}}_{0})\cdots\alpha_{Q_{2}}(\widetilde{\boldsymbol{H}}_{Q_{2}+1}-\widetilde{\boldsymbol{H}}_{0})\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}+\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}},$
(43)

where $\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$ can be selected to
be any $L\times L$ matrix in the left null-space of
$\widetilde{\boldsymbol{\mathcal{H}}}$. Thus,
$\mathrm{rank}(\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}})\leq
L-Q_{2}(K-L)$, and $\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$ would
vanish in case of equality in (26) ($\widetilde{\boldsymbol{\mathcal{H}}}$
square). We can now note that the first term in the sum from the RHS of (43)
has rank $Q_{2}(K-L)$ with probability 1. The reason is that it is the
multiplication of an $L\times Q_{2}(K-L)$ matrix with a $Q_{2}(K-L)\times L$
matrix, so its rank cannot be above $Q_{2}(K-L)$, while, if we multiply
$\widetilde{\boldsymbol{\mathcal{H}}}$ from the right, which cannot increase
the rank, we get an $L\times Q_{2}(K-L)$ randomly chosen matrix (full-rank
with probability 1). On the other hand,
$\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$ adds its rank to the other
term of the sum, since they are in perpendicular spaces (left null-space and
row-space are perpendicular). Therefore, by selecting any
$\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$ spanning the whole null-
space of $\widetilde{\boldsymbol{\mathcal{H}}}$, i.e., having rank
$L-Q_{2}(K-L)$, we get a full-rank $\widetilde{\boldsymbol{W}}_{1}^{-1}$.

We now show that full-rank solutions for
$\widetilde{\boldsymbol{W}}_{i}^{-1}$, $i>1$, are also available as long as
$\alpha_{i}\neq\alpha_{j}$ for $i\neq i$. Substituting
$\widetilde{\boldsymbol{W}}_{1}^{-1}$ from (42) in the first equation of (41)
gives a solution for each $\widetilde{\boldsymbol{W}}_{i}^{-1}$ of the form

$\displaystyle\widetilde{\boldsymbol{W}}_{i}^{-1}=$
$\displaystyle\alpha_{i}\mathbf{I}_{L}-\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$
(44)
$\displaystyle\\!-\\!\begin{bmatrix}\alpha_{1}(\widetilde{\boldsymbol{H}}_{2}-\widetilde{\boldsymbol{H}}_{0})&\cdots&\alpha_{Q_{2}}(\widetilde{\boldsymbol{H}}_{Q_{2}+1}-\widetilde{\boldsymbol{H}}_{0})\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}.$

Let us define $\tilde{\alpha}_{j}=\alpha_{j}-\alpha_{i}$. We then have

$\displaystyle\widetilde{\boldsymbol{W}}_{i}^{-1}=$
$\displaystyle\alpha_{i}\left(\mathbf{I}_{L}-\widetilde{\boldsymbol{\mathcal{H}}}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}\right)-\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$
(45)
$\displaystyle-\begin{bmatrix}\tilde{\alpha_{1}}\widetilde{\boldsymbol{H}}_{2}&\cdots&\tilde{\alpha}_{Q_{2}}\widetilde{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}$
$\displaystyle+\begin{bmatrix}\alpha_{1}\widetilde{\boldsymbol{H}}_{0}&\cdots&\alpha_{Q_{2}}\widetilde{\boldsymbol{H}}_{0}\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}.$

However, it can be checked that

$\widetilde{\boldsymbol{\mathcal{H}}}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}=\boldsymbol{U}\mathrm{diag}\left(\mathbf{I}_{Q_{2}(K-L)},\boldsymbol{0}_{L-Q_{2}(K-L)}\right)\boldsymbol{U}^{\mathrm{H}},$

where $\boldsymbol{U}$ corresponds to the left unitary matrix from the
singular value decomposition of $\widetilde{\boldsymbol{\mathcal{H}}}$. Thus,
we get

$\displaystyle\widetilde{\boldsymbol{W}}_{i}^{-1}=\alpha_{i}\boldsymbol{U}\mathrm{diag}\left(\boldsymbol{0}_{Q_{2}(K-L)},\mathbf{I}_{L-Q_{2}(K-L)}\right)\boldsymbol{U}^{\mathrm{H}}-\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$
(46)
$\displaystyle-\begin{bmatrix}\tilde{\alpha_{1}}\widetilde{\boldsymbol{H}}_{2}&\cdots&\tilde{\alpha}_{Q_{2}}\widetilde{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}$
$\displaystyle+\begin{bmatrix}\alpha_{1}\widetilde{\boldsymbol{H}}_{0}&\cdots&\alpha_{Q_{2}}\widetilde{\boldsymbol{H}}_{0}\end{bmatrix}\widetilde{\boldsymbol{\mathcal{H}}}^{\dagger}$
.

We then note that the first two matrices on the RHS of (46) are both in the
null-space of $\widetilde{\boldsymbol{\mathcal{H}}}$, and, since we have
freedom in selecting $\mathbf{N}_{\widetilde{\boldsymbol{\mathcal{H}}}}$ as
long as it spans the whole null-space, we can choose it so that the rank from
the first matrix is not reduced after subtracting. Therefore, the first two
matrices will always add rank $L-Q_{2}(K-L)$ to the last two, which lay in the
row-space of $\widetilde{\boldsymbol{\mathcal{H}}}$. On the other hand, after
multiplying $\widetilde{\boldsymbol{\mathcal{H}}}$ to the last two matrices in
the RHS of (46) we get one matrix of rank $N_{\tilde{\alpha}}(K-L)$, with
$N_{\tilde{\alpha}}$ the number of non-zero $\tilde{\alpha}_{j}$, and one
matrix of rank $(K-L)$. Note that $N_{\tilde{\alpha}}\leq(Q_{2}-1)$ since
$\tilde{\alpha}_{i}=0$ by definition. The sum of the latter two matrices would
then be

$\begin{bmatrix}\alpha_{1}\widetilde{\boldsymbol{H}}_{0}\\!-\\!\tilde{\alpha_{1}}\widetilde{\boldsymbol{H}}_{2}\\!&\cdots&\\!\alpha_{i}\widetilde{\boldsymbol{H}}_{0}&\\!\cdots&\\!\alpha_{Q_{2}}\widetilde{\boldsymbol{H}}_{0}\\!-\\!\tilde{\alpha}_{Q_{2}}\widetilde{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}.$

The rank is then121212$\widetilde{\boldsymbol{H}}_{0}$ and
$\widetilde{\boldsymbol{H}}_{i}$ do not destroy rank since they are blocks
from a randomly chosen matrix, and they are full-rank with probability 1.
$(N_{\tilde{\alpha}}+1)(K-L)$ so, in order to have (46) full-rank, we need
$N_{\tilde{\alpha}}=Q_{2}-1$ for each $i$, which means all
$\tilde{\alpha_{j}}$ ($j\neq i$) should be non-zero. Considering
$\widetilde{\boldsymbol{W}}_{i}^{-1}$ $\forall i$, this translates to having
$\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. Hence, Proposition 4 is proved.

## Appendix C: Proof of Proposition 5

Selecting
$\widetilde{\boldsymbol{A}}_{\mathrm{T}}=\mathbf{I}_{T_{\mathrm{P}}}$ and
$\widetilde{\boldsymbol{A}}_{\mathrm{B}}$ as in (27) leads to

$\widetilde{\boldsymbol{B}}=\begin{bmatrix}\boldsymbol{1}_{\Phi\times
1}&\left[\boldsymbol{1}_{Q_{2}\times
1}\otimes\mathbf{I}_{J}\right]_{1:\Phi,:}&\underbrace{\mathbf{I}_{\Phi}\;\;\cdots\;\;\mathbf{I}_{\Phi}}_{Q_{1}}\;\;-\mathbf{I}_{\Phi}\end{bmatrix}^{\mathrm{T}}.$

Note the similarity of the previous $\widetilde{\boldsymbol{B}}$ with (37),
which for $\alpha_{i}=1$ only the $Q_{1}$ extra $\mathbf{I}_{\Phi}$ would be
added. Applying similar arguments as in the proof of Proposition 4, which
include fixing $\boldsymbol{W}_{1}$ to an arbitrary full-rank matrix, we can
transform the equivalent formulation of the WAX decomposition (17) into a
series of independent equations of the form

$\displaystyle\boldsymbol{1}_{1\times
Q_{2}}\otimes\widehat{\boldsymbol{H}}_{0}=\begin{bmatrix}\widehat{\boldsymbol{W}}_{1}^{-1}\\!&\\!\cdots\\!&\\!\widehat{\boldsymbol{W}}_{(Q_{1}+1)Q_{2}+1}^{-1}&\end{bmatrix}$
(47)
$\displaystyle\times\begin{bmatrix}\widehat{\boldsymbol{H}}_{1}&\cdots&\widehat{\boldsymbol{H}}_{1}\\\
\widehat{\boldsymbol{H}}_{2}&&\\\ &\ddots&\\\
&&\widehat{\boldsymbol{H}}_{Q_{2}+1}\\\ &\vdots&\\\
\widehat{\boldsymbol{H}}_{Q_{1}Q_{2}+2}&&\\\ &\ddots&\\\
&&\widehat{\boldsymbol{H}}_{(Q_{1}+1)Q_{2}+1}\\\ \end{bmatrix}$ ,

where we have relaxed notation by removing the blocks of zeros. Again, only
$\widehat{\boldsymbol{H}}_{0}$ is shared among the different independent
equations. Let us have

$\widehat{\boldsymbol{H}}_{m}=\begin{bmatrix}\widehat{\boldsymbol{H}}_{m,\mathrm{sq}}&\widehat{\boldsymbol{H}}_{m,\mathrm{r}}\end{bmatrix},\;m=0,\dots,(Q_{1}+1)Q_{2}+1,$

where $\widehat{\boldsymbol{H}}_{m,\mathrm{sq}}$ and
$\widehat{\boldsymbol{H}}_{m,\mathrm{r}}$ are $L\times L$ and $L\times(K-L)$
blocks from a randomly chosen matrix, respectively. We can then have
$\widehat{\boldsymbol{H}}_{1,\mathrm{r}}=\boldsymbol{0}_{L,(K-L)}$ by
absorbing the corresponding right unitary matrix in the rest of
$\widehat{\boldsymbol{H}}_{m}$ as before. We then get the set of equations

$\left\\{\begin{array}[]{l}\widehat{\boldsymbol{W}}_{1}^{-1}\widehat{\boldsymbol{H}}_{1,\mathrm{sq}}+\sum_{q=0}^{Q_{1}}\widehat{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widehat{\boldsymbol{H}}_{i+1+qQ_{2},\mathrm{sq}}=\widehat{\boldsymbol{H}}_{0,\mathrm{sq}}\\\
\sum_{q=0}^{Q_{1}}\widehat{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widehat{\boldsymbol{H}}_{i+1+qQ_{2},\mathrm{r}}=\widehat{\boldsymbol{H}}_{0,\mathrm{r}}\end{array}\right.,$
(48)

where $i=1,\dots,Q_{2}$. Let us isolate
$\widetilde{\boldsymbol{W}}_{i+1+Q_{1}Q_{2}}^{-1}$ in the first equation of
(48)

$\displaystyle\widehat{\boldsymbol{W}}_{i+1+Q_{1}Q_{2}}^{-1}$
$\displaystyle=\bigg{(}\widehat{\boldsymbol{H}}_{0,\mathrm{sq}}-\widehat{\boldsymbol{W}}_{1}^{-1}$
(49) $\displaystyle-$
$\displaystyle\sum_{q=0}^{Q_{1}-1}\widehat{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widehat{\boldsymbol{H}}_{i+1+qQ_{2},\mathrm{sq}}\bigg{)}\widehat{\boldsymbol{H}}_{i+1+Q_{1}Q_{2},\mathrm{sq}}^{-1},$

which, assuming full-rank $\widehat{\boldsymbol{W}}_{i+1+qQ_{2}}$ for
$q<Q_{1}$, corresponds to a random combination of full-rank matrices, so it
will lead to full-rank $\widehat{\boldsymbol{W}}_{i+1+Q_{1}Q_{2}}^{-1}$ with
probability 1. Substituting in the second equation from (48), absorbing some
square randomly chosen matrices (full-rank with probability 1) in the
corresponding $\widehat{\boldsymbol{W}}_{i}$, and renaming blocks, we get

$\displaystyle\widetilde{\boldsymbol{W}}_{1}^{-1}\widetilde{\boldsymbol{H}}_{1i}+\sum_{q=0}^{Q_{1}-1}\widetilde{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widetilde{\boldsymbol{H}}_{i+1+qQ_{2}}=\widetilde{\boldsymbol{H}}_{0}+\widetilde{\boldsymbol{H}}_{1i},$
(50)

where all $\widetilde{\boldsymbol{H}}_{m}$ (or
$\widetilde{\boldsymbol{H}}_{mn}$) correspond again to blocks of size
$L\times(K-L)$ from a randomly chosen since they come from sums and products
of different blocks from a randomly chosen matrix. Multiplying both sides by
$\widetilde{\boldsymbol{V}}_{1i}$, where
$\widetilde{\boldsymbol{V}}_{1i}^{\mathrm{H}}$ corresponds to the right
unitary matrix of $\widetilde{\boldsymbol{H}}_{1i}$, we reach

$\displaystyle\scalebox{0.95}{$\begin{bmatrix}\\!\widetilde{\boldsymbol{H}}_{01}+\widetilde{\boldsymbol{H}}_{11}\\!&\\!\cdots\\!&\\!\widetilde{\boldsymbol{H}}_{0Q_{2}}+\widetilde{\boldsymbol{H}}_{1Q_{2}}\end{bmatrix}\\!=\\!\begin{bmatrix}\widetilde{\boldsymbol{W}}_{1}^{-1}\\!&\\!\cdots\\!&\\!\widetilde{\boldsymbol{W}}_{Q_{1}Q_{2}\\!+\\!1}^{-1}\end{bmatrix}$}$
(51)
$\displaystyle\times\begin{bmatrix}\widetilde{\boldsymbol{H}}_{11}&\cdots&\widetilde{\boldsymbol{H}}_{1Q_{2}}\\\
\widetilde{\boldsymbol{H}}_{2}&&\\\ &\ddots&\\\
&&\widetilde{\boldsymbol{H}}_{Q_{2}+1}\\\ &\vdots&\\\
\widetilde{\boldsymbol{H}}_{(Q_{1}-1)Q_{2}+2}&&\\\ &\ddots&\\\
&&\widetilde{\boldsymbol{H}}_{Q_{1}Q_{2}+1}\\\ \end{bmatrix}$ ,

where
$\widetilde{\boldsymbol{H}}_{0i}=\widetilde{\boldsymbol{H}}_{0}\widetilde{\boldsymbol{V}}_{1i}$,
and
$\widetilde{\boldsymbol{H}}_{1i}=[\widetilde{\boldsymbol{H}}_{1i,\mathrm{sq}}\;\;\boldsymbol{0}_{L\times(K-2L)}]$.
We then reach the following set of equations for $i=1,\dots,Q_{2}$

$\left\\{\begin{array}[]{l}\widetilde{\boldsymbol{W}}_{1}^{-1}\widetilde{\boldsymbol{H}}_{1i,\mathrm{sq}}\\!+\\!\sum_{q=0}^{Q_{1}\\!-1}\\!\widetilde{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widetilde{\boldsymbol{H}}_{i+1+qQ_{2},\mathrm{sq}}\\!=\\!\widetilde{\boldsymbol{H}}_{0i,\mathrm{sq}}\\\
\sum_{q=0}^{Q_{1}-1}\widetilde{\boldsymbol{W}}_{i+1+qQ_{2}}^{-1}\widetilde{\boldsymbol{H}}_{i+1+qQ_{2},\mathrm{r}}=\widetilde{\boldsymbol{H}}_{0i,\mathrm{r}}\end{array}\right.\\!,$
(52)

where
$\widetilde{\boldsymbol{H}}_{m}=[\widetilde{\boldsymbol{H}}_{m,\mathrm{sq}}\;\;\widetilde{\boldsymbol{H}}_{m,\mathrm{r}}]$,
with $\widetilde{\boldsymbol{H}}_{m,\mathrm{sq}}$ being square blocks as
before. Note that (52) is almost like (48), but the dimensions have been
reduced, as well as the number of sum elements, and we have now different
$\widetilde{\boldsymbol{H}}_{1i,\mathrm{sq}}$ and
$\widetilde{\boldsymbol{H}}_{0i,\mathrm{sq}}$. If we follow the same steps as
before, isolating $\widetilde{\boldsymbol{W}}_{i+1+(Q_{1}-1)Q_{2}}$ instead,
we would reach an expression as (51) with one less diagonal block where each
$\widetilde{\boldsymbol{H}}_{m}$ (still randomly chosen) has reduced the
column dimension by $L$. We can thus perform these reductions inductively
until we reach

$\displaystyle\scalebox{0.95}{$\begin{bmatrix}\\!(\widecheck{\boldsymbol{H}}_{01}\\!+\\!\widecheck{\boldsymbol{H}}_{11}\\!)\\!&\\!\cdots\\!&\\!(\widecheck{\boldsymbol{H}}_{0Q_{2}}\\!+\\!\widecheck{\boldsymbol{H}}_{1Q_{2}}\\!)\\!\end{bmatrix}\\!=\\!\begin{bmatrix}\widecheck{\boldsymbol{W}}_{1}^{-1}\\!&\\!\cdots\\!&\\!\widecheck{\boldsymbol{W}}_{Q_{2}\\!+\\!1}^{-1}\end{bmatrix}$}$
(53)
$\displaystyle\times\begin{bmatrix}\widecheck{\boldsymbol{H}}_{11}&\cdots&\widecheck{\boldsymbol{H}}_{1Q_{2}}\\\
\widecheck{\boldsymbol{H}}_{2}&&\\\ &\ddots&\\\
&&\widecheck{\boldsymbol{H}}_{Q_{2}+1}\end{bmatrix}$ ,

where
$\widecheck{\boldsymbol{H}}_{0i}=\widecheck{\boldsymbol{H}}_{0}\widetilde{\boldsymbol{V}}_{i}$,
with $\widecheck{\boldsymbol{V}}_{i}$ being a unitary matrix coming from a
product of unitary matrices from randomly chosen blocks,
$\widecheck{\boldsymbol{H}}_{1i}=[\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}\;\;\boldsymbol{0}_{L\times(K-(Q_{1}+1)L)}]$
with $\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}$ randomly chosen, and
$\widecheck{\boldsymbol{H}}_{m}$ for $m=0,2,\dots,Q_{2}+1$ are different
$L\times(K-Q_{1}L)$ randomly chosen blocks. It only remains to show that (53)
is solvable with full-rank $\widecheck{\boldsymbol{W}}_{i}^{-1}$ for
$i=1,\dots,Q_{2}+1$. If we compare (53) with (39) we can note that they have
the same structure, but the changes in the blocks, which will allow to have
$\alpha_{i}=1$, require a new proof.

Let us now prove that (53) allows for a solution with full-rank
$\widecheck{\boldsymbol{W}}_{i}^{-1}$ if (29) is fulfilled. By trivial linear
algebra, we immediately note that (29) follows from the need to have at least
as many rows as columns in the matrix multiplying the RHS of (53), since said
matrix will be full-rank with probability 1. We should then check we can have
full-rank $\widecheck{\boldsymbol{W}}_{i}^{-1}$ given (29). Proceeding as
before we express the set of equations

$\left\\{\begin{array}[]{l}\widecheck{\boldsymbol{W}}_{1}^{-1}\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}+\widecheck{\boldsymbol{W}}_{i+1}^{-1}\widecheck{\boldsymbol{H}}_{i+1,\mathrm{sq}}=\widecheck{\boldsymbol{H}}_{01i,\mathrm{sq}}\\\
\widecheck{\boldsymbol{W}}_{i+1}^{-1}\widecheck{\boldsymbol{H}}_{i+1,\mathrm{r}}=\widecheck{\boldsymbol{H}}_{0i,\mathrm{r}}\end{array}\right.,i=1,\dots,Q_{2}$
(54)

with
$\widecheck{\boldsymbol{H}}_{m}=[\widecheck{\boldsymbol{H}}_{m,\mathrm{sq}}\;\;\widecheck{\boldsymbol{H}}_{m,\mathrm{r}}]$,
where $\widecheck{\boldsymbol{H}}_{m,\mathrm{sq}}$ are again square, and
$\widecheck{\boldsymbol{H}}_{01i,\mathrm{sq}}=\widecheck{\boldsymbol{H}}_{0i,\mathrm{sq}}+\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}$.
Isolating $\widecheck{\boldsymbol{W}}_{i+1}^{-1}$ in the first line of (54),
substituting it in the second line, and solving for
$\widecheck{\boldsymbol{W}}_{1}^{-1}$,131313 Note that reaching from (54) to
(55) corresponds to the same set of steps as reaching from (41) to (43) in the
proof of Proposition 4, with the only difference that in the first line of
(54) each term has an invertible (with probability 1) matrix multiplying from
the right. we reach

$\widecheck{\boldsymbol{W}}_{1}^{-1}\\!\\!=\\!\widecheck{\boldsymbol{\mathcal{H}}}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}\\!\\!+\begin{bmatrix}(\boldsymbol{\Theta}_{1}\\!-\\!\widecheck{\boldsymbol{H}}_{0i,\mathrm{r}})\\!&\\!\cdots\\!&\\!(\boldsymbol{\Theta}_{Q_{2}}\\!-\\!\widecheck{\boldsymbol{H}}_{0i,\mathrm{r}})\end{bmatrix}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}\\!\\!+\mathbf{N}_{\widecheck{\boldsymbol{\mathcal{H}}}},$
(55)

where
$\boldsymbol{\Theta}_{m}=\widecheck{\boldsymbol{H}}_{0m,\mathrm{sq}}\widecheck{\boldsymbol{H}}_{m+1,\mathrm{sq}}^{-1}\widecheck{\boldsymbol{H}}_{m+1,\mathrm{r}}$,
which has dimensions $L\times(K-Q_{1}L)$,
$\mathbf{N}_{\widecheck{\boldsymbol{\mathcal{H}}}}$ is an $L\times L$ matrix
to be selected from the left null-space of
$\widecheck{\boldsymbol{\mathcal{H}}}$,

$\widecheck{\boldsymbol{\mathcal{H}}}=\begin{bmatrix}\widecheck{\boldsymbol{H}}_{11,\mathrm{sq}}\widecheck{\boldsymbol{H}}_{2,\mathrm{sq}}^{-1}\widecheck{\boldsymbol{H}}_{2,\mathrm{r}}\\!&\cdots&\\!\widecheck{\boldsymbol{H}}_{1Q_{2},\mathrm{sq}}\widecheck{\boldsymbol{H}}_{Q_{2}\\!+\\!1,\mathrm{sq}}^{-1}\widecheck{\boldsymbol{H}}_{Q_{2}\\!+\\!1,\mathrm{r}}\end{bmatrix},$

and $\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}$ is the left pseudo-
inverse of $\widecheck{\boldsymbol{\mathcal{H}}}$. Note that the existence of
said pseudo-inverse also leads to the condition (29). By similar arguments as
in the proof of Proposition 4, we have a sum between a matrix in the row space
of $\widecheck{\boldsymbol{\mathcal{H}}}$, having rank $Q_{2}(K-Q_{1}L)$ with
probability 1, and a matrix free to choose in the left null-space of
$\widecheck{\boldsymbol{\mathcal{H}}}$, so $\widecheck{\boldsymbol{W}}_{1}$ is
full-rank with probability 1 as long as
$\mathbf{N}_{\widecheck{\boldsymbol{\mathcal{H}}}}$ is selected such that its
rows span the whole left null-space of dimension $L-Q_{2}(K-Q_{1}L)$ (with
probability 1). We then substitute the expression of
$\widecheck{\boldsymbol{W}}_{1}^{-1}$ obtained in (55) into the first equation
from (54) and get

$\displaystyle\scalebox{0.97}{$\widecheck{\boldsymbol{W}}_{i+1}^{-1}\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}^{-1}\widecheck{\boldsymbol{H}}_{i+1,\mathrm{sq}}\\!=\widecheck{\boldsymbol{H}}_{0i,\mathrm{sq}}\\!+\\!\Big{(}\begin{bmatrix}\widecheck{\boldsymbol{H}}_{01,\mathrm{r}}\\!&\cdots&\\!\widecheck{\boldsymbol{H}}_{0Q_{2},\mathrm{r}}\end{bmatrix}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}$}$
(56)
$\displaystyle\scalebox{0.97}{$+\mathbf{I}_{L}-\\!\widecheck{\boldsymbol{\mathcal{H}}}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}-\mathbf{N}_{\widecheck{\boldsymbol{\mathcal{H}}}}-\begin{bmatrix}\boldsymbol{\Theta}_{1}\\!&\cdots&\\!\boldsymbol{\Theta}_{Q_{2}}\end{bmatrix}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}\Big{)}\widecheck{\boldsymbol{H}}_{1i,\mathrm{sq}}$}$
,

where we only need to check that the RHS is full-rank, since multiplying
square randomly chosen blocks cannot reduce the rank (with probability 1).
Reasoning as in the proof of Proposition 4,
$(\mathbf{I}_{L}-\\!\widecheck{\boldsymbol{\mathcal{H}}}\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger})$
is a matrix in the null-space of $\widecheck{\boldsymbol{\mathcal{H}}}$ which
gives rank $L-Q_{2}(K-Q_{1}L)$, and
$\mathbf{N}_{\widecheck{\boldsymbol{\mathcal{H}}}}$ can be selected so as to
not destroy said rank. Then, the other two matrices multiplying
$\widecheck{\boldsymbol{\mathcal{H}}}^{\dagger}$ can add the remaining rank.
Furthermore, $\widecheck{\boldsymbol{H}}_{0i,\mathrm{sq}}$ only shares space
with $\boldsymbol{\Theta}_{i}$ (the rest are made of different randomly chosen
blocks), so it can at most reduce rank $(K-Q_{1}L)$, which would be then
compensated with the rank added by
$\widecheck{\boldsymbol{H}}_{0i,\mathrm{r}}$, which does not share randomly
chosen blocks with either $\widecheck{\boldsymbol{H}}_{0i,\mathrm{sq}}$ or
$\boldsymbol{\Theta}_{i}$. Hence, we have proved that we can obtain full-rank
$\widecheck{\boldsymbol{W}}_{i+1}^{-1}$, and this concludes the proof of
Proposition 5.

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