arXiv:1904.04376v1 [eess.SP] 8 Apr 2019 1 On Kaczmarz Signal Processing Technique in Massive MIMO Victor Croisfelt Rodrigues ∗ and Taufik Abr˜ ao ∗ ∗ Department of Electrical Engineering (DEEL), State University of Londrina (UEL), Londrina, Brazil E-mail: [email protected], taufi[email protected]Abstract To exploit the benefits of massive multiple-input multiple-output (M-MIMO) technology in scenarios where base stations need to be inexpensive and thus possibly equipped with simple hardware, the computational complexity of classical signal processing schemes for spatial multiplexing must be reduced. This calls for suboptimal or ”relaxed” schemes that perform well the required combining/precoding steps and simultaneously achieve low computational complexities. An approach based on the iterative Karczmarz algorithm (KA) has been recently investigated with this intention, assuring to execute well without the knowledge of any a priori statistics of the channel responses. Herein, modifications are proposed on this first KA-based attempt, aiming to improve its rate of convergence with a rising performance-computational complexity trade-off solution for the M-MIMO combining/precoding problems. Such improvements are motivated by observed negative effects of pathloss and shadowing, originally presented here, over the rate of convergence granted by the KA, which consequently worsens its achievable spectral efficiency (SE). Specifically, the randomized version of KA (rKA) is used when conceiving the signal processing schemes due to its global convergence properties, where our adaptations can be seen as a fairness policy for algorithm initialization that guarantees the converge of all combining/precoding vectors, whether they be from center located users or those located at the edge of the cell. Two different rKA-based schemes are in fact discussed, of which only one is considered applicable, given the constraints placed by the underlying scenario. To ensure that the proposed method outclasses the original proposal, a mathematical characterization is presented along with numerical results that employ more realistic system and channel conditions. More precisely, the simulations embrace the least-squares and minimum mean-squared error estimators of the channel responses, also taking into account uncorrelated and correlated Rayleigh fading channels. The spatial correlation of the latter is modeled using the exponential correlation model combined with large-scale fading variations over the array, which compound is demonstrated to be prejudicial to algorithm performance. The computational complexity of the modified, most promising rKA- based scheme, which intends to obtain the receive combining and transmit precoding matrices, is derived
Transcript
arX
iv:1
904.
0437
6v1
[ee
ss.S
P] 8
Apr
201
91
On Kaczmarz Signal Processing Technique in
Massive MIMO
Victor Croisfelt Rodrigues∗ and Taufik Abrao∗
∗ Department of Electrical Engineering (DEEL), State University of Londrina (UEL),
and columns vectors, respectively. The superscript (·)H denotes the Hermitian transpose and NC(·, ·)stands for the circularly symmetric complex Gaussian distribution. E {x} holds for the expected value
of a random variable x. The N × N identity matrix is indicated by IN , whereas a vector of N and
a matrix of N × N zero elements are represented, respectively, as 0N and 0N×N . ‖·‖2 and ‖·‖F are
the l2-norm and Frobenius norm, respectively. The inner product between two vectors is denoted as:
〈·, ·〉. The concatenation is represented by [·, ·] and the selection of the row i and column j of A is
performed as [A]i,j . In particular, the ith element of a vector a can be seen as ai.
II. SYSTEM MODEL
In this section, a canonical single-cell M-MIMO is presented, aiming to introduce the SE bound
utilized to evaluate the signal processing techniques and to state the signal estimation as an optimization
problem to be solved later through the Kaczmarz’s premises. The setup considered here is comprised of
a BS equipped with M antennas serving K single-antenna UEs. Presupposing that the coherence blocks
with a length of τc (coherence interval) symbols are statistically independent over their realizations
(block-fading model) [1], [2], let gk ∈ CM denote the channel from UE k to the BS. In addition, it is
adopted a correlated Rayleigh fading model, whereby the channel can be written as gk ∼ NC(0,Rk).
The covariance matrix Rk ∈ CM×M thus embodies effects such as pathloss and spatial channel
correlation, where both correspond to large-scale propagation phenomena, while the complex Gaussian
distribution stands for the small-scale fading [11]. At this moment, one can assume a pilot training
phase to obtain the CSI that follows the features of the TDD scheme. The pilot sequences attributed to
the UEs are regarded as mutually orthogonal, since it is desired to combat the intra-cell interference.
As a consequence of this process, the BS acquires the estimated channel matrix G ∈ CM×K being
gk ∈ CM its kth column, which represents the estimated channel vector of UE k.
A. Uplink Data Transmission
In the UL, the BS receives the data symbols transmitted by all UEs, yielding the following linearly
combined signal: y =√
ρul∑K
i=1 gisi + n; where ρul is the normalized UL transmit power, si ∼NC(0, 1) stands for the data symbol sent by UE i, and n ∈ CM ∼ NC(0M , IM) denotes the receiver
noise. The performance of a selected receive combining vector, vk ∈ CM , will be evaluated using the
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UL SE metric obtained through the application of the so-called use-and-forget technique2. The net
ergodic and lower bounded UL SE for UE k is then [1], [2]
SEulk =
τulτc
log2
(
1 + γul
k
)
[bit/s/Hz], (1)
with the effective signal-to-interference-plus-noise ratio (SINR) given as
γul
k=
ρul|E{vHkgk}|2
ρul∑K
i=1 E{|vHkgi|}2 − ρul|E{vH
kgk}|2 + E{‖vk‖22}, (2)
where τul is the fraction of the coherence interval spent in UL. Supposing that only the UL is occurring,
τul = τc − τp being τp the length of the pilot training phase. The expectations in (2) are taken with
respect to all the randomness related to the channel estimates and combining schemes.
B. Downlink Data Transmission
The BS sends data to its respective UEs during the DL, where each data bearing has to be precoded
with the aim of spatially direct it towards the desired UE. This directional beam control is performed
through a transmit precoding vector given as wk ∈ CM for the kth UE. Thus, the first stage of
DL is to steer the information signal to be transmitted by the BS via a precoding process stated as
x =∑K
i=1wiςi; where ςi ∼ NC(0, 1) is the message signal sent by the BS for UE i. An important
remark to be made is that ‖wi‖22 = 1 such that the transmit precoding process does not interfere in
the signal power. The precoded signal is then transmitted by the BS, while the UEs are awaiting for
its reception. Recall that once the TDD scheme is assumed, the UL and DL channels are reciprocal
within each τc-block. Keeping this in mind, the UE k receives: yk =√
ρdl∑K
i=1 gHi x + n; where ρdl
is the normalized DL transmit power. It can be seen that each UE is affected by all UEs’ precoding
vectors and, for this reason, the selection of wi becomes a difficult problem to solve. An heuristic
way to resolve the selection of the precoding vectors across the UEs is through the use of the UL-DL
duality, which will be better discussed and deeply exploited in the sequence. The UE k can estimate
its respective message signal by computing the average precoding channel E{gHkx}. This procedure
allows the derivation of a hardening bound for the DL SE similar to (1) with an alike definition of τdl,
which is given in [2, p. 317] and extremely relies on attaining the channel hardening effect. The SE
bound of the DL is not directly evaluated here, since the UL evaluation is sufficient to demonstrate
the effectiveness of the work proposals without any loss of generality.
2This denomination stems from the exploitation of the channel estimates only to compute the receive combining vector, whereas the
CSI is not used in the detection process.
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C. Combining/Precoding Schemes: Problem Statement
Throughout this section, the ZF and RZF receive combining schemes are presented and then,
relying on the UL-DL duality, their respective transmit precoding vectors are obtained. For ease of
exposition, the receive combining and transmit precoding matrices are defined as V ∈ CM×K and
W ∈ CM×K with vk and wk being their kth columns, respectively. It is furthermore demonstrated
that the introduced canonical schemes can be seen as the solution of optimization problems. As we
will see, the optimization problems can be treated as SLEs and thus can be naturally solved through
rKA, as conducted in Section IV, giving rise to rKA-based signal processing schemes.
The RZF is a sophisticated approach that takes into account the regularization of the estimated
channel matrix by the signal-to-noise ratio (SNR). This regularization factor can mathematically aid
the inversion of G and, physically, it weights the rates of interference suppression and desired signal
maximization provided by this scheme. The RZF is given as [2]
VRZF = G(
GHG+ ξIK
)−1
, (3)
where ξ = 1ρul
is the regularization factor, defined as the inverse of the SNR. This approach is
recommended to be used in scenarios where a good estimation performance is achieved, and the
inter-cell interference is not duly strong. The latter only valid for a multi-cell case.
Notice that if the SNR is high, ξ → 0, and/or a large M is applied, tr(GHG)≫ tr (ξIK), RZF can
be approximated as [1], [2]
VZF = G(
GHG)−1
. (4)
The term ZF is provident from the fact that GHVZF = IK , noting that VZF is the pseudo-inverse of
GH. Consequently, the ZF aims to reduce all the intra-cell interference, while it assures the power
level of the desired signals. When the receive combining is applied, namely, (VZF)Hy, however, the
real channels are different from the estimated ones, whose fact translates into residual interference [2].
In addition, note that not even all the UEs have high SNR, hence, the ZF is expected to give lower
performance results as compared to RZF. It should be clear that the ZF can be comprehended as the
RZF case with ξ = 0.
In order to separate and estimate the signals of each UE, the receive combining scheme is applied
over the received signal in the BS. This can be expressed as
s = VHy, (5)
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where s ∈ CK is the vector with the estimates of the data symbols transmitted by all UEs in one
communication instant and V can be either the RZF or the ZF filter. In view of that, the RZF scheme
can be stated as the optimal solution of the signal estimation problem handled above, as it is presented
in Corollary 1 and proved in the sequel, based on [5].
Corollary 1. From (3) and (5), the RZF signal estimate is
s = (VRZF)Hy =(
GHG+ ξIK
)−1
GHy, (6)
thus, it is possible to infer that s is an optimum solution for the estimation problem, since it is the
optimal answer of argmin∈CK‖G− y‖22 + ξ‖‖22.
Proof. The first derivative in relation to of the minimum cost function above is 2GH(G−y)+2ξ.
The optimal solution that minimizes the given derivative is therefore ⋆ = (GHG+ ξIK)−1GHy; then,
one can contemplate that the given answer corresponds to s.
The same can be done for the ZF when considering ξ = 0, however, an analysis focused on the
more general case characterized by the RZF scheme will be assumed hereafter. The optimization
problem given in Corollary 1 can be rewritten compactly as ‖B − y0‖22, where B = [G;√ξIK ] is
an (M +K) ×K matrix with the estimated channel matrix and the factorized regularization factor,
and y0 = [y; 0] is an (M +K)-dim vector with the received signal and a zero vector. Thus, s can be
expressed as
s = (BHB)−1BHy0 = (GHG+ ξIK)−1GHy. (7)
These definitions will be used to properly apply the rKA as an alternative way to solve the optimization
problem established in Corollary 1.
As anticipated before, the selection of the transmit precoding vectors is a very complex problem
to be solved, owing to its self-dependence, i.e., a precoding vector is affected by all the precoding
vectors of the system. An heuristic way to solve it is the UL-DL duality, a direct consequence of
considering TDD mode, by which the transmit precoding vector of UE k can be computed as [2]
wk =vk
‖vk‖2(8)
for any receive combining scheme. Recall that the normalization is related to the desired fact that the
transmit precoding vector should not interfere with the transmitted power. The above definition allows
us to obtain the transmit precoding vectors for ZF, RZF, and also KA- or rKA-based schemes without
the exact solution of the precoding problem. This result is deeply explored in the next sections.
9
III. KACZMARZ AND RANDOMIZED KACZMARZ ALGORITHMS
The mathematical concepts behind the KA are now briefly described based on the Kaczmarz’s
seminal work [6], in addition to those underlying the rKA, an extension of KA provided by Strohmer
and Vershynin in [9], which has a strict rate of convergence that allows more reliable results when
compared to the classical KA. For this purpose, we consider, throughout this section, the canonical
and generic SLE Ax = b, where A ∈ Cm×n is the matrix of constant coefficients, x ∈ Cn is the
vector of unknowns, and b ∈ Cm is the vector of known offset coefficients. Note that, under the
consideration of an underdetermined (UD) SLE (m < n), A is a full row-rank matrix, whereas for
an OD SLE (m > n), A is a full column-rank matrix. Besides that, the SLE is deemed consistent,
which means that it possesses only one possible solution.
The KA can be written as [6], [9]
xt+1 = xt +bi − 〈ai,x
t〉‖ai‖22
ai, (9)
where t is the iteration index and ai = ([A]i,:)H ∈ Cn is the ith round-robin selected row of A; similarly,
bi is the ith element of b. It is important to stay clear that the selected row i is a deterministic variable
for the classical KA and can then be computed circularly, for instance, as i = mod(t,m) + 1. We
describe the KA in detail as follows. First, the KA is initialized with an arbitrary solution, x0, seeking
to find the genuine answer, x. It then takes a row i in a specific KA iteration t that is successively
chosen from the order of the equations in A, i.e., sweeping from a1 to am. After that, KA performs
the computation bi − 〈ai,xt〉, where notice that if 〈ai,x
t〉 = bi, the solution is not updated, resulting
in xt+1 = xt; if not, the KA projects the current approximate solution, xt, onto the hyperplane
defined by 〈ai,xt〉 = bi, thus securing the solution consistency and performing the update of xt+1.
These successive orthogonal projections tend to lead x0 to x for a suitable number of KA iterations.
In summary, the KA is an efficient manner to find the solution of an SLE, which strives to guide
a random guess in the direction of the desired answer, obeying the maximum energy perturbation
principle [5].
Despite the KA be an extremely powerful solver, mainly for OD SLEs, its rate of convergence
is not duly known and is therefore not satisfactorily explored. This occurs due to the KA’s rate of
convergence be a difficult metric to be estimated, since there is a strong dependence on the way the
equations are arranged in A; which fact affects the so-called KA’s update schedule, namely, the manner
the equations are selected as the number of iterations grows. Eventually, some works identified that the
10
random choice of the rows of A can guarantee a rigorous rate of convergence for the KA. From this
observation, [9] proposed the rKA scheme that assures an expected exponential rate of convergence,
which is of capital importance to ensure a reliable KA solution in terms of reproducible results.
The rKA randomly and independently chooses the rows of A in its update schedule, founded on a
metric that measures the relevance of the rows. In this case, the random KA-variant model described
in [9] can be expressed as
xt+1 = xt +br(t) − 〈ar(t),x
t〉∥
∥ar(t)
∥
∥
2
2
ar(t), (10)
where the random picked row in rKA iteration t is denoted as r(t) ∈ {1, 2, . . . , m}. In particular, each
r(t) has a sample probability given as Pr(t) ∈ [0, 1] that embraces a vector with the specified sampling
distribution of all rows equal to p = (P1, . . . , Pm) ∈ Rm+ . It should be observed that
∑mr(t)=1 Pr(t) = 1.
This sample probability is defined to minimize the expected mean-squared error (MSE) at a given
rKA iteration t, which fact occurs when [9]
Pr(t) :=‖ar(t)‖22‖A‖2F
. (11)
In fact, the above Pr(t) is suboptimal3 and can be interpreted as an amount that measures the fraction
of energy of a given row with respect to the total energy of the system. Although suboptimal, this
probability distribution has demonstrated effectiveness and, hence, will be adopted in this work. The
main convergence result of the rKA established in [5] and [9] is summarized by the following corollary.
Corollary 2. The expected rate of convergence of the rKA is
E{‖xt − x⋆‖22} ≤ (1− κX (A))t‖x0 − x⋆‖22, (12)
with x⋆ being the optimal solution and x0 an arbitrary initial guess. Moreover, κX (A) is nominated
as the average gain and defined as
κX (A) := minϑ∈X ,ϑ6=0
‖Aϑ‖22‖A‖2F‖ϑ‖22
, (13)
where one should observe that the above metric is totally independent of the sample probability and
is proportional to the condition number4 of A.
3Optimization of the probability may not be useful in M-MIMO scenario, since the CSI changes every τc. Indeed, as it shall be
demonstrated, A is directly linked to G. Altogether, it will be required to solve optimization problems for each τc, wherein these
solutions require unwanted hardware load [5], [12], given the posed BS restrictions.
4Remember that ‖A‖22‖A−1‖2F is usually defined as the condition number of A.
11
Proof. The proof can be found in [5, p. 11].
To gain further insights, the above corollary defines in (12) the quantity of rKA iterations (pro-
jections) required to take the initial solution x0 closest to x⋆, based on an average gain provided by
the rKA. This convergence is exponentially reaching the average inaccuracy limit that is given by the
left-hand side of (12). Note that the higher the (1−κX (A)), more rKA iterations are needed to lower
the initial error ‖x0 − x⋆‖22 towards the coveted boundary. It is thus desirable to have a high value
of κX (A) so as to shrink the number of rKA iterations necessary to achieve, suitably, the expected
error limit. This means that there is an irrefutable connection between the κX (A) and the convergence
of the rKA. As a result, the best-case of convergence is represented by the maximum value that the
average gain can assume, while its minimum stands for the worst-case scenario of convergence. It is
also worth mentioning explicitly that the number of rKA iterations is inversely proportional to the
average gain. Because of the above convergence result, which brings additional reliability, our focus,
from now on, is the use of rKA in the design of signal processing schemes based on the Kaczmarz’s
methodology5.
IV. OBTAINING THE COMBINING/PRECODING MATRICES VIA RANDOMIZED KACZMARZ
ALGORITHM
The rKA is now presented as a valid approach to solve the receive combining problem established
in Corollary 1, which can be interpreted as if the rKA was emulating the solution offered by the RZF
scheme. More precisely, two rKA-based RZF strategies are presented to obtain the receive combining
matrix, by which the transmit precoding matrix can be derived using the UL-DL duality, described
in (8), without loss of generality. An alternative way to solve the optimization problem considered in
Corollary 1 is its statement through the resolution of an OD SLE given as B = y0. Notice that this
SLE is OD, since the number of equations, M +K, is greater than the number of unknowns, K, in
a typical M-MIMO scenario [13]. Moreover, observe that this problem is not consistent, because of
the arbitrariness inserted by the receiver noise in the signal received by the BS, y, which makes that
the OD SLE possesses an infinite number of solutions. If the rKA is applied to solve this problem,
a convergence bias or residual error is eventually introduced in the solution due to the inconsistency.
In [5], this problem was circumvented by splitting B = y0 with ⋆ = s into two stages. The first
5One must stay clear that the receive combining and transmit precoding based therefore on the rKA can also be implemented with
the KA, however, the converge will immensely rely on the way that the equations are ordered in the SLE to be solved.
12
strives to remodel the OD SLE to a consistent form, leading to an UD SLE; then, another OD SLE
is derived based on the genuine stated problem (B = y0). We present this in depth below.
A. The Two Steps in Solving the SLE
The first goal is to obtain an estimate of y0, which term embodies the inconsistency. To this end,
y0 ∈ C(M+K) is first defined as
y0 := Bs(a)= B(BHB)−1GHy, (14)
where in (a) the relation exposed in (7) was applied. By making a parallel with the original SLE,
BHy0 = , the y0 can be estimated via the following SLE:
BHy0 = (BHB)(BHB)−1GHy = GHy. (15)
The above SLE is UD inasmuch as the number of equations, K, is smaller than the unknowns amount,
M +K. Even more, it is consistent in view of all the solutions are within the subspace generated by
the columns of BH. Regardless of not being optimum, y0 can be estimated through rKA. That being
the case, it is natural to derive a second SLE given as Bs = y0, wherein both OD and consistency
conditions are comprised. The solution of this second SLE based on rKA, however, is not strictly
necessary, seeing that the estimate of s can be acquired through the K last rows of y0. This means
that solely part of the UD SLE has to be solved, which can be easily seen when y0 is replaced in
Bs = y0 as its definition in (14), doing that we get
Bs = y0 = B(BHB)−1GHy
s = BHB(BHB)−1GHy(a)= BHy0,
(16)
where in (a) it was used the equality in (15).
B. Parallel rKA-Based RZF Scheme
The receive combining matrix can be obtained through the implementation of K rKAs in parallel
of which the kth one is solving an SLE of the form: BHc = ek. By already adopting the Kaczmarz’s
notation, ct ∈ C(M+K) = [ut; zt] with ut ∈ CM and zt ∈ CK . Further, ek ∈ CK is the canonical
basis vector with [ek]k = 1 and 0 otherwise. This analogous problem is essentially founded in the UD
SLE exposed in (15) and the property given in (16), being its parallel resolution properly described
in Algorithm 1. It is remarkable to see that, in the parallelized form, ek is replacing GHy from its
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underlying SLE in (15), which only activates the resolution for the equations related to the kth rKA. In
other words, this aspect suggests that the kth rKA is solving the receive combining vector associated
with UE k. In fact, the kth rKA solution does not actually yields vk, but it provides the vector dk ∈ CK
that is given solely by the last K rows of ct, i.e., zt; thereby, vk = Gdk.
Notably, the execution of each rKA can be envisioned to run concurrently in different hardware
units, where each unit is independently or quasi-independently reaching its own receive combining
vector. This independence is very related to the way that the randomness of the update schedule in
(10) is interpreted at a given context, i.e., how the random selection of rows are associated across the
hardware units. With this in mind, two possibilities are plausible to give satisfactory results: (a) all the
K rKAs can share the random rows, r(t), and (b) each rKA has its self-realized arbitrariness. This
discussion is better deepened in Section V, which treats about computational complexity; however,
realize that both approaches provide suitable convergence, as established in Corollary 2.
In spite of the fact that Algorithm 1 was indirectly6 assessed in [5], some changes are included here.
To put them in context, it is reasonable to infer that the sample probability Pr(t) is generally impacted
by the pathloss and shadowing due to their relation with the estimated channel powers (see Pr(t) in
step 10 of Algorithm 1). If p is detrimentally affected, the rKA’s update schedule is also degraded. At
this moment, one must be noted that if row k is not selected in the kth rKA, the solution update will
not even occur, making the desired convergence unattainable. This also happens because of the use
of zero vectors to initialize the state variables in c0. The zero vectors, however, are desirable, since
other arbitrary values, like a vector of ones, can cause a power bias over the rKA-estimated receive
combining vectors. In summary, the UEs close to the BS (center-UEs) have significant higher values
of Pr(t) than those for UEs located at the edge of the cell (edge-UEs). This disparity leads to a poor or
impossible convergence of the combining vectors of edge-UEs obtained through Algorithm 1. More
insightful results about the consequences of large-scale effects over p are carried out in Section VI-A.
Striving to overcome this negative effect, we observed that a suitable and non-random manner to
initialize the kth rKA and thus guarantee its updating consists of forcing the selection of the kth row
in the first run; as performed in step 8 of Algorithm 1. This proceeding can also provide a better
average initial guess, which translates into a better average gain that improves the rate of convergence
presented in Corollary 2, as shall be evidenced in the sequel. What makes the proposed procedure an
6Recall that the authors of [5] discuss the rKA as an approach to directly obtain s, motivated by slow mobility scenarios. Herein, we
analyze a most generic case, where obtaining individually the combining/precoding matrices is of paramount importance.
14
Algorithm 1 Parallel (PARL) approach of rKA to estimate the RZF receive combining matrix
1: Input: Estimated channel matrix G ∈ CM×K , inverse level of SNR ξ, number of UEs K , and number of rKA
iterations TrKA.
2: Initialization: Specify the state vectors ut ∈ CM and zt ∈ CK with u0 = 0M and z0 = 0K ; either DRZF ∈ CK×K
as the factorized version of GHVRZF.
3: Procedure:
4: for i← 1 to K do
5: Compute the canonical basis, ei ∈ CK , where [ei]i = 1 and [ei]j = 0, ∀j 6= i.
6: for t← 0 to TrKA − 1 do
7: if t = 0 then
8: Pick the ith row of GH, gH
r(t) ∈ C1×M , for r(t) = i.
9: else
10: Pick the r(t)th row of GH, gH
r(t) ∈ C1×M , with r(t) ∈ {1, 2, . . . ,K} drawn based on p with Pr(t) =‖gr(t)‖
22+ξ
‖G‖2F+Kξ
.
11: end if
12: Compute the residual ηt :=[ei]r(t)−〈gr(t),u
t〉−ξzt
r(t)
‖gr(t)‖22+ξ
.
13: Update ut+1 = ut + ηtgr(t).
14: Update zt+1r(t) = ztr(t) + ηt and repeat the other positions: zt+1
j = ztj , ∀j 6= r(t).
15: end for
16: Update [DRZF]:,i = zTKA−1.
17: end for
18: Output: VRZF = GDRZF.
attractive way to initialize Algorithm 1 is the fact that edge-UEs can now fairly obtain their receive
combining vectors, even with the tough preference given to the center-UEs in the computation of
Pr(t), making the parallel (PARL) rKA-based RZF scheme more robust to practical regimes and,
consequently, enhancing its convergence towards the aimed solutions. One can additionally conjecture
that the proposed initialization approach that can be seen as a hybrid algorithm between KA and
rKA is not strictly necessary to UEs that have high values of Pr(t), for the simple reason that they
are more probable to occur when drawing r(t). As a consequence, an optimization problem to find
which one of the K rKAs has to be force-initialized could be suggested. Note that the gain provided
by this probable optimization, however, is merely one more iteration for each rKA, and its finding
may produce a considerable and undesired computational complexity. This unattractive optimization
procedure was therefore not considered in this work.
15
The convergence performance of Algorithm 1 is defined in Theorem 1. Through this result, some
comments about the average gain bounds are made in Remarks 1 and 2. Recall that the limits of
κX (BH) are inversely proportional to the respective bounds of the number of rKA iterations. These
remarks will hence be important for the computational evaluation of the rKA-based RZF schemes.
Theorem 1. Consider the kth UD SLE BHc = ek, where BH ∈ CK×(M+K) is the matrix of constant
coefficients, c ∈ C(M+K) is the vector of unknowns, and ek ∈ CK is the canonical basis. By applying
the rKA to solve this system, as done in Algorithm 1, it converges towards the optimal solution c⋆
bounded by the following average error:
E{‖ct − c⋆‖22} ≤ (1− κX (BH))t‖c(1) − c⋆‖22, (17)
with κX (BH) =
λmin(BHB)
‖B‖2F(a)=
λmin(GHG) + ξ
‖G‖2F +Kξ, (18)
where in (a), B is replaced by its definition stated in (7). Note that c(1) can be seen as the initial
solution of the rKA, which also provides an average gain based on the KA:
κ(1)X (BH) = min
ϑ∈X ,ϑ6=0
‖bH
kϑ‖22‖bk‖22‖ϑ‖22
, (19)
that is very dependent on the row index k. One should observe that the gap c(1) − c⋆ can thus be
made smaller on average than the provided by the adoption of a zero or any other random solution
as the initial one.
Proof. The proof is given in Appendix B.
Remark 1 (Average Gain Bounds: Generic Correlated Covariance Matrices with Gaussian Estimated
Channel Matrix). If the elements of G are distributed as NC(0, 1)7, the eigenvalues of G are upper
limited by 1 and then, if all eigenvalues are 1, we have ‖G‖2F = K. Thereby, the average gain stated
in (18) is limited as the following manner [5]:
λmin(GHG)
‖G‖2F≤ λmin(G
HG) + ξ
‖G‖2F +Kξ≤ 1
K, (20)
where ξ ≥ 0. From the above inequality, it is possible to presume that the higher the ξ, the faster
the convergence of Algorithm 1, since κX (BH) also turns out to be higher. Consequently, the ZF
filter case, i.e., ξ = 0 is expected to support a slower convergence than the provided by emulating
7For the canonical channel estimators analyzed in the numerical results, this distribution does not hold, since the estimated powers
are certainly smaller than the true ones.
16
the RZF scheme through rKA8. In addition to that, one should be observed that edge-UEs have a
better average gain than center-UEs; namely, even though the edge-UEs are poorly selected by the
randomized approach, they reach the solution more quickly than center-UEs.
Remark 2 (Average Gain Bounds: Uncorrelated Rayleigh Fading with Gaussian Estimated Channel
Matrix). When the uncorrelated Rayleigh fading channel modeling is considered, i.e., Rk = βkIM ,
where βk is the average large-scale fading coefficient for UE k, and the elements of G are distributed
as NC(0, 1), the limits of κX (BH) can be strictly established based on the random matrix theory. This
is done by solving the characteristic polynomial of GHG, finding its minimum, and then applying the
statistical characteristic of G, yielding [5]:
κX (BH) ≥ λmin(G
HG)
‖G‖2F=
(√M −
√K)2
MK=
(
1−√
KM
)2
K, (21)
where KM
is the UE-BS antenna ratio, also known as the loading factor in M-MIMO. From this result,
the number of rKA iterations is therefore definitively associated with the value of the loading factor.
Interestingly, as the value of KM
is close to 1, the average gain tends to be zero. This means that the
number of rKA iterations needed to comprise the worst-case convergence condition goes to infinity,
i.e., the convergence can be considered unattainable in that case. One needs to keep in mind, however,
that the UE-BS antenna ratio in M-MIMO is typically KM∈ [0, 0.5] [1], [2].
C. Linear Operator rKA-Based RZF Scheme
The following key property of rKA has to be introduced so that we can present the second rKA-based
method for obtaining the RZF receive combining matrix.
Corollary 3. Supposing any given SLE in its trivial form Ax = b, where A ∈ Cm×n is the matrix
of constant coefficients, x ∈ Cn is the vector of unknowns, and b ∈ Cm is the vector of known offset
coefficients with m and n representing the number of equations and unknowns, respectively. One can
solve this SLE via rKA with an initial guess x0, yielding in the tth rKA iteration an approximate
solution denoted as xt. This estimate can be expressed in terms of a linear operator (LO) realized
over A as xt = Lt(A)b, being Lt(A) ∈ Cn×m only dependent on A and the randomness attained to
the rKA convergence; however, it is apart from b.
8The results that will be obtained for the rKA emulating the RZF scheme can be easily expanded to the ZF case, but, since ZF has
a worse rate of convergence than RZF, the expected computational gains must be reduced for the former scheme.
17
Proof. The proof follows [5, p. 20].
As a result of the consistent OD SLE Bs = y0 and Corollary 3, it is easy to see that st = Lt(B)y0
is similar to the signal estimation expression in (5) with the proviso that Lt(B) is a K × (M +K)
matrix, whereas VH is K ×M ; the same can be said for y0 and y. It is therefore possible to realize
that the M first columns of the LO tend to approach the RZF receive combining matrix hermitian
for a suitable number of rKA iterations. This is the concept behind Algorithm 2, but the proposed
procedure was conceived aiming at estimating the LO over BH; by founding on BHs′ = y′0, where
s′ = Bs ∈ C(M+K) and y′0 = BHy0 ∈ CK . In this case, the control of p does not depend on M +K
rows, but on K rows by which the UEs are represented. Again, this management is desirable, since
the large-scale effects negatively impact the distribution range of p. An attempt to reduce this harmful
influence was to use the first K iterations to initialize the rKA, as realized in step 7 of Algorithm 2.
Theorem 2 establishes the convergence performance of Algorithm 2, where its proof is quite similar
to the conducted for Theorem 1. The Remarks 1 and 2 are also valid here.
Theorem 2. Let BHs′ = y′0 be an OD SLE with BH ∈ CK×(M+K) denoting the matrix of constant
coefficients, s′ ∈ C(M+K) the vector of unknowns, and y′0 ∈ CK the vector of known consistent
estimates of the received signal. As can be seen from Algorithm 2, the given SLE can be solved
through rKA. The convergence of this procedure is limited by the average error between the solution
obtained in rKA iteration t and the optimal (s′)⋆, given as
E{
‖(s′)t − (s′)⋆‖22}
≤ (1− κX (BH))t ‖(s′)(K) − (s′)⋆‖22, (22)
with κX (BH) =
λmin (BHB)
‖B‖2F. (23)
The initial solution (s′)(K) also carries an average gain, which is
κ(K)X (BH) = min
ϑ∈X ,ϑ6=0
K∑
k=1
‖bHkϑ‖22
‖bk‖22‖ϑ‖22. (24)
The above average gain can be better than that the provided by any other arbitrary solution.
V. COMPUTATIONAL COMPLEXITY ANALYSIS
Since canonical and proposed schemes based on rKA are adequately described, we now turn our
attention to find a fair way to analyze and compare their computational complexities. This investigation
aids to answer the ad hoc question that desires to verify if the rKA-based strategies pave the way for
18
computationally light signal processing schemes. Our main goal is basically to define functions that
describe the costs of computational complexity of the signal processing schemes in view of matrix-
to-matrix multiplication and matrix inversion, together with the consideration of the same operations
for vectors. This computational finding strategy is motivated by the work conducted in [2, App. B,
p. 558], wherein the authors give a very detailed framework upon the costs of those elementary
matrix/vector operations. To emphasize, complex multiplications/divisions are then taken into account
in the computational complexity measuring functions, while additions and subtractions are neglected
with the premise that the latter are not as heavy as the former from hardware point of view. With this
in mind, the examination of the computational cost will be divided into the UL and DL phases.
Algorithm 2 Linear Operator (LO) approach of rKA to estimate the RZF receive combining matrix
1: Input: Estimated channel matrix G ∈ CM×K , inverse level of SNR ξ, number of UEs K and number of rKA iterations
TKA.
2: Initialization: Define the linear operator Lt(BH) ∈ C(M+K)×K with L0(BH) = 0(M+K)×K .
3: Obtain B = [G;√ξIK ].
4: Procedure:
5: for t← 0 to TKA − 1 do
6: if t < K then
7: Pick the kth row of BH, bH
r(t) ∈ C1×(M+K), where r(t) = k for t ∈ {1, 2, . . . ,K}.8: else
9: Pick the r(t)th row of BH, bH
r(t) ∈ C1×(M+K), with r(t) ∈ {1, 2, . . . ,K} drawn based on p with Pr(t) =‖br(t)‖
22
‖B‖2F
.
10: end if
11: Compute the canonical basis, er(t) ∈ CK , where [er(t)]r(t) = 1 and [er(t)]j = 0, ∀j 6= r(t).
12: Update Lt+1(BH) =
(
I(M+K) −br(t)b
Hr(t)
‖br(t)‖22
)
Lt(BH) +br(t)e
Hr(t)
‖br(t)‖22
.
13: end for
14: Output: VRZF = [LTKA−1(BH)]1:M,:.
A. Computational Cost: Uplink
Since its inception the idea behind the PARL rKA-based RZF scheme is to detach the solution of
the receive combining matrix onto K low complexity hardware unit points, i.e., each unit point has a
simple, cheap, and dedicated hardware responsible to obtain the receive combining vector relative to its
respective UE through the rKA. From Algorithm 1, it can be observed that most of the computational
load executed by each dedicated hardware is related to the computation of the sampling distribution
19
p in step 10 and of the residual ηt in step 13. In particular, the computational cost assigned to p is
considered to be the knowledge of all the sample probabilities, Pr(t), in such a way the generation
of some r(t)s comprises of drawing random numbers based on p. This generation can be done in a
distributed or centralized way, as already mentioned. The former stands for a scenario in which each
kth hardware unit is computing its own p and, hence, independently generates its values of r(t); while
the latter denotes a scenario where a central hardware point is quantifying p and afterwards sharing
the draw of r(t)s across the other K hardware units (hardware units are free from computing p). As
each hardware unit has to compute its p, the distributed fashion is said to be the most computationally
heavy. The centralized method, however, can lead to a substantial loss of time, which represents a
fraction of the available coherence interval, by communicating with the central hardware point. We
should therefore expect an impact on performance by adopting the centralized way, especially in
overcrowded scenarios where the value of the loading factor tends to be close to the upper limit
defined here as KM
= 0.5. A computational comparison between both manners of generating r(t)
values is carried out in Section VI-D.
Another key discussion point for the PARL rKA-based RZF scheme is related to the signal recovery
problem described in (5). In view of the fact that each kth unit is actually computing dk and not vk
in Algorithm 1, it is appropriate to perform the following computations in a central hardware point
so as to obtain the signal estimate on a given communication instant:
VPARL = GD, (25a)
s = (VPARL)Hy. (25b)
where (25a) is computed only one time and takes MK2 complex operations, while the signal recovery
in (25b) leads to MK and has to be computed for all τul instants. A centralized fashion is considered
for these computations, because the time spent in obtaining the information needed to perform the
calculations in a distributed or centralized way seems to be approximately equal. A brief note on the
computational complexity of Algorithm 2 is now in order. The linear operator (LO) rKA-based RZF
scheme has the computation of p in step 9 and the update of the LO in step 14 as the main hardware
loads. Different from the PARL approach, Algorithm 2 is essentially co-processed (centralized).
Table I summarizes the obtained complexity measuring functions for rKA-based schemes emulating
the classical RZF strategy, in addition to the canonical results from [2], where both were obtained based
on the computational framework provided by the same work. Remember that this mathematical struc-
20
ture takes into account only complex multiplications and divisions from elementary matrix-to-matrix
(and vector-to-vector) operations. The reception column entails the computational cost associated with
the signal estimation expressed in (5). Some remarks can then be made regarding the given results. The
computation of the receive combining matrix for Algorithm 1 seems to be the most computationally
light, but this value is very dependable on the number of rKA iterations, TrKA. The PARL rKA-based
RZF scheme shows, however, the greatest reception complexity and the fact that, when considering the
distributed fashion, the calculation of the sampling distribution adds a considerable extra computational
floor. The LO procedure (Algorithm 2) is by far the most costly in computational terms, thus it will
not be considered in the following analysis. It is important to emphasize, however, that this procedure
plays a good way into the design of networks envisioning to apply rKA-based techniques. This is
corroborated by its rate of convergence be fairly similar to the one obtained for the PARL rKA-based
RZF scheme, as we observe through simulation results, and also by the fact that simulation tools, like
Matlab, have powerful built-in functions to well execute matrix operations.
TABLE I
COMPUTATIONAL COMPLEXITY FOR RECEIVE COMBINING SCHEMES PER COHERENCE INTERVAL.
SchemeReceive combining matrix Reception Sampling distribution
