Randomized Kaczmarz Algorithm Applied D’Hondt Method for

Extremely Massive MIMO Wireless Communication Systems

TATSUKI FUKUDA,

Department of Data Science, Faculty of Data Science,

Shimonoseki City University,

2-1-1 Daigakucho, Shimonoseki City, Yamaguchi, 751-8510,

JAPAN

Abstract: - Extremely massive MIMO (Multiple-Input Multiple-Output) is a technique to enable the spatial di-

versity. The systems employ a large number of antennas at the base stations, resulting in high computational

complexity in various processes of wireless communications. The precoding process is one of them because the

process requires the calculation of matrix inversion. The randomized Kaczmarz algorithm(rKA) is an iterative

method to obtain the approximation so the computational time of precoding can be decreased. Some improve-

ments of rKA were proposed so far, the iteration number required to obtain the approximation of inverse matrix

is not so small. In this paper, we propose a new rKA method that applies the D’Hondt method, typically used for

seat allocation in elections. In rKA process, the row vector is selected to use for updating approximation. Our

method selects the row vector based on the D’Hondt method while the conventional rKA methods select the row

vector probabilistic. Some results of simulation showed that the bit error ratio (BER) performance of our method

is superior to other rKA methods at higher normalized transmit powers (NTP). The results also showed that the

BER performances of our method with small number of iterations are more accurate than the others especially at

high NTPs. That means our method can achieve the same BER performance with smaller number of iterations as

the others, so the computational complexity of precoding with rKA is decreased.

Key-Words: - Massive MIMO, Precoding, rKA, RZF, D’Hondt method.

Received: March 2, 2024. Revised: September 3, 2024. Accepted: October 3, 2024. Published: November 15, 2024.

1 Introduction

The COVID-19 pandemic, which began around 2020,

has led to a rapid advancement in the online transfor-

mation of societal infrastructure. The increasing net-

work traffic has further escalated due to the explosive

growth in the user base of various services, such as

video streaming platforms, and the widespread use of

online tools for work and education. Wireless connec-

tivity is now more prevalent than ever, with devices

such as smartphones and Internet-of-Things (IoT) de-

vices, [1], [2], [3], [4], requiring reliable connections.

To meet the demands of this immense network

traffic, high-speed and stable wireless communication

is critical. Technologies like Multiple-Input Multiple-

Output (MIMO), which enable the use of multiple

antennas to transmit multiple signals simultaneously,

have been key to achieving this. MIMO systems have

become a cornerstone of 5G networks, providing im-

proved spectral efficiency and system capacity.

As we transition towards 6G, the next generation

of wireless communication systems, even more ad-

vanced techniques are needed to handle the growing

data demands.

Extremely massive MIMO, [5], an extension of

conventional MIMO, involves the deployment of

hundreds or even thousands of antennas at base sta-

tions, allowing for enhanced spatial diversity and

beamforming capabilities. This technique promises

to further increase spectral and energy efficiency,

[6], making it a crucial technology for 6G sys-

tems.However, the computational complexity of pro-

cesses such as precoding also grows significantly as

the number of antennas increases, [7], [8], [9], [10],

[11], [12], [13].

The linear tensor zero-forcing (ZF) method, [14],

is a widely used technique for signal detection in

receivers and precoding in transmitters. However,

while ZF can effectively eliminate inter-user inter-

ference, its bit error ratio (BER) characteristics are

often not practical in large-scale systems due to its

sensitivity to noise. To address this, improved meth-

ods such as the regularized ZF (RZF) method, [15],

have been proposed, but the matrix inversion required

by these methods becomes a major bottleneck in ex-

tremely massive MIMO systems.

To reduce this complexity, iterative methods like

the randomized Kaczmarz algorithm (rKA) have been

proposed, [16]. rKA approximates the solution to the

large linear systems involved in precoding, without

the need for direct matrix inversion. However, while

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rKA reduces computational cost, the number of iter-

ations required for convergence is still a challenge.

Several improved variants of rKA have been devel-

oped, [17], but they still face limitations, especially

in extremely massive MIMO systems. Most conven-

tional methods for precoding, such as those used in

uplink scenarios, [18], [19], [20], assume that the

channel is stationary, [21]. However, in extremely

massive MIMO systems, the long antenna arrays lead

to non-stationary channel conditions, where the chan-

nel varies significantly over different antennas, [22].

This study focuses on precoding in the downlink en-

vironment, where non-stationary channels pose a sig-

nificant challenge.

In this paper, we propose a novel approach that ap-

plies the D’Hondt method, typically used for seat allo-

cation in elections, to the randomized Kaczmarz algo-

rithm for precoding in extremely massive MIMO sys-

tems. By improving the selection process of row vec-

tors during iterations, our method achieves faster con-

vergence and better BER performance, particularly

at higher normalized transmit powers (NTP). This

contribution reduces the computational complexity

of precoding and demonstrates superior performance

compared to other existing rKA-based methods.

Organization: The rest of this paper is organized

as follows. In Section II, the preliminary knowledges

including the model assumed in this study are ex-

plained. In Section III, the Kaczmarz Algorithm and

improved methods of it are shown for precoding pro-

cess. Our proposed method is also described in the

section. In Section IV, the BER performances of the

methods are shown used results of some computer

simulations. The conclusions are drawn in Section V.

2 Preliminaries

First, let us provide a summary of the fundamental

aspects relevant to this study.

2.1 Model

We establish the framework for the Massive MIMO

wireless communication system investigated in this

study. Initially, the base station is equipped with M

antennas, catering to Kuser devices, with each device

having a single antenna. The Mantennas at the base

station are partitioned into Ssubarrays, where the s-th

subarray comprises Msantennas, and each subarray

accommodates a distributed set of user devices. The

number of user devices connected to subarray jis de-

noted as Kj, as expressed in (1).

K= ΣjKj.(1)

In this context, let us denote yj,k as the received

signal by the k-th user device Uj,k connected to the

j-th subarray. The transmitted signal from the s-th

subarray is represented by xs, while hs

j,k signifies the

channel vector between the s-th subarray and Uj,k.

The channel matrix between the s-th subarray in the

base station and Kjuser devices in the j-th subarray

is denoted as Hs

j= [hs

j,1,hs

j,2,··· ,hs

j,Kj]. The re-

lationship between the received signal and the trans-

mitted signal is expressed in (2).

yj,k = Σs(hs

j,k)Hxs+nk

j.(2)

Here, (·)Hrepresents the complex conjugate trans-

pose, nk

jdenotes the additive circularly symmetric

complex Gaussian noise at Uj,k with zero mean and a

covariance of σ2.

In this research, it is assumed that the base sta-

tion possesses imperfect channel information. More

precisely, the estimated value of the communication

channel vector, denoted as ˜

j,k, is obtained through

the process described in (3).

j,k =(1 −τ2)hs

j,k +τns

j,k,(3)

where hs

j,k and ns

j,k denote the true values

of the communciation channel vector and the in-

dependent error vector, respectively. ˜

[˜

j,1,˜

j,2,··· ,˜

j,Kj]dentoes the estimated channel

matrix between s-th subarray in the base station and

Kjuser devices in the j-th subarray.

For the non-stationarity channel assumed in this

paper, the communication channel vector between

Uj,k and Msantennas in the s-th subarray is repre-

sented by (4) and (5).

j,k =Ms(Φs

j,k)1

2zs

j,k ∈CMs×1,(4)

Φs

j,k = (Ds

j,k)1

2Rs

j,k(Ds

j,k)1

2∈CMs×Ms,(5)

where zs

j,k ∈CMs×1follows a Gaussian distribution

with mean zero and covariance 1

MsIMs,IMsdentoes

the identity matrix of order Ms,Rs

j,k ∈CMs×Msden-

toes the spatial correlation matrix between Uj,k and s-

th subarray in the base station, and Ds

j,k ∈CMs×Ms

denotes a diagonal matrix and has Ds

j,k non-zero di-

agonal elements between Uj,k and s-th subarray in the

base station.

2.2 Precoding in RZF method

Precoding refers to the process in which signal pro-

cessing is applied to the transmitted signal at the trans-

mitter side in order to increase the signal gain at the re-

ceiver. Precoding is closely related to signal detection

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processing, and the selection of precoding depends on

the detection method being employed. Here, we will

explain the precoding technique when using the reg-

ularized zero-forcing (RZF) method as the signal de-

tection approach, [23].

The information vector expected to be transmit-

ted from the j-th subarray is represented as sj=

[sj,1, sj,2,··· , sj,Kj]T, where sj,k denotes the infor-

mation transmitted from the j-th subarray to Uj,k.

Note that sj,k follows a Gaussian distribution with

mean 0 and covariance pj,k. The transmitted signal

vector xjis generated as (6).

xj= ΣKj

k=1gj,ksj,k =Gjsj,(6)

where Gj= [gj,1,gj,2,··· ,gj,Kj]is the precoding

matrix for user devices in the j-th subarray, and gj,k

represents the precoding vector for Uj,k.gj,k satisfies

the condition ∥gj,k∥2= 1. Now, (2) is transformed

into (7).

yj,k = (hj

j,k)Hgj,ksj,k + ΣKj

i=1,i=k(hj

j,i)Hgj,isj,i

+ ΣS

s=1,s=jΣKi

i=1(hs

s,i)Hgs,iss,i +nj,k,(7)

The first term in (7) is a desired signal, the second

one is intra-subarray interference, and the third one

is inter-subarray interference. One of the linear pre-

coding schemes is RZF, [13], whose precoding matrix

GRZF

jis denoted by (8).

Fj=Hj

j((Hj

j)HHj

j+ξIKj)−1,

GRZF

j=P/Tr(FH

jFj)·Fj,(8)

where ξ=σ2

P,Pdentoes average signal power, and

Tr(·)denotes the trace operator.

While the ZF method carries the risk of ampli-

fying the noise component, the RZF method miti-

gates this issue by incorporating a regularization term

ξinto the equation to prevent noise enhancement.

However, the inverse operator of matrix to obtain Fj

makes RZF method unpractical for extremely mas-

sive MIMO systems.

3 Methods

Although the RZF method can cancel the interference

between users and avoid noise enhanced, it requires a

large complexity for inverse operation. Therefore, it-

erative methods have been proposed to obtain approx-

imate solutions. In this chapter, various approaches

based on the Kaczmarz Algorithm (KA) are shown,

and the proposed improvements in this paper will be

explained.

3.1 Kaczmarz Algorithm

The KA is an iterative algorithm to solve the linear

equation systems represented as Ax =b. The equa-

tion can be considered as a set of mlinear equations

aix=bifor i= 1,2,··· , m where aiand bidenote

i-th row of Aand b, respectively. The algorithm up-

dates the approximation of xwith i-th row of Aand

bas shown in (9).

xk+1 =xk+bi− ⟨ai,xk⟩

∥ai∥2¯

ai,(9)

where i=kmod m,xkdenotes k-th approximation

of x,⟨·⟩ denotes inner product operation. ¯

aidenotes

complex conjugation of ai.

3.2 Randomized Kaczmarz Algorithm

In order to meet the overdetermined linear systems,

the randomized KA (rKA) was proposed, [16]. The

difference between KA and rKA is the method of

selecting row for the update of xk. The rKA se-

lects randomly based on probability proportional to

∥ai∥2while the KA selects in order from the first row.

Specifically, in rKA, the probabilities of selecting i-th

row from m rows are 1

3.3 RKA for RZF Precoding

In RZF precoding process, the matrix GRZF

jis

needed to calculate. For that, kjrKA processes are

run to solve a set of linear equations as shown in (10),

[22].

(Aj)Hzj=sj,(10)

where Aj= [Hj

j;√ξIKj]∈C(Mj+Kj)×Kj.Kjpro-

cesses of rKA to solve (10) are run in parallel. Specif-

ically, the procedure of rKA for RZF precoding is

shown in Algorithm 1.

Algorithm 1 rKA for RZF precoding

1: Input n-th canonical basis ek∈CKjas sjfor

k-th process.

2: Assume the solution of (10) zj= [mj∈

CMj×1;√ξnj∈CKj×1].

3: Update the approximation of zjwith iterations of

(9). The i-th row in (9) is selected with the prob-

ability PrKA

j,i = 1/Kj.

4: Divide last Kjcomponents of zjobtained in k-th

process by ξand denote it as wj,k.

5: GRZF

j=βHj

jWjis obtained where Wj=

[wj,1,wj,2,··· ,wj,Kj]∈CKj×Kj.

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3.4 Sampling without Replacement rKA for

RZF

The Sampling without replacement rKA, [22], re-

ferred to as SwoR-rKA, is a method in which the

probability of row selection differs from the original

rKA. SwoR-rKA can improve the convergence speed

and spectral efficiency compared to the original. The

probability of selecting i-th row in Algorithm 1 is

changed into (11).

PSwoR

j,i =∥hj

ji∥2+ξ

∥Hj

j∥2

F+Kjξ,(11)

where ∥·∥Fdentoes Frobenius norm.

In SwoR-rKA, the row is selected based on the

channel quality, so SwoR-rKA intends to select the

row corresponding to the user in good communication

channel. This approach can allocate more resources

to users in better channel than the others and that leads

to enhanced SE and faster convergence speed com-

pared to the original rKA.

3.5 Reverse Policy of SwoR-rKA for RZF

In SwoR-rKA, the selection of row is based on the

channel condition. The row corresponding to good

channel is likely to be selected. However, this ap-

proach leads to that only a limited set of linear equa-

tions can be solved well because approximation cor-

responding to worse channel is hardly solved due to

small number of updates.

When the channel conditions are favorable, there

is no need to allocate a significant computational load

to the estimation process. Conversely, when the chan-

nel conditions are poor, a larger computational load is

required. Therefore, the reverse policy of SwoR-rKA,

referred to as RSwoR-rKA, was proposed, [24] , in

which the probability shown as (12) is used to select

i-th row in Algorithm refalg1.

PRSwoR

j,i = 1 −∥hj

ji∥2+ξ

∥Hj

j∥2

F+Kjξ.(12)

(12) means that the columns corresponding to

worse channel are tends to be selected.

3.6 Proposed Method: D’Hondt method in

rKA for RZF

In Algorithm 1, it is believed that the estimation val-

ues can converge more quickly if the i-th row to be

updated is appropriately selected. Therefore, we fo-

cused on the candidate selection system used in elec-

tions.

The D’Hondt system is one of the methods used

to determine the allocation of seats in an election. In

this system, voters cast their votes for choices such as

political parties or candidates, and seats are allocated

based on the election results.

The specific procedure involves voters first cast-

ing their votes for the choices, and the number of fa-

vorable votes obtained by each choice (political party

or candidate) is aggregated.

Then, the determination of seat allocation takes

place. Following the algorithm 2, the allocation of

seats is repeated until the remaining number of seats

reaches zero.

Algorithm 2 The original D’Hondt method

1: Let V(i)represent the number of votes obtained

by political party i(i= 1,2,···), and W(i)rep-

resent the number of seats obtained by the party

at the current stage.The number of seats to be al-

located is denoted as S.

2: Calculate V(i)/(W(i) + 1) for each political

party and find the party jwith the highest value.

3: Increment W(i)by one and decrease Sby one.

4: Go back 2.

In the D’Hondt method, the allocation of seats

is carried out in order of the highest number of fa-

vorable votes. In other words, choices that have re-

ceived greater support from voters are given priority

in obtaining seats. This system is widely utilized as a

method for determining seat allocation in elections in

many countries and regions, as it allows for a direct

and fair reflection of the election results.

We apply the D’Hondt method into the selection

of i-th row in Algorithm 1, we call the method as D-

rKA. Specifically, D-rKA select rows by Algorithm

Algorithm 3 Select rows with the D’Hondt method

1: Calculate PRSwoR

j,i for i= 1,2,··· , Kj.

2: Vote for each row probabilistically according to

PRSwoR

j,i .

3: Select row by D’Hondt method according to the

number of votes.

D’Hondt method try keeping the average number

of votes per seat equally, so the number of updates

with each row in D-rKA are more stable than the prob-

abilistic selection method such as SwoR-rKA.

4 Simulation

We conducted some simulation whose aims are to

confirm the changes of the BER performance with the

normalized transmit powers and with the number of

iterations in rKA method.

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4.1 Parameters

The environment of developing the simulation pro-

grams is shown in Table 1. The source code of this

simulation was written with Python.

The simulation program in this section used the

fixed parameters and precoding methods shown in Ta-

ble 2. Other parameters are described in the com-

ments of each figure.

Table 1: Softwares and libraries to conduct the simulation.

Software/Library Version

Python 3.10.6

NumPy 1.24.3

SciPy 1.10.1

Matplotlib 3.7.1

Table 2: Parameters used in the simulation

Parameter Value Explanation

Kj16 The number of users

in a subarray.

M256 The number of antennas

at base station.

S16 The number of subarrays.

Mj16 The number of antennas

in a subarray.

T16 The length of timeslot

to keep channel unchanged.

m 64QAM Digital modulation method.

trials 105The number of trials

with fixed channel states.

τ0.3 The quality of channel state

information at base station.

σ21 dBm The power of noise.

4.2 Results

We show the results in two viewpoints.

• The BER performances against the normalized

transmit power (NTP) with fixed number of it-

erations in rKA algorithm.

• The BER performances against the number of it-

erations in rKA algorithm with fixed NTP.

4.2.1 BER performace against NTP

The rKA method is an iterative method, so the num-

ber of iterations is important to make the accuracy of

approximation higher. However, too many iterations

are wasteful because residuals remain. Therefore, we

conducted the simulation that compare the BER per-

formance against the number of iterations. The re-

sults are shown in Figure 1, Figure 2, Figure 3, Fig-

ure 4, Figure 5, Figure 6, and Figure 7. The horizon-

tal line in each figure indicates the normalized trans-

mit power and vertical one indicates the BER perfor-

mances. The number of iterations in rKA process are

20, 40, 60, 80, 100, 120, and 140, respectively. The

channels in all simulations are non-stationary and the

base stations in all simulations have imperfect chan-

nel state information.

Figure 1: BER performances against normalized transmit power

with 20 times iterations in rKA algorithm. The channel

is non-stationary one and the base stations have imper-

fect channel state information.

Figure 2: BER performances against normalized transmit power

with 40 times iterations in rKA algorithm. The channel

is non-stationary one and the base stations have imper-

fect channel state information.

4.2.2 BER performace against the number of

iterations

Next, we compare the results of Figure 1, Figure 2,

Figure 3, Figure 4, Figure 5, Figure 6, and Figure 7,

according to changes in the number of iterations. The

results are shown in Figure 8, Figure 9, and Figure 10.

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Figure 3: BER performances against normalized transmit power

with 60 times iterations in rKA algorithm. The channel

is non-stationary one and the base stations have imper-

fect channel state information.

Figure 4: BER performances against normalized transmit power

with 80 times iterations in rKA algorithm. The channel

is non-stationary one and the base stations have imper-

fect channel state information.

Figure 5: BER performances against normalized transmit power

with 100 times iterations in rKA algorithm. The chan-

nel is non-stationary one and the base stations have im-

perfect channel state information.

Figure 6: BER performances against normalized transmit power

with 120 times iterations in rKA algorithm. The chan-

nel is non-stationary one and the base stations have im-

perfect channel state information.

Figure 7: BER performances against normalized transmit power

with 140 times iterations in rKA algorithm. The chan-

nel is non-stationary one and the base stations have im-

perfect channel state information.

These figures illustrate the BER performance at NTP

values of 20, 30, and 40, respectively. The horizontal

line in each figure indicates the number of iterations

in rKA algorithm and vertical one indicates the BER

performances.

4.3 Consideration

As illustrated in Figure 1, Figure 2, Figure 3, Figure

4, Figure 5, Figure 6, and Figure 7, our method shows

significantly better BER performance with a small

number of iterations, especially when the normalized

transmit power (NTP) exceeds approximately 10 dB.

This result suggests that our method achieves a more

accurate approximation with fewer iterations com-

pared to other methods. The difference between the

methods becomes smaller as the number of iterations

increases, but even at lower NTPs and higher itera-

tion counts, our method continues to outperform the

others.

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Figure 8: BER performances against the number of iteration in

rKA algorithm. NTP is 20 dB and the channel is non-

stationary one. The base stations have imperfect chan-

nel state information.

Figure 9: BER performances against the number of iteration in

rKA algorithm. NTP is 30 dB and the channel is non-

stationary one. The base stations have imperfect chan-

nel state information.

Figure 10: BER performances against the number of iteration in

rKA algorithm. NTP is 40 dB and the channel is

non-stationary one. The base stations have imperfect

channel state information.

One of the key findings, as seen in Figure 8, Figure

9, and Figure 10, is that our method maintains a su-

perior BER performance, particularly with fewer iter-

ations. In extremely massive MIMO systems, where

a large number of antennas is employed, reducing the

number of iterations is critical to minimizing compu-

tational complexity. The results demonstrate that our

method is effective in achieving low BER while sig-

nificantly reducing the required number of iterations.

For instance, at 20, 30, and 40 dB NTP , our method

converges to a lower BER than the other methods

even with fewer iterations.

In practical applications, the reduced number of

iterations has important implications. It can lead to

faster processing times and lower energy consump-

tion in real-time communication systems. Massive

MIMO systems, particularly in future 6G networks,

will require efficient algorithms to handle the in-

creased computational demands. The ability of our

method to achieve better BER with fewer iterations

makes it highly suitable for such large-scale systems.

However, despite the clear advantages, some lim-

itations remain. While our method performs well at

higher NTPs, residual errors limit its performance

at very high transmit powers. Exploring how the

method performs under different channel conditions,

such as those with more severe non-stationarity or in-

terference, would also be valuable.

5 Conclusion

The proposed method, which innovatively applies the

D’Hondt method for row selection in the randomized

Kaczmarz algorithm, stands out by offering faster

convergence and better BER performance. Unlike

conventional methods that rely on probabilistic selec-

tion, our approach ensures a more balanced and ef-

ficient update process, making it particularly advan-

tageous in scenarios with high normalized transmit

power.

Our simulation results demonstrated that our pro-

posed method significantly improves BER perfor-

mance compared to existing rKA-based methods, par-

ticularly in scenarios with imperfect channel state in-

formation at the base stations. The ability to achieve

better BER with fewer iterations highlights the com-

putational efficiency of our method, making it a suit-

able solution for real-world applications in massive

MIMO systems, where reducing complexity is cru-

cial.

The main contribution of this research is the in-

novative application of the D’Hondt method, which

helps balance the update process in rKA, leading to

more stable and accurate results. This approach is

particularly useful in large-scale systems where com-

putational efficiency is key.

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With these findings, we recommend that the pro-

posed method be implemented in large-scale wire-

less communication systems, such as future 6G net-

works, where the reduction of computational com-

plexity in precoding is critical. The faster conver-

gence and improved BER performance, especially at

higher normalized transmit powers, make our method

well-suited for real-time applications requiring high

reliability and low latency. Additionally, the pro-

posed method could be further optimized for hard-

ware implementation, allowing for even faster pro-

cessing in practical systems.

For future work, we plan to address the residual er-

rors that limit the performance of our method at very

high transmit powers. Moreover, testing the method

in real-world hardware environments will be essen-

tial to ensure that the computational savings trans-

late into tangible performance improvements. Inves-

tigating the performance of our method under differ-

ent channel conditions, such as perfect channel state

information or more severe non-stationary environ-

ments, will play an essential role in further validating

and enhancing the robustness of the method.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author solely contributed in the present research,

at all stages from the formulation of the problem to

the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The author is affiliated with Shimonoseki City Uni-

versity, a public university corporation located in Shi-

monoseki City, which is the subject of this research.

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DOI: 10.37394/232018.2024.12.48

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