An impulse response is present in long-length wireless communication channels in communication systems such as massive MIMO or 5G systems that use block coding, such as Low-Density Parity-Check (LDPC) code. To deal with it, it is favorable to use block adaptive filtering instead of sequential or traditional algorithms. In this study, a block adaptive algorithm called the Speedy Euclidean Direction Search (SEDS) algorithm is proposed for a Space Division Multiple Access Base Station (SDMA) to perform adaptive beamforming operation of the system. An investigation and analysis of the performance of the SEDS algorithm were conducted using SDMA with a Multiple Input Single Output (MISO) downlink scheme that uses block processing of the data samples for a single-cell downlink. Moreover, a novel and fair comparative study was carried out between SEDS and the Block adaptive Least Mean Square (BLMS) algorithm. As far as we know, no research has used the BLMS algorithm for the SDMA system, which is another contribution of this study. The simulation results demonstrate that the SEDS algorithm has fast convergence and a very accurate estimation compared with BLMS. Moreover, the SEDS algorithm has better performance in indoor environments compared with BLMS. On the other hand, SEDS starts to be unstable for a large value of block length (L), whereas BLMS stays stable especially for outdoor environments.

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## 1. Introduction

Each mobile user within a wireless communication network is connected to and effectively
communicates with at least one base station in a geographical region called a cell
^{[1]}. The antennas of these cellular base stations are Space Division Multiple Access
(SDMA) antennas (also called smart antennas), which have an adjustable radiation pattern
according to the environment. The SDMA’s antenna radiation patterns direct the main
beam towards a desired user and attenuate or cancel out undesired users or interference
depending on the channel environments and the user's location, as shown in Fig. 1 ^{[2]}. An SDMA antenna can update the antenna weight coefficients by using an adaptive
algorithm that beamforms or steers the radiation patterns towards a desired user and
cancels out interference in a cell ^{[3]}.

SDMA has two different beamforming arrangements: one for producing downlink beams
and a second one for producing uplink beams ^{[1]}. In this study, single-cell downlink beams are used such that a base station having
M elements in a uniform linear array (ULA) is connected with different users, such
that each user has a single antenna. In this case, Multiple-Input Single-Output (MISO)
links are obtained. A block adaptive algorithm is applied to perform the smart operation
of the system. This is done to manage the long channel impulse response of some wireless
communication channels such as massive MIMO using a block coding for a 5G system,
such as Low-Density Parity-Check (LDPC) code, which is the best error-correcting code
^{[4]}.

To our knowledge, there is no study or analysis on the performance of a time-domain
block adaptive LMS algorithm in SDMA. All previous research compared between the SEDS
(also known as FEDS) algorithm and sequential or traditional algorithms like Recursive
Least Square (RLS), Least Mean Square (LMS), and Normalized Least Square (NLMS) ^{[5-}^{8]}. The SEDS algorithm is a block adaptive algorithm. Therefore, it is fair to compare
its performance with block algorithms like BLMS. An SDMA antenna was simulated using
MATLAB, and we modeled three different multipath fading wireless communications channels,
which represent indoor and outdoor environments.

Section 2 presents a literature review, while section 3 provides an overview of time-domain block adaptive algorithms. Section 4 presents the two-block adaptive algorithms BLMS and SEDS and how to choose the block length. In section 5, simulation results are presented, and finally, section 6 concludes the study.

## 2. Literature Review

As mentioned previously, very few works in the literature studied the performance
of the SEDS algorithm applied to an SDMA system and compared it with other algorithms
that use sequential time processing (i.e., not block processing). The authors of article
in ^{[5]} compared the performance of SEDS algorithm with classical algorithms like LMS, RLS,
and NLMS. Another research ^{[6]} was study the effect of choosing the window length parameter (L) of the SEDS algorithm
over the AWGN channel only. In another study ^{[7]}, the author repeats the previous study ^{[6]} but for a Rayleigh channel. Finally, the authors in another study ^{[8]} used different channels to demonstrate the results of the outcome in the previous
study ^{[7]}. All previous studies used both classical sequential time processing and a traditional
SDMA system, while in this study, block data processing and a single-cell MISO downlink
channel were used.

## 3. Method of Time -Domian Block Adaptive Algorithms

Some wireless communications channels such as massive-MIMO channels (or those using
an LDPC for a 5G system) have a long impulse response, so the traditional or sequential
adaptive algorithms like LMS or RLS are not sufficient or are expensive to use ^{[9]}. In order to reduce the computational complexity of adaptive ﬁlters, block implementation
or block processing was proposed because using parallel processing of data samples
can lead to less computational complexity. Moreover, high computational efficiency
can be obtained due to sharing of the processing time among the samples in each block.
The computational complexity of block adaptive filtering can be defined as the operations
needed to process one block of data divided by the block length ^{[9]}.

Due to a block of the input signal, samples must be gathered before starting block
processing, and then processing time delay will arise at the system output and should
be minimized. This processing delay increases with the block length. Of the possible
options for the limitations on the block size relevant to the length M of FIR filter,
choosing a block length equal to the filter order (M) represents the best choice from
the viewpoint of minimum computational complexity. A second option considers the reduction
of processing delay when choosing a block length less than the filter order. The third
choice is when the block length is larger than the filter order, which means processing
data will be redundant ^{[9]}. In this study, two-block adaptive algorithms were used: block LMS (BLMS) and SEDS
algorithms. The SEDS and BLMS algorithms are least square and mean square error algorithms,
respectively.

## 4. Block Adaptive Algorithms

### 4.1 Block Samples Processing

Fig. 2 shows the concept of block processing for block adaptive filtering, where the input
and desired output signals of the block adaptive filter are composed in order to have
a block of the output signal ^{[10]}. The symbol L refers to the length of the data blocks, which will be used to update
or modify the main filter’s parameters. Other symbols like u and W represent the input
and filter weight coefficients, respectively.

The parameters that are defined at time index n = k L + I are as follows: - u(k L + i) (the input signal); y(k L + i ) = wT (k) u(k L + i) (the output of the filter), and finally, e(k L + i) (the error signal). The only parameter that is defined at time instants Kl is the weight vector, W(k).

### 4.2 The BLMS Algorithm

Fig. 3 illustrates the adaptive block filter idea. It is clear that the input sequence (u)
enters the serial-to-parallel converter to divide into blocks of length L, and one
block at a time is used by the M-order FIR filter. After gathering each block of u
samples, the weight coefficients of the FIR filter are adjusted block by block instead
of conventional LMS, which uses a sample-by-sample process ^{[11]}.

The block length index is k, and the sample time n is:

The input signal vector at time n for block k is defined by a set:

The output of the FIR filter will be calculated as:

##### (3)

$y\left(kL+i\right)=~ \hat{w}^{T}\left(k\right)u~ \left(kL+i\right) \\ =\sum _{j=0}^{M-1}\hat{w}_{j}\left(k\right)u\left(kL+i-j\right),\,\,i=0,1,\ldots ,L-1. $where w ̂^T (k) denotes the tap-weight vector of the filter at time k. Let d(kL + i) denote the desired response. Then, an error signal can be calculated as:

In a synchronous way with the input end of the FIR filter, the error signal is divided
into L point blocks to be used later to update the filter weight coefficients. Different
values of error signals will be used for each block of samples in the BLMS adaptive
algorithm. The weight coefficient vector of the BLMS algorithm will be adjusted over
all possible values of i according to the following formula after summation of the
product process u (kL+i)e(kL+i) ^{[11]}.

##### (5)

$ \hat{w}\left(k+1\right)=\hat{w}\left(k\right)+u~ \sum _{i=0}^{L-1}u\left(kL+i\right)e\left(kL+i\right), $where u is the BLMS step size. Table 1 shows the step sequence of BLMS.

##### Table 1. Step sequence of BLMS algorithm.

### 4.3 Direction Search (DS) Algorithm

The first direction search (DS) algorithm was developed by Powell and Zangwill in
1964 and 1967 and is a least square algorithm ^{[12]}. Recently, a modified version of the DS algorithm called the Euclidean Direction
Search (EDS) algorithm was developed by Xu and Bose in 1999 ^{[13]}. When the N weights are updated by cyclically searching through the N Euclidean directions,
then the EDS algorithm is called the Fast EDS algorithm.

This fast EDS was named as Speedy Euclidean Direction Search (SEDS) in this study
^{[5-}^{8]}. Fig. 4 shows how DS family algorithms find a preferable estimated weight by searching in
the duration of each direction among a set of linearly independent directions. Then,
with every block of data, the weights will be reduced exponentially instead of with
discrete time or sample by sample as in conventional least square algorithms like
RLS (where ${\lambda}$ is the forgetting factor) ^{[13]}. Then, for every sample of data, only one Euclidean direction search is processed.
The SEDS algorithm steps are listed in Table 2. For each block index, k, the formulas in Table 2 are carried out. There are N data within each block. Items (1) through (8) are performed
for each data (index i). At the end of each block, both items (A) and (B) are processed.

For each Euclidean direction search (i.e., for each i), one row of Q (item (l) is
the first part of Q), one element of w, and one element of r (item (2) is the first
part of r) are updated. Items (3) and (4) are accumulations of Q and r within one
block. The update items of one row of Q and r are shown in items (5) and (6), respectively,
such that this one-row update is q$^{\mathrm{(i)}}$, and q~$^{\mathbf{(i)}}$for Q
and (r + r$^{\mathbf{\sim }}$) for r. The step size parameter a is set in items 5,
6, and 7. One element of the weight vector was updated in item (8). Finally, items
(A) and (B) represents an update of Q and r, and we also reset Q~ and r~ at the end
of each block ^{[13]}.

## 5. Results and Discussion

### 5.1 SDMA Base Station Simulation

As shown in Fig. 5, the SDMA base station with MISO (i.e., 16 X1) has a uniform linear antenna array
that consists of 16 receiving antennas separated by half a wavelength. Four users
transmit at a specified elevation angle such that the desired angle of the desired
user is 0$^{\circ}$, while the three other undesired or interference angles are 30$^{\circ}$,
-30$^{\circ}$, and 60$^{\circ}$ respectively. Two time-domain block adaptive algorithms
were used: BLMS and SEDS with a different block index K and different data length
L. Three different multipath fading propagation channel models were used according
to 3GPP ^{[14]}.

##### Table 2. SEDS algorithm[13].

### 5.2 EPA Wireless Channel Model

Extended Pedestrian A model (EPA) has seven paths with gain equal to [0 -1 -2 -3 -8
-17.2 -20.8], and the corresponding delay for each path is [0 30 70 90 110 190 410]
*1e-9/Ts, such that the sampling frequency fs equals 7.68e6 Hz, and the sampling time
is Ts equals 1/fs ^{[14]}. This channel represents indoor environments with small cell sizes and a low delay
spread environment. Fig. 6 shows the EPA frequency response ^{[14]}.

Fig. 7 shows MSE for both algorithms with L values and K equal to 100. It is clear that the performance of the SEDS algorithm is better than BLMS in terms of fast convergence and accurate estimation with minimum error. Also, we can notice that this performance (both SEDS and BLMS) is enhanced when L is increased, and we have an accurate estimation output signal when the input desire signal is a pure cosine wave, as illustrated in Fig. 8.

### 5.3 EVA Wireless Channel Model

Extended Vehicular A model (EVA) has nine paths with gain equal to [0 -1.5 -1.4 -3.6
-0.6 -9.1 -7 -12 -16.9], and the corresponding delay for each path is [0 30 150 310
370 710 1090 1730 2510] *1e-9/Ts. This channel represents urban environments with
large cells and has a medium delay spread model ^{[14]}. Fig. 9 shows the EVA frequency response.

Figs. 10 and 11 show the performance of two algorithms in terms of the MSE and estimation output signal, respectively.

As shown in the aforementioned figures, when L is less than 40, the performance of the SEDS algorithm is better than BLMS, but it starts to become unstable when L has a value more than 40. Therefore, the performance of SEDS for L equal to 60 is omitted. On the other hand, the performance of BLMS remains stable even when L has values more than 40. This fact is also repeated for the next channel type (ETU). Therefore, one way to solve the instability problem of the SEDS algorithm is to control or minimize its step size.

### 5.4 ETU Wireless Channel Model

The Extended Typical Urban model has high delay spread environments, so it is used
to model urban environments with large cells ^{[14]}. Fig. 12 shows the ETU frequency response. ETU has nine paths with gain equal to [-1 -1 -1
0 0 0 -3 -5 -7], and the corresponding delay for each path is [0 50 120 200 230 500
1600 2300 5000] *1e-9/Ts ^{[14]}. The figure below shows the ETU frequency response ^{[14]}.

Figs. 15-17 depict beam patterns for both algorithms with different values of block lengths (L) for the ETU channel. It is clear that the beam pattern of SEDS is better than the BLMS in terms of null beamforming at interfering angles with no deviations from these nulling angles (30$^{\circ}$, -30$^{\circ}$, and -60$^{\circ}$), while both algorithms steer the main beam towards the desired user angle (0$^{\circ}$). Moreover, the beampattern of BLMS is enhanced with increasing block length (L).

## 6. Conclusion

The main contribution of this paper is to propose two adaptive time-domain blocks for long-length wireless communication channels impulse response for future communications systems. There has been no earlier works that employ or implement adaptive time-domain block algorithms (such as BLMS or SEDS) for the SDMA system that we are aware of. To our knowledge, no research has been done to evaluate or analysis the performance of the time-domain block adaptive LMS algorithm in SDMA, or to compare it to the SEDS method. These schemes are a mean and least squared error which includes BLMS and SEDS algorithms respectively.

The performance of the base station SDMA system with MISO downlink channel (16 X 1) was investigated using the above algorithms. The simulated results show that both BLMS and SEDS algorithms have good performance when block size (L) increased. Moreover, although the SEDS algorithm has accurate and better estimation compared with BLMS in indoor environments SEDS’s performance is degraded for outdoor environments when the block size was set to 60. On another side, the BLMS algorithm has stable performance and is enhanced with increasing block length.

### REFERENCES

## Author

Vian S. Al-Doari is one of the staff at Al-Rafidain university college, Baghdad, Iraq. She was graduated and had B.Sc., M.Sc. and Ph.D. degree awards in modern communications systems from Al-Nahrian University, Engineering college, Baghdad, Iraq. Her field study interest includes Adaptive Signal Processing, MIMO and OFDM systems.

Thamer M. Jamel graduated from the University of Technology with a Bachelor's degree in electronics engineering. He received a Master's and a Doctoral degree in communi-cation engineering in 1997. His scientific degree is Professor and currently, he is one of the staff of the communication engineering department at University of Technology, Baghdad, Iraq. His Research Interests are Adaptive Digital Signal Processing (Algorithms and Applications) for modern and future Communications system.

Bashar M. Mansoor received the M.Sc. and B.Sc. degrees in communication engineering from Electrical and Electronic Engineering (EEE) department at University of Technology, Baghdad, Iraq, in 2008 and 2013, respectively. He gained a Ph.D. degree in Electrical engineering from Engineering college at University of Baghdad in 2019. His research interests are in Adaptive filtering, Digital wireless communication and Coding