Beamforming in cognitive radio networks using partial update adaptive learning algorithm


ACTA IMEKO 
ISSN: 2221-870X 
March 2022, Volume 11, Number 1, 1 - 8 

 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 1 

Beamforming in cognitive radio networks using partial 
update adaptive learning algorithm 

Md Zia Ur Rahman1, P. V. S. Aswitha1, D. Sriprathyusha1, S. K. Sameera Farheen1 

1 Dept. of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur-522502, Andhra  
  Pradesh, India 

 

 

Section: RESEARCH PAPER  

Keywords: Adaptive learning, bandwidth, cognitive radio, frequency, power transmission 

Citation: Md Zia Ur Rahman, P. V. S. Aswitha, D. Sriprathyusha, S. K. Sameera Farheen, Beamforming in cognitive radio networks using partial update 
adaptive learning algorithm, Acta IMEKO, vol. 11, no. 1, article 30, March 2022, identifier: IMEKO-ACTA-11 (2022)-01-30 

Section Editor: Md Zia Ur Rahman, Koneru Lakshmaiah Education Foundation, Guntur, India 

Received December 4, 2021; In final form February 18, 2022; Published March 2022 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: S. K. Sameera Farheen, e-mail: sksameera667@gmail.com  

 

1. INTRODUCTION 

In medical telemetry wireless communication techniques are 
used widely in the present generation. Generally, these types are 
used to monitor the condition of a person via pulse, respiration, 
etc., This wireless medical telemetry service was first established 
by the federal communications commission by allocating some 
of the frequency bands separately for this wireless purpose so 
that it will not have any discrepancies while allocating frequency 
bands while cross-checking the patient's condition [1]. Instead of 
that, we can use the existing spectrum for a medical telemetry 
application by using cognitive radio. In this cognitive radio 
spectrum sensing technique would be there for this spectrum 
sense we use the beamforming technique. The efficient 
utilization of spectrum is the most important thing in cognitive 
radio. cognitive radio offers a spectrum allocation for unlicensed 
users simultaneously with licensed users. So, the secondary users 
need to detect the spectrum availability using spectrum sensing 
by sensing the primary user in the same frequency [2], [3]. The 
main purpose to introduce beamforming technique is to remove 

the unwanted noise in different systems which we are using in 
this process like military SONAR and RADAR systems. This will 
be separating the sources when the overlapping frequency 
content which originates at different spatial locations [4]-[6]. This 
technique is mainly used for sensing purposes, in this phase, the 
licensed user transmitter will be estimated by the secondary user 
transmitter according to that estimation the spectrum allocation 
will be done [5]-[8]. This technique is used to provide an 
exchange of information within the cells and the remaining near 
cells will be in a secured manner.  

The interference which is occurred in signals is mainly due to 
the imperfection of spectrum sensing, Due to that secondary 
user are freely accessing the primary user channels [9], [10]. At 
the same time, this technique should return to the channel before 
the secondary users. This beamforming would be very effective 
while sensing the spectrum without any interference. A cognitive 
radio network is one of the important systems in the broadband 
communication system. The beamforming technique [11], [12] 
which we are proposing in this paper is used to connect the 
information inside the cells and the outer cells which are in use 

ABSTRACT 
Cognitive radio technology is a promising way to improve bandwidth efficiency. Frequency which is not used in any aspect will be utilized 
by using some of the most powerful resources in this cognitive radio. One of the main advantages of cognitive radio signal is to detect 
the different channels which are there in the spectrum and it can modify the frequencies which is utilized frequently. It allows the 
licensed users to gain the licensed bandwidth under the condition to protect the licensed users from harmful interference i.e., from 
secondary users. In this paper, we would like to implement cognitive radio using the beamforming technique, by using power allocation 
as a strategy for the unlicensed transmitter which is purely form on the result of sensing. It is on the state of the primary user in a various 
cognitive radio network whereas the unlicensed transmitter gives a single antenna and it modify its power transmission. For the cognitive 
radio setup, we have used normalized adaptive learning algorithms. This application would be very useful in medical telemetry 
applications. Nowadays wireless communication plays a vital role in healthcare applications for that we have to build a separate base. 
It reduces the effort of the building of separate infrastructure for medical telemetry applications. 

mailto:sksameera667@gmail.com


 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 2 

from all the users. It has proved that the beamforming technique 
is the smooth technique for spectrum sensing analytically too. In 
that paper, they explained some statistical tests between 
Maximum-to-Minimum Eigenvalue and Maximum-to-Minimum 
Beam Energy algorithms based on the statistical values they 
conclude that beamforming is a smooth sensing technique [13]. 

The sensors in secondary users are used to assume their 
particular PU DoAs. Always the secondary users access the 
primary user's accounts freely and simultaneously, licensed users 
also intend to get back the mechanism before the unlicensed user 
acquire expires. At last, the binding centre collaborate with DoAs 
into a Primary User localization estimation. The result of our 
final implementation gives that to make possible to make a 
localization system with a poor complexity and a good primary 
user localization capability in this cognitive network. The 
capabilities which are present in PU localization are used to 
enhance the PU interference [14]. Generally, in services of 
wireless communication, it has rectifiers at every junction point 
to manage every event of the upcoming trending technologies 
which gives profit from abstraction property. It has gained from 
different latest antenna sets and also from algorithms that form 
adaptive beams [15]. The process which we have used is purely 
based on Least Mean Square (LMS) algorithm, which gives a 
brief and proper care to the signal model which we have used for 
the beamforming technique. To get the accuracy convergence 
rate more than the expected one for the LMS which we have 
used in the smart antenna system, we have used BBNLMS 
(Block-Based Normalized Least Mean Square) algorithm. The 
presence of this algorithm gives a lot of difference in 
convergence rate. Mainly this algorithm will be performed only 
in the existence of different effects and many users which is 
scanned using MATLAB simulations using different white 
signals. The LMS formula was used only in the sensible antenna 
systems in between the adjustive beam forming algorithms [16]. 
In place of quick exploitation, the traditional square of the error 
vector is used [17]. Although many methods have been 
implemented and implemented with the new ideas for the DOA 
(Direction of Arrival) of signal. 

The adaptive array was first discovered by VAN Atta in the 
year 1959 which is defined as a self-phased array [18]. It reflects 
all the incident signals in the arrival direction using a clever 
phasing scheme. These beamforming algorithms are divided into 
various methods they are mainly fixed weight beamforming and 
adaptive beamforming algorithm. This Adaptive algorithm will 
update the array weights continuously which is based on 
optimization of changing signal environment. This section 
briefly discusses LMS, recursive least square, conjugate gradient 
algorithm, and quasi-Newton algorithm [19]. Data transmission 
in a communication channel with a low probability of bit error is 
possible up to a certain bit rate with a given SNR (signal-to-noise 
ratio). Therefore, in addition to the adaptive algorithm, to 
increase the noise immunity of the wireless system, let us 
consider the use of channel coding schemes. Such coding is a 
series of transformations of the original bit sequence, as a result 
of which the transmission of information flow becomes more 
resistant to the deterioration of the quality of the transmitted 
information caused by noise, interference, and signal fading. 
Channel coding allows reaching a compromise between 
bandwidth and the probability of bit error [20]. This paper gives 
a different aspect in the direction of arrival of signals with a 
different error rate and at last, in this paper, we provide the 
simulation results which provides better evidence for the 
technique which we have provided in this paper. 

2. METHODOLOGY 

The networks that we are using in this paper are Cognitive 
Radio Network (CRN) and Non-Orthogonal Multiple Access 
(NOMA) are widely used system in the 5G broadband 
communication system. There is one advantage for the users 
who are using a cognitive radio network is to protect the 
information from different devices like multiple-input multiple-
output-NOMA. In our paper, we are trying to implement the 
substitute of beam forming technique which is already available 
to protect the data exchange inside the cells and outer cells from 
different users and their system model is shown in Figure 1. The 
intervention was caused by a faulty signal of the unlicensed users. 
The unlicensed users are intended to get the availability of 
licensed users. Though the licensed user repays the channel 
before the licensed user access terminates. Adaptive antenna 
systems include a mixture of different antenna components with 
a signal-processing ability to improve its radiation or acceptance 
pattern instinctively in response to the signal environment. The 
method which we are proposing in our paper on a Partial Update 
Least Mean Square (PU-LMS) algorithm is an algorithm that is 
used to control the overload and less power consumption in the 
implementation of an adaptive filter. Thus, the problem of 
adaptive filtering algorithm improves the filter coefficients is 
shown in Figure 2. Hence, we provide a better result from our 
proposed technique.  

LMS algorithm is an algorithm admired in adaptive beam 
formers in which we use the antenna arrays. It is also used for 
channel levelling to conflict the inter-symbol interference. 
Simultaneously, few applications of LMS can incorporate 
interference, echo cancellation, space-time modulation, and 
coding, and the signals which are in observation. Whereas, the 
already algorithms have high faster convergence rate like 
recursive least square and least mean square which is already 
admired because of their implementation and its computational 
costs. It is one of the effective methods for power consumption 
and for reducing the computational load in the adaptive filter 
implementations, which is appealing in mobile communications. 

 

Figure 1. Cognitive system mode.  

 

Figure 2. Overview of project implementation. 



 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 3 

Block diagram and overview of proposed algorithm is shown in 
Figure 3 and Figure 4, respectively. There are many mobile 
communication devices and those applications like channel 
equalization as well as echo cancellation which requires the 
adaptive filter to get a high number of coefficients. To modernize 
the entire signal vector which is costly regarding RAM, capacity, 
and computation and sometimes it does not fit mobile units. 
Finally, in this paper, we present the analysis of the confluence 
of partial update adaptive learning (PUAL) algorithm lesser than 
different types of suppositions, and when the diversions are 
visible then it can be prevented by different coefficient updates 
accidentally, which is also known as sequential partial update 
adaptive learning (SPU-AL). 

2.1. Cyclostationary input signals 

Cyclostationary is one of the processes which is having 
different analytics which differ cyclically with respect to meter. It 
is the one that can be showed as a various interleaved stationary 
process. It is also having a detection method for estimating and 
for spectral autocorrelation function technique to analyse the 
spectrum sensing. It can be detected whether it is active or not 
and whether the signal is used by licensed users or not it can 
sense these easily because this technique is robust in the sense. 
Sequential PU-AL algorithm, shows how the coefficients are 
updating in simulation and associated samples of the signals 
which are used in every overhaul (x power 1/4)(n) being the value 
of the retreat signal at the present instant and its flow chart is 
shown in Figure 4. It simulates as a higher bound for each step 
size of PU AL compatible algorithm when the signal is given as 
an input signal as a periodic reference which consists of many 
euphonious 

𝐸{ℎ(𝑙 + 𝑇)} = 𝐸{ℎ(𝑙)𝐸{ℎ(𝑙 + 𝑇)ℎ(𝑙 + 𝑇 + 𝑛)}
= 𝐸{ℎ(𝑙)ℎ(𝑙 + 𝑛)} 

(1) 

𝑦(𝑙) = [𝑦(𝑙)𝑦(𝑙 − 1)𝑦(𝑙 − 2) … . , 𝑦(𝑙 − 𝑁 + 1)]T 

𝐵𝑦 (𝑙) = 𝐸{𝑦(𝑙)𝑦
T(𝑙)}

= diag[𝜎𝑦
2(𝑙), … . , 𝜎𝑦

2(𝑙 − 𝑁 + 1)] 

(2) 

𝜎𝑦
2 = {

𝑀1 for 𝑖 𝑇 < 𝑙 ≤ 𝑖 𝑇 + 𝛼 𝑇

𝑀2 for 𝑖 𝑇 + 𝛼 𝑇 ≤ (𝑖 + 1) 𝑇
 (3) 

or 0 < 𝛼 < 1 and 𝑖 = ⋯ , −4, −3, −2, −1, 0, 1, 2, 3, 4, … and 
sinusoidal power of a variation in time. 

𝜎𝑦
2 = 𝛽(1 + sin(𝜔𝑜 𝑙)) . (4) 

Here, is larger than zero, and 𝜔o falls between 0 and , with 

𝜔o/(2 ) being a rational integer. If the sinusoidal power of a 
variation in time period is more than three, then 𝑦(𝑙) is not an 
input signal. 

2.2. Sequential Partial Update Adaptive Learning Algorithm 

The needed signal is denoted by 𝑑(𝑙), the desirable weight 
vectors are denoted by 𝜔o, and the noise measurement is 
denoted by 𝑣(𝑙) 

𝑑(𝑙) = 𝑦T(𝑙)𝜔o + 𝑣(𝑙) (5) 

𝑒(𝑙) = 𝑒(𝑙) − 𝑦T(𝑙) 𝑢(𝑙) (6) 

𝑢(𝑙 + 1) = 𝑢(𝑙) + 𝜇𝑒 (𝑙)𝐼𝐾 (𝑙)𝑦(𝑙) . (7) 

Here 𝑢(𝑙) is the weight vector (adjustable filter coefficient 

vector), 𝑑(𝑙) = 𝑦T(𝑙) 𝑢(𝑙) indicates fallacy of method 
(whatever algorithm we have taken) and 𝐼 coefficient preference 
matrix is: 

𝐼k(𝑙) diag[𝑖1(𝑙), 𝑖2(𝑙), … … , 𝑖𝑁 (𝑙)  (8) 

with 𝐴 = 𝑁 𝐾⁄ . Here, 𝐴 is taken as integer %(𝑙, 𝐴) represents 
the ‘%’ operation which results the remainder after making 

division 𝑙 by 𝐴.  

∑ 𝑖𝑗 (𝑙)

𝑁

𝑖=1

= 𝐾, 𝑖𝑗 ∈ {0,1}  

The coefficient subsets 𝐸i are different up to they reach the 
following requirements: 

1. 𝑈i=1
A 𝐸i = 𝑍 where 𝑍 = {1,2, … , 𝑁} 

2. 𝐸𝑖 ∩ 𝐸𝑗 = ∅, ∀𝑗 , 𝑗 ∈ {1,2,3, … , 𝐴} and 𝑖 ≠ 𝑗 . 

2.3. Performance analysis for the input signal 

�̌�(𝑙 + 1) = �̌�(𝑙) − 𝜇𝑒 (𝑙)𝐼𝐾 (𝑙)𝑦(𝑙) (9) 

Here is 

�̃�(𝑙) = 𝜔o − 𝑢(𝑙). (10) 

Using (11),  

𝑔(𝑙) = 𝑦T�̃�(𝑙) + 𝑣(𝑙)  (11) 

is obtained. Putting (11) into (9), then 

�̃�(𝑙 + 1)

= 1 − 𝜇 𝐼𝑘 (𝑙) 𝑦(𝑙) 𝑦
T(𝑙) �̃�(𝑙)

− 𝜇 𝑣(𝑙) 𝐼𝐾 (𝑙) 𝑦(𝑙) 

(12) 

Taking the exception on both sides of (12), we get  

𝐸{�̃�(𝑙 + 1)} = (𝐼 − 𝜇 𝐸{𝐼𝐾 (𝑙) 𝑦(𝑙) 𝑦
T(𝑙)}) 𝐸{�̃�(𝑙)} (13) 

Time varying variance model 

𝜎𝑦
2 = {

𝑀1  if mod(1, 𝑇) = 1
…

𝑀𝑇−1  if mod(1, 𝑇) = 𝑇 − 1

𝑀𝑇    if mod(1, 𝑇) = 0

 . (14) 

We took the set 𝑀1, 𝑀2, …, 𝑀𝑇 , which has one large value 
(like 1) and one tiny value (like 0.001). It's to make sure that (3) 
and (4) are both true (14). Between the SU-partial update 
parameter 𝐴 and the input signal period 𝑇, the six instances 
reflect all potential scenarios. 

2.3.1. Case Study 1 

𝑇 ≤ 𝐴 and %(1, 𝑇) = 0. In case of (13) can be rework as 

𝐸{�̃�(𝑙 + 1)} = 1 − 𝜇𝑖𝑗 (𝑙)𝜎𝑦
2(𝑙 − 𝑗 + 1)𝐸{𝑢�̃�(𝑙)} (15) 

 

Figure 3. Overview of partial update algorithm.  



 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 4 

Here 𝑢�̃�(𝑙) is the j th entry of �̃�𝑗 (𝑙). By verifying (7), the 

rechecking (15) is changed for every 𝐴 iteration. By adding the 𝐴 
iterations of (15) 

𝐸{𝑢�̃�(𝑙 + 𝐵)} = (1 − 𝜇𝜎𝑦
2(𝑙𝑗 − 𝑗 + 1)) 𝐸{𝑢�̃�(𝑙)} (16) 

Here 𝑙 is taking as integer satisfying 𝑙 where 𝑙𝑗 < 𝑙 + 𝐴. Let 

declare the parameter 𝑑𝑗 = 𝑙𝑗 − [𝑙𝑗 𝐴⁄ ]𝐴 to indicate which entry 

of 𝐸{𝑢�̃�(𝑙)} where 𝑙𝑗  is satisfies 𝑙𝑗 < 𝑙 + 𝐴. Let declare the 

parameter like 𝑑𝑗 = 𝑙𝑗 − [𝑙𝑗 𝐴⁄ ]𝐴, to represents the coming of 

𝐸{𝑢�̃�(𝑙)} is upgraded. The function [𝑦] converts 𝑦 to largest 

integer ≤ 𝑦. By giving the sequential PU variable 𝐴, 𝑑𝑗  only 

depends on 𝑗 value. 

Here %(𝐴, 𝑇) = 0, we have the equation 𝜎𝑦
2(𝑙𝑗 − 𝑗 + 1) =

𝑀𝑡𝑗  here 𝑡�̃� = 𝑐(𝑡�̃�) and 𝑑𝑗 + 𝑁 − 𝑗 + 1 − [𝑑𝑗 + 𝑁 − 𝑗 +
1

𝑇
] 

and 𝑐(𝑦) = 𝑦 − 𝑇|sign(𝑦)| + 𝑇. By putting this 𝜎𝑦
2(𝑙𝑗 − 𝑗 +

1) = 𝑀𝑡𝑗  in to (16), we get 

𝐸{𝑢�̃�(𝑙 + 𝐵)} = (1 − 𝜇𝜎𝑦
2𝑀𝑡𝑗 ) 𝐸{𝑢�̃�(𝑙)}    (17) 

Here %(𝐴, 𝑇) = 0 and 𝑡𝑗 is like take as an integer, by looping 
(17), we get 

𝐸{𝑢�̃�(𝑙 + 𝑏 + 1)𝐴} = (1 − 𝜇𝑀𝑡𝑗 ) 𝐸{𝑢�̃�(𝑙 + 𝑏𝐴)} . (18) 

In this instance, 𝒃 is a positive integer. It would be the 
combination of (17) and (18), according to (18). 𝑬{(𝒖�̃�(𝒍 +

(𝒃 + 𝟏)𝑨} depends on 𝑴𝒕𝒋 . If 𝑴𝒕𝒋  in the input signal which is 

cyclostationary is very small, is explained in [11]. Input signal will 

change partially for every iteration, it effects the 𝒖�̃�(𝒍)to 
converge and moving towards update. Hence, in this sequential 
PU-AL will certainly met with those tough conditions. 

2.3.2. Case Study 2 

𝑇 ≤ 𝐴 and %(𝐴, 𝑇) ≠ 0 and Greatest Common Divisor 
(GCD) (𝐴, 𝑇) is equal to 1. Here GCD(𝐴, 𝑇) represents the GCD 

of 𝐴 by 𝜎𝑦
2(𝑙𝑗 − 𝑗 + 1) = 𝑀𝑡𝑗(𝑙) here 𝑡𝑗 (𝑙) is taken like an 

integer declared as 𝑇𝑗 (𝑙) = 𝑐(𝑡�̃�(𝑙)) and 𝑡�̃�(𝑙) = 𝑙𝑗 − [𝑙𝑗 𝑇⁄ ]𝑇. 

Then, 𝑇𝑗 (𝑙) is depends on the values of both 𝑗 and 𝑙. Hence, (17) 
becomes 

𝐸{𝑢�̃�(𝑙 + 𝐵)}

= (1 − 𝜇𝑀𝑇𝑗(𝑙) ) 𝐸 {1 − 𝜇𝑀𝑇𝑗(𝑙)) 𝐸{𝑢�̃�(𝑙)} 
(19) 

Looping the equation (19) for 𝐴 number of times, we get  

𝐸{𝑢�̃�(𝑙 + 2𝐵)} = (1 − 𝜇𝑀𝑓(𝑇𝑗(𝑙))
) 𝐸{𝑢�̃�(𝑙 + 𝐴)} (20) 

𝑓 (𝑇𝑗 (𝑙))  

= {
𝑇𝑗 (𝑙) + (%(𝐴, 𝑇)),   𝑇𝑗 (𝑙) + (%(𝐴, 𝑇) ≤ 𝑇

𝑇𝑗 (𝑙) + (%(𝐴, 𝑇) − 𝑇,                    otherwise .
 

(21) 

Since 𝑇 ≤ 𝐴, %(𝐴, 𝑇) is not equal to 0 and GCD(𝐴, 𝑇) is equal 
to 1, looping  (21) for 𝑇 times, we get 

𝐸{𝑢�̃�(𝑙 + 𝑇𝐵)}

= ⟦1 − 𝜇𝑀
𝑓… 𝑓(𝑇𝑗(𝑙))

⟧ 𝐸{𝑢�̃�(𝑙 + (𝑇 − 1)𝐴}  

= (1 − 𝜇𝑀1) … (1 − 𝜇𝑀𝑇 )𝐸{𝑢�̃�(𝑙)} 

(22) 

where 𝑓 …  𝑓 (𝑇𝑗 (𝑙)) represents the configuration of 𝑓(. ) in 𝑇 

times. In (22), here we can observe the updates the process of 

input signal is declared by 𝑇 variances {𝑀1, 𝑀2, … 𝑀𝑇 }. As a 
consequence, the strategy (serial PU-AL) will not interfere with 
toughness in this scenario. If the step-size meets the 
requirements, the PU-LMS is stable. 

0 < 𝜇 ≤ 2/max (𝑀1, 𝑀2, … , 𝑀𝑇 ) (23) 

2.3.3. Case Study 3 

𝑇 ≤ 𝐴, %(𝐴, 𝑇) ≠ 0 and GCD(𝐴, 𝑇) is greater than one. The 
Least Common Multiple (LCM) of 𝑇 and 𝐴 is represented by 
LCM(𝐴, 𝑇). Clearly LCM(𝐴, 𝑇) < 𝐴, 𝑇. Here, the variance 

would become, 𝜎𝑦
2(𝑙𝑗 − 𝑗 + 1) is given that 𝜎𝑦

2(𝑙𝑗 − 𝑗 + 1) =

𝑀𝜉𝑗(𝑙), where 𝜉𝑗 (𝑙) = 𝑐 (𝜉�̃�(𝑙)) and 𝜉�̃�(𝑙) = 𝑙𝑗 − [
𝑘𝑗

𝑇
] 𝑇. Then, 

(17) becomes 

 

Figure 4. Flow Chart of a partial update adaptive learning algorithm.  



 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 5 

𝐸{𝑢�̃�(𝑙 + 𝐵)} = (1 − 𝜇 𝑀𝜉𝑗(𝑙) ) 𝐸{𝑢�̃�(𝑙)} (24) 

Looping equation (24), then we get 

𝐸{𝑢�̃�(𝑙 + 2𝐵)} = (1 − 𝜇 𝑀𝑓(𝜉𝑗(𝑙))
) 𝐸{𝑢�̃�(𝑙)} (25) 

Though, 𝑇 ≤ 𝐴 and %(𝐴, 𝑇) is not equal to zero and 
GCD(𝐴, 𝑇) is greater than one, looping the equation (25) 
LCM(𝐴, 𝑇) times, then  

𝐸{𝑢�̃�(𝑙 + LCM(𝐴, 𝑇))}

= (1 − 𝜇 𝑀
𝑓……𝑓(𝜉𝑗(𝑙))

) × …

× (1 − 𝜇 𝑀𝜉𝑗(𝑙)) 𝐸{𝑢�̃�(𝑙)} , 

(26) 

where LCM(𝐴, 𝑇) 𝐴⁄  is less than 𝑇, there is a one or more than 
one parameter in the set {𝑀1, 𝑀2, … … , 𝑀𝑇 } that is not matched 

to input signal i.e., 𝐸{𝑢�̃�(𝑙)}. The parameters is with the input 

signal i.e., 𝐸{𝑢�̃�(𝑙)} all are tiny values, the up-dation purpose of 

the input signal of 𝐸{𝑢�̃�(𝑙)} will be very slow, while resulting the 
interference. 

2.3.4. Case Study 4 

𝑇 > 𝐴 and %(𝐴, 𝑇) = 0 

In this case the divergence would become 𝜎𝑦
2(𝑙𝑗 − 𝑗 + 1) =

𝑀𝜉𝑗(𝑙) in this equation 𝜉𝑗 (𝑙) would be taken as positive integer 

and having an equal probability for taking on those values of 

[𝑠�̃� , 𝑠�̃� + 𝐴, … , 𝑠�̃� + 𝑇 − 𝐴] where 0 < 𝑑�̃� < 𝐴 is related to 𝑑𝑗  

through 𝑑�̃� − 𝑑𝑗 = 𝑧, where 𝑧=0, 1, 2,…. Then we get 

𝐸{𝑢�̃�(𝑙 + 𝐵)} = (1 − 𝜇𝑀(𝜉𝑗(𝑙))
) 𝐸{𝑢�̃�(𝑙)} (27) 

Looping equation (27), we get 

𝐸{𝑢�̃�(𝑙 + 2𝐵)} = (1 − 𝜇𝑀𝑞(𝜉𝑗(𝑙))
) 𝐸{𝑢�̃�(𝑙)}  (28) 

𝑞 (𝜉𝑗 (𝑙)) = {
𝜉𝑗 (𝑙) + 𝐴            𝜉𝑗 (𝑙) + 𝐴 ≤ 𝑇

𝜉𝑗 (𝑙) + 𝐴 − 𝑇     otherwise .       
  

Here 𝑇 is greater than 𝐴 and %(𝐴, 𝑇) is equal to zero, looping 
equation (28) 𝑇 𝐴⁄  number of times, we get 

𝐸{𝑢�̃�(𝑙 + 𝐵)}

= (1 − 𝜇𝑀
𝑞…..𝑞(𝜉𝑗(𝑙))

) 𝐸{𝑢�̃�(𝑙 + 𝑇 − 𝐴)} 

= (1 − 𝜇𝑀
𝑔……..𝑔(𝜉𝑗(𝑙))

) 𝐸{𝑢�̃�(𝑙 + 𝑇 − 𝐴)} 

(29) 

If all values in the taken set {𝑀𝑞(𝜉𝑗(𝑙),… ,𝑀𝑔…𝑔(𝜉𝑗(𝑙))
} have very 

tiny values, up-dation for input signal i.e., 𝐸{𝑢�̃�(𝑙)} might be 
slightly slow, sequential PU-LMS might show very low 
interference. 

2.3.5. Case Study 5 

𝑇 > 𝐴, %(𝐴, 𝑇) ≠ 0 and GCD(𝐴, 𝑇) is equal to one. 
Likewise coming to case 2, the Sequential PU-LMS is stable if 

equation-24 obeys the step-size. 𝑇 > 𝐴, %(𝐴, 𝑇) ≠ 0 and 
GCD(𝐴, 𝑇) is greater than one. Likewise looking to case-3, the 
sequential PU-LMS faces the low interference. Note: We learned 
from this that for input signals with regularly time-varying 

variance (observe equations-2, 3, 4, and 15), the sequential Partial 
Update-LMS method would not confront the low intersection 

challenging condition in case-2 or case-5. 𝐴 and 𝑇 become co-
prime integers in just two situations, with the GCD(𝐴, 𝑇) equal 
to 1. 𝐴 and 𝑇 are not co-primes, and the sequential Partial 
Update-LMS algorithm may demonstrate a very sluggish 
intersection, depending on how the repeating power levels were 
balanced. 

3. SIMULATION RESULTS 

In this simulation part, the response of a signal with one DOA 
comes at the base station with an angle of 60 degrees. Threshold 
values of 0.1, 0.5 and 1 simulation results are conducted. For each 
threshold point interference rate is considered in terms of the 
samples which we have used to reach the steady state. From the 
simulation results clearly, we can know that the received signal 
converges at a mean square error of 0.0007. Generally, if we 

observe the simulation results for 𝛼 = 0.5 and 1, steady state 
converged faster when compared to the conventional LMS 
algorithm and also, we can clearly observe the delay period for 
these threshold points. This delay corresponds to the samples 
which are observed before the adaptive antenna is ready to adapt. 
The improvement in interference rate purely depends on the 
number of taps adapted. 

3.1. One white signal three DOAs 

In this module effects of multipath in antenna systems are 
studied for various threshold conditions. Three multipaths with 
the three different directions of arrivals of 60, 30 and -20 degrees 
are transmitted to a base station with the different sampling 
periods. Hence, three signals are arrived with time differences of 

𝑡, 𝑡 − 1, and 𝑡 − 2 each with amplitudes of 0.6, 0.75 and 1.0. 
Using the PU-AL algorithm three different weight updating 
equations are used for processing each multipath signal. These 

 

Figure 5. Beam Pattern of one white signal with 3 DOAs using PU-AL for 
α = 0.1.  

 

Figure 6. Received Error Signal of one white signal with three DOAs using PU-AL 
for α = 0.1.  



 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 6 

simulation results are conducted at threshold values of 0.1, 0.2 
and 1. Interference tables are for each multipath signal is shown 
in terms of samples to reach the steady state. For each multipath 
signal, the mean square error is approximately lies between 0.006, 
0.0014 and 0.00036 respectively and their simulation output is 
shown in Figure 5. Beam patterns of the proposed PU-AL 
algorithm are shown in Figure 6. From that, we can clearly say 
that the proposed algorithm is having a better ability to steer the 
beams in different directions, and nulls are placed in the place of 
interferences. Gain of the beam is obtained corresponding to 
gain introduced by each multipath signal. 

3.2. Two white signals with one DOA each 

Transmitting of one signal with two multipath signals and two 
white signals with one DOA effect which is similar. In those 
cases, two signals are uncorrelated to each other, and it is divided 
by one sample period. Firstly, the signal with an amplitude of 0.5 
and the second signal with an amplitude of 1.0 is considered. The 
interference table of each signal in terms of the number of 
samples is shown. From this, we can know that smaller gain 
amplitudes lead to longer responses to adapt taps and for 

estimated signals. For 𝛼 = 0.1 value, it will converge faster when 
compared to 0.2 and 1. From this delay is created for these values 
so it takes a longer time to the response before adaption of taps. 
These threshold points can adapt to the limited number of taps 
and it affects the system performance and its simulation output 
in Figure 7. The beam pattern is shown in Figure 8 with two 
beams corresponding to the DOAs at 60 and -25 degrees. This 
demonstrates a smart antenna system with desired signals and 
interfering ones. 

Beamforming technique as sensing technique in cognitive 
radio network Inside this work, we attempted to explain the 
beamforming approach in cognitive radio as spectrum sensing. 
This beamforming approach may be used in two ways: 

centralized or distributed. We've used a dispersed strategy in this 
case. Cognitive radio may use the beamforming approach to 
target a specific beam onto a specific receiver while reducing 
involvement in surrounding directions, improving network 
performance. This distributed method gives each user a separate 
antenna, and several of them broadcast the signal together by 
manipulating the transmitter's carriers. The authorized users' 
interference is lessened when this method is used. Cognitive 
radio would be able to expand the range of communication by 
employing this beamforming approach in the signal beam is 
vigorous to the appropriate direction by this spectrum. Another 
benefit of this beamforming technology is that it reduces delay 
spread, multipath fading, and radio transmission on the same 
frequency channel as other interferences, among other things. 

The diagram depicted in the illustration is a geometrical 
representation of a cognitive radio network at the intended 
receiver location, which includes licensed users. There are K 
cognitive users who are uniformly scattered over a disc with a 
radius of R and a centre of P. Assume the cognitive radio users' 
position is (SK, $k), which is polar coordinates. Similarly, use the 
specified spherical coordinates to represent the receiver. We 
made the assumption that cognitive radio nodes are uniformly 
dispersed across a disc with a radius of R. A single antenna is 
provided for each node. Because the channel between the users 
and the receiver is in line of sight, there is no shadowing. The 
principal users, also known as licensed users, the transmitters are 
in the far zone of the beam pattern, while the receivers are in the 
near zone. The simulation depicts the statistical distribution of 
the radiation pattern in the lobe that includes the direction in 
which radiation strength is greatest (major lobe) and lobes that 
do not contain the main lobe (sidelobes). These are generally rays 
that are aimed in an unfavourable direction. It is used to examine 
these power levels by running 10,000 trials to create a beam 
pattern. The radius of the disc is R/(lambda) = 2, azimuth 
angle = 0 degrees, and elevation angle = pi/2, all of which are 
regularised by wavelength. These cognitive users employ 
uniform distribution, with numbers such as 4, 7, 16, 100, and 256 
being used. The signal-to-noise ratio in the loop (SNR) of the 
phase loop locked output variance is expected to be 2 dB, 3 dB, 
and 10 dB as shown in Figure 9. At an angle of 20 degrees and 
30 degrees, two licensed users or principal users are presumed to 
be present. 

 

Figure 7. Received Error Signal of two white signal with one DOA using PU-AL 
for α = 0.1. 

 

Figure 8. Beam pattern of two white signals with one DOA using PU-AL for 
α = 0.1. 

 

Figure 9. Average power vs. Direction of Antenna. 



 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 7 

The beampattern Phase-only distributed beamforming 
(PODB) approach does not have a perfect phase in the diagram 
above. In this situation, the number of users is 100. The average 
gain for loop SNR 10 dB is 0 dB at an angle of 0 degrees, whereas 
the gain marginally reduces for loop SNR 2 dB and 3 dB. Because 
of the incomplete phase, the main lobe is at its apex. The users 
are at a 20-degree and 30-degree angle, respectively, therefore the 
sidelobe power is about equal to -20 dB. 

The cumulative complementary distribution function 
(CCDF) of PODB with phase offset is shown in the figure above 
for K values of 3, 8 and 15. The proportion of equal to or more 
than a certain power level beampattern is shown in Figure 10. 
For the CCDF computation, we ran 10,000 simulations. For a 
loop SNR of 10 dB, the angle would be set to 0 degrees. For 
values of 4, 7 and 16, the major lobe is at its highest. Finally, we 
employed the sequential partial update in this case. The least-
square technique is used to repair errors caused by altering 
antenna element placements. The error between the actual and 
expected replies is reduced with this approach. 

4. CONCLUSION 

This paper has analysed the Spectrum sensing issue using the 
beamforming technique in that we have used an algorithm for 
the input signal is sequential PU-AL. Here we have taken the 
input signal as cyclostationary signal with white Gaussian noise 
and we have analysed the input signal using PU-AL in different 
case studies and these case studies also represent how the 
sequential PU-AL tolerate the convergence related issues. We 
have implemented many new things as a result of our project, to 
keep up with technological advancements. Nowadays, we rely on 
online sources for everything instead of utilizing the spectrum 
that is available not only now but also in the future to save 
money. By employing this approach in the future, stricter bounds 
on the rate of convergence can be achieved. For stationary 
signals, a mean update equation of PU-AL can be developed. It 
can also be looked at the techniques that can be used to analyse 
the performance of the Max PU-AL algorithm. 

REFERRENCES 

[1] S. Surekha, Md Zia Ur Rahman, Navarun Gupta, A Low Complex 
Spectrum Sensing Technique for Medical Telemetry System, 
Journal of Scientific & Industrial Research, Vol. 80, pp. 449-456, 
May 2021. 

[2] J. Divya Lakshmi, Rangaiah. L, Cognitive Radio Principles and 
Spectrum Sensing, Blue Eyes Intelligence Engineering & Sciences 
Publication, Volume-8, August 2019, pp. 2249 – 8958. 

[3] Omi Sunuvar, A study of distributed beamforming in cognitive 
radio networks, 2013. 

[4] Numa Joy, Anu Abraham Mathew, Beamforming for Sensing 
Based Spectrum Sharing in Cognitive Radio Network, 
International Journal of Engineering Research & Technology 
(IJERT), 2015. 

[5] Hyils Sharon Magdalene Antony, Thulasimani Lakshmanan, 
Secure Beamforming in 5G-Based Cognitive Radio Network, 
Symmetry, 2019.   
DOI: 10.3990/sym11101260  

[6] Kais Bouallegue, Matthieu Crussiere, Iyad Dayoub, On the impact 
of the covariance matrix size for spectrum sensing methods: 
beamforming versus Eigen values, IEEE, 2019.  
DOI: 10.1109/ISCC47284.2019.8969741  

[7] Aki Hakkarainen, Janis Werner, Nikhil Gulati, Damiano Patron, 
Doug Pfeil, Henna Paaso, Aarne Mammela, Kapil Dandekar, 
Mikko Valkama, Reconfifigurable Antenna Based DoA 
Estimation and Localization in Cognitive Radios: Low Complexity 
Algorithms and Practical Measurements, 2014. 

[8] Yu Sing Xiao, Danny H. K. Tsang, Interference Alignment 
Beamforming and Power Allocation for Cognitive MIMO-
NOMA Downlink Networks, IEEE, 2019.   
DOI: 10.1109/WCNC.2019.8885714  

[9] Md. Zia Ur Rahman, V. Ajay Kumar and G V S Karthik, A Low 
Complex adaptive algorithm for Antenna beam steering, IEEE, 
2011.   
DOI: 10.1109/ICSCCN.2011.6024567  

[10] Janis Werner, Jun Wang, Aki Hakkarainen, Danijela Cabric,Mikko 
Valkama, Performance and Cramer-Rao Bounds for DoA/RSS 
Estimation and Transmitter Localization Using Sectorized 
Antennas, IEEE Transactions on Vehicular Technology,Volume: 
65, May 2016.  
DOI: 10.1109/TVT.2015.2445317  

[11] N. J. Bershad, E. Eweda, and J. C.M. Bermudez, Stochastic 
analysis of the LMS and NLMS algorithms for cyclostationary 
white Gaussian inputs, IEEE Trans. Signal Process., vol. 62, no. 
9, pp. 2238–2249, May 2014. 

[12] M. Z. U. Rahman, S. Surekha, K. P. Satamraju, S. S. Mirza, A. Lay-
Ekuakille, A Collateral Sensor Data Sharing Framework for 
Decentralized Healthcare Systems, IEEE Sensors Journal, 
November 2021.   
DOI: 10.1109/JSEN.2021.3125529  

[13] S. Surekha, Md Zia Ur Rahman, Spectrum Sensing for Wireless 
Medical TelemetrySystems Using a Bias Compensated 
Normalized Adaptive Algorithm, International Journal of 
Microwave and Optical Technology, vol. 16, 2021, no.2, pp. 1-10 

[14] S. Surekha, A. Lay-Ekuakille, A. Pietrosanto, M. A. Ugwiri, Energy 
Detection for Spectrum Sensing in Medical Telemetry Networks 
using Modified NLMS algorithm, 2020 IEEE International 
Instrumentation and Measurement Technology Conference 
(I2MTC), 2020, pp. 1-5,   
DOI: 10.1109/I2MTC43012.2020.9129107  

[15] Shafi Shahsavar Mirza, Nagesh Mantravadi, Sala Surekha, Md Zia 
Ur Rahman, Adaptive Learning Based Beamforming for Spectrum 
Sensing in Wireless Communications, International Journal of 
Microwave and Optical Technology, vol. 16, 2021, no.5. 

[16] Armando Coccia, Federica Amitrano, Leandro Donisi, Giuseppe 
Cesarelli, Gaetano Pagano, Mario Cesarelli, Giovanni D'Addio, 
Design and validation of an e-textile-based wearable system for 
remote health monitoring, ACTA IMEKO, vol.10, no.2, pp. 1-10, 
2021.  
DOI: 10.21014/acta_imeko.v10i2.912  

[17] Ayesha Tarannum, Zia Ur Rahman, L. Koteswara Rao, T. 
Srinivasulu, Aimé Lay-Ekuakille, An Efficient Multi-modal 
Biometric Sensing and Authentication Framework for Distributed 
Applications, IEEE Sensor Journal, vol. 20, no. 24, 2020, pp. 

 

Figure 10. CCDF vs. Instantaneous Power. 

https://doi.org/10.3990/sym11101260
http://dx.doi.org/10.1109/ISCC47284.2019.8969741
https://doi.org/10.1109/WCNC.2019.8885714
https://doi.org/10.1109/ICSCCN.2011.6024567
https://doi.org/10.1109/TVT.2015.2445317
https://doi.org/10.1109/JSEN.2021.3125529
https://doi.org/10.1109/I2MTC43012.2020.9129107
http://dx.doi.org/10.21014/acta_imeko.v10i2.912


 

ACTA IMEKO | www.imeko.org March 2022 | Volume 11 | Number 1 | 8 

15014-1515025. 
DOI: 10.1109/JSEN.2020.3012536    

[18] S. Y. Fathima, K. Murali Krishna, Shakira Bhanu, S. S. Mirza, Side 
Lobe Suppression in NC-OFDM Systems Using Variable 
Cancellation Basis Function, IEEE Access, Vol.5, no.1, 2017, pp. 
9415-9421.  
DOI: 10.1109/ACCESS.2017.2705351  

[19] Imran Ahmed, Eulalia Balestrieri, Francesco Lamonaca, IoMT-
based biomedical measurement systems for healthcare 

monitoring: a review, Acta IMEKO, vol. 10, 2021, no.2, pp. 174 - 
184.  
DOI: 10.21014/acta_imeko.v10i2.1080  

[20] K. Murali Krishna, K. Krishna Reddy, M. Vasim Babu, S.S. Mirza, 
S.Y. Fathima, Ultra-wide band Band-Pass Filters Using Plasmonic 
MIM Wave guide Based Ring Resonators, IEEE Photonics 
Technology Letters, vol.30, 2018, no.9, pp. 1715-1718.  
DOI: 10.48084/etasr.4194  

 
 

https://doi.org/10.1109/JSEN.2020.3012536
https://doi.org/10.1109/ACCESS.2017.2705351
http://dx.doi.org/10.21014/acta_imeko.v10i2.1080
https://doi.org/10.48084/etasr.4194