# Polygon Number Algorithm for Peak-to-Average Ratio Reduction of Massive 5G Systems Using Modified Partial Transmit Sequence Scheme

Authors: Hayder Alkatrani; Muhammad Ilyas; Salam Alyassri; Ali Nahar; Fadi Al-Turjman; Jawad Rasheed; Ali Alshahrani; Basil Al-Kasasbeh
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

The high peak-to-average power ratio (PAPR) of the transmit signal is a major shortcoming of OFDM systems, which results in band radiation and distortion due to the nonlinearity of the high-power amplifier (H.P). To resolve the traditional OFDM high-PAPR issue, where the transmit sequence is designed to avoid similar data from being sent in the same order to reduce PAPR, there are numerous conventional ways for lowering the PAPR for OFDM system, such as selective mapping, tone reservation, block coding, filtering, clipping, and partial transmit sequence (PTS). This study proposes a new method called polygon number algorithm (PN) with conventional partial transmit sequence (C-PTS). This method (PN-PTS) processes the entered data before sending it, taking advantage of the number nonsimilarity according to the geometry of the number to prevent direct sending of similar data via PTS, and thus, this improved the level of PAPR rise in the proposed system. The amount of reduction that can be achieved in PAPR is up to 8 dB by different techniques. The best result obtained was the amount of reduction between the conventional method and the proposed method is 4.5683 whereN=64. Besides this, there is no transmission of side information (SI), which improves transmission efficiency. Finally, this method is easy in the calculation process and the ciphering and deciphering of data, which adds a few calculations.


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## Body

## 1. Introduction

The orthogonal frequency division multiplexing (OFDM) is a frequency division scheme in which the carrier frequencies are very close yet do not interfere with each other. The frequency spacing that is used in OFDM is very small, and its symbol period is bigger than or equal to the reciprocation of this spacing; OFDM usually has a high symbol duration, which reduces intersymbol interference (ISI) [1]. Multimedia communication systems necessitate a high data rate, which necessitates a large amount of transmission power and bandwidth. As a result, high compression efficiency is required to transmit wireless multimedia information over a limited available bandwidth [2]. A common problem with wireless communications is especially in a multipath environment so that the ISI prevents high-speed communication. Therefore, OFDM allows high-productivity communication by increasing the number of carrier frequencies and maintaining high spectral efficiency. Nonetheless, it has challenges to implementation. OFDM signals have trouble with high peak-to-average power ratio (PAPR), to convert digital-to-analog (DAC) signal with high PAPR requires a high-speed and large dynamic range amplifier to forget perfect signal. These components are very expensive and lead to energy loss, shortening battery life in portable devices; besides, heat loss is not desirable in a fixed device [3].One of the things to consider in the multicarrier transmission is the high-power ratio (PAPR) to normal signal transmission, as this is one of the OFDM signal negatives [4]. Therefore, this minimizes the range of the OFDM signal transmission. Furthermore, to avoid defects of the OFDM signal as a correction for burial between out-of-band radiation and subcarriers, the transmit power enhancer must be worked in its direct district. It is necessary to improve the transmission capacity in its immediate area. Therefore, the OFDM signal needs widespread backing, and this conversion is a waste of power [5]. OFDM is insensitive to the nonlinear effects of power amplifiers, due to the high peak-to-average power ratio (PAPR), and this is one of the major problems in this system [6]. There are a lot of problems caused by high PAPR as it limits its wide use in some communications devices, and it is not preferred to use it in uplink transmission due to the restrictions in it [7]. It also leads to the consumption of large amounts of energy as the power amplifiers operate in the saturation region, and this leads to the deterioration of the bit error rate (BER) [8]. Also, high PAPR results in intermodulation and out-of-band radiation due to the nonlinearity in the power amplifier [9]. Therefore, many PAPR reduction schemes such as clipping and filtering [10, 11], block coding [12, 13], partial transmit sequence (PTS) [14], selected mapping (SLM) [15], tone injection (TI) [16], tone reservation (TR) [17], active constellation extension (ACE) [18], and random phase injection algorithms are used [19].Despite the large number of methods used to reduce PAPR, where each method has pros and cons, there is still a problem of reducing complexity in the methods used, If we take, for example, the PTS and SLM methods, they have better performance in reducing PAPR and BER, but they have a problem with high computational complexity [20]. Among these theories that were used for reducing complexity in PTS is the ‘rotating phase shift technique,’ in which the signal was divided into groups, and this group was selected randomly. Moreover, choosing the low level of the PAPR depends on the different phase shifts according to a combination of factors [21]. The primary goal of source coding is to compress the original data intended for wireless transmission. Source coding techniques are used to greatly compress multimedia data [22]. A new method to reduce the PAPR value and complexity from the conventional PTS schemes is used. Polygon number is used to minimize the PAPR in the OFDM system, and good results are achieved when compared to the old method where the result was 4.7205, where N=64 and no.ofsignal=1000. This method is one of the more recent methods used to improve OFDM defects. Moreover, when we compare the PN-PTS to the C-PTS or SLM, we can see how simple this algorithm is in comparison to the previous methods.This paper is divided into seven sections. Section1 outlines the introduction. Section 2 exposes the concept of the OFDM system and the peak-to-average power ratio (PAPR) and overviews the PAPR reduction scheme. Section 3 deals with the PTS scheme in the OFDM system, and Section 4 presents the concept of the number theory, while in Section 5 the proposed PTS scheme is indicated, Section 6 discussed the simulation results, and the conclusion and future work are discussed in Section 7.


## 2. OFDM System and PARP

In an OFDM scheme, the data symbols used by theN orthogonal subcarriers in the transmitter where X=X0,X1⋯XN−1 time domain sequence are as shown in Figure 1. An OFDM signal can be generated and written as in (1) for IFFT:
(1)xn=1N∑K=0N−1XKej2πknN,n=0,1⋯N−1.Figure 1
OFDM block diagram transmitter and receiver.Xn is the discrete signal for the N orthogonal subcarriers. This is in the time domain, so the OFDM symbol is often the variations in the envelope will be as of PAPR, where PAPR is the ratio of the actual signal power to the mean signal power [17].The OFDM signal series x PAPR is named as(2)PAPR=max0≤n<NXn2EXn2,where E· stands for expectation. Complementary cumulative distribution function (CCDF) is used for the calculation value of PAPR level reduction performance in the OFDM system. PAPR CCDF can be described to given clip level PAPR as in (3) and (4).
(3)CDFPAPR=PrPAPR>PAPR0,(4)CCDFPAPR=1−1−e−PAPR0N,whereN is subcarrier numbers and PAPR0 is a certain threshold value.


## 3. PTS Scheme in OFDM System

In the conventional PTS scheme for the OFDM system, an input symbols vectors in the OFDM representX=X0,⋯.XN−1T. The OFDM signal is generated by subcarriers N. The subcarrier generates a set of symbols. The subcarrier is separated from the adjacent subcarrier by the orthogonality of N when ∆f=1/T, where T is the signal duration of OFDM. The OFDM signal is expressed mathematically as follows:
(5)xn=1LN∑i=0LN−1X¯lej2πln/N,0≤n≤LN−1.X here is a subcarrier vector for N constellation symbols and ε constellation, according to quadrature phase shift keying modulation (QPSK). The ratio of the maximum instantaneous power to the average power of the OFDM signal is known as the PAPR, and this can be expressed as in
(6)PAPR=10log10max∣Xt∣20≤t<T∣Pav.A sampling of the continuous-time signalXt produces the separate transmitted signal Xn and often the Nyquist rate is taken; some information is lost at the height of Xt, and therefore, Xn is exceeded by a factor L to obtain the signal Xt low value for PAPR. Factor L helps to sequence LN-point data by inserting NL−1 zeros in the center of the N-point frequency domain signal and this is by LN-point IFFT and this gives IFFT, which can be expressed as
(7)xn=1LN∑i=0LN−1X¯lej2πln/N,0≤n≤LN−1,whereX¯=X0,⋯,XN/2−1,0,⋯,0⏟L−1N,XN/2,⋯,XN_1T. It shows that L=4 is enough to take the peak of the information Xt.The disjointedM submasses are the result of dividing the input data block into X, and this relationship can be represented by vectors according to Equation (8), and this is in the PTS system to reduce PAPR in OFDM: Xm,m=1,2..,M. Therefore, we can get
(8)X=∑m=1MXm.The sum of the subcarriers must be zero for the original signal which is displayed in another block. Then, the input symbol subblocksX are transformed into (M) time-domain partial transmit sequence. Therefore, X can be represented as
(9)Xm=IFFTLN×NXm.By incorporating phase factorsP=P1,P2,⋯Pm, this results in a group of candidates when PTS rotated independently. It can be represented as
(10)X=∑m=1MPmXm=∑m=1MIFFTPmʘXm.It is possible to observe the special chart no. (4) for C-PTS and to know that the choice of the candidate with the lowest PAPR is by searching the previous candidate for transmission, as to retrieve the original symbol X by sending the side information bits to indicate the improved phase sequence and then to the receiver, as seen in Figure 2.Figure 2
Block diagram of the conventional PTS scheme [23].


## 4. Number Theory

Number theory is a branch of pure mathematics mainly dedicated to studying the integral functions and integer. German mathematician Carl Friedrich Gauss (1777–1855) said: “ A Queen of Sciences is Arithmetic — and the Queen of Arithmetic is the theory of numbers” [24]. Polygon numbers are a type of number theory. The diagrams in Figure 3 show how polygon numbers are geometrically constructed. The illustrated polygonal numbers are, respectively, triangular, square, pentagonal, and hexagonal [25]. In the proposed algorithm, we took advantage of this theory in preventing data similarities and reducing the value of PAPR. According to this theory, the numbers of polygons are characterized by their preventing similarity, and this leads to sequence similarities being prevented. This feature makes us think about applying it to our work to reduce PAPR.Figure 3
Type of polygon number: triangular, square, pentagonal, and hexagonal [25].


## 5. Proposed PTS Scheme

In this method, the geometric shape theory of numbers was used, and three sets of numbers 3, 5, and 6, for triangular, pentagonal, and hexagonal numbers, respectively, were used. The following are the stages that have been implemented in applying this algorithm to the conventional scheme.


### 5.1. Polygon Number Theory for Numbers 3, 5, and 6

Three types of groups have been taken, and these groups of numbers are not equal ifN is equal, as they depend on the geometry of the number. Below are the equations that were used in this proposed method in the form of each of these numbers:
(11)tn=12nn+1,(12)pn=12n3n+1,(13)hn=2n−1,wheretn is a triangular number, pn refers to the pentagonal number, and hn is a hexagonal number.We took advantage of this feature in the input data when sending it; as this feature is highlighted on the input data, it prevents similar data. For example, when three consecutive values come one after the other equal in value, and when multiplying each of these equal values by a number from the group chosen (3, 5, and 6), the result of these numbers will be unequal according to the succession property, as seen in Figures4 and 5. This drawing, for the proposed method, shows the values that are not overlapping and preventing similar values from appearing. Moreover, Figure 6 depicts the amount of fluctuation that occurred due to similarities in numbers while they are progressing.Figure 4
Input sequent numbers with different values.Figure 5
Processing number with autocorrelation.Figure 6
Autocorrelation with different values.


### 5.2. Effect of Theory

After knowing the effect of the theory on similar numbers, we made an experiment to analyse the effect on the shape of these three groups when increasing the value. This increase can be seen in Figure7, whereas the amount of change that occurs when changing the value of each of these groups is shown in Figure 8.Figure 7
Demeanour of polygonal numbers for triangular, pentagonal, and hexagonal.Figure 8
Demeanour of polygonal numbers when changing the value of a pentagonal number.


### 5.3. Set of Periods

In this step, we made an experiment of a set of periods, which were divided into 10 values. After that, we chose a specific period from these periods, which is the selection of a portion of the abovementioned schemes, where we began to experiment with a period of (1.2-2.2) and another (2.2-3.2), etc. Meaning that the amount of the increase is 1, and we noticed that there was no big difference when changing these values between one group and another. This confirms the increase in fluctuation that was mentioned in the previous clause, which was processed in the similarity of data. Figures9–12 show these periods are taken.Figure 9
Data input with a range of magnitude 1.2-2.2.Figure 10
Data processing with a range of magnitude 1.2-2.2.Figure 11
Data input with a range of magnitude 2.2-3.2.Figure 12
Data processing with a range of magnitude 2.2-3.2.These figures further demonstrate to us how to process similar data at different periods. Furthermore, it is clear from the figures that despite the fluctuation, it happens in all periods, but it differs in values. Therefore, to avoid amplifying the signal, we will choose one group over an average period for the rest of the work, which is (3.2-2.2).


### 5.4. Mechanism for the Proposed Method

After studying the effect of periods on data fluctuation, and to prevent amplification, the pentagonal number group was reversed (see Figures9–12) to indicate the mechanism for the work of this algorithm by the proposed method. This can be observed in Figure 13, when the value of N is equal, the method is based on the premise that the three terms are not equal, and when the value of N is not equal, the algorithm is based on the progression property. According to Equations (11)–(13), the data do not overlap among them. We shed these data on the proposed algorithm, which can be observed in the sequence of this method as in Figure 14, where the data is entered on the algorithm, and then, it alternately changes similar values to different values. The data generated is considered encrypted, and the resulting data is ready for the transmission process according to the conventional method steps in PTS shown in Figure 15. Upon receipt, decryption is performed to obtain the original data.Figure 13
Principle of the polygonal numbers with data input and output to prevent similar data.Figure 14
Principle block diagram of the proposed method.Figure 15
Block diagram of the proposed method with C-PTS scheme.These figures show how to process similar data at different periods, and it is clear from the figures that despite the fluctuation, it happens in all periods, but it differs in values. Therefore, to avoid amplifying the signal, we choose one group over an average period for the rest of the work, which is (3.2-2.2).Similarly, Figure15 is showing the proposed method that was applied to the (C-PTS) method, as the old method characteristics (mentioned before) used modified PTS to improve the PAPR. The first block is the input data source to the OFDM system, which is synthesized by subcarriers represented by a sequence of input codes in X vector. The resultant is then converted into the M group by subblocks, Xm, 1≤m≤M.Then, the time field signal is converted to IFFT multiplied by the phase factors; after that, we get the OFDM and then pick the bestX signal with the lowest PAPR point.


## 5.1. Polygon Number Theory for Numbers 3, 5, and 6

Three types of groups have been taken, and these groups of numbers are not equal ifN is equal, as they depend on the geometry of the number. Below are the equations that were used in this proposed method in the form of each of these numbers:
(11)tn=12nn+1,(12)pn=12n3n+1,(13)hn=2n−1,wheretn is a triangular number, pn refers to the pentagonal number, and hn is a hexagonal number.We took advantage of this feature in the input data when sending it; as this feature is highlighted on the input data, it prevents similar data. For example, when three consecutive values come one after the other equal in value, and when multiplying each of these equal values by a number from the group chosen (3, 5, and 6), the result of these numbers will be unequal according to the succession property, as seen in Figures4 and 5. This drawing, for the proposed method, shows the values that are not overlapping and preventing similar values from appearing. Moreover, Figure 6 depicts the amount of fluctuation that occurred due to similarities in numbers while they are progressing.Figure 4
Input sequent numbers with different values.Figure 5
Processing number with autocorrelation.Figure 6
Autocorrelation with different values.


## 5.2. Effect of Theory

After knowing the effect of the theory on similar numbers, we made an experiment to analyse the effect on the shape of these three groups when increasing the value. This increase can be seen in Figure7, whereas the amount of change that occurs when changing the value of each of these groups is shown in Figure 8.Figure 7
Demeanour of polygonal numbers for triangular, pentagonal, and hexagonal.Figure 8
Demeanour of polygonal numbers when changing the value of a pentagonal number.


## 5.3. Set of Periods

In this step, we made an experiment of a set of periods, which were divided into 10 values. After that, we chose a specific period from these periods, which is the selection of a portion of the abovementioned schemes, where we began to experiment with a period of (1.2-2.2) and another (2.2-3.2), etc. Meaning that the amount of the increase is 1, and we noticed that there was no big difference when changing these values between one group and another. This confirms the increase in fluctuation that was mentioned in the previous clause, which was processed in the similarity of data. Figures9–12 show these periods are taken.Figure 9
Data input with a range of magnitude 1.2-2.2.Figure 10
Data processing with a range of magnitude 1.2-2.2.Figure 11
Data input with a range of magnitude 2.2-3.2.Figure 12
Data processing with a range of magnitude 2.2-3.2.These figures further demonstrate to us how to process similar data at different periods. Furthermore, it is clear from the figures that despite the fluctuation, it happens in all periods, but it differs in values. Therefore, to avoid amplifying the signal, we will choose one group over an average period for the rest of the work, which is (3.2-2.2).


## 5.4. Mechanism for the Proposed Method

After studying the effect of periods on data fluctuation, and to prevent amplification, the pentagonal number group was reversed (see Figures9–12) to indicate the mechanism for the work of this algorithm by the proposed method. This can be observed in Figure 13, when the value of N is equal, the method is based on the premise that the three terms are not equal, and when the value of N is not equal, the algorithm is based on the progression property. According to Equations (11)–(13), the data do not overlap among them. We shed these data on the proposed algorithm, which can be observed in the sequence of this method as in Figure 14, where the data is entered on the algorithm, and then, it alternately changes similar values to different values. The data generated is considered encrypted, and the resulting data is ready for the transmission process according to the conventional method steps in PTS shown in Figure 15. Upon receipt, decryption is performed to obtain the original data.Figure 13
Principle of the polygonal numbers with data input and output to prevent similar data.Figure 14
Principle block diagram of the proposed method.Figure 15
Block diagram of the proposed method with C-PTS scheme.These figures show how to process similar data at different periods, and it is clear from the figures that despite the fluctuation, it happens in all periods, but it differs in values. Therefore, to avoid amplifying the signal, we choose one group over an average period for the rest of the work, which is (3.2-2.2).Similarly, Figure15 is showing the proposed method that was applied to the (C-PTS) method, as the old method characteristics (mentioned before) used modified PTS to improve the PAPR. The first block is the input data source to the OFDM system, which is synthesized by subcarriers represented by a sequence of input codes in X vector. The resultant is then converted into the M group by subblocks, Xm, 1≤m≤M.Then, the time field signal is converted to IFFT multiplied by the phase factors; after that, we get the OFDM and then pick the bestX signal with the lowest PAPR point.


## 6. Simulation Results

In this research paper, we consider several simulation results to evaluate the performance in terms of PAPR reduction to demonstrate the effectiveness of the PN-PTS method. We used the MATLAB simulation program; the simulation results are based on the transmission of104 randomly generated OFDM symbols with N=64 carriers and an oversampling factor L=4. Table 1 lists the most important variables used in the simulation program to represents the PN-PTS method in which PTS was improved in reducing the PAPR.Table 1
The parameters for the simulation.
ParameterValueNumber of OFDM signal10000Number of subcarriersN64Modulation4, 32 QPSKIFFT & FFT size256Oversampling factor (L)4Number of partition p4The simulation results in Figure16 show that when compared the PN-PTS with C-PTS, for 4QPSK, N=32, no.ofOFDMsignal=1000 symbols, and the original signal was 15.1263 dB. It was processed in the C-PTS, and the amount of reduction was calculated 6.4068 dB, The proposed scheme’s (PN-PTS) PAPR performance is 4.6961 dB.Figure 16
PAPR performance of the PN-PTS method with C-PTS for 4 QPSK with 1000 symbols.In Figure17, the following are seen: no.ofOFDMsignal=10000 symbols and the original signal was 15.1287 dB. The PAPR reduction with C-PTS is 6.3157 dB, and the PAPR performance for PN-PTS is 4.6826 dB. This is only the first stage of minimization; the rest of the enhancements is shown in Table 2, which demonstrates the amount of progress made in reducing the PAPR.Figure 17
PAPR performance of the proposed scheme with C-PTS for 4 QPSK with 10000 symbols.Table 2
Computational complexity between the proposed and conventional schemes with a different number of subbands whenN=8, 32, and 64.
NNo. of OFDM signalOriginal signal (dB)C-PTSPN-PTS8100015.12636.40684.69611000015.12876.31574.682632100015.13426.44734.70591000015.12886.41694.712564100015.12686.43764.56831000015.13796.45034.7205In Figure18 for simulation, 4 QPSK, the original signal is 15.1268 dB, the no.ofOFDMsignal=10000 symbols, where the C-PTS gives 6.4376 dB, while PAPR performance for PN-PTS gives 4.5683 dB. In addition, when using 32 QPSK, the original signal is 17.9169 dB, where we see the most improvement between the two methods is clear where C-PTS gives worse PAPR performance 6.4363 dB, while the PN-PTS gives 5.5072 dB. Table 3 displays the results.Figure 18
PAPR performance of the proposed scheme with C-PTS, 32 QPSK.Table 3
Computational complexity between the proposed and conventional schemes with different modulations.
QPSK modulationOriginal signal (dB)C-PTSPN-PTS415.12686.43764.56833217.91696.43635.5072


### 6.1. Discussion

When comparing the benefits of the proposed method to the conventional method and the previous methods that were used, we notice that this method is distinguished by an important feature, which has resulted in a reduction in the level of PAPR, which is the prevention of similar data when transmitting. In the conventional method, the selection of the candidate with the lowest PAPR is by searching for the previous candidate for transmission, which is accompanied by bits of the side information of the signal when retrieving the signal sent in the reception. The advantage of PN-PTS being simple to encrypt and simple to decrypt when returning data for transmission. Furthermore, it is less complexity.Many methods have been used to reduce PAPR, including SLM, clipping, tone reservation, rotating phase shift, and hybrid methods, but the method we are used had never been used before. Simulation results have shown that the proposed scheme is reliable to estimate the selection of the PAPR.


## 6.1. Discussion

When comparing the benefits of the proposed method to the conventional method and the previous methods that were used, we notice that this method is distinguished by an important feature, which has resulted in a reduction in the level of PAPR, which is the prevention of similar data when transmitting. In the conventional method, the selection of the candidate with the lowest PAPR is by searching for the previous candidate for transmission, which is accompanied by bits of the side information of the signal when retrieving the signal sent in the reception. The advantage of PN-PTS being simple to encrypt and simple to decrypt when returning data for transmission. Furthermore, it is less complexity.Many methods have been used to reduce PAPR, including SLM, clipping, tone reservation, rotating phase shift, and hybrid methods, but the method we are used had never been used before. Simulation results have shown that the proposed scheme is reliable to estimate the selection of the PAPR.


## 7. Conclusion

In this paper, we proposed a PAPR reduction scheme for OFDM signals based on an improved PTS with polygon number algorithm (PN-PTS). To achieve the best PAPR performance possible, the proposed method is based on preventing similar data from being input during transmission. This has improved transmission by preventing the transmission of side information and lowering the increase in the PAPR. In addition, the PN-PTS algorithm has the advantage of being simple to encrypt and simple to decrypt when returning data for transmission. It is clear from the above results that using PN-PTS reduces PAPR. In addition, the amount of the reduction was good as the reduction between the proposed method and the conventional method was done by a good percentage from (6.4068 to 4.6961).In the future, it is planned to apply this algorithm to other methods such as the ‘rotating phase shift technique.’



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