Research on Grid Scroll Chaotic Extended Sequence Algorithm and Its Circuit Implementation for F-OFDM System

High peak-to-average power ratio (PAPR), a main problem of 5G Filtered-Orthogonal Frequency Division Multiplexing (FOFDM) system, seriously aects the system’s transmission eciency. Aiming at the shortcomings of the traditional selective mapping (SLM) algorithm, the grid scroll chaotic extended sequence (GSCES) algorithm is proposed to solve the high PAPR problem.e proposed grid multiscroll chaotic maps are used as random phase sequence vectors instead of the traditional pseudo random sequence, which provide a large number of random sequences with good correlation characteristics, random characteristics, and high security. e proposed random phase sequence vectors can meet the needs of 5G mass connections. e original input signals are divided into real part and imaginary part, which are, respectively, dot produced with the proposed grid multiscroll chaotic maps. e numerical simulation results show that the proposed GSCES algorithm eectively increases the number of candidate sequences, reduces the PAPR values, and improves the transmission eciency and security. Moreover, the hardware experimental results produce at 2× 2× 2-grid multiscroll chaotic attractor, which further proves the physical realization of the proposed system. e proposed schemes have a broad application prospect in 5G multicarrier modulation technology.


Introduction
In 2015, Filtered-OFDM (F-OFDM) technology was rst put forward publicly at the World Mobile Communication Congress.It is an adaptive airport waveform modulation technology whose subcarrier bandwidth can be adjusted according to speci c application scenarios.F-OFDM technology has many advantages, such as no need for strict synchronization, reduced out-of-band energy leakage, the di erent waveform modulation, multiple access technology and frame structure based on the mobile communication application scene and business requirements, the dynamic soft interface parameter con guration according to the 5G business needs, and packet tra c transmission suitable for Internet of things [1][2][3][4][5].In the F-OFDM system, when subcarriers of sub-band are superimposed after IFFT operation, multiple subcarriers will reach the peak value at the same time, resulting in higher peak-to-average power ratio (PAPR) of synthetic signal, which seriously a ects the system performance and limits its application in 5G. e characteristics of selective mapping (SLM) scrambling technology are simple structure and obvious e ect, but there are a large side band information and high computational complexity.Many researchers have improved the traditional SLM algorithm.e cyclic-SLM scheme is proposed to omit the side information (SI) and apply only one IFFT.In the time domain, the original signal and its periodic shift signal are added and the minimum PAPR is selected as the candidate signal [6].
e time-frequency domain encryption technique is proposed based on the multichaos and SLM PAPR suppression algorithm.e Lozi and Logistic maps are used to interfere the carrier in the time-frequency domain.In a 100 kilometer single-mode ber, 8.9 Gb/s encrypted OFDM signal is transmitted safely [7].A method is proposed to decrease the PAPR of the 16-QAM OFDM system, which combines the chaotic map and SLM algorithm so that the phase rotation factors are controlled by chaotic sequence [8].e improved SLM algorithm based on the Lehmer random phase sequence vector is proposed to embed side band information into the 16-PSK transmit signal [9].
e improved SLM algorithm is proposed to reduce PAPR, combining interlaced IDCT transform with pulse forming technology [10].
e low complexity improved SLM algorithm and μ-law compression scheme are proposed to reduce the PAPR and computational complexity of the OFDMA system [11].e low-complexity SLM improvement algorithm without SI is proposed to reduce the PAPR of the coherent light OFDM system [12].e blind differential improved SLM algorithm is proposed to reduce the PAPR of Alamouti DSFBC-OFDM system [13].
In this paper, a model of the new system is constructed to produce gird multiscroll chaotic attractors, using different sign functions acted as nonlinear function.Moreover, the design of the hardware circuit produces at 2 × 2 × 2-grid multiscroll hardware experimental results.
e produced grid multiscroll chaotic maps are used as random phase sequence vectors instead of the traditional pseudo random sequence.
e grid scroll chaotic extended sequence (GSCES) algorithm is proposed to solve the high PAPR problem.
e numerical simulation results show that the proposed GSCES algorithm effectively increases the numbers of candidate sequences, reduces the PAPR values, and improves the transmission efficiency and security.

F-OFDM System
Modes.F-OFDM system dynamically configures suitable waveform parameters such as bandwidth, cycle prefix (CP) length, subcarrier spacing, and FFT points for each sub-band according to service requirements, which improves the flexibility and scalability of 5G communication system.e downlink of the F-OFDM system is shown in Figure 1.
As shown in Figure 1, the frequency bands of the system are divided into M sub-bands, which are subcarrier mapping, IFFT transformation, and plus CP operation, respectively, and each sub-band is filtered through the subband filter for coupling transmission.e biggest difference between the F-OFDM system and traditional OFDM system is the sub-band filter added at the transmitter and receiver.For each sub-band, at the transmitter, the input signal of the sub-band filter is the traditional OFDM signal.e function of the sub-band filter is to complete the baseband shaping of each sub-band, and its output signal is the F-OFDM signal.At the receiving end, the function of sub-band filter is to filter each sub-band signal and complete sub-band decoupling.
e data symbol of the ith sub-band of the F-OFDM system is given by where g i (n) is the ith sub-band symbol before filtering, f i (n) is the impulse response of the ith sub-band filter at the beginning, and M is the number of sub-band.e transmit signal of the F-OFDM system is as follows: e channel output signal is given by (3)

Construction of Grid Multiscroll Chaotic System.
In this paper, the new grid multiscroll chaotic system is constructed with the combination of sign functions, which act as nonlinear functions.So, the dimensionless equation of the state is constructed by the following equation: , and e � 3.9, the grid multiscroll chaotic system is in a state of chaos.

Grid Scroll Chaotic Extended Sequence Algorithm.
e traditional SLM algorithm is a kind of PAPR suppression algorithm without signal distortion.However, the receiver needs to know the selected random phase sequence to demodulate, so it also needs to transmit the corresponding random phase sequence information in addition to the data information.It is restricted in practical application.In this paper, the pseudo random sequences are generated by the proposed grid multiscroll chaotic maps.
e pseudo random sequences are generated by the initial value iterations.It only needs to transmit its initial value, which reduces the complexity of the operation and the side band information transmission.Aiming at the shortcomings of the traditional SLM algorithm, the grid scroll chaotic extended sequence (GSCES) algorithm is proposed in this paper.e original signals are divided into real part and imaginary part, the proposed grid multiscroll chaotic maps are used as random phase sequence vectors, the IFFT transforms are performed, respectively, the PAPR is calculated by linear superposition, and the minimum PAPR is selected for transmission as shown in Figure 2.
As shown in Figure 2, the 5G F-OFDM data block is x(t), the serial input signal x(t) is symbol mapped, which is divided into 2M subchannels with serial-to-parallel conversion, the signal x(t) is divided into the real part and the imaginary part, and the transmission is carried out on the M 2 Complexity subchannel, respectively.
e proposed grid multiscroll chaotic maps are used to generate random phase sequence vector P with different N lengths, and then the mth random phase sequence vector is P m � (P m,0 , P m,1 , . . ., P m,N− 1 ), 0 ≤ m ≤ M − 1. e lth data block x l (t) is divided into the real part x l_r (t) and the imaginary part x l_i (t), which are, respectively, dot produced with the mth random phase sequence vector P m , then IFFT transforms are performed, and the candidate transmission sequences s m l r (t) and s m l i (t) are produced: where ⊗ is dot product operation between two vectors, IFFT is discrete Fourier inverse transformation, s m l r (t) is the real part of the candidate sequence, and s m l i (t) is the imaginary part of the candidate sequence.e candidate sequence s k,q l (t) are obtained by linear combination of the real part s k l r (t) and imaginary part s q l i (t): If α k P k ± jβ g P g is the unit phase sequence vector, that is , and the positive and negative polarity have the same PAPR, then According to equation ( 7), the number of expanded candidate transmission sequences s el (t) increases from original M to M 2 , and it is expressed as

Real part
Parallel-toserial conversion Complexity 3 In the candidate transmission sequence of lth data block, the minimum PAPR is selected as D l , namely, So, the output sequence is as follows:

Experiment
According to the modular circuit design method, equation (4) has been carried out by the differential-integral transforms; we have the following equation: According to equation (11), the design of the circuit is shown in Figure 3. e model of the operational amplifier is TL082.
e power supply voltage of all active devices is ±15 V.All resistors are precision adjustable resistors or precision adjustable potentiometers.

Results and Discussion
4.1.Dynamics of Lyapunov Exponents.For the system corresponding equation ( 4), the Runge-Kutta integration method is used with the length of time t � (0, 200), initial values (x, y, z) � (0.02, 0.01, 0.03), and the numerical simulation result of the Lyapunov exponents is shown in Figure 4.
e Lyapunov exponents of the proposed system are given as LE 1 � 0.055545, LE 2 � 0.025606, and LE 3 � − 0.881151, also LE 1 + LE 2 + LE 3 � − 0.8.Moreover, the fractal dimension is given by e proposed system has positive Lyapunov exponent and the dimension of Lyapunov is fractional, which further confirms that the system is in the state of chaos and the calculated Lyapunov exponent is reasonable.Moreover, this system produces strange attractor dimension LD � 2.0921, in line with the geometric characteristics of the chaotic attractor.

Poincare Map of Grid Multiscroll Chaotic System.
For the system corresponding equation ( 4), the Runge-Kutta integration method is used with the length of time t � (0, 2600) and initial values (x, y, z) � (0.02, 0.01, 0.03).e Poincare map of the grid multiscroll chaotic system is shown in Figure 5.
A Poincare section is selected in the phase space, which is used to cut off all the motion trajectories for observing section cut point.From Figure 5, we can see that the Poincare map of the grid multiscroll chaotic system is a piece of dense point set and has a hierarchical structure, which verifies the system's chaotic motion state.

Hardware Results.
e hardware circuits shown in Figure 3 can be obtained, and experimental results are shown in Figure 6.
From Figure 6, it can be seen that the hardware circuit experiment results of the analog oscilloscope generate 2 × 2 × 2-grid multiscroll chaotic attractors, which verifies the physical feasibility of the proposed system.e phase rails of the proposed grid multiscroll chaotic attractor are separated in more directions and have more complex dynamic characteristics.erefore, the grid multiscroll chaotic map is used as random phase sequence vectors.Compared with the traditional pseudo random sequences, the proposed grid multiscroll chaotic sequence can provide a large number of random sequences with good correlation characteristics, random characteristics, and high security, which meet the requirement of 5G mass connection.

PAPR Performance Simulation Results
. In this paper, the QPSK modulation is adopted, the subcarrier number equals to 128, and the complementary cumulative distribution function (CCDF) is used to describe the PAPR distribution of the proposed system.In paper [6], the selected data utilization (SDU) algorithm is proposed to rotate the phase factor with specific selection conditions, which improves the PAPR performance of the traditional SLM algorithm.e PAPR performance comparison between the proposed GSCES algorithm and the traditional SLM algorithm and the SDU algorithm is shown in Figure 7.
In Figure 7, the variable original is the PAPR value calculated by 1 × 128 original signal.
In the traditional SLM algorithm, the 1 × 128 original signal is input data, L groups input data are copied and are dot product of random phase vectors, IFFT transforms are calculated, and the PAPR values are obtained, and the variable SLM is the minimum PAPR value of the L candidate sequences.
e traditional SLM algorithm using random phase vectors requires the transmission of phase information of subchannel, so the side band information is very huge, which not only increases the complexity but also reduces the efficiency of the system.
In the proposed GSCES algorithm, the 1 × 128 original signal is the input data, L groups input data are copied, the real part and the imaginary part of input data are separated, L random phase vectors are generated by the proposed grid multiscroll chaotic maps, the real part and the imaginary part are separately dot product with the random phase vectors, IFFT transforms are calculated, and real and imaginary data are linearly added.
e proposed GSCES algorithm extends the candidate sequence to L 2 groups, and 4 Complexity variable LSLM is the minimum PAPR value of L 2 candidate sequences.
In Figure 7, compared with the traditional SLM algorithm and SDU algorithm, the proposed GSCES algorithm e ectively reduces the PAPR of the 5G F-OFDM system, increases the candidate sequences, reduces calculation, and transmits data more e ciently.

BER Performance Simulation Results.
e F-OFDM signal divides the frequency band of the system into several di erent sub-bands, which can be con gured with di erent link parameters according to the actual service requirements, such as di erent subcarrier spacing, IFFT/FFT size, and CP length.In this section, we evaluate the bit error rate (BER) performance of the proposed GSCES algorithm in the F-OFDM system.In the additive white Gaussian noise (AWGN) channel, the system performance test is carried out, and the speci c simulation parameter con guration of the F-OFDM system is shown in Table 1.
e con guration of link parameters in this paper is shown in Table 1.In the F-OFDM system, the BER performance of the proposed GSCES algorithm is shown in Figure 8.
From Figure 8, it can be seen that the BER performance of the proposed GSCES algorithm has a certain degree of degradation compared with the original signal, but the degree of degradation is not large.Combining with Figure 7,

Complexity
it can be seen that the proposed GSCES algorithm can effectively improve the PAPR performance of the system, and when the transmission signal meets certain signal-to-noise ratio (SNR) conditions, it will not affect the BER performance of the system.4.6.Computational Complexity Analysis.Assuming that both the SLM algorithm and the proposed GSCES algorithm generate L 2 candidate sequences, the number of subcarriers is R, the oversampling rate is Q, and the IFFT transformation of RQ points requires (RQ/2)log 2 RQ complex multiplications and RQ log 2 RQ complex additions.e SLM algorithm needs L 2 IFFT transforms to generate L 2 phase sequences, i.e., a total of (L 2 RQ/2)log 2 RQ complex multiplications and L 2 RQ log 2 RQ complex additions.For the proposed GSCES algorithm, the Lth IFFT transform generates the real and imaginary parts of L candidate sequences.
at is to say, (LRQ/2)log 2 RQ complex multiplications and LRQ log 2 RQ complex additions are needed.en, the linear combination of the real and imaginary parts of L candidate sequences is used to generate L 2 candidate sequences, which requires L 2 RQ complex additions.
When the number of the candidate sequence equals to L 2 , the computational complexity of the proposed GSCES algorithm is compared with that of the SLM algorithm as shown in Table 2.
Table 2 shows that when the numbers of candidate sequences are all L 2 , the number of complex multiplication needed by the proposed GSCES algorithm is 1/L of that of the SLM algorithm and the number of complex addition is ((1/L) + (1/log 2 RQ)) of the SLM algorithm.With the increase of the number of candidate sequences, the ratio decreases continuously.So, the proposed GSCES algorithm expands the number of candidate sequences by using the segmentation method and linear combination transformation.Compared with the SLM algorithm, the computational complexity is greatly reduced and the data can be transmitted more efficiently.

Conclusions
Aiming at the shortcomings of the traditional SLM algorithm, the GSCES algorithm is proposed to solve the high PAPR problem of 5G F-OFDM systems.e proposed grid multiscroll chaotic maps instead of the traditional pseudo random sequence are used as random phase sequence vectors, which provide a large number of random sequences with good correlation characteristics, random characteristics, and high security.e proposed random phase sequence vectors can meet the needs of 5G mass connections.
e original input signals are divided into real part and imaginary part, which are, respectively, multiplied by the proposed grid multiscroll chaotic sequences.
e IFFT transforms are performed, respectively, the PAPR values are calculated by linear superposition, and the minimum PAPR value is selected for transmission.e numerical simulation results show that the proposed GSCES algorithm effectively increases the numbers of candidate sequences, reduces the PAPR values, the redundant information transmission, and the side band information transmission, and improves the transmission efficiency and security of the 5G F-OFDM system.Moreover, the hardware experimental results produce at 2 × 2 × 2-grid multiscroll chaotic attractor, which further proves the physical realization of the proposed system.
e proposed schemes have a broad application prospect in 5G multicarrier modulation technology.

Figure 2 :Figure 1 :
Figure 2: Schematic diagram of the grid scroll chaotic extended sequence algorithm.

Figure 5 :
Figure 5: Poincare map of the grid multiscroll chaotic system.

Table 2 :
Computational complexity comparison of different algorithms.