Hybrid Peak-to-average-power-ratio Reduction Method Based on Filter Bank Multicarrier in Wireless Sensor Networks

Filter bank multicarrier (FBMC) technology is considered a suitable solution for replacing orthogonal frequency division multiplexing (OFDM) technology for the transmission of the fifth-generation wireless system (5G) multicarrier. In this study, we investigate the problem of a high peak-to-average-power ratio (PAPR) of FBMC in wireless sensor networks (WSNs). To solve the problem of imaginary interference in the FBMC system, we use an auxiliary pilot. Then, on the basis of the pilot-assisted FBMC system, a hybrid method of the Vandermondelike matrix (VLM) and clipping with filtering is applied to reduce the PAPR. The input data is precoded by the VLM of Chebyshev polynomials to reduce the autocorrelation of the input signal. Then, the signal is sent to the clipping and filtering module to suppress the PAPR of the FBMC signal further. The pilot-assisted FBMC was applied to the multipath channel model. The performance was evaluated using the PAPR and bit error rate (BER) curves. The simulation results indicate that the hybrid method can improve the PAPR suppression performance of the FBMC system compared with the precoding method or the clipping and filtering method alone. From the results, we can also see that the hybrid method improves the BER performance.


Introduction
Wireless sensor networks (WSNs) are one of the supporting technologies of the Internet of things industry, which is of great significance for promoting economic development and scientific and technological progress. Because of its rapid deployment, convenient networking, and unrestricted nodes, it is widely used in battlefield detection, disaster rescue, and environmental perception. Therefore, WSNs have become a hot topic of academic and industrial research. WSNs transmit information to users through sensor nodes; this requires the use of sensing, collection, embedded distribution, and communication technologies to complete information-related collection, processing, and transmission tasks. However, owing to the dense distribution and large quantity of WSNs, it is easy to be affected by problems such as uncertain topological structure, insufficient energy consumption guarantee, and limited computing and storage capacity, among which, the cost and energy consumption of nodes are the key factors restricting the extensive application and promotion of nodes. (1) Currently, there are two IEEE standards for information transmission in WSNs. IEEE802 is designed for low-speed wireless personal LAN, and the IEEE802 standard, which provides the ultrawide-band (UWB) wireless personal area network (WPAN) physical layer in the 7 GHz band between 3 and 10 GHz to WSNs, allows multiple signals to jointly occupy the entire bandwidth. Because of the existence of different subcarriers in the multiband, the subcarriers can use different data modulation modes, and filter bank multicarrier (FBMC) technology is suitable for the over-bandwidth wireless communication system. (2) In future wired and wireless communication standards, (3,4) FBMC will be a promising technology that avoids the use of redundant samples as the cyclic prefix (CP). By using the FBMC system, spectral efficiency and robustness to synchronization requirements can be improved. Moreover, in a noise environment, such as in a powerline channel, FBMC has a stronger robustness. (5) One problem with FBMC is imaginary interference. To deal with the problem, Javaudin et al. (6) used one symbol per pilot, the so-called auxiliary pilot symbol, to cancel the imaginary interference. (7) The main disadvantage of this method is the power offset. To solve this problem, Nissel and Rupp (8) used two auxiliary symbols per pilot to reduce the power offset from 4.3 to 0.8 and to increase the achievable capacity for low signal-to-noise ratio (SNR) values by approximately 5%. Therefore, we study the peak-to-average-power ratio (PAPR) in the FBMC system with an auxiliary pilot, which we refer to as the pilot-assisted FBMC system.
One of the biggest problems, however, is the PAPR of the transmitted FBMC signal, as this large peak can cause a significant performance degradation when the signal passes through a nonlinear high-power amplifier (HPA). The nonlinearity of HAP leads to in-band distortion, resulting in increased bit error rate (BER) and adjacent channel interference owing to out-ofband (OOB) radiation. (9) For this reason, several methods, such as selective level mapping (SLM), (10) partial transmit sequence, (11) square-rooting companding, absolute exponential companding, tone reservation (TR), clipping, and active constellation extension, (9,12,13) have been proposed to reduce the PAPR of the FBMC signal.
Because of the overlapping signal structure of the FBMC system, the direct application of the above scheme originally adjusted for orthogonal frequency division multiplexing (OFDM) is infeasible. To adapt it to the signal structure of FBMC, some signal processing operations must be introduced into these PAPR reduction schemes. There are some works on the PAPR reduction of FBMC systems in the literature. In Ref. 14, Hrasnica et al. described the use of the sliding window algorithm to improve the TR scheme and used it for the PAPR reduction of FBMC signals. This advanced scheme, known as sliding window tone retention (SWTR), eliminates the peak value of the FBMC signal in the window by using a series of continuous blocks of peak scaling tones. SW-TR can properly reduce the PAPR of FBMC signals. Zimmermann and Dostert (15) and Krongold and Jones (16) proposed overlapping SLM to reduce the PAPR of OFDM signals. Dongjun and Choi proposed a low-PAPR scheme and resolved a pending issue of FBMC, i.e., OOB spectrum regrowth due to HPA nonlinearity. (17) In this paper, the Vandermonde-like matrix (VLM) and clipping with filtering-based PAPRreduction schemes aided to reduce the PAPR of FBMC signals are presented. The pilot-assisted FBMC was transmitted to the PLC channel model and the performance was evaluated using PAPR and BER curves.
The paper is organized as follows. In Sect. 2, the main features of FBMC and PLC channel models are introduced, and the form of FBMC receiving signal transmitted through a multipath channel is derived. In Sect. 3, the PAPR reduction methods based on the pilot-assisted FBMC, including clipping, filtering, VLM, and their combinations are discussed. In Sect. 4, simulation and experimental results are given. Finally, we summarize this paper in Sect. 5.

FBMC system
The implementation structure of FBMC is shown in Fig. 1. In the literature, this structure has often been referred to as OFDM-OQAM, where OQAM stands for offset quadrature amplitude modulation (QAM).
FBMC filters each subcarrier modulated signal in a multicarrier system. The prototype filter is the one used for the zero-frequency carrier and is the basis of the other subcarrier filters. The filters are characterized by the overlapping factor K, which is the number of multicarrier symbols that overlap in the time domain. The prototype filter order can be chosen as 2 × K − 1 where K = 2, 3, or 4 and is selected as per PHYDYAS project.
The current FBMC implementation uses frequency spreading and an N × K-long inverse fast Fourier transform (IFFT) with symbols overlapped with a delay of N/2, where N is the number of subcarriers. This design choice makes it easy to analyze FBMC and compare the results with those of other modulation methods.
To achieve a full capacity, OQAM processing is employed. The real and imaginary parts of a complex data symbol are not transmitted simultaneously, as the imaginary part is delayed by half the symbol duration.
The modulation is achieved digitally using an IFFT. P/S and S/P represent parallel-to-serial and serial-to-parallel conversions, respectively.
In the FBMC transmission system, a series of data signals are transformed firstly and concatenated through source and channel codes, and then mapped into a certain space to improve the transmission efficiency of information bits. Complex number sequences are divided into real and imaginary parts. In the FBMC system, OQAM is used to preprocess symbols of the real and imaginary parts. The preprocessed symbols are then modulated by prototype filters, applying filter bank multicarrier technology, and superimposed to the channel in the time domain. With the help of the IFFT algorithm, the modulation of multicarrier technology can be realized rapidly. At the receiving end, a set of symmetric prototype filters are also used, which have exactly the same performance as the original filter banks at the sending end.
The processing includes matched filtering followed by OQAM separation to form the received data symbols. These data symbols are demapped into bits and the resultant BER is determined. In the presence of a channel, linear multitap equalizers may be used to mitigate the effects of frequency-selective fading. Demodulation is performed by a fast Fourier transform (FFT).
Mathematically, the transmitter signal x(t) of the multicarrier system in the time domain can be expressed as where g m,n (t) is the time and frequency-shifted version of the prototype filter g(t) with τ 0 denoting the time spacing and 1/F 0 the frequency spacing.
is the adjustment of the phase, which ensures real orthogonality between the systems.

Multipath channel model
In this paper, we use the practical multipath channel model introduced by Zimmermann and Dostert, (15) which is suitable for describing the transmission behavior of power line channels. On the basis of the actual measurement of the actual powerline network, the model is given by the channel transfer function (12) where g i is the weighting factor representing the product of the reflection and transmission factors for each path i. The attenuation parameters a 0 (offset of attenuation), a 1 (increase in attenuation), and β (exponent of attenuation) are derived from the measured frequency response. (12) d i represents the length of the cable. v p is the phase velocity.

FBMC in multipath channel model
After transmission over the channel, the received symbols r(t) of the FBMC can be expressed as where h(t, τ) represents the impulse response of the multipath channel model. Δ represents the maximum delay extension of the channel. Substituting Eq. (1) into Eq. (4) gives Assuming that the delay spread Δ of the multipath channel is much smaller than one symbol period, it can be approximated in the time span of 0 ≤ τ ≤ Δ: In this case, Eq. (6) can be simplified as After the received signal passes through the receive filter, the demodulated signal at point (n 0 , m 0 ) can be expressed as

Definition of PAPR in FBMC system
In general, the ratio between the maximum peak power and the average power over a period of time is called the PAPR. For a continuous time baseband signal x(t), PAPR within a transmission period t is defined as Most PAPR reduction schemes are performed on frequency domain signals, so we need to sample the FBMC signal in the continuous time domain. The ordinary Nyquist rate sampling will lose some peak values, so we need to use oversampling technology in the actual operation. The PAPR expression of the baseband modulation discrete-time signal x(n) is The complementary cumulative distribution function (CCDF) is a common method of measuring and calculating the multicarrier PAPR. CCDF is defined as the probability that the PAPR of the mth modulation symbol exceeds a given threshold, and its expression is In the following sections, we use the CCDF function to measure the PAPR distribution of the FBMC system.

Clipping and filtering
where A is the prespecified clipping level. Let us define the clipping ratio (CR) as the clipping level normalized by the RMS value of the FBMC signal such that It is known that = N σ and = 2 N σ in the baseband and passband FBMC signals with N subcarriers, respectively.
Using clipping to suppress PAPR will cause in-band noise and increased OOB radiation owing to distortion caused by the direct limitation of the signal amplitude, thus reducing the BER performance of the system. Because clipping can cause OOB spectrum dispersion, clipping and filtering can eliminate OOB interference by filtering the limited data and hence improve the system BER performance of the system.

Hybrid PAPR suppression method
The input data is precoded by the VLM of the Chebyshev polynomial to reduce the autocorrelation of the input signal. Then, the signal is sent to the clipping and filtering module to suppress the PAPR of the signal further.
Suppose a 1 , a 2 , …, a n are n real or complex numbers, and T k (x) (k = 0, 1, …, n − 1) is a polynomial of degree k and satisfies the following recursive relation: The n-order square matrix B composed of T k (x) is called the VLM.  T  T  T  T  T  T   T  T When T k (x) = x k (k = 0, 1, ..., n − 1), λ k = 1, α k = 0, and β k = 0, B is the Vandermonde matrix. In the VLM, the polynomial T k (x) is called the Chebyshev polynomial, that is, T k (x) = cos(karccos(x)) satisfies Assuming that the parallel input signal after encoding and modulation is X and x is obtained after the IFFT, the instantaneous power is (20) λ(k) is the aperiodic autocorrelation function of input signal X.
The average power of the signal is fixed. Let the average power P av = E{|x n | 2 }; then, the PAPR is Chebyshev polynomials have orthogonality, (18) which can reduce the autocorrelation of input data by linear precoding input signal sequences through the VLM composed by them. According to Eq. (21), PAPR is related to the input signal's aperiodic autocorrelation function. The smaller the sum of the modules of the signal's aperiodic autocorrelation function is, the smaller PAPR will be after the signal is transformed into the time domain. Therefore, the input signal can be precoded by the Chebyshev-Vandermonde matrix to reduce the autocorrelation of the input signal and thus reduce the PAPR of the FBMC signal.
The specific process of the hybrid method is as follows. At the transmitting end of the system, after encoding, modulating, and concatenating the input serial data, linear precoding is carried out using the VLM of the Chebyshev polynomial, and the time domain data is obtained by oversampling and IFFT. After limiting the time domain data in accordance with the set limiting ratio, the frequency domain data is obtained through the FFT. The IFFT is carried out again to obtain the time domain data. Finally, the data is sent after parallel string conversion, the insertion of the CP, digital-to-analog conversion, and other processes. The receivers in the system undergo a corresponding process. The received signal removes the CP after the analogto-digital conversion. After series-to-parallel conversion, the frequency domain data is obtained by FFT. Then, the frequency domain data is decoded by the inverse matrix of the Chebyshev-VLM. Finally, after decoding and demodulation, the original information is obtained.

Simulation Results and Discussion
In the simulation, we use OQAM, where the size of the FFT/IFFT block is 128. The BER is tested by simulation experiments in a multipath channel. x m . Figure 3 shows that the amplitude of the clipped signal is distributed below the clipping level. Finally, it can be seen from Fig. 3 that the filtered signal shows a peak value beyond the clipping level. It can also be seen that the OOB spectrum increases after clipping, but decreases again after filtering. Figure 4 shows the CCDFs of the crest factor for the clipped and filtered FBMC signals. The following are the parameters used for the simulation of clipping and filtering: 4-QAM signal constellation, N = 256 subcarriers, f = 1 MHz sampling frequency, and L = 8 oversampling factor. Recall that the CCDF of the crest factor can be considered to approximate the PAPR distribution since the crest factor is the square root of the PAPR. It can be seen from Fig. 4 that the PAPR of the FBMC signal decreases significantly after clipping and increases slightly after filtering. Note that the smaller the CR is, the greater the PAPR reduction effect is. Figure  5 shows the BER performance when the clipping and filtering technique is used. In Fig. 5, "C" and "C&F" denote the cases with clipping only and with both clipping and filtering, respectively.
Clipping also causes OOB radiation, which imposes OOB interference signals to adjacent channels. Although the OOB signals caused by clipping can be reduced by filtering, it may affect high-frequency components of the in-band signal (aliasing) when clipping is performed with the Nyquist sampling rate in the discrete-time domain. However, if clipping is performed for the sufficiently oversampled FBMC signals in the discrete-time domain before a lowpass filter (LPF) and the signal passes through a band-pass filter (BPF), the BER will be less degraded.  Figure 6 shows the PAPR performance when the VLM and clipping with filtering are used. It can be seen from this figure that when CCDF = 10 −2 , the Z 0 values of clipping and VLM, clipping and filtering and VLM, VLM, and clipping and filtering are 4.4, 4.8, 8.6, and 8.8 dB, respectively. Compared with the VLM precoding method and clipping and filtering methods, the hybrid method can improve the PAPR suppression effect of the FBMC system. Figure 7 shows the BER performance when the VLM and clipping with filtering are used. It can be seen from this figure that the BER is highest when clipping is used only, and the BER decreases when clipping and filtering are used. Clipping causes in-band signal distortion, resulting in the BER performance degradation. Clipping also causes OOB radiation, which imposes OOB interference signals to adjacent channels. The OOB signals caused by clipping can be reduced by filtering. It can also be seen from this figure that the BER performance increases when the hybrid method is used. The VLM coding technique is used to select such codewords that minimize or reduce the PAPR. It causes no distortion and creates no OOB radiation.     Figure 8 shows the CCDF of the PAPR for a 4-QAM signal constellation and N = 256 subcarriers. The PAPR's CCDF curves of the pure FBMC and the DFT-spread schemes are also included for comparison. FBMC has the same PAPR as the simple DFT-spread FBMC. ITSM-DFT-spread FBMC (17) has the same PAPR as the hybrid method, but with an additional advantage of much lower OOB emission. Moreover, ITSM-DFT-spread FBMC is approximately 1.5 dB better than FBMC and the simple DFT-spread FBMC. This is because in addition to DFT spreading, ITSM-DFT-spread FBMC has another factor for further PAPR reduction, i.e., selection among the four waveform variations with different peak powers. Figure 9 shows the BER for a 16-QAM signal constellation, N = 256 subcarriers, F = 15 kHz subcarrier spacing, and f c = 2.5 GHz carrier frequency. The simple DFT-spread FBMC has the same BER as the hybrid method. Only for high signal noise ratios, small deviations can be observed because the channel-induced interference is not Gaussian-distributed.

Conclusions
A hybrid method of VLM and clipping with filtering was applied to reduce the PAPR of pilot-assisted FBMC in WSNs. The correlation was reduced by the linear precoding of the encoded modulated data, and the PAPR of the FBMC signal was further reduced by clipping and filtering. From the simulation results, we observed that the hybrid method has a lower PAPR than the unclipped system and a lower BER than the case with clipping only. CR also affects the PAPR performance of the FBMC system: the smaller the CR is, the greater the PAPR reduction effect is.