Abstract
Orthogonal frequency division multiplexing (OFDM) is the highly spectrally well-organized method that has the difficulty of excessive peak power to average power ratio (PAPR), which ultimately imposes constraints on the high-power amplifier. Many practices have been projected to lessen PAPR of the OFDM systems. Amongst all the practices, the selected mapping (SLM) method has drawn more attention because of distortion-less behaviour. This technique uses unique phase sequences. It has been learnt that phase formation for SLM is very tedious. In the proposed work, the SLM method has been used, but phase arrangement formation is based on the usage of discrete cosine transform (DCT) matrix. In this proposed work, discrete cosine transform matrix has been chosen based on the requirement of optimization so that the arrangement with lowest PAPR can be nominated for the transmission. MATLAB simulation depicts that the remarkable gain is achieved as compared with the existing technique. In the proposed work, scheming of phase sequences are very informal due to the use of a DCT matrix which has a definite structure and can be generated at the receiver side with the help of side information of the phases and communicated from the transmitter to the receiver.
1. Introduction
OFDM is favourable for speedy communication of the information. It has many advantages like high spectral efficiency because of orthogonality among subcarriers, strong impact on intersymbol interference (ISI), and easier channel estimation. Besides all these, it has the disadvantage of high PAPR. Many PAPR depletion techniques like partial transmitted sequence (PTS), coding, clipping, selected mapping (SLM), tone injection, and tone reservation [1, 2, 3] have been proposed. Among all these techniques, SLM has been found to be very efficient due to its distortion-less behaviour. It chooses the phase sequence which leads to the generation of minimum PAPR. Selected mapping proposed in [4] uses Riemann matrix where Riemann matrix is nominated as the phase sequences and thus remarkable gain has been achieved. Selected mapping proposed in [5] uses only randomly generated phase sequences. But designing of random phase sequences up to length equal to data length is very difficult. In [6], the authors proposed a SLM system that leads to the reduction of PAPR and improvement in bit error rate (BER), and the SLM system also does not necessitate side information transmission. Chaotic sequence projected for the lessening of PAPR outperforms conventional methods like Shapiro–Rudin and Walsh–Hadamard sequences [7]. The pseudo random interferometry codes have been preferred as the phase sequences for the SLM to lessen PAPR and also to enhance BER performance that yields better results as compared to Walsh–Hadamard arrangement and Golay arrangement [8]. Monomial phase sequence for SLM for the lessening of the PAPR has been proposed in [9], and the cubic phase sequence is found suitable for the PAPR reduction. A new method for controlling the PAPR has been demonstrated in [2] in which the authors have used standard array of linear codes. This organization can be understood as a revised SLM algorithm which is ultimately a probabilistic system for diminishing the PAPR only after opting the signal with less PAPR from many candidates for transmission. Since the coset leader of the linear block codes is preferred in this method, no information is needed as syndrome decoding can be used for the received signals recovery. In [3], the hybridization of higher-order segregated PTS with Bose–Chaudhuri–Hocquenghem codes (BCH) has been suggested to lessen PAPR. This method diminishes the PAPR by choosing the signal with small PAPR as compared with many signals. For the recovery of transmitted signal, syndrome decoding is preferred. The simulation results of suggested work are superior as related with OFDM system and existing PTS method. In Reference [10], the authors suggested an ICF algorithm for minimization of PAPR of OFDM. ICF is implemented with window which is rectangular in domain of frequency and needs iterative condition for the particular threshold in CCDF. In this study, the authors have developed an ICF-optimized technique which finds out filter frequency response for every ICF repetition using convex optimization. For the reduction of signal distortion, the designing of optimal filter is required. After 1 or 2 repetitions, the suggested technique obtains decline in CCDF graph and diminishes in PAPR whereas conventional ICF needs 8 to 16 repetitions to get same lessening in PAPR. Moreover, the obtained symbol of OFDM has less out of band radiation and lower distortion as compared with available techniques.
In Reference [11], the authors concentrated on the effective technique preferred for diminishing of the PAPR is PTS. In the C-PTS, many IFFT operations are performed to get optimum sequence of phase which leads to the increment of complexity. In this research, the authors have proposed a method for the reduction of half IFFT calculations but at the small cost of degradation of PAPR. The simulation results are obtained with modulation of QPSK and Saleh power amplifier. The examinations of digital predistortion (DPD) to enhance the efficiency of the Saleh power amplifier (PA) are also performed.
The research by Tan and Beaulieu [12], which focused on the calculation of BEP of DCT-based OFDM with AWGN channels in the existence of frequency offset, is considered. These consequences of BEP performance of the discrete cosine transform-OFDM method and the discrete Fourier transform-OFDM technique are related. Many digital techniques such as BPSK, QPSK, and 16-PSK are considered. The zero padded discrete cosine transform-OFDM method and zero padded discrete Fourier transform-OFDM technique are compared with the MMSE detections and the MMSE decisions feedback finding over Rayleigh channel. Simulation results and analysis show that DCT-OFDM outperforms DFT-OFDM.
Han and Lee [13] suggested the improved selected mapping system for PAPR lessening of coded OFDM. In this method, the authors have embedded phases which reduce the PAPR of information blocks of coded OFDM. Improvement in error performance and PAPR can be easily obtained with absolutely no damage in data rates. Furthermore, CCDF of PAPR is originated and related with the simulation results.
In [14], there exist several benefits of OFDM such as good spectral efficiency, but it has many disadvantages. The main issue in the OFDM system is very high PAPR. There are numerous techniques present to diminish the PAPR such as partial transmitted sequence (PTS), tone reservation (TR), selected mapping (SLM), and interleaving. During large number of subcarriers, the clipping technique is not suitable at all. The important methods which are used for PAPR lessening are SLM and PTS. In this study, the authors have proposed the hybrid combination of the PTS technique and SLM technique. It diminishes PAPR from 6 to 5, and at last the results obtained from the hybrid technique obtains better results as compared with existing technique.
In [15], a novel phase updating algorithm has been proposed. With the help of random increment, the phase of several subcarriers is reorganized till the PAPR goes below the particular near threshold. There exists the examination of distinct distributions for the increment in phase and variance for distribution on both the mean and variance for the PAPR. In this, after changing the shifts of phases, threshold is found to be reduced. This random phase updating algorithm delivers best results and diminishes mean power variance of the OFDM.
In [16], Hosseini et al. proposed the algorithm which is dependent on companding for diminishing the PAPR. In this algorithm, IFFT works with a compressing polynomial at both sides i.e., transmitter and receiver. The reverse extending function with Jacobi’s method works with FFT. This procedure has very low complexity as related with other systems. It needs fewer increases in the signal to noise ratio (SNR) for bit error rate (BER) as compared with another companding approaches. There exists a compromise between the performances and complexities which can establish the polynomial compressing order and the number of repetition.
In [17], the usage of partial transmitted sequence (PTS) has been shown for diminishing the PAPR. Generally, the original PTS technique involves a finding over several phase factors, which ultimately results in the increment of computation complication with many subblocks. In order to find perfect phase factor, it undergoes combinatorial optimization with some constraint. Now, the authors presented an approach which depends on simulated annealing, and it is characterized as a nonlinear optimization. This proposed technique works with very low complexity. To justify the outcomes, several simulations have been performed which represent that the proposed technique can obtain less complexity with good PAPR lessening. In [18], the SLM algorithm has been studied with Hadamard matrix for lessening the PAPR. Due to the use of Hadamard matrix, phase sequence generation and choice have become very easy. Numerous outcomes clearly depicted the outstanding performance of suggested algorithm with conventional system of OFDM.
In this current proposed work, the discrete cosine transform (DCT) matrix has been used for SLM as a phase arrangement for the lessening of the PAPR. The arrangement of this paper is as follows: Section 1 presents a general overview of related work. Section 2 incorporates description of OFDM and SLM Technique. Section 3 presents proposed scheme whereas simulation results exhibition is followed by conclusion inferred in Section 5 based upon the opinion from the simulation results.
2. Description of OFDM
2.1. The Symbol of OFDM
Let be the OFDM symbol of frequency domain and be the value of symbol supported by the subcarrier. Then, OFDM symbol of time domain, associated with c with l times oversampling is given as [10]where is the time index.
2.2. Parameters of OFDM System
Let us describe two parameters associated with OFDM arrangement: PAPR and complementary cumulative distribution function (CCDF). For suitability, we will consider and to signify symbols of domain of frequency and symbols of domain of time, respectively, and and for OFDM symbol of the frequency domain and the time domain.
2.2.1. PAPR
The PAPR of OFDM, , can be well defined as [10]
2.2.2. CCDF
In order to calculate the depletion in , CCDF of is employed and is given as [11]
Equation (3) represents the probability that of symbols tops the threshold .
The IDCT vector of vector and DCT of vector , can be given as [12]respectively, where the unitary matrix is the matrix of DCT, and IDCT matrix is . The of DCT matrix is given aswhere
2.3. SLM Technique
In conventional SLM technique [13], at first, the data are divided into the information block having size . Then, the information block of OFDM is multiplied with phase sequence , to generate the phase alternated OFDM blocks of information where , and phase sequence , . All of the several phase rotation blocks of OFDM signify the similar information as the unchanged information of OFDM block given that the phase sequence is well recognized. To contain unchanged block of information in the given set of phase interchanged OFDM blocks, is chosen as single vector having size . After application of SLM method to , equation (1) develops with the assistance of inverse fast Fourier transform (IFFT) aswhere and .
PAPR is designed for phase rotation blocks of OFDM information by using equation (2):
Among the several phase rotation data blocks of OFDM blocks, the single smallest PAPR is picked and communicated. The data about the chosen phase sequence must be conveyed as information to the receiver. Converse operation at receiver side must be performed to recover unchanged OFDM blocks of data. The phase sequences are chosen so that phase rotation OFDM blocks are adequately different.
3. Proposed Scheme
The proposed scheme uses discrete cosine transform matrix as the phase sequence for SLM for the lessening of the PAPR (Figure 1).
Discrete cosine matrix (M) is an orthogonal matrix and follows , where is an identity matrix. Due to the definite structure of DCT matrix, it has been opted for proposed work. Steps for the offered procedure are as follows:(1)Data sequence having length N, after passing through modulator, is represented by , being number of subcarriers. For better approximation of exact PAPR, oversampling by of each signal is required. To upsample a signal zeros are added in the information vector which is for .(2)In order to optimize and reduce PAPR, the data blocks , are multiplied with phase sequence which is represented as where is the number of rows as a phase sequence of DCT matrix which is designed for phase rotation of blocks of OFDM system.(3)Phase alternated OFDM symbol is generated with the help of multiplication of modulated information and phase sequence and is given as(4)Time-domain transformation of the altered data block of OFDM () is implemented with the assistance of IFFT and is characterized by(5)Now, PAPR is obtained as(6)Among several phase sequence rotated OFDM blocks of data, the data block which generates tiniest PAPR is chosen, i.e., for the transmission. In this way, the side information about phases will be transmitted to the receiver.(7)In order to compute the lessening of PAPR, CCDF of PAPR is calculated as which represents the probability that the PAPR of symbols tops the threshold .
4. Simulation Results
MATLAB was used for the simulation of OFDM system taking subcarriers (N) = 128, 64, and 32, phase sequence (U) = [1, 2, 4, 8, 16, 32, 64, 128], modulation scheme = BPSK, 4 ary-PSK, 8-PSK, 16-PSK, and 32 PSK. CCDF curve was utilized for the calculation of PAPR of the OFDM.
Figure 2 depicts the comparison of conventional SLM given in [5] with the proposed SLM technique. Comparable result is obtained at U = 1 due to consideration of less number of phase sequences. As the phase sequences are improved from U = 1 to U = 8, considerable growth in the gain was observed. Particularly at U = 8, gain of 1.35 dB at CCDF = .001% has been achieved as compared to [5]. Hence, it can be concluded that growth in the quantity of phase sequences led to the enhancement of performance of the proposed work.
Figure 3 shows the comparison of conventional SLM [5] with the proposed SLM for higher number of phases U = 16, 32, 64, and 128. Remarkable gain has been achieved by the proposed scheme in comparison with the [5], i.e., 1.36, 1.25, 1.4, and 1.27 dB while comparing at CCDF = .001% for U = 16, 32, 64, and 128 phase sequences, respectively.
Figures 4 and 5 depict PAPR for conventional SLM and proposed SLM for U = 1, 2, 4, and 8 under N = 32 and 64. The comparison is also shown in Table 1. For N = 32 and N = 64, the presentation of projected SLM system for U = 8 is tremendous in comparison with the proposed SLM for U = 1, 2, and 4 and the conventional SLM scheme for U = 1, 2, 4, and 8 (Figures 4 and 5).
Here, it is perceived that PAPR (dB) required to accomplish for N = 32 at 4-PSK is 10.60, 10.30, 10.15, and 8.21 dB, respectively, for conventional SLM for U = 1, 2, 4, and 8 whereas it is 10.60, 9.25, 8.20, and 7.10 dB, respectively, for the proposed SLM for U = 1, 2, 4, and 8 (Table 1). Similarly, we can perceive for N = 64 and N = 128 in order to estimate the PAPR lessening of the proposed SLM system in contrast with the conventional SLM system (Table 1).
Figures 6 and 7 depict the PAPR performance analysis for conventional SLM and proposed SLM for U = 16, 32, 64, and 128 under N = 32 and 64. The PAPR value for comparison purpose is also shown in Table 2. Close observation of Table 2 reveals that for N = 32, the performance of the proposed SLM scheme for U = 128 is tremendous in comparison with conventional SLM scheme for U = 16, 32, 64, and 128 and also with the proposed SLM for U = 16, 32, and 64.
Here, we have perceived that PAPR (dB) required to accomplish for N = 32 at 4-PSK is 7.90, 7.00, 6.26, and 5.82 dB for conventional SLM for U = 16, 32, 64, and 128, respectively, whereas it is 6.25, 5.96, 5.65, and 5.36 dB for the proposed SLM for U = 16, 32, 64, and 128, respectively (Table 2). Similarly, we can perceive the performance of PAPR lessening of the proposed SLM scheme in comparison with conventional SLM scheme for N = 64 and N = 128 (Table 2).
Figures 8–12 depict PAPR performance analysis of conventional SLM and the proposed SLM for U = 1, 2, 4, and 8 under N = 128 for various higher-order modulation schemes such as BPSK, 4-PSK, 8-PSK, 16-PSK, and 32-PSK. The performance of PAPR for the conventional SLM and the proposed SLM is displayed in Table 3. It is obvious from Table 3 that the performance of proposed SLM scheme, for N = 128 and BPSK, for U = 8, is marvelous in contrast with the conventional SLM system for U = 1, 2, 4, and 8 and also with the proposed SLM for U = 1, 2, and 4.
Here, we have perceived that PAPR (dB) required to accomplish for N = 128 and BPSK modulation is 10.90, 10.45, 10.75, and 9.10 dB, respectively, for conventional SLM for U = 1, 2, 4, and 8 whereas it is 10.60, 9.10, 8.35, and 7.30 dB, respectively, for the proposed SLM for U = 1, 2, 4, and 8. Similarly, we can perceive for higher order of modulation in order to estimate the performance of PAPR lessening of the proposed SLM system in comparison with conventional SLM scheme (Table 3).
Figures 13–17 depict the PAPR performance analysis for conventional SLM and the proposed SLM for U = 16, 32, 64, and 128 under N = 128 for various higher-order modulation schemes such as BPSK, 4-PSK, 8-PSK, 16-PSK, and 32-PSK. The PAPR performance comparison is shown in Table 4. For N = 128 and BPSK modulation scheme, the performance of the proposed SLM scheme for U = 128 is impressive in comparison with the performance of proposed SLM for U = 16, 32, and 64 and also with the conventional SLM scheme for U = 16, 32, 64, and 128 (Table 4).
Here, we have perceived that PAPR (dB) required to accomplish for N = 128 at BPSK is: 8.35, 7.61, 7.30, and 7.15 dB, respectively, for conventional SLM for U = 16, 32, 64, and 128 whereas it is 6.57, 6.40, 5.95, and 5.67 dB for the proposed SLM for U = 16, 32, 64, and 128, respectively (Table 4). Similarly, we can perceive for higher order of modulation in order to estimate the performance of PAPR lessening of the proposed SLM system in contrast with the conventional SLM system (Table 4).
5. Conclusion
The proposed SLM method using DCT for the selection of phase sequences is found to be better than the work presented in the literature. In conventional SLM, randomized phase sequence criterion has been adopted whereas in the proposed work, designing of phase sequence is based on the use of DCT matrix which needs less computational efforts. Moreover, in the proposed work, very small information about phase is required to be sent because of specific structure of DCT matrix so that the data can be easily reproduced at the receiver. In future, the proposed work can also be simulated using other modulation schemes such as QAM.
Data Availability
The input data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Prof. B. Arun Kumar, School of Electronics and Electrical Engineering (SEEE), Lovely Professional University, Punjab, India, for his valuable suggestions.