A Novel PAPR Reduction Scheme for Low-Cost Terminals in 6G

Abstract

This paper presents a low peak-to-average power ratio (PAPR) modulation scheme and demodulation method for low-cost terminals in sixth-generation (6G), which is called one plus delay (1 + D) pi/2(N + 1) BPSK. Explicitly, the modulation scheme is first generated based on pi/2 BPSK, then proceeded by interpolation and (1 + D) transformation. Simulation results show that the proposed (1 + D) pi/2(N + 1) BPSK scheme has lower PAPR and smaller out-of-band leakage in discrete Fourier transform spread orthogonal frequency division multiplexing (DFT-s-OFDM). Moreover, better performance can be attained by increasing the value of N. Finally, similar block error ratio (BLER) performance can be obtained at the same spectral efficiency of the conventional pi/2 BPSK scheme.

1. Introduction

Mobile communications have been updated every decade or so, and now we have entered the era of fifth-generation (5G). However, 5G cannot meet all the requirements for 2030 and beyond [1]. In 5G, three typical application scenarios are considered: Enhanced Mobile Broadband (eMBB), massive Machine Type Communications (mMTC) and Ultra Reliable Low Latency Communications (URLLC). Specifically, eMBB is characterized by a high data rate, wide coverage, high frequency efficiency and support for mobility. In order to meet these requirements, new waveforms are expected to achieve high order modulations, large Fourier transform points and high data rates. Moreover, low out of band emission and low peak-to-average power ratio (PAPR) are required. In 5G, the support for massive connections is the basic requirement of mMTC, where devices deployed in this scenario require low transmission power and long-term operation. In this case, energy efficiency is one of the key requirements for the waveform design in mMTC. Finally, URLLC is a scenario with the target of enabling vertical industry applications, such as robot cooperation and autonomous vehicles. In URLLC, low latency and high reliability are the typical requirements. In order to meet these requirements, new techniques of frame structure, resource allocation and optimization are expected to minimize the transmission delay under the constraints of the reliability threshold. For the waveform design, lower complexity and the ability of interference mitigations are required.

In comparison with 5G, sixth-generation (6G) wireless communications may be at least characterized by the enhanced versions of eMBB, mMTC, and URLLC [2]. The enhanced version of eMBB focuses on the potential of enriching human communications with extremely high data rates, where waveforms used in the enhanced version of eMBB are expected to achieve higher order modulation applied in higher frequency band. The enhanced version of mMTC focuses on the support of machine communications with extremely larger connections and high reliability. Since the machine devices are power limited, the PAPR reduction is of great importance in the enhanced version of mMTC. Finally, the enhanced version of URLLC in 6G aims at enabling some new vertical industries, e.g., body sensing, sensors or interactive devices around the environment, etc. In this case, low latency and high reliability are required in the enhanced version of URLLC.

In order to support the above-mentioned scenarios in 6G, PAPR reduction techniques of waveforms play an important role. In particular, waveforms used in the enhanced version of eMBB are expected to introduce PAPR reduction techniques in order to remain stable and avoid jitter caused by nonlinear devices during high data rate transmissions [3]. In the enhanced version of mMTC, the deployment of PAPR reduction techniques is urgent, since high energy efficiency is critical in massive new types of devices enabled by 6G. In the enhanced version of URLLC, the requirements of low latency and high reliability make it ever challenging to design low PAPR reduction techniques that can be compatible with new frame structures [4].

The remainder of this paper is organized as follows. In Section 2, an overview of existing PAPR reduction schemes is detailed. In Section 3, we discuss hardware design perspectives of low PAPR waveforms. The design of a low PAPR modulation scheme is proposed in Section 4. In Section 5, the soft demodulation algorithm and the block error ratio (BLER) performance evaluation are shown. The conclusions are presented in Section 6.

2. An Overview of Existing PAPR Reduction Schemes

Orthogonal frequency division multiplexing (OFDM) is widely used due to its ability to overcome the frequency selective fading in multipath channels with relatively low complexity in both transmitter and receiver [5]. However, it suffers from high PAPR because of the superposition of subcarriers on the same time domain symbol, which degrades the system performance severely when high power amplifiers (HPAs) of both transmitter and receiver are used [3]. Other multicarrier waveforms, such as filter bank multicarrier with offset quadrature amplitude modulation (FBMC-OQAM) and universal filtered multicarrier (UFMC), have similar problems [6]. Recently, some PAPR reduction schemes for multicarrier waveforms have been investigated [6,7,8,9,10,11,12,13,14]. In [7], the combination of continuous phase modulation (CPM) and direct sequence spread spectrum (DSSS) was considered to improve the PAPR performance of OFDM. However, the synchronization of CPM–DSSS is hard to implement and the system is sensitive to channel conditions. A design of complementary set sequence based on OFDM was proposed in [8] to achieve low PAPR under the spectral constrains. Then, a method called lexicographical symbol position permutation was proposed for PAPR reduction in [9]. In the MIMO–OFDM downlink system, a low complexity algorithm for PAPR suppression was proposed, which is called as the accelerated proximal gradient method [10]. Another approach based on constructive interference was designed for MIMO–OFDM systems by considering the tradeoff between PAPR reduction and complexity in [11]. In addition to traditional approaches, a PAPR reduction method based on neural networks (NNs) was proposed for OFDM systems in [12]. In order to maintain the BER performance, a PAPR decompression module and a PAPR reduction module were jointly trained. However, the system complexity increases as the number of subcarriers increases. Nevertheless, these PAPR reduction schemes for OFDM cannot be directly applied to FBMC due to its overlapping structure. In [13], a conversion vector-based dispersive selection mapping approach was designed to reduce PAPR for FBMC, which has relatively low complexity. In order to achieve lower PAPR, an approach that removes side information burden in FMBC was proposed in [6], where DFT-spread was also applied for further reducing PAPR. In [14], a low complexity selected mapping approach was proposed for UFMC to reduce PAPR by alleviating its complexity to the level approximates to OFDM.

On the other hand, single carrier waveforms, such as the discrete Fourier transform spread OFDM (DFT-s-OFDM), the guard interval DFT-s-OFDM (GI DFT-s-OFDM) and the unique word DFT-s-OFDM (UW DFT-s-OFDM), are usually characterized by low PAPR due to the DFT processing before OFDM modulation [15]. However, GI/UW DFT-s-OFDM requires high orthogonality of subcarriers and is sensitive to phase jitter and spectrum offset. Hence, the PAPR performance of single carrier waveforms is closely related to the modulation mode employed. For example, in the R16 version of 5G New Radio (NR) [16], pi/2 BPSK (in this paper, pi is equivalent to π) is the modulation mode with the lowest modulation order and the lowest bit rate in the DFT-s-OFDM waveform. Nevertheless, the PAPR of existing pi/2 BPSK modulation mode of 5G NR is not low enough to meet the demand of 6G [17]. Therefore, some novel modulation schemes with low PAPR were put forward [18,19,20]. Most of these schemes were designed according to the principle that both the modulus and the adjacent phases of modulation symbols remain close or even equal to each other. Some of the existing low PAPR modulation schemes for single carrier waveforms mainly include 8-BPSK [18], continuous phase modulation (CPM) [19] and one plus delay (1 + D) pi/2 BPSK [20], etc. The 8-BPSK maps the information bits to the constellation points of 8PSK by using a trellis diagram so that the phase changes of adjacent symbols are small [18]. Continuous phase modulation (CPM) uses the accumulative phase and the instantaneous phase to form the modulation symbol in order to guarantee that the phase variation be approximately continuous [19]. The (1 + D) pi/2 BPSK scheme is obtained by using the (1 + D) filter to pi/2 BPSK, where the phase difference of pi/2 BPSK was changed from pi/2 to zero or unchanged [20]. Both the 8-BPSK and the CPM can achieve a very low PAPR, but they have high demodulation complexity and poor demodulation performance at the receiving side. Although (1 + D) pi/2 BPSK has low demodulation complexity and the same demodulation performance as that of pi/2 BPSK in additive white Gaussian noise (AWGN) channel, its PAPR is higher than those of 8-BPSK and CPM. Thus, further improvement for (1 + D) pi/2 BPSK is required to achieve low PAPR, as well as at the same time keeping low demodulation complexity and the same demodulation performance. Finally, an overview of existing PAPR schemes is summarized in

Table 1. Characteristics of PAPR reduction schemes.

Waveform	Techniques	Advantages	Limitations
OFDM	Companding	Low PAPR	High complexity
	Piecewise Linear Companding	Low PAPR Better BER performance Low complexity	High OOBE
	Complementary set sequences	Low PAPR Low spectral nulls	Hard to implement
	Constructive interference	Low complexity	High PAPR
	lexicographical symbol position permutation	Low PAPR	/
	Neural network	Low PAPR Better BER performance	Higher complexity
FBMC	Dispersive selection mapping	Low PAPR Low complexity	Higher PAPR incompatible with MIMO.
	LP-FBMC	Low PAPR Better SI solution	/
UFMC	Modified selected mapping	Low PAPR	High complexity
DFT-s-OFDM	(1 + D) filter	Low PAPR Low OOBE	High demodulation complexity
	Chirp signal	Better BER Better OOBE	High complexity
	CPM	Low PAPR	Require new hardware design

.

Generally speaking, the requirement for data rate is relatively low for the low-cost terminals in 6G. Therefore, in this paper, we focus our attention on a low PAPR modulation design, which is able to reduce the phase difference of adjacent symbols by uniform interpolation of pi/2 BPSK. Although this operation will reduce the transmission efficiency, we found that at the same data rate the performance is the same as that of pi/2 BPSK in AWGN channel. Even in the fading channel, we can enhance the performance through the method of frequency domain selective resource scheduling.

3. Hardware Design Perspectives of Low PAPR Waveforms

Recently, the hardware design of various PAPR reduction methods has drawn considerable attention in relevant fields. In [21], an encoder that combines the circuit characteristics with the channel response was proposed to accomplish encryption after the modulation process, which can provide security while reducing the PAPR of OFDM signals. Specifically, this approach can achieve a reduction of PAPR by 3.56 dB in a reality test. In [22], an encoder based on complementary sequences was proposed by considering a maximum-likelihood recursive decoder, where the corresponding PAPR is reduced to less than 3 dB. In [23], an iterative dichotomy PAPR (IDP) was proposed, where the PAPR for any dichotomy order M was computed by using a recurrent method. As a result, the IDP technique achieves a reduction of PAPR by 2.2 dB for the dichotomy order M = 2 and 4 dB for the dichotomy order M = 4. In addition, the output signal power of the IDP approach is also better than that of classical OFDM. Another method was proposed to achieve higher energy efficiency by considering Doherty power amplifiers with output power back-off (OBO) in [24]. This approach utilizes multiple OBO parameters via multiple methods, such as complex combining load, virtual stub and out-phased current combining. Another hybrid PAPR reduction method was proposed in [25], showing that a reduction of PAPR by 7 dB is achievable compared with OFDM.

In single carrier waveforms, a transmitter design of CPM module was proposed, which allows a more accurate signal processing with modular arithmetic instead of memory resources [19]. This approach sets the cumulated phase term of CPM signals as points in a constellation diagram, where a reduction of 15% in logic and memory resources can be achieved. In order to alleviate hardware resource burden, a reconfigurable transmitter based on single h and multi-h CPM modulations was proposed in [26]. This approach is capable of generating Gaussian minimum shift keying (GMSK), continuous phase frequency shift keying (CPFSK), tamed frequency modulation and other modulation schemes with a manageable cost of resource increments. As a result, this scheme can reduce 50 kbits memory resources under the Bluetooth standard. In [27], a new design of GFSK receiver was proposed, where the in-phase and quadrature baseband signal are transformed and two kinds of new metrics for constellation are generated. However, it yields 3 dB bit error rate (BER) degradation. Finally, some hardware design perspectives of low PAPR waveforms are summarized in

Table 2. Hardware design perspectives of different waveforms.

Hardware	PAPR Reduction Schemes	Waveform	Advantages
Modulator	MA-DSP Multi-h CPM	Multicarrier waveforms	Low memory resources
	CPM	Single carrier	Low memory cost
	SOQPSK	Single carrier	Low hardware cost
Encoder	Encryption encoder and decoder	OFDM	Low PAPR
	Partitioned complementary sequences	OFDM	Low PAPR
Power amplifiers	Iterative Dichotomy	OFDM	Low PAPR
	Doherty power amplifiers	OFDM	Low PAPR Low power consumption
	Hybrid PAPR reduction method based on HPA	OFDM	Low PAPR
Receiver	New metrics	Single carrier	Low resources cost

.

4. Modulation and Performance Evaluation

4.1. (1 + D) pi/2(N + 1) BPSK Modulation

It needs to be emphasized that the low PAPR modulation schemes designed in this paper are for single carrier waveforms. Without loss of generality, we consider the DFT-s-OFDM waveform for the following analysis [28]. Generally, each low PAPR modulation scheme has two empirical characteristics: (1) the modulus difference of modulation symbols is relatively small or even non-existent; (2) the phase change of the adjacent symbol is relatively small. Based on these characteristics, we propose a new modulation method named (1 + D) pi/2(N + 1) BPSK.

Before providing our signaling expression, we first show some examples to help readers understand the (1 + D) pi/2(N + 1) BPSK scheme. We assume that bit 0 maps to −1 or −j, and bit 1 to 1 or j in pi/2 BPSK modulation, where j denotes the imaginary unit. Based on pi/2 BPSK modulation symbols, as shown by the red and blue points in Figure 1a, N symbols shown as the yellow points are inserted between the adjacent pi/2 BPSK modulation symbols, where the modulus of each inserted is 1 and the corresponding phase shift is equal to pi/2(N + 1). For convenience, it is named pi/2(N + 1) BPSK. For example, Figure 1b shows the constellation diagram for N = 1, i.e., pi/4 BPSK. We can readily show that the phase difference between the adjacent symbols of pi/2(N + 1) BPSK is pi/2(N + 1), as the name indicates.

On the other hand, (1 + D) filter can be used to further reduce PAPR for pi/2(N + 1) BPSK symbols. In the time domain, the (1 + D) filter amounts to convolution with h_1+D, correspondingly; in the frequency domain, it amounts to dot product with the DFT of h_1+D, where h_1+D can be expressed as

$${\mathrm{h}}_{1+\mathrm{D}}=\mathsf{\alpha }{\left[1,1,0,\dots ,0\right]}^{}$$

(1)

where α is the power normalization factor, and the length of h_1+D is the same as the total length of the pi/2(N + 1) BPSK symbol block. Figure 2a,b show the constellations of pi/2(N + 1) BPSK and the proposed (1 + D) pi/2(N + 1) BPSK for N = 0 and 1, respectively. In particular, (1 + D) pi/2 BPSK can be regarded as a special case of the new modulation mode (1 + D) pi/2(N + 1) BPSK when N = 0. By inputting pi/2(N + 1) BPSK symbols into (1 + D) filter, the phase difference of adjacent symbols changes from pi/2(N + 1) to 0 or pi/2(N + 1).

In fact, the production process of (1 + D) pi/2(N + 1) BPSK modulation can also be regarded as pi/2 BPSK performing zero interpolation and convolving with a filter in the time domain. For example, we can first insert one zero for every two symbols in pi/2 BPSK, and then convolve with [1,cos(pi/4), 0,…,0,cos(pi/4)] to obtain pi/4 BPSK. Finally, the pi/4 BPSK symbol is convolved with h_1+D to generate (1 + D) pi/2(N + 1) BPSK symbols. According to the associative law of convolution, [1,cos(pi/4),0,…,0,cos(pi/4)] and h_1+D can be combined, which can be expressed as

$$\begin{array}{cc}\hfill {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/4 \mathrm{BPSK}}=& \frac{1}{\sqrt{2+\sqrt{2}}}[1,\cos (\mathsf{\pi }/4),0,\dots ,0,\cos (\mathsf{\pi }/4)]\hfill \\ & \otimes [1,1,0,\dots ,0]\hfill \end{array}$$

(2)

where $\otimes$ represents the convolution. More generally, for any N, we can present the expression of the filter coefficient of (1 + D) pi/2(N + 1) BPSK as

$$\begin{array}{cc}\hfill {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/2(\mathrm{N}+1) \mathrm{BPSK}}=& \left[1,\cos (\frac{1\cdot \mathsf{\pi }}{2(\mathrm{N}+1)}),\cos (\frac{2\cdot \mathsf{\pi }}{2(\mathrm{N}+1)})\right. ,\dots ,\hfill \\ & \cos (\frac{\mathrm{N}\cdot \mathsf{\pi }}{2(\mathrm{N}+1)}),0,\dots ,0, \cos (\frac{\mathrm{N}\cdot \mathsf{\pi }}{2(\mathrm{N}+1)}),\hfill \\ & \left.\cos (\frac{(\mathrm{N}-1)\cdot \mathsf{\pi }}{2(\mathrm{N}+1)}),\dots ,\cos (\frac{1\cdot \mathsf{\pi }}{2(\mathrm{N}+1)})\right]\hfill \\ & [1,1,0,\dots ,0]\cdot \frac{1}{\sqrt{2+2\cos (\frac{\mathsf{\pi }}{2(\mathrm{N}+1)})}}\hfill \end{array}$$

(3)

Based on the above-mentioned examples, given a vector of pi/2 BPSK symbols $a=\left[{\mathrm{a}}_1,{\mathrm{a}}_2,\dots ,{\mathrm{a}}_{\mathrm{n}}\right]$, the (1 + D) pi/2(N + 1) BPSK symbols can be expressed as

$$s=a^{\prime }\otimes {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/4 \mathrm{BPSK}}$$

where we have $a^{\prime }=\left[{\mathrm{a}}_1,0,{\mathrm{a}}_2,0,\dots ,{\mathrm{a}}_{\mathrm{n}},0\right]$. Finally, the obtained (1 + D) pi/2(N + 1) BPSK symbols can be sent into IFFT to generate the time domain signal vector denoted as x = [x[0],…, x[N − 1]], where the corresponding PAPR and PSD are analyzed below.

4.2. PAPR and PSD Performance

The simulation parameters are shown in

Table 3. Simulation parameter.

Description	Value
Waveform	DFT-s-OFDM
FFT Number	4800
Available Carrier Number	600
Symbol Number	15
Transfer Length	500 subframes
Oversampling Ratio	8 times

. The PAPR comparisons of pi/2(N + 1) BPSK and (1 + D) pi/2(N + 1) BPSK are shown in Figure 3, where the CPM scheme is generated by 2-ary symbols and raised cosine phase smoothing function with the signal memory L = 3, and the digital modulation index h is 2/3 or 2/5. The PAPR of the time domain signal x[n] can be defined as

$$\mathrm{PAPR}\{\mathrm{x}[\mathrm{n}]\}=\frac{\underset{0\le \mathrm{n}\le \mathrm{N}-1}{\max }\mathrm{x}{[\mathrm{n}]}^2}{\mathrm{E}[{\left|\mathrm{x}[\mathrm{n}]\right|}^2]}$$

(4)

Furthermore, the filter coefficients of several modulations are shown in Equation (5), which can be easily obtained according to the generating principle of several modulation schemes. Since the time domain convolution is equivalent to the frequency domain dot product, the PSD of (1 + D) pi/2(N + 1) BPSK can be obtained by the dot product of the PSD of pi/2 BPSK and the PSD of the filter F_{1+D pi/2(N+1) BPSK}.

$$\begin{array}{cc}\hfill {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/2 \mathrm{BPSK}}=& \frac{\sqrt{2}}{2}[1,1,0,\dots ,0]\hfill \\ {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/4 \mathrm{BPSK}}=& \frac{1}{\sqrt{2+\sqrt{2}}}[1,\cos (\mathsf{\pi }/4),0,\dots ,0,\cos (\mathsf{\pi }/4)]\hfill \\ & \otimes [1,1,0,\dots ,0]\hfill \\ {\mathrm{F}}_{1+\mathrm{D} \mathrm{pi}/6 \mathrm{BPSK}}=& \frac{1}{\sqrt{2+\sqrt{3}}}[1,\cos (\mathsf{\pi }/6),\cos (\mathsf{\pi }/3),0,\dots ,0,\hfill \\ & \cos (\mathsf{\pi }/3),\cos (\mathsf{\pi }/6)]\otimes [1,1,,0,\dots ,0]\hfill \end{array}$$

(5)

We can see that compared with pi/2 BPSK, the PAPR of other modulation schemes is much lower at CCDF = 10⁻⁴. It can be clearly found that as N increases, both the PAPR of pi/2(N + 1) BPSK and (1 + D) pi/2(N + 1) BPSK decreases monotonously, and (1 + D) pi/2(N + 1) BPSK is better than that of pi/2(N + 1) BPSK, so we will focus on (1 + D) pi/2(N + 1) BPSK in the following part of this paper. It is worth noting that the PAPR of (1 + D) pi/6 BPSK is very close to that of 1 + D pi/8 BPSK, that is when N > 3, it is difficult to further decrease PAPR. In addition, it can be found in Figure 3 that the new modulation has lower PAPR than CPM modulation, without requiring higher demodulation complexity.

Figure 4 presents the PSD comparison of in-band and out-of-band between (1 + D) pi/2(N + 1) BPSK symbols and pi/2 BPSK symbols through simulations. The simulation parameters are shown in

Table 4. Simulation parameter.

Description	Value
Waveform	DFT-s-OFDM
FFT Number	300, 500
Available Carrier Number	300
Symbol Number	15
Transfer Length	500 sub-frames
Oversampling Ratio	5/3 times

. It can be seen from Figure 4 that the power spectral density (PSD) of (1 + D) pi/2(N + 1) BPSK modulation symbols within the transmission bandwidth is larger in the middle and smaller on both sides, which is different from pi/2 BPSK. We can find that (1 + D) pi/2(N + 1) BPSK scheme can reduce out-of-band leakage, and the larger N is, the more the reduction. This is mainly because with the increase of N, the power in the frequency domain is more concentrated in the middle, reducing the power outside the band.

5. Demodulation and Performance Evaluation

For convenience, we set N = 1, that is, the (1 + D) pi/4 BPSK scheme is used as an example to illustrate the demodulation scheme of (1 + D) pi/2(N + 1) BPSK symbols. It should be noted that for other values of N, the demodulation scheme is similar, hence we neglect the corresponding details in this paper.

As shown in Figure 5, the modulation process of the constellation points of (1 + D) pi/4 BPSK can be summarized as follows: (1) the bit information b is modulated by pi/2 BPSK to obtain the modulation symbol a; (2) the modulation symbol a is interpolated to obtain the modulation symbol c of pi/4 BPSK and (3) the modulation symbol c is transformed by (1 + D) to obtain the modulation symbol s of 1 + D pi/4 BPSK. Below, log likelihood ratio (LLR) algorithm is applied to demodulate and input the output soft information to the decoding module.

5.1. Log Likelihood Ratio-Based Soft Demodulation

The LLR algorithm is commonly used as a soft demodulation method for representing soft information [29]. Here, LLR is used for demodulation of (1 + D) pi/2(N + 1) BPSK by finding the modulation symbol containing the corresponding bit information and then conducting the LLR calculation. Assuming that bit b_i is mapped to the real pi/2 BPSK (the derivation of the imaginary number line is the same), it can be seen from Figure 5 that the value of b_i will affect four (1 + D) pi/4 BPSK modulation symbols, namely s_2i−2, s_2i−1, s_2i and s_2i+1; then, the soft information of LLR of b_i can be written as

$$\begin{array}{cc}\hfill \mathrm{LLR}({\mathrm{b}}_{\mathrm{i}})& =\ln \frac{{\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^1\cdot {\mathrm{P}}_{2\mathrm{i}-1,\mathrm{i}}^1\cdot {\mathrm{P}}_{2\mathrm{i},\mathrm{i}}^1\cdot {\mathrm{P}}_{2\mathrm{i}+1,\mathrm{i}}^1}{{\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^0\cdot {\mathrm{P}}_{2\mathrm{i}-1,\mathrm{i}}^0\cdot {\mathrm{P}}_{2\mathrm{i},\mathrm{i}}^0\cdot {\mathrm{P}}_{2\mathrm{i}+1,\mathrm{i}}^0}\hfill \\ & =\ln \frac{{\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^1}{{\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^0}+\ln \frac{{\mathrm{P}}_{2\mathrm{i}-1,\mathrm{i}}^1}{{\mathrm{P}}_{2\mathrm{i}-1,\mathrm{i}}^0}+\ln \frac{{\mathrm{P}}_{2\mathrm{i},\mathrm{i}}^1}{{\mathrm{P}}_{2\mathrm{i},\mathrm{i}}^0}+\ln \frac{{\mathrm{P}}_{2\mathrm{i}+1,\mathrm{i}}^1}{{\mathrm{P}}_{2\mathrm{i}+1,\mathrm{i}}^0}\hfill \end{array}$$

(6)

where, ${\mathrm{P}}_{2\mathrm{i},\mathrm{i}}^1$ represents the probability that the ith bit is 1 according to the received symbol information of the (2i)th, and the rest is the same.

Table 5. Relationship between symbols and bits.

(a) bi
Bit\Symbol	s2i−2	s2i−1	s2i	s2i+1
0	3, 6	4, 5	4, 5	3, 6
1	2, 7	1, 8	1, 8	2, 7
(b) bi+1
Bit\Symbol	s2i	s2i+1	s2i+2	s2i+3
0	5, 8	6, 7	6, 7	5, 8
1	1, 4	2, 3	2, 3	1, 4

shows the relationship between symbols and bits at the receiver side. For convenience, the (1 + D) pi/4 BPSK modulation symbol is numbered; then, ${\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^1$ and ${\mathrm{P}}_{2\mathrm{i}-2,\mathrm{i}}^0$ can be represented by

$$\begin{array}{cc}{\mathrm{P}}_{2\mathrm{i}-2}^1=& \mathrm{p}({\mathrm{s}}_{2\mathrm{i}-2}^2)+\mathrm{p}({\mathrm{s}}_{2\mathrm{i}-2}^7)\hfill \\ \hfill =& \exp (-\frac{{\left(\mathrm{real}({\mathrm{s}}_{2\mathrm{i}-2})-\mathrm{u}\right)}^2+{\left(\mathrm{imag}({\mathrm{s}}_{2\mathrm{i}-2})-\mathrm{v}\right)}^2}{2{\mathsf{\sigma }}^2})\hfill \\ & +\exp (-\frac{{\left(\mathrm{real}({\mathrm{s}}_{2\mathrm{i}-2})-\mathrm{u}\right)}^2+{\left(\mathrm{imag}({\mathrm{s}}_{2\mathrm{i}-2})+\mathrm{v}\right)}^2}{2{\mathsf{\sigma }}^2})\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathrm{P}}_{2\mathrm{i}-2}^0=& \mathrm{p}({\mathrm{s}}_{2\mathrm{i}-2}^3)+\mathrm{p}({\mathrm{s}}_{2\mathrm{i}-2}^6)\hfill \\ \hfill =& \exp (-\frac{{\left(\mathrm{real}({\mathrm{s}}_{2\mathrm{i}-2})+\mathrm{u}\right)}^2+{\left(\mathrm{imag}({\mathrm{s}}_{2\mathrm{i}-2})-\mathrm{v}\right)}^2}{2{\mathsf{\sigma }}^2})\hfill \\ & +\exp (-\frac{{\left(\mathrm{real}({\mathrm{s}}_{2\mathrm{i}-2})+\mathrm{u}\right)}^2+{\left(\mathrm{imag}({\mathrm{s}}_{2\mathrm{i}-2})+\mathrm{v}\right)}^2}{2{\mathsf{\sigma }}^2})\hfill \end{array}$$

(8)

where $\mathrm{u}=\frac{\cos (\mathsf{\pi }/4)}{2\cos (\mathsf{\pi }/8)}$, $\mathrm{v}=\frac{\cos (\mathsf{\pi }/4)+1}{2\cos (\mathsf{\pi }/8)}$, $\mathrm{p}({\mathrm{s}}_{2\mathrm{i}-2}^2)$ is the probability that the (2i − 2)th receiving symbol is the symbol 2 at the transmitter side, and σ² is the variance of noise. Substituting Equations (7) and (8) into the first term of Equation (6), we derive

$$\ln \frac{{\mathrm{P}}_{2\mathrm{i}-2}^1}{{\mathrm{P}}_{2\mathrm{i}-2}^0}=\frac{2\mathrm{u}}{{\mathsf{\sigma }}^2}\mathrm{real}({\mathrm{s}}_{2\mathrm{i}-2})$$

(9)

Similarly, other terms of Equation (6) can be obtained. Finally, we have

$$\mathrm{LLR}({\mathrm{b}}_{\mathrm{i}})=\frac{2}{{\mathsf{\sigma }}^2}\mathrm{real}(\mathrm{u}\cdot {\mathrm{s}}_{2\mathrm{i}-2}+\mathrm{v}\cdot {\mathrm{s}}_{2\mathrm{i}-1}+\mathrm{v}\cdot {\mathrm{s}}_{2\mathrm{i}}+\mathrm{u}\cdot {\mathrm{s}}_{2\mathrm{i}+1})$$

(10)

where if b_i+1 is mapped to the imaginary of pi/2 BPSK, the similar result can be obtained:

$$\mathrm{LLR}({\mathrm{b}}_{\mathrm{i}+1})=\frac{2}{{\mathsf{\sigma }}^2}\mathrm{imag}(\mathrm{u}\cdot {\mathrm{s}}_{2\mathrm{i}}+\mathrm{v}\cdot {\mathrm{s}}_{2\mathrm{i}+1}+\mathrm{v}\cdot {\mathrm{s}}_{2\mathrm{i}+2}+\mathrm{u}\cdot {\mathrm{s}}_{2\mathrm{i}+3})$$

(11)

It can be found that if N is another value, Equations (10) and (11) can also be used for soft demodulation by changing the coefficient values. As a summary, we can formulate the soft information as

$$\mathrm{y}({\mathrm{b}}_{\mathrm{i}})=\mathrm{real}({\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}+1}{\mathrm{t}}_{\mathrm{n}}\cdot {\mathrm{s}}_{2\mathrm{i}-\mathrm{n}}}+{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}+1}{\mathrm{t}}_{\mathrm{n}}\cdot {\mathrm{s}}_{2\mathrm{i}+\mathrm{n}-1}})$$

(12)

$$\mathrm{y}({\mathrm{b}}_{\mathrm{i}+1})=\mathrm{imag}({\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}+1}{\mathrm{t}}_{\mathrm{n}}\cdot {\mathrm{s}}_{2\mathrm{i}-\mathrm{n}+2}}+{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}+1}{\mathrm{t}}_{\mathrm{n}}\cdot {\mathrm{s}}_{2\mathrm{i}+\mathrm{n}+1}})$$

(13)

$${\mathrm{t}}_1=\frac{\cos \frac{\mathrm{N}\mathsf{\pi }}{2(\mathrm{N}+1)}+1}{2\cos (\frac{\mathsf{\pi }}{4(\mathrm{N}+1)})}$$

(14)

$$\begin{array}{l}{\mathrm{t}}_{\mathrm{n}}=\frac{\cos \frac{\mathsf{\pi }(\mathrm{N}+2-\mathrm{n})}{2(\mathrm{N}+1)}+\cos \frac{\mathsf{\pi }(\mathrm{N}+1-\mathrm{n})}{2(\mathrm{N}+1)}}{2\cos (\frac{\mathsf{\pi }}{4(\mathrm{N}+1)})}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\mathrm{n}=2,3\cdots ,\mathrm{N}+1\end{array}$$

(15)

for $\forall \mathrm{N}\ge 2$. The process of calculability is omitted due to the page limit.

5.2. BLER Performance

Next, we analyze the performance comparison of BLER with pi/2 BPSK and (1 + D) pi/4 BPSK in AWGN. It is important to note that in order to compare performance and ensure fairness, the same spectral efficiency is used for both modulation schemes; that is, since N = 1 of (1 + D) pi/4 BPSK (one symbol interpolated in every two symbols), the bit rate is two times the other two modulation schemes.

The simulation parameters are shown in

Table 6. Simulation parameter.

Description	Value
Waveform	DFT-s-OFDM
FFT Number	1024, 512
RB Number	56, 25
Channel Model	AWGN TDL-D 10 Hz 300 ns, TDL-A 10 Hz 300 ns
Modulation Mode	pi/2 BPSK, 1 + D pi/4 BPSK
Coding Scheme	Low Density Parity Check Code (LDPC)
Channel Estimation	Ideal Channel Estimation

and the simulation results are shown in Figure 6. It can be found that the performance of both modulation schemes is the same, that is, there is no performance loss in (1 + D) pi/4 BPSK. For the (1 + D) pi/4 BPSK, this is mainly because it is equivalent to sending data repeatedly; by increasing the bit rate, the performance is reduced. These two concerns affect each other, leading to the final performance unchanged. It should be mentioned that in some cases, different coding gains may be generated due to different bit rates, but this is not our main concern. Therefore, it is generally believed that (1 + D) pi/4 BPSK is recommended under low bit rate and AWGN channel.

6. Conclusions

In this paper, a new modulation scheme was proposed with low PAPR for low-cost terminals in 6G. We showed the principle of pi/2(N + 1) BPSK and (1 + D) pi/2(N + 1) BPSK, as well as the characteristics of the corresponding PAPR performance. We demonstrated that with an increase of N, PAPR can be reduced. Furthermore, our proposed scheme has the advantage of a relatively low out-of-band leakage. For convenience and without loss of generality, we derived LLR soft demodulation for N = 1, i.e., (1 + D) pi/4 BPSK. The performance results showed that a comparable performance can be attained in terms of BLER for our proposed scheme.