An Optimized Algorithm for Peak to Average Power Ratio Reduction in Orthogonal Frequency Division Multiplexing Communication Systems: An Integrated Approach

OFDM has emerged as a widely adopted method for multicarrier modulation, acclaimed for enhancing spectrum efficiency while concurrently improving immunity to inter-symbol interference (ISI) and multipath interference [1], [2], [3]. However, the nature of OFDM signals, comprising a superimposition of several sub-channel signals, often leads to an escalated PAPR when these signals converge in phase. Such a phenomenon necessitates the operation of the power amplifier in a system within a broad linear range to avert signal distortion, as illustrated in Figure 1.

The challenge of reducing the PAPR in OFDM signals has given rise to three prevalent approaches: pre-distortion, encoding, and probability methods. The pre-distortion method entails preliminary non-linear processing of signals, causing high-power signals to undergo pre-distortion. This method is characterized by its simplicity and straightforwardness, despite the introduction of in-band noise and out-band interference from the non-linear transformation [4], [5].

The encoding method deploys adjustable encoding patterns to generate multiple code groups, selecting the code group with the smallest PAPR to transmit OFDM symbols. However, the complexity and tediousness of encoding and decoding processes mar its usability [6], [7], [8], [9]. Furthermore, the drastic reduction in the information rate rendered by this method restricts its application to fewer sub-carriers.

Lastly, the probability method aims to reduce the peak signal occurrence probability, although it doesn't ensure PAPR reduction and may introduce some information redundancy [10], [11], [12]. Noteworthy examples of probability methods include the PTS and SLM, which modify the phase distribution of the original signals to decrease the likelihood of in-phase superposition, effectively reducing the PAPR of signals without distortion.

This study is organized as follows: Section II provides a theoretical analysis, defining PAPR, and detailing the principles of traditional PTS and µ-law Companding Transformation. Section III presents the proposed PTS-Companding method, and the subsequent sections IV and V respectively demonstrate simulation results and conclude the study.

The PAPR is defined as the ratio between the maximum power and the average power of the signal over a symbol period. The PAPR of an OFDM signal can be expressed as shown in Eq. (1):

where, $a_n$ represents the OFDM signal post the influence of C-point Inverse Fast Fourier Transform (IFFT), as illustrated in Eq. (2):

where, $\mathrm{C}$ denotes the sub-carrier index, while $A_i(0 \leq i \leq C-1)$ signifies the $\mathrm{i}^{\text {th }}$ complex modulated symbol in the OFDM system [13], [14]. The CCDF, illustrating the probability of the PAPR exceeding a particular threshold, is often employed to gauge the PAPR reduction capacity of a communication system [15], [16], [17], [18].

The probability method primarily employs linear transformation, given in Eq. (3):

where, $A_n$ represents the original input data before IFFT in the frequency domain, and $O_n$ symbolizes the output following the linear transformation. The probability method endeavors to identify a C-point vector $\mathrm{V}$ and $\mathrm{W}$ that can diminish the peak value occurrence probability of the transmission signals post IFFT. The PTS method initially sets vector $\mathrm{W}$ to zero and subsequently selects an apt vector $\mathrm{V}$ to reduce peak signal incidence probability. The linear transformation of PTS consists only of phase rotation, limiting the vector V's amplitude to unit amplitude [19], [20], [21], [22].

In a PTS-OFDM system, three data partitioning methods are prevalent: adjacent partition, interleaved partition, and random division. Evidently, random division exhibits superior PAPR performance, albeit interleaved partitioning presents the least complexity. Furthermore, the PAPR in a PTS system is influenced by the number of groups V and phase factors F; an increase in either improves the PAPR, albeit at the expense of system complexity [23], [24], [25], [26], [27], [28].

The Companding Transformation employs non-uniform quantization functions, such as the $\mu$-law and $A$-law companding functions, to compress high-power signals and amplify low-power signals, thus sustaining average power. This method enhances the resistance of low power signals and reduces PAPR. A critical step in this process involves the transformation of signals post-orthogonal modulation in the time domain using the Companding Transformation, a step less computationally intensive than PTS. The received signals are restored via an equivalent inverse Companding Transformation before demodulation [29], [30], [31], [32].

A conventional Companding Transformation is the $\mu$-law, using a non-linear transform function based on the $\mu$-law non-uniform quantization in speech processing. It should be noted that this method introduces in-band noise and out-band interference at the receiver since the inverse Companding Transformation amplifies large signals and their noise while attenuating small signals and their noise, leading to an inferior BER [33].

This study explores the development of a RIA that effectively merges both the PTS and Companding Transformation to alleviate the occurrence of maximum peak signals [34]. The architecture of the proposed system is displayed in Figure 2.

The original data, denoted as "$A$" is partitioned into "$S$" non-overlapping sub-blocks of length "$N$", represented as $A^s, s=1,2,3,4, \ldots \ldots S$, where s ranges from 1 to $S$. This process is outlined in Eq. (4).

After the application of the C-point IFFT, each data block is multiplied by its corresponding phase factor, $W_s$. The phase factors across different data blocks are statistically independent, resulting in statistically independent phase factors within the rotation vector, as expressed in Eq. (5).

The optimization of the phase vector involves selecting suitable weighted coefficients $\left[w_1, w_2, w_3, w_4 \ldots \ldots w_s\right]$ that minimize the peak values of the powers. The optimization condition for these coefficients is given in Eq. (6).

where, argmin(.) is the condition that allows the function to reach its lowest value [35]. Subsequently, $\mu$-law Companding Transformation is applied to the signal “a” improving the PAPR. This transformation is represented by Eq. (7).

At the receiver end, the received signal "$y$" is first transformed using the $\mu^{\prime}$-law inverse Companding Transformation where $\mu^{\prime}$ denotes the inverse companding coefficient and A’ represents the normalization constant, taken as $\max (|o|)$ [36], [37]. The signal $O^{\prime}$, divided into sub vector signals of length C, is used to determine carrier signal side information, represented by $\left\{w_1, w_2, w_3, w_4 \ldots \ldots . w_s\right\}$. A N-point FFT is then applied after dividing the $O^s$ by the rotation factor $W_s$. Finally, the data blocks are added as outlined in Eq. (8).

This integrated approach significantly improves PAPR performance without compromising BER or transmission rate. Methods to enhance BER and maintain transmission rate are discussed in parts 1) and 2), respectively [38], [39], [40].

To counteract the poor BER performance induced by the Companding Transformation, a companding coefficient $\mu^{\prime}$ at the receiver end is chosen to be less than that of the transmitter, i.e., $\mu^{\prime}=I_\mu(I<1)$. This selection reduces the noise gain of larger signals and ensures a satisfactory BER.

PTS employs an appropriate rotation vector to minimize the peak signal occurrence. The exhaustive search required to traverse all phase combinations $\left[w_1, w_2, w_3, w_4 \ldots \ldots w_s\right]$ is $F$, where "$S$" is the group number and "$F$" is the phase factor number. As "$S$" and "$F$" increase, the system complexity escalates [41], [42], [43], [44], [45], [46], [47].

To alleviate computational demands, a simple search is recommended when the number of phase combinations is extensive. Initially, all weighted coefficients $W_S$ are set to $1(s=1,2,3,4, \ldots \ldots S)$ and $b 1$ is allowed to traverse every vector in the phase space. An optimal vector is selected such that the current PAPR reduction is maximized [48], [49], [50], [51], [52], [53], [54]. This selection process is repeated for the remaining weighted coefficients $w_s(s=1,2,3,4, \ldots \ldots S)$. This iterative search is typically performed three times, with negligible PAPR performance improvement beyond three iterations. While this simple search compromises PAPR reduction, it notably enhances the transmission rate when "$F$" or "$S$" is high [55], [56], [57], [58], [59], [60], [61], [62].

The proposed RIA underwent comprehensive simulations using MATLAB, adopting the parameters delineated in Table 1. Subsequently, PAPR and BER performances were compared between the traditional PTS and $\mu$-law Companding Transformation. An audio signal of 12KB size was employed as the input source for these simulations.

In Figure 3, a comparative analysis of PAPR performance between the proposed method and existing strategies is illustrated. Observations from Table 2 indicate reductions in both BER and elapsed time. It is evident from Figure 3 that variations in PAPR between the novel technique and established techniques such as clipping, SLM, TR, PTS, and Companding Transformation are +0.90dB, -0.80dB, -3.4dB, -0.47dB, and -7 dB, respectively, at a probability of $10^{-2}$. Although clipping has superior PAPR performance, its BER is approximately six times greater. When the novel method's factor $F$ is adjusted to 4, the Complementary Cumulative Distribution Function (CCDF) distribution curve closely aligns with that of clipping, and the novel system's BER only increases by 0.02%. Table 2 reaffirms that the novel system can ensure a precise BER and transmission rate.

Figure 4 presents an analysis of PAPR performance under varying conditions of $\mu$, specifically, when $\mu$ is 4, 10, 28, and 82. With the conventional $\mu$-law Companding Transformation, the PAPR performance improves as $\mu$ increases, albeit at the expense of signal quality. At $\mu$ =82, the BER reaches 5.969%. Conversely, the novel system generally demonstrates a positive relationship between $\mu$ and PAPR performance, although PAPR reduction is less noticeable. At $\mu$ =82, the BER of the novel system is 5.32%, showing the signal distortion level comparable to that of the Companding Transformation. As illustrated in Figure 4, when $\mu$ is 4, and the inverse companding scale I is 0.4, 0.7, and 1, the BER is 0.13%, 0.18%, and 0.25%, respectively, even though PAPR remains relatively stable, indicating that a smaller inverse companding coefficient $\mu$' at the receiver can improve signal quality.

Figure 5, Figure 6 and Figure 7 showcase PAPR performance due to various factors involved in PTS, namely, data partitioning methods, group number $V$, and phase factor number $P$. The results in Figure 5 indicate that optimal PAPR performance can be achieved via random division for the novel system and PTS. According to Figure 6, the novel method can efficiently reduce PAPR of Orthogonal Frequency Division Multiplexing (OFDM) signals; the greater the group number $S$, the better the PAPR, but with increased system complexity.

In Figure 7, a comparison is made with a varying phase factor number $P$. As $F$ increases, more options for weighted coefficients become available, improving PAPR but compromising the system's transmission rate. As $S$ or $F$ increases, more combinations need to be traversed to find the optimal solution, thus increasing time. However, by adopting a simplified search, PAPR reduction is somewhat weakened. The exhaustive search elapsed times are 7.19s and 104.11s for $S$=4/$F$=4 and $S$=4/$P$=8, respectively, while the elapsed times for the simplified search are 2.10s and 3.25s. This indicates a trade-off between complexity and PAPR performance.

To provide a more comprehensive understanding of the proposed technique, future work could include a detailed breakdown of the PTS method, including further elucidation on the effect of group number $V$ and phase factor number $P$ on PAPR performance. This could also be supplemented by an extensive discussion on why the simple search strategy affects PAPR reduction and how this trade-off impacts real-world applications of the system.

This study prioritized the mitigation of PAPR in an OFDM system utilizing the proposed RIA. Analyses indicate a lower CCID of the proposed RIA compared to the traditional PTS method, given $S$=8,4,2. Additionally, with respect to $\mu$=4,10,28,82, the RIA has demonstrated superior performance over the conventional CT technique in terms of CCID.

Considering the methods of interleaved partition, adjacent partition, and random partition for determining the CCDF between the proposed RIA and conventional PTS, simulation analyses have demonstrated the superior performance of RIA. The RIA algorithm has effectively integrated the PTS and CT to decrease PAPR to 0.55dB and 0.656dB compared to the respective PTS and CT algorithms. A reduction of signal distribution by 0.02% relative to conventional methods was observed.

Elapsed times for the exhaustive PAPR search were determined as 7.65s and 104.12s for S=4/F=4 and S=4/F=8, respectively. For the simplified search, elapsed times were recorded as 2.10s and 3.25s.

Looking ahead, the proposed algorithm may offer significant enhancements to the BER and transmission rates of future wireless communication technologies such as 5G and 6G. As a rising number of urban environments transition to smart city infrastructures, an increasing volume of data communication will occur via IoT devices. These innovations necessitate highly accurate and rapid communication processes. The proposed methods, albeit necessitating more parallel operations, demonstrate potential compatibility with 6G communication processes.

Limitations were identified in operating the OFDM communication system within 6G technology, where parallel operations can generate jitters and hazards, leading to signal attenuation. These effects could compromise the accuracy of long-distance communication and result in a loss of original information.

Future research may benefit from a comprehensive examination of how the proposed RIA algorithm performs under different real-world conditions, especially within the context of smart city infrastructures. Furthermore, studies could delve into mitigating the limitations identified in this research, particularly those related to long-distance communication accuracy and the handling of information loss. These advancements could support the adaptation of the proposed RIA algorithm for future wireless communication technologies, ensuring accurate and fast communication for increasingly data-driven environments.