## 1 Introduction  
The high-speed data rate technologies such as 4G and 5G vastly use Orthogonal Frequency Division Multiplexing (OFDM) and expected to be used in Beyond 5G (B5G). Also, OFDM has been combined with most emerging communication techniques such as Cognitive Radio (CR), Massive Multiple-Input and Multiple-Output (mMIMO) [1,2]. However, OFDM has a high Peak-to-Average Power Ratio (PAPR) problem. This problem leads to excessive distortion in the OFDM signal due to the nonlinear High Power Amplifier (HPA) in the transmitter chain. Distortion in the OFDM signal causes degradation in the Bit Error Rate (BER) of the OFDM systems and grows the out-of-band (OOB) radiation. This problem can be solved by amplifying the signal with high PAPR with a large back-off or using a highly linear amplifier at the transmitter. The first solution causes HPA to work inefficiently, while the second solution is expensive. Thus, we need to reduce the PAPR of the OFDM signal [3]. Several PAPR reduction methods are introduced, such as clipping [4], Partial Transmit Sequence (PTS) [5], Selective Mapping (SLM) [6], Interleaving [7], Active Constellation Extension (ACE) [8], Tone Reservation (TR) [9], Tone Injection (TI) [9], companding techniques [10], and precoding techniques [11]. There are mainly three categories for PAPR reduction techniques: (i) coding techniques, (ii) Multiple Signal Representation (MSR) techniques, and (iii) adding-signal techniques [12]. Different metrics are used to compare PAPR reduction techniques, such as PAPR reduction gain, BER enhancement, OOB-radiation reduction, excess average power, rate loss, computational complexity, and downward compatibility. Generally, none of the PAPR reduction techniques can be claimed as the best PAPR reduction technique. Of course, each PAPR reduction technique has its advantages and disadvantages [3; 13; 14]. For example, MSR techniques have large PAPR reduction gain and well reduce BER and OOB-radiation. However, they require excessive computational complexity in addition to rate loss [15]. On the other hand, companding techniques have small PAPR reduction gain with trivial computational complexity [10]. Also, precoding techniques have moderate PAPR reduction gain with small computational complexity [11]. Hybridization between two or three PAPR reduction techniques is used to gain the advantages of the combined techniques. In the literature, different hybrid techniques combine different PAPR reduction techniques from different categories, or the same category [16]. For example, PTS-SLM [17], and PTS-Interleaving [18] are examples of hybridizing techniques that belong to the same category (i.e., MSR). On the other hand, ACE-PTS [19], precoding-clipping [20], PTS-companding [21], and precoding-companding [22] are examples of hybridizing different PAPR reduction techniques from different categories. Hybrid precoding-companding techniques are attractive as they require small computational complexity to achieve large PAPR reduction gain. In the literature, many hybrid precoding-companding techniques exist. Walsh-Hadamard Transform-based precoding hybridized with \(\mu\)-law (WHT-\(\mu\)) [22] is the first precoding-companding technique introduced in the literature. In [23] WHT hybridized with exponential (\(exp\)) companding (WHT-\(exp\)) was compared with WHT-\(\mu\). However, the parameter of \(\mu\)-law was not optimized to enhance its BER performance in [23]. The authors in [24] evaluated the performance of Discrete Hartley Transform hybridized technique with \(\mu\)-law (DHT-\(\mu\)) considering different companding profiles of \(\mu\)-law. However, all the companding profiles showed the same performance in Rayleigh fading channel. The authors in [25] hybridized DHT with Piecewise Linear Companding (PLC). However, PLC is impractical. It requires many side information bits to inform the receiver with the position of data samples above or below the inflection point [10]. Recently, DHT-A introduced in [26] without comparison to any previous precoding-companding technique.