## 1 Introduction
The authors in [27; 28] showed that Discrete Fourier Transform hybridized with \(\mu\)-law (DFT-\(\mu\)) is better than DHT-\(\mu\) from PAPR reduction and BER points of view. However, their model considered only the AWGN channel and neglected HPA. The authors in [29] introduced DFT-PLC. However, PLC is impractical, as discussed before. The authors in [30] showed that DST-\(\mu\) is better than the WHT-\(\mu\), DHT-\(\mu\), and DCT-\(\mu\) from the PAPR reduction point of view. The model in [30] considered only the AWGN channel and neglected HPA. The authors in [31] introduced DST-A without any comparison with the previous precoding-companding techniques. The authors in [32] showed that DST-\(\mu\) is better than DST-A and DCT-A from PAPR reduction and BER points of view. Recently, DCT-\(\mu\) has been compared with DCT-\(\mu\)-clipping in [33], only from the PAPR reduction point of view. The authors in [34] showed that Zadoff-Chu Transform hybridized with \(\mu\)-law (ZCT-\(\mu\)) is better than WHT-\(\mu\) from PAPR reduction point of view. The authors in [35] hybridized ZCT with the Piecewise Exponential Companding technique. However, the solution is impractical because it uses inflection point and side information data as PLC. Finally, Square root Raised Cosine-based precoding was hybridized with \(\mu\)-law (SRC-\(\mu\)) in [36; 37] without compared to the previous precoding-companding techniques. The limitations of the discussed works are summarized as follows:  
* The authors in [22; 24; 27; 28; 30; 32; 33; 34; 36; 37] hybridized \(\mu\)-law with different precoding matrices due to its high PAPR reduction gain. However, BER and OOB radiation are more important metrics than PAPR reduction gain when working with PAPR reduction techniques that cause BER degradation, such as companding techniques. * Some works hybridized A-law companding [26; 31] and \(exp\) companding [23] with different precoding matrices. Although A-law and \(exp\) companding techniques have PAPR reduction gain and BER performance worse than the \(\mu\)-law companding, as shown in [10]. * The authors in [25; 29] hybridized piecewise based companding techniques with different precoding matrices. However, piecewise based companding techniques are impractical. * Many precoding-companding techniques [26; 29; 31; 36; 37], are not compared with any previous works. * HPA was neglected in the models [22; 26; 27; 28; 30; 31; 33; 34; 35; 36; 37], although HPA is the source of nonlinear distortion. * The authors in [22; 26; 28; 30; 31; 34; 36; 37] have not considered Rayleigh channel. However, OFDM was designed originally for Rayleigh channels. * Most importantly, the previous works did not clarify why the precoding matrices are selected, and the companding transforms are hybridized. Although this is the key challenge in the hybridization process. This paper proposes a new hybrid precoding-companding technique to reduce the BER and OOB radiation in the presence of HPA hybridizing Log companding and SRC precoding. The Log companding is selected because it is better than other practical companding techniques in terms of BER enhancement and OOB radiation reduction [10]. Also, the SRC precoding outperforms all other techniques [11]. Hence, hybrid SRC-Log can better reduce the BER and OOB radiation than the previous precoding-companding techniques. The proposed SRC-Log is compared with the ten state of the art precoding-companding techniques (i.e., DHT-\(A\)[26], SRC-\(\mu\)[37], DCT-\(\mu\)[33], DFT-\(\mu\)[28], DST-\(\mu\)[32], DHT-\(\mu\)[24], WHT-\(\mu\)[22], WHT-\(exp\)[23], DST-\(A\)[31], and ZCT-\(\mu\)[34]) in the presence of HPA from PAPR reduction gain, OOB radiation, Error Vector Magnitude (EVM), and BER points of view.