Orthogonal frequency division multiplexing (OFDM) continues to be deployed in 5G communication systems and is likely to be used in beyond 5G (B5G) communication systems as well, due to its many advantages. However, one major drawback with OFDM systems is its high peak-to-average power ratio (PAPR), especially with large bandwidth transmissions. In this paper, we have provided a regularization optimization based flexible hybrid companding and clipping scheme (ROFHCC) used for PAPR reduction in OFDM systems. To reduce the design complexity, the companding function has two parts. It restrains the signal samples with amplitudes over a given value to a constant value for both peak power reduction and small power compensation. For signals with samples less than a given amplitude, they are expanded by a linear companding function. We build a regularization optimization model to jointly optimize the companding distortion as well as the continuity of the companding function for bit error rate (BER) performance as well as power spectral density (PSD) performance. Simulation results indicate that for the same PAPR performance, the proposed companding scheme has an advantage over the referenced companding schemes. For example, when the average signal power is normalized to be 1, we choose both PAPR for ROFHCC scheme and two-piecewise companding (TPWC) scheme as 4 dB, then we can find that at $BER=10^{-4}$ , the minimum required $E_{b}/{N_{0}}$ for ROFHCC scheme is around 2.3 dB lower than TPWC scheme.
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