Estimation and Cancellation of Nonlinear Companding Noise for Companded Multicarrier Transmission Systems

Abstract

Nonlinear companding transform (NCT) is an efficient method to reduce the high peak-to-average power ratio (PAPR) of multicarrier transmission systems. However, the introduced companding noise severely restrains the bit-error-rate (BER) performance. In this paper, a general and simple companding noise cancellation (CNC) technique is proposed to mitigate the nonlinear companding noise at the receiver. By exploiting the Bussgang theorem and reconstructing the companding process at the transmitter, the estimated approximate companding noise can be used to refine the received signals. Furthermore, by employing the proposed approach to a typical exponential companding (EC), our results indicate that the proposed scheme can greatly relieve the conventional bottleneck, i.e. the so-called trade-off between the PAPR reduction and BER performance, of NCTs. It shows that for a 512-subcarrier and quadrature phase shift keying modulated orthogonal frequency division multiplexing system, the gap of the signal-to-noise ratio is no more than 0.3 dB at ${P_e} = 1 \times {10^{ - 5}}$ between the ideal performance bound and EC-CNC regardless of the companding degree ($d=1$ or $d=2$) over additive white Gaussian noise channel.

Appendix: Analysis of the BER Performance with HPA over Fading Channel

Let ${s_n}$ denotes the output after the companded signal $y_n$ passing through the SSPA, then, ${s_n}$ can be expressed as

$$\begin{aligned} {s_n}& = y_n+ {w_n} \nonumber \\& = \alpha {x_n} + {d_n} + {w_n}, \end{aligned}$$

(24)

where ${w_n}$ is the equivalent noise caused by the SSPA. When transmitted through the fading channel, then, the received signal is

$$\begin{aligned} {r_n} = {h_n} * \left( {\alpha {x_n} + {d_n} + {w_n}} \right) + {v_n}. \end{aligned}$$

(25)

In order to obtain the decided sequence $\left\{ {{{{\hat{X}}}_k}} \right\} _{k = 0}^{k = N - 1}$, the channel estimation and equalization should be performed before step 1, hence, the equalized signal $\left\{ {{{{\tilde{r}}}_n}} \right\} _{n = 0}^{n = JN - 1}$ is therefore (assume with ideal channel estimation)

$$\begin{aligned} {{{\tilde{r}}}_n}& = {h_n}^{ - 1} * \left\{ {{h_n} * \left( {\alpha {x_n} + {d_n} + {w_n}} \right) + {v_n}} \right\} \nonumber \\& = \alpha {x_n} + {d_n} + {w_n} + {h_n}^{ - 1} * {v_n}, \end{aligned}$$

(26)

where ${h_n}^{ - 1}$ is the set of tap coefficients of the equalizer.

Since the channel equalization will often enhance the channel noise, thus, the decision errors of $\left\{ {{{{\hat{X}}}_k}} \right\} _{k = 0}^{k = N - 1}$ will be increased compared with the ideal AWGN channel. After employing the CNC technique at the receiver, the refined channel observation in step 6 is therefore

$$\begin{aligned} {{\hat{r}}_n} = \alpha {x_n} + \left( {{d_n} - {{{\hat{d}}}_n}} \right) + {w_n} + {h_n}^{ - 1} * {v_n}. \end{aligned}$$

(27)

Different from the ideal AWGN channel with soft limiter or the practical SSPA, in fading channel, the dominant interference will become the component ${w_n} + {h_n}^{ - 1} * {v_n}$ as well as the enhanced estimation error among $\left( {{d_n} - {{{\hat{d}}}_n}} \right)$, hence, the BER performance improvement for the CNC technique with SSPA over fading channel will not be so optimistic as that in the AWGN channel.