A low complexity peak-to-average power ratio reduction scheme based on selected mapping and radix-II IFFT in OFDM systems

Abstract

Because of its lack of feedback process and the simplicity of its searching algorithm, conventional selected mapping (CSLM) is an efficient peak-to-average power ratio (PAPR) reduction technique in orthogonal frequency division multiplexing systems compared to the present techniques such as partial transmit sequence and active constellation extension. The requirement for large numbers of inverse fast Fourier transform (IFFT) blocks to provide desired PAPR reduction performance is introduced as the most significant drawback of the CSLM. This paper uses the special structure of an N-point radix-II IFFT in the CSLM and proposes a low complexity method to reduce the redundant calculations with almost the same PAPR reduction, bit-error rate, and power spectral density performances as those of the CSLM. The simulation results show that the computational complexity is reduced by at least 46.8% compared to that of the CSLM with approximately the same PAPR performance.

Appendix

In this appendix, the proposed scheme and some equations are described for an N-point radix-2 IFFT where $N=8$ and $K=4$. As mentioned above, K is a variable that is specified by the user and determines the computational complexity and PAPR reduction value. If $m^{{\prime }}=0,1$, then:

$$\begin{aligned} m_1= & {} 4(N/4-m^{{\prime }})-1=\left\{ {7,3} \right\} \nonumber \\ m_2= & {} 4(N/4-m^{{\prime }})-2=\left\{ {6,2} \right\} \nonumber \\ m_3= & {} 4(N/4-m^{{\prime }})-3=\left\{ {5,1} \right\} \nonumber \\ m_4= & {} 4(N/4-m^{{\prime }})-4=\left\{ {4,0} \right\} \end{aligned}$$

(36)

Therefore, the input signal is partitioned into the 4 groups by the proposed approach. The required matrices are defined as

$$\begin{aligned} B^{{\prime }}= & {} \left[ {{\begin{array}{cccccccc} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ {-1}&{} {-1}&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ j&{} j&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ {-j}&{} {-j}&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ \end{array} }} \right] _{4\times 8} = \;b_{ 1} \end{aligned}$$

(37)

$$\begin{aligned} X^{\prime }= & {} \left[ {{\begin{array}{cccccccc} {X_3 }&{} {X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {X_3 }&{} {X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {X_3 }&{} {X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {X_3 }&{} {X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ \end{array} }} \right] _{4\times 8} \end{aligned}$$

(38)

$$\begin{aligned} Y_1= & {} \left[ {{\begin{array}{cccccccc} {X_3 }&{} {X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {-X_3 }&{} {-X_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {jX_3 }&{} {jX_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ {-jX_3 }&{} {-jX_7 }&{} {X_6 }&{} {X_2 }&{} {X_5 }&{} {X_1 }&{} {X_4 \quad X_0 } \\ \end{array} }} \right] _{4\times 8} \end{aligned}$$

(39)

The matrix $Y_1$ shows that by changing the phase sequences 4 times, only 2 input samples are changed, and the other samples are fixed. Hence,$R_l$, $R_{l+N/8}$, $G_d$, and $H_d$ have to be calculated once, saved, and used for the subsequent rows, whereas in the CSLM method, all similar calculations should be performed during each searching procedure. According to Table 1, for $q=3$ and $l=0$, the saved parameters are $G_0 ,\;G_1 ,\;G_2 ,\;G_3 ,\;R_0$, and $R_1$ where $R_0 =\alpha _0 +W^{0}\beta _0$ and $R_1 =\alpha _0 -W^{0}\beta _0$ $(\alpha _0 =X_5 , \beta _0 =X_1 )$. $T_{K,_{\Pr o-Mul} }$ has 2 parts. The first part is related to the calculations of the first row, and the second part is for the other rows of the selected matrix. $T_{K,_{\Pr o-Mul} }$ and $T_{K,_{CSLM-Mul} }$ with $K=4$ and $N=8$, are evaluated as

$$\begin{aligned}&T_{K,_{\Pr o-Mul} } =4\log _2 8+(3)\underbrace{(\log _2 8+4)}_7 \end{aligned}$$

(40)

$$\begin{aligned}&T_{K,_{CSLM-Mul} } = 4(4\log _2 8) \end{aligned}$$

(41)

Referring to (40) the second part is shown by bold circles in Fig. 4 (i.e., 7 multipliers for each row, except the first row). The following matrix is generated, whit $N=8$ and $K=8$:

$$\begin{aligned} B^{{\prime }} =\left[ {{\begin{array}{l} {b_1 } \\ {b_3 } \\ \end{array} }} \right] =\left[ {\begin{array}{cccccccc} {\begin{array}{cccccccc} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ {-1}&{} {-1}&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ j&{} j&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ {-j}&{} {-j}&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ \end{array} } \\ {\begin{array}{cccccccc} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1&{} 1 \\ 1&{} 1&{} 1&{} 1&{} {-1}&{} {-1}&{} 1&{} 1 \\ 1&{} 1&{} 1&{} 1&{} j&{} j&{} 1&{} 1 \\ 1&{} 1&{} 1&{} 1&{} {-j}&{} {-j}&{} 1&{} 1 \\ \end{array} } \\ \end{array}} \right] _{\;8\times 8} \end{aligned}$$

(42)