Low-complexity PTS PAPR reduction scheme for UFMC systems

Abstract

In this study, a low-computational-complexity partial transmit sequence (PTS) is proposed for reducing peak-to-average power ratios (PAPRs) of universal filtered multi-carriers (UFMCs). First, we analyze the differences in PAPR between a UFMC system and an orthogonal frequency division multiplexing system. Second, we characterize the threshold of the minimum peak power of the UFMC symbol and delete the candidate signal samples with power values below the threshold. This allows for a reduction in the number of time-domain samples that are multiplied by the corresponding phase-rotating vectors; in addition, it can be used for estimating the PAPR. Third, considering the relationship between the phase-weighting sequences, the computational complexity of the candidate signals is simplified. Numerical results show that the proposed PTS scheme achieves nearly the same PAPR reduction as that achieved by the conventional PTS scheme; meanwhile, it has considerably low computational complexity when the probability, $\beta$, of the maximum peak power being more than the possible minimum peak power is greater than or equal to 0.5.

Appendix

Equation (4) is a nonlinear programming problem with a constrained condition. The Lagrange multiplier scheme can be used to solve nonlinear programming problems with restrictions. The basic idea of the Lagrange multiplier scheme is to transform nonlinear programming problems with restrictions into non-boundary extremum problems. The scheme can solve non-linear optimal control problems with complex constraints of equality.

A nonlinear programming problem can be expressed as follows:

$$\begin{aligned} \begin{array}{lll} \min &{} f(x)&{} \\ st. &{} h_i (x)=0 &{} {i=1,2,\ldots ,m} \\ \end{array} \end{aligned}$$

(A.1)

where f(x) is an objective function, and $h_i (x)$ is a set of equations with constraints.

First, multiply m equalities with constraints by Lagrange multipliers $\lambda _1 ,\lambda _2 ,\ldots ,\lambda _m$, respectively. Then, merge it with the objective function. We can get the Lagrange equation, which can be expressed as follows:

$$\begin{aligned} L=f(x)-\sum \limits _{i=1}^m {\lambda _i h_i (m)} \end{aligned}$$

(A.2)

The objective function L can be thought to have $m+n$ variables. Compute $m+n$ derivatives of L and set those to zero. Then, the equation group can be expressed as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial L}{\partial x_j }=\frac{\partial f}{\partial x_j }-\sum \nolimits _{i=1}^m {\lambda _i } \frac{\partial h_i }{\partial x_j } &{} {j=1,2,\ldots ,n} \\ \frac{\partial L}{\partial \lambda _i }=h_i (x)=0 &{} {i=1,2,\ldots ,m} \end{array} \right. \end{aligned}$$

(A.3)

Based on Eqs. A.3, (4) can be expressed as follows:

$$\begin{aligned} \forall (i,j)\in \{0,\ldots ,M-1\}^{2},i\ne j, \frac{\beta _i^2 }{(1-e^{\gamma \beta _i })}=\frac{\beta _j^2 }{(1-e^{\gamma \beta _j })}\nonumber \\ \end{aligned}$$

(A.4)

Define $f(\beta )$ as $f(\beta )=\frac{\beta ^{2}}{e^{\beta r}-1}$ and the Lambert function $LW(x)e^{LW(x)}=x$ . Then, $f(\beta )$ is increases monotonically in the interval $[{\begin{array}{ll} 0&{} {\beta _0 } \\ \end{array} }]$ based on the property of the Lambert function. $\beta _0$ can be defined as follows:

$$\begin{aligned} \beta _0 =\frac{1}{\gamma }[2+LW(-2e^{-2})]{\dot{=}} \frac{1.59}{\gamma } \end{aligned}$$

(A.5)

Hence, if $x\in [{\begin{array}{ll} 0&{} {\beta _0 } \\ \end{array} }]$, Eq. (A.4) can be expressed as follows:

$$\begin{aligned} \forall (i,j)\in \{0,\cdots ,M-1\}^{2},i\ne j, {\beta _{i}} = {\beta _{j}} =\beta \end{aligned}$$

(A.6)

To optimize the CCDF solution, a few sufficient conditions must be satisfied. The Lagrange function of the $M *M$ Hessian matrix H can be expressed as follows:

$$\begin{aligned} H = \left[ {{ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {a(\gamma )}&{} {b(\gamma )}&{} \cdots &{} {b(\gamma )} \\ {b(\gamma )}&{} {a(\gamma )}&{} \ldots &{} {b(\gamma )} \\ \vdots &{} \vdots &{} \ddots &{} {b(\gamma )} \\ {b(\gamma )}&{} {b(\gamma )}&{} \cdots &{} {a(\gamma )} \\ \end{array} }} \right] \end{aligned}$$

where

$$\begin{aligned} \left\{ {\begin{array}{l} a(\gamma ) = \frac{\partial L(B,\lambda ,\gamma )}{\partial \beta _{_i }^2 } = \gamma e^{-\gamma }(\gamma -1)(1-e^{-\gamma })(1-e^{-\gamma })^{M-1} \\ b(\gamma ) = \frac{\partial L(B,\lambda ,\gamma )}{\partial \beta _i \partial \beta _j } = -\gamma ^{2}e^{-2\gamma }(1-e^{-\gamma })^{M-2} \end{array}} \right. \end{aligned}$$

Then, we need to assess the positive properties of the Hessian matrix H

$$\begin{aligned}&\forall Z=(z_1 ,\ldots ,z_{N+L-1} ), \,and \, {z_1 + \cdots + z_M \ne 0} \\&if \, {ZHZ^{T}>0} \\&then\, H \,is \, positive \end{aligned}$$

We have:

$$\begin{aligned}&ZHZ^{T} = \gamma (1-e^{-\gamma })^{M-2}e^{-\gamma }[e^{-\gamma }+\gamma -1]\\&\quad \times (z_1^2 +\cdots +z_M^2 )>0 \end{aligned}$$

Thus, the Hessian matrix H is a positive matrix. We can obtain the minimum CCDF as Eq. (5).