# Differential Evolution in Waveform Design for Wireless Power Transfer
## 1 Introduction  
A lot of research has been recently carried out on the Internet of Things (IoT) concept. IoT is an evolutionary approach that is promising to make a great impact on everyday life and business activities. It is believed that its wide deployment will also bring economic growth, but this requires the advancement of various enabling technologies [1]. One of these technologies is Radio Frequency (RF) Wireless Power Transfer (WPT), which is used for powering up low-power devices without the need of batteries or cables [2; 3; 4]. In a traditional radiative WPT system, a transmitter device generates a continuous electromagnetic field, which transmits power across the medium to a receiver device (rectenna). However, the transmission of pulsed electromagnetic fields that demonstrate higher peak-to-average power ratio (PAPR) has proved to be advantageous. Pulsed wave signals are multi-tone signals that exploit the nonlinear behavior of the receiver's rectifying circuit and increase its RF-DC conversion efficiency [5; 6]. When the propagation between the transmitter and the receiver is characterized by a multipath channel function, Channel State Information (CSI) should be taken into account for the design of multi-tone signals. Towards this direction, multi-tone waveforms constructed by Time Reversal (TR) processing are proposed for WPT in [7; 8]. TR waves are designed in a predetermined way, according to the CSI, and are capable of focusing on the receiver with the highest peak voltage. Consequently, they exploit both the multipath propagation channel and the nonlinear behavior of the receiver'srectifier. The same waveform design technique was proposed in [9, 10] as an Adaptive Matched Filter technique and Frequency-Maximal Ratio Transmission technique, respectively. Another advancement in waveform design for WPT is the optimization of multi-tone waveforms according to an objective function. This function incorporates both the CSI and the rectifier's nonlinear behavior through the analytical models of a WPT system and a simple rectenna introduced in [9]. It has been verified by numerical simulations that the transmission of these waveforms is the most efficient way to transfer power wirelessly [9]. An improvement of the objective function was proposed in [10] for the same optimization problem. The authors in both of these publications denoted that the non-convex nature of the problem is the cause of obtaining non-guaranteed globally optimal solutions by the proposed Sequential Convex Programming (SCP) methods. Many works utilizing the basic framework of [9] for waveform optimization have been published. Experimental verification of the observations made in [9] is provided in [11, 12]. Low complexity optimization methods are proposed in [10, 13, 14]. An extension of the analytical model for application in wirelessly powered backscatter communications is introduced in [15]. In such systems, the Signal to Noise Ratio (SNR) value of the backscattered signal is equally important and is taken into consideration by adding one more constraint to the problem. The model is further extended by considering multi-tone waveforms with modulated carriers and adding a constraint to ensure a minimum rate of information transfer [16]. The waveform optimization problem has been examined for different types of systems like Single Input Single Output (SISO) in [9, 10, 15], Multiple Input Single Output (MISO) in [9, 12, 13, 16], multiple users MISO in [9, 13], Multiple Input Multiple Output (MIMO) in [9] and multi-receiver SISO in [17]. In artificial intelligence, Evolutionary Algorithm (EA) is a subset of evolutionary computation, i.e., a population-based metaheuristic optimization algorithm. EAs are commonly used in real-world problems and are capable of tracking the global optimum in challenging objective functions.