# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
###### Abstract
Based on the points discussed above, it is interesting to discuss the performance of an adaptive dual-function MIMO radar-communications system that combines the adaptive FH waveform optimization scheme stated in [38] and PM-based information embedding strategy stated in [31]. To further adapt to the dynamic radar environment, we consider the problem of adaptive waveforms design and propose a two-step waveform optimization scheme, which provides better target detection performance and high data-rate communication capability between the proposed dual-function systems. The proposed scheme is summarized as follows:  
Step 1. Waveform Design: this part includes the design of the FH waveform for the individual MIMO transmit antennas. The primary goal is to maximize the MI between the target-scattering signal and the estimated target response. The method ensures that the target-scattering signal at each time instant is dependent on the target response. Step 2. Waveform Selection: after the best waveform ensemble is gained, part two is to choice the suitable PM-based FH sequences for emission. The principle of this module is to minimize the MI between consecutive target scattering signals. This part ensures that we constantly obtain target returns that become independent of each other in time, with the purpose of achieving more information about the target characteristics at each time instant of reception. The premise of the FH waveform optimization scheme is channel estimation. The target feature estimation can be performed by the MIMO radar receiver through observations implemented in the previous time instant. A feedback loop enables the delivery of the estimates to the dual-function MMO radar-communications transmitter. At a result, the optimization strategy allows the MIMO radar transmitter to constantly adjust FH codes to suit the time-varying channel scene. We choose the FH waveform for the following reasons:  
1. The channel environment is complicated to wireless communications due to densely populated scatterers. However, FH waveforms are immune to multipath channel fading under the circumstances. 2. The FH waveforms are robust to antagonistic environments by offering low interception probability. Furthermore, FH waveforms are immune to clutter interference. 3. The constant-modulus waveforms have the property of high transmission power efficiency. FH waveforms enjoy the constant-modulus feature and are easy to generate. The primary innovations of our work are summarized as follows:We define the PM-based FH waveforms in dual-function MIMO radar-communications configuration and derive the associated MIMO ambiguity function;
2. We develop a two-step waveform optimization scheme in the adaptive PM-based dual-function MIMO radar-communications framework;
3. We evaluate the performance of the proposed scheme in terms of target response estimation, delay-Doppler resolution and communication symbol error rate (SER). 4. We compare the proposed scheme with other radar systems through analysis of the target detection and receiver operating characteristics (ROC) in an interference noise environment. The organization of this paper is as follows. In Section 2, we describe the dual-function MIMO radar-communications system and the PM-based FH signal model. In Section 3, we derive the MIMO radar ambiguity function of the PM-based FH waveform. In Section 4, we present an adaptive approach to optimizing the proposed information embedding waveform. The transmit waveforms are designed at step 1 of the algorithm and selected based on the criterion presented in step 2. The simulation results demonstrating the proposed scheme are presented in Section 5. Finally, our conclusions and directions for possible future work are drawn in Section 6. Throughout this paper, the following notations will be used.