# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
###### Abstract  
Multiple-input multiple-output (MIMO) radar has acquired considerable attention as it offers an additional degree of freedom which results in performance gains when contrasted with the regular single antenna element radar system. Waveform optimization in MIMO radar is essential as it can offer tremendous improvements in target detection which are quantified in terms of reductions in the symbol error rate and improvements in target detection probability. In this work, we foster a strategy for the optimization of transmitter and receiver waveform in a collocated MIMO radar by only considering the second order statistics, thus relaxing the information of the instantaneous target states. Our contributions are primarily two-fold. First, we find a closed-form expression of the outage probability of an unknown target under clutter environment. For this prospect, we model the signal-to-interference-plus-noise ratio in a canonical quadratic structure, and then utilize the modern residue theory approach to characterize the distribution function. Secondly, we propose constrained and unconstrained optimization problems for the reduction in outage probability using algorithmic techniques such as interior-point, sequential-quadratic programming, and the active-set method for the optimization of the transmitter and receiver waveform. We also provide simulated re-enactments to validate our hypothetical deductions. MIMO radar; waveform design; indefinite quadratic forms; outage probability; constrained optimization; SINR analysis +
Footnote †: journal: _Mathematics_  
orthogonal waveform designing methods have been proposed in [9; 10; 11] and they deal with the minimization of auto-correlation and cross-correlation of the waveform. Furthermore, authors in [12; 13; 14; 15; 16; 17] deliberate on the waveform synthesis in creating beam patterns, and several works including [18; 19; 20; 21] deal with the maximization of the signal-to-interference-plus-noise-ratio (SINR). It is becoming evident that unlocking the true possibilities offered by modern collocated MIMO radars is achieved via waveform designs as well as detection and estimation approaches [22]. While the last decade has seen a greater interest and thus greater research output in the MIMO radar, there exist significant challenges which are need to be thoroughly addressed. Firstly, the assumption of a known target in [18; 23] is rather limiting. Secondly, the randomization procedure can often require undesirably large computational resources [18; 24]. Moreover, the exact characterization and minimization of outage probability for the collocated MIMO radar is rather involved [18; 25; 26; 27]. Lastly, several existing works utilize the SINR statistics where signal and interference powers are evaluated separately by assuming both the target and interferers as either moving slowly or completely still [24; 28; 29]. Motivated by the aforementioned discussion, we investigate herein the probability of outage based on the waveform optimization of collocated transmit antenna MIMO radar systems by relaxing the assumption of a known target and removing the constraint of the slow-moving or stationary target and interferers. We also provide a unified characterization methodology for the system by employing recent advancements in the indefinite quadratic forms (IQF) approach [30; 31; 32; 33]. The IQF approach enables us to achieve a closed-form expression for the cumulative density function (CDF) and the corresponding probability density function (PDF) in a simple form. Next, using the closed-form expression of the CDF, also referred to as the outage probability, we define several non-linear constrained and unconstrained waveform optimization techniques for collocated MIMO radar and reflect on the convergence, time complexity, and, most importantly, outage probability minimization. Table 1 provides selected papers on collocated MIMO radar and describes the key contrasting features of the proposed work.