# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
## 3 Characterization of Outage Probability  
This section deals with the exact characterization of outage probability for the collocated antenna MIMO radar system. The notion of outage probability herein is that the instantaneous value of SINR expression, which is a random variable, falls below a predefinedthreshold value, thus resulting is a monotonic CDF function of SINR. Hence, we proceed by imposing a condition on the CDF of SINR given in (10), i.e., \(Pout(\gamma|\zeta)=Pr(\eta(f,\mathbf{f})<\gamma)\), where \(\zeta=\|\mathbf{c}\|_{\mathbf{B}}^{2}\) defines the condition and \(\gamma\) is a predefined threshold value. Hence, the outage probability is expressed as:  
\[Pout(\gamma|\zeta)= Pr\left(\frac{||\mathbf{\bar{t}}||_{\mathbf{A}}^{2}}{\sigma^{2} \|\boldsymbol{h}\|^{2}+\zeta}<\gamma\right),\] \[= Pr\Big{(}(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma-|| \mathbf{\bar{t}}||_{\mathbf{A}}^{2}>0\Big{)},\] \[= \int_{-\infty}^{\infty}f(\mathbf{\bar{t}})u\Big{(}(\sigma^{2}\| \boldsymbol{h}\|^{2}+\zeta)\gamma-||\mathbf{\bar{t}}||_{\mathbf{A}}^{2}\Big{)} d\mathbf{\bar{t}}, \tag{11}\]  
where \(f(\mathbf{\bar{t}})\) is the PDF of an \(L\)-dimensional \(\{L=N_{R}(L_{R}+1)N_{T}(L_{T}+1)\}\) circular white Gaussian random vector \(\mathbf{\bar{t}}\), mathematically, \(f(\mathbf{\bar{t}})=\frac{1}{\pi^{L}}e^{-||\mathbf{\bar{t}}||^{2}}\). Moreover, using the Fourier representation of unit step function as in [30], the CDF in (11) is expressed as  
\[Pout(\gamma|\zeta) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\times\frac{1}{\pi^{L}}\int _{-\infty}^{\infty}\frac{e^{(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma( \omega+\beta)}}{(j\omega+\beta)}e^{-||\mathbf{\bar{t}}||_{\mathbf{A}}^{2}+ \boldsymbol{\Lambda}(j\omega+\beta)}dtdw, \tag{12}\] \[= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{(\sigma^{2}\| \boldsymbol{h}\|^{2}+\zeta)\gamma(\omega+\beta)}}{(j\omega+\beta)|\mathbf{I}+ \boldsymbol{\Lambda}_{A}(j\omega+\beta)|}dw,\] \[= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{(\sigma^{2}\| \boldsymbol{h}\|^{2}+\zeta)\gamma(\omega+\beta)}}{(j\omega+\beta)(1+ \lambda_{A}(j\omega+\beta))}dw,\]  
where we achieve the second equality by using the multi-dimensional Gaussian integral solution, and \(\boldsymbol{\Lambda}_{A}\) is a diagonal matrix having eigenvalues of matrix \(\mathbf{A}\) as elements. The third equality is obtained owing to rank one of matrix \(\mathbf{A}\); thus, such a matrix has only one eigenvalue which is denoted as \(\lambda_{A}\). Next, we remove the condition in (11) by evaluating the CDF, i.e., \(F(\zeta)\) by using the quadratic formulation as  
\[F(\zeta) = Pr\Big{(}\|\mathbf{\bar{c}}\|_{\mathbf{B}}^{2}<\zeta\Big{)}, \tag{13}\] \[= Pr\Big{(}\zeta-\|\mathbf{\bar{c}}\|_{\mathbf{B}}^{2}>0\Big{)},\] \[= \int_{-\infty}^{\infty}f(\mathbf{\bar{c}})u\Big{(}\xi-\|\mathbf{ \bar{c}}\|_{\mathbf{B}}^{2}\Big{)}d\mathbf{\bar{t}},\] \[= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{\xi(j\omega+\beta)} }{(j\omega+\beta)|\mathbf{I}+\boldsymbol{\Lambda}_{B}(j\omega+\beta)|}dw,\]  
where \(f(\mathbf{\bar{c}})\) is the PDF of \(\mathbf{\bar{c}}\). As in other notable works [37; 38], clutter is considered as a Gaussian random vector.