# Statistical Method Based Waveform Optimization in Collocated MIMO Radar Systems
## 3 Characterization of Outage Probability
Now, differentiation of \(F(\zeta)\) yields the PDF \(f(\zeta)\) which is simplified using the indefinite quadratic forms approach [30] as follows  
\[= Pr\Big{(}(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma-|| \mathbf{\bar{t}}||_{\mathbf{A}}^{2}>0\Big{)},\]0 \[= Pr\Big{(}(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma-|| \mathbf{\bar{t}}||_{\mathbf{A}}^{2}>0\Big{)},\]1 \[= Pr\Big{(}(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma-|| \mathbf{\bar{t}}||_{\mathbf{A}}^{2}>0\Big{)},\]2  
where \(\boldsymbol{\Lambda}_{B}\) is the diagonal matrix containing the eigenvalues of Hermitian matrix \(\mathbf{B}\), \(\lambda_{B}\) is the only non-zero eigenvalue resulting in second equality, whereas the third equality is from the definition of residue theory [39], i.e.,\[= Pr\Big{(}(\sigma^{2}\|\boldsymbol{h}\|^{2}+\zeta)\gamma-|| \mathbf{\bar{t}}||_{\mathbf{A}}^{2}>0\Big{)},\]3  
Next, the outage probability with the aforementioned aspects simplifies to  
\[ \begin{split} Pout(\gamma)=&\int_{-\infty}^{\infty} Pout(\gamma|\zeta)f(\zeta)d\zeta,\\ =&\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ \sigma^{2}\|h\|^{2}\gamma(j\omega+\beta)}}{(j\omega+\beta)(1+\lambda_{A}(j \omega+\beta))}\frac{1}{\lambda_{B}}\times\int_{-\infty}^{\infty}e^{\frac{- \zeta}{\lambda_{B}}+\zeta\gamma(j\omega+\beta)}u\Big{(}\frac{\zeta}{\lambda_{ B}}\Big{)}d\zeta dw.\end{split}  \tag{16}\]  
Since \(\mathbf{B}\) is a positive semi-definite matrix, \(\lambda_{B}\) will have a non-negative value. Therefore, the inner integral of variable \(\zeta\) is solved as follows  
\[\int_{-\infty}^{\infty}e^{\frac{-\zeta}{\lambda_{B}}+\zeta\gamma(j\omega+\beta )}u\Big{(}\frac{\zeta}{\lambda_{B}}\Big{)}d\zeta=\int_{0}^{\infty}e^{-\zeta( \frac{1}{\lambda_{B}}-\gamma(j\omega+\beta))}d\zeta=\frac{1}{\frac{1}{ \lambda_{B}}-\gamma(j\omega+\beta)}. \tag{17}\]  
\[ \begin{split} Pout(\gamma)=\frac{1}{\lambda_{B}}& \times\quad\frac{1}{2\pi}\int_{-\infty}^{\infty}\underbrace{\frac{1}{(j \omega+\beta)\big{(}1+\lambda_{A}(j\omega+\beta)\big{)}\big{(}\frac{1}{ \lambda_{B}}-\gamma(j\omega+\beta)\big{)}}}_{D1}\\ &\times\quad e^{\sigma^{2}\|h\|^{2}\gamma(j\omega+\beta)}dw, \end{split}  \tag{18}\]  
where we employ partial fraction expansion on the under-brace term \(D1\) as  
\[D1= \lambda_{B}\frac{1}{(j\omega+\beta)}-\frac{\lambda_{B}\gamma}{1+ \frac{\lambda_{A}}{\lambda_{B}\gamma}}\frac{1}{\big{(}(j\omega+\beta)-\frac{ 1}{\lambda_{B}\gamma}\big{)}}-\frac{\lambda_{B}}{1+\frac{\lambda_{B}\gamma}{ \lambda_{A}}}\frac{1}{\big{(}(j\omega+\beta)-\frac{1}{\lambda_{A}}\big{)}}. \tag{19}\]  
Next, by plugging (19) in (18), we obtain  
\[Pout(\gamma) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{(j\omega+\beta)} \ e^{\sigma^{2}\|h\|^{2}\gamma(j\omega+\beta)}dw \tag{20}\] \[- \frac{\gamma}{1+\frac{\lambda_{A}}{\lambda_{B}\gamma}}\frac{1}{ 2\pi}\int_{-\infty}^{\infty}\frac{1}{\big{(}(j\omega+\beta)-\frac{1}{\lambda_ {B}\gamma}\big{)}}\ e^{\sigma^{2}\|h\|^{2}\gamma(j\omega+\beta)}dw\] \[- \frac{1}{1+\frac{\lambda_{B}\gamma}{\lambda_{A}}}\frac{1}{2\pi} \int_{-\infty}^{\infty}\frac{1}{\big{(}(j\omega+\beta)+\frac{1}{\lambda_{A}} \big{)}}\ e^{\sigma^{2}\|h\|^{2}\gamma(j\omega+\beta)}dw.\]  
Eventually, applying the residue theory (15), yields a closed-form expression for the CDF of collocated antenna MIMO radar system for an unknown random target as follows  
\[Pout(\gamma)= 1-\bigg{[}\frac{\gamma}{1+\frac{\lambda_{A}}{\lambda_{B}\gamma}}e^{ \frac{\sigma^{2}\|h\|^{2}}{\lambda_{B}}}u\bigg{(}-\frac{\sigma^{2}\|h\|^{2}}{ \lambda_{B}}\bigg{)}+\frac{1}{1+\frac{\lambda_{B}\gamma}{\lambda_{A}}}e^{- \frac{\sigma^{2}\|h\|^{2}\gamma}{\lambda_{A}}}u\bigg{(}\frac{\sigma^{2}\|h\|^{ 2}\gamma}{\lambda_{A}}\bigg{)}\bigg{]}. \tag{21}\]