# Covert Surveillance Video Transmission with QAM Constellations Modulation
## 3 Performance Analysis of Cover Surveillance Video Transmission
### Analysis of Covertness Performance  
In Equation (13), we measure the covertness performance with KL divergence. For any QAM modulation, the KL divergence can be expressed as  
\[\mathcal{D}(Q_{1,w}^{n}\|Q_{0,w}^{n})=n\mathcal{D}(Q_{1,w}\|Q_{0, w}). \tag{28}\]  
Then, we have  
\[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\] \[=\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,w}(x,y)\log\frac{ \sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}q_{j,k,w}(x,y)}{q_{0}(x,y)}\mathrm{d}x \mathrm{d}y, \tag{30}\]  
where  
\[q_{j,k,w}(x,y) =\frac{1}{2\pi\sigma_{w}^{2}}\Bigg{[}\exp\bigg{(}-\frac{(x+g_{w} \mathbf{d}[j]\beta)^{2}}{2\sigma_{w}^{2}}-\frac{(y+g_{w}\mathbf{d}[k]\beta)^{ 2}}{2\sigma_{w}^{2}}\bigg{)}\Bigg{]}, \tag{31}\] \[q_{0}(x,y) =\frac{1}{2\pi\sigma_{w}^{2}}\Bigg{[}\exp\bigg{(}-\frac{x^{2}+y^{ 2}}{2\sigma_{w}^{2}}\bigg{)}\Bigg{]}. \tag{32}\]  
For ease of notation, we define the functions as  
\[\mathbf{T}(\beta) \triangleq\log\Bigg{[}\frac{\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k,w} q_{j,k,w}(x,y)}{q_{0}(x,y)}\Bigg{]} \tag{33}\] \[\triangleq\log\Bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k,w}\exp \bigg{(}K_{j,k}(\beta)\bigg{)}\Bigg{]}, \tag{34}\]  
where  
\[K_{j,k} \triangleq-\frac{2\pi\mathbf{d}[j]g_{w}\beta+2y\mathbf{d}[k]g_{w} \beta+\beta^{2}g_{w}{}^{2}(\mathbf{d}[j]^{2}+\mathbf{d}[k]^{2})}{2\sigma_{w} ^{2}}. \tag{35}\]  
Differentiating with respect to \(\beta\) in the function \(g(\beta)\), we can obtain  
\[\mathbf{T}^{\prime}(\beta) =\frac{1}{\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k,w}\exp\big{(}K_{j,k} \big{)}}\sum_{i=1}^{N}\Bigg{[}p_{j,k}\exp\big{(}K_{j,k}\big{)}K_{j,k}{}^{ \prime}\Bigg{]}, \tag{36}\] \[\mathbf{T}^{\prime\prime}(\beta) =-\frac{1}{\big{[}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k,w}\exp \big{(}K_{j,k}\big{)}\big{]}^{2}}\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k,w} \exp\big{(}K_{j,k}\big{)}K_{j,k}{}^{\prime}\Bigg{]}^{2}\] \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]0Given the constraints imposed by space limitations, the nuanced expressions for \(K_{j,k}\arcsec(\beta)\) and \(\mathbf{T}\arcsec(\beta)\) concerning the derivative with respect to \(\beta\) are regrettably omitted. Consequently, we proceed to derive the expressions for \(\mathbf{T}^{\prime}(0)\), \(\mathbf{T}^{\prime\prime}(0)\), \(\mathbf{T}^{\prime\prime\prime}(0)\), and \(\mathbf{T}^{\prime\prime\prime\prime}(0)\). Leveraging the Taylor series expansion, we arrive at  
\[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]1 \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]2 \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]3  
Based on (39), we can calculate the first and second terms, which are given by  
\[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]4 \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]5  
and  
\[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]6 \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]7 \[\mathcal{D}(Q_{1,w}\|Q_{0,w}) =\iint\mathcal{Q}_{1,w}(x,y)\log\frac{Q_{1,w}(x,y)}{Q_{0,w}(x,y)} \mathrm{d}x\mathrm{d}y \tag{29}\]8  
Then, we consider two cases.