# Covert Surveillance Video Transmission with QAM Constellations Modulation
## 3 Performance Analysis of Cover Surveillance Video Transmission
### Analysis of Transmission Rates  
When a surveillance video is transmitted with QAM modulation, the transmission rate can be measured with mutual information. Recalling the probability density function given in (4), the mutual information can be expressed as  
\[R_{b} =n\mathrm{H}(Z_{b})-n\mathrm{H}(Z_{b}|X) \tag{14}\] \[=-n\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,b}(x,y)\log \left[\frac{\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}q_{j,k,b}(x,y)}{q_{t,s,b}(x,y)} \right]\mathrm{d}x\mathrm{d}y. \tag{15}\]  
Now, we define  
\[\mathrm{F}(\beta) \triangleq\log\bigg{[}\frac{\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}q_ {j,k,b}(x,y)}{q_{t,s,b}(x,y)}\bigg{]} \tag{16}\] \[\triangleq\log\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}\exp \bigg{(}G_{j,k}(\beta)\bigg{)}\bigg{]}, \tag{17}\]  
Figure 3: Comparison between previous methods and our proposed covert surveillance video transmission performance analysis method.  
where  
\[G_{j,k} \triangleq-\frac{2\pi\beta g_{b}(\mathbf{d}[j]-\mathbf{d}[t])+2y\beta g _{b}(\mathbf{d}[k]-\mathbf{d}[s])}{2\sigma_{b}^{2}}\] \[-\frac{\beta^{2}g_{b}^{2}(\mathbf{d}[j]^{2}-\mathbf{d}[t]^{2}+ \mathbf{d}[k]^{2}-\mathbf{d}[s]^{2})}{2\sigma_{b}^{2}}. \tag{18}\]  
Taking the derivative of \(G_{j,k}\) with respect to \(\beta\) and setting \(\beta\) to \(0\), we can obtain  
\[G_{j,k}(0) =0, \tag{19}\] \[G_{j,k}{}^{\prime}(0) =-\frac{xg_{b}(\mathbf{d}[j]-\mathbf{d}[t])+yg_{b}(\mathbf{d}[k]- \mathbf{d}[s])}{\sigma_{b}^{2}},\] (20) \[G_{j,k}{}^{\prime\prime}(0) =-\frac{g_{b}{}^{2}(\mathbf{d}[j]^{2}-\mathbf{d}[t]^{2}+\mathbf{ d}[k]^{2}-\mathbf{d}[s]^{2})}{\sigma_{b}^{2}}. \tag{21}\]  
Taking the derivative of \(\mathbf{F}(\beta)\) and setting \(\beta\) to \(0\), we can obtain  
\[\mathbf{F}^{\prime}(0) =\sum_{j=1}^{N}\sum_{k=1}^{N}\bigg{[}p_{j,k}G_{j,k}{}^{\prime}(0 )\bigg{]}, \tag{22}\] \[\mathbf{F}^{\prime\prime}(0) =-\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}G_{j,k}{}^{\prime}(0 )\bigg{]}^{2}+\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}\bigg{[}(G_{j,k}{}^{\prime}(0 ))^{2}+G_{j,k}{}^{\prime\prime}(0)\bigg{]}. \tag{23}\]  
Using the Taylor series expansion, when \(\beta\to 0\), we have  
\[\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,b}(x,y)\log\bigg{[} \frac{\sum_{j=1}^{N}\sum_{k=1}^{N}p_{j,k}q_{j,k,b}(x,y)}{q_{t,s,b}(x,y)}\bigg{]} \mathrm{d}x\mathrm{d}y\] \[=\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,b}(x,y)\bigg{[} \beta\mathbf{F}^{\prime}(0)+\frac{\beta^{2}}{2}\mathbf{F}^{\prime\prime}(0)+ \mathcal{O}(\beta^{3})\bigg{]}\mathrm{d}x\mathrm{d}y. \tag{24}\]  
Then, we can obtain  
\[\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,b}(x,y)\bigg{[} \beta\mathbf{F}^{\prime}(0)\bigg{]}\mathrm{d}x\mathrm{d}y\] \[=\beta^{2}g_{b}{}^{2}\frac{\sum_{t=1}^{N}\sum_{s=1}^{N}\sum_{s=1} ^{N}p_{t,s}\Big{(}\mathbf{d}[j]\mathbf{d}[t]+\mathbf{d}[k]\mathbf{d}[s]- \mathbf{d}[t]^{2}-\mathbf{d}[s]^{2}\Big{)}}{\sigma_{b}^{2}}, \tag{25}\]  
and  
\[\iint\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}q_{t,s,b}(x,y)\bigg{[} \frac{\beta^{2}}{2}\mathbf{F}^{\prime\prime}(0)\bigg{]}\mathrm{d}x\mathrm{d}y\] \[=\beta^{2}g_{b}{}^{2}\frac{\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s} \Big{(}\mathbf{d}[t]^{2}+\mathbf{d}[s]^{2}\Big{)}}{2\sigma_{b}^{2}}\] \[-\beta^{2}g_{b}{}^{2}\frac{\Big{(}\sum_{t=1}^{N}\sum_{s=1}^{N}p_{ t,s}p_{j,k}\mathbf{d}[t]\Big{)}^{2}+\Big{(}\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s} \mathbf{d}[s]\Big{)}^{2}}{2\sigma_{b}^{2}}. \tag{26}\]Combining (15), (23), (25), and (26), the covert transmission rate of a surveillance video can be expressed as  
\[\mathbb{I}(Z_{b}^{n};X^{n}) =n\beta^{2}g_{b}{}^{2}\frac{\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s} \Big{(}\mathbf{d}[t]^{2}+\mathbf{d}[s]^{2}\Big{)}}{2\sigma_{b}^{2}}\] \[-n\beta^{2}g_{b}{}^{2}\frac{\Big{(}\sum_{t=1}^{N}\sum_{s=1}^{N}p_{ t,s}\mathbf{d}[t]\Big{)}^{2}+\Big{(}\sum_{t=1}^{N}\sum_{s=1}^{N}p_{t,s}\mathbf{d}[s] \Big{)}^{2}}{2\sigma_{b}^{2}}. \tag{27}\]