# On the Energy Efficiency of On-Off Keying Transmitters with Two Distinct Types of Batteries
## 4 Theoretical Analysis
### Impact of \(B\) on BECPB  
From Equation (19), the derivative of \(E_{\text{bat,I}}\) with respect to \(B\) is given by  
\[E_{\text{bat,I}}^{\prime}=\frac{\gamma\lambda^{2}B^{2}+2v\lambda\gamma B-v( \eta-\gamma v)}{2B^{2}[\eta-\gamma(\lambda B+v)]^{2}}-\frac{P_{\text{d}}}{2B^ {2}(\eta-\gamma P_{\text{d}})} \tag{24}\]  
where \(\lambda=\zeta d^{k}\), \(v=P_{\text{a}}+P_{\text{d}}\). Note that \(\frac{P_{\text{d}}}{2B^{2}(\eta-\gamma P_{\text{d}})}\) is much smaller than the first term in Equation (24), therefore we can make the approximation:  
\[E_{\text{bat,I}}^{\prime}\approx\frac{\gamma\lambda^{2}B^{2}+2v\lambda\gamma B -v(\eta-\gamma v)}{2B^{2}[\eta-\gamma(\lambda B+v)]^{2}}. \tag{25}\]  
After mathematical derivation, it can be shown that the equation \(E_{\text{bat,I}}^{\prime}=0\) has two solutions expressed as  
\[B_{1,I}=\frac{-\gamma v-\sqrt{\gamma v\eta}}{\gamma\lambda}<0,B_{2,I}=\frac{- \gamma v+\sqrt{\gamma v\eta}}{\gamma\lambda}>0. \tag{26}\]  
When \(0<B<B_{2,I}\), \(E_{\text{bat,I}}^{\prime}<0\), \(E_{\text{bat,I}}\) decreases with \(B\). When \(B>B_{2,I}\), \(E_{\text{bat,I}}^{\prime}>0\), \(E_{\text{bat,I}}\) increases with \(B\). As a result, \(B_{2,I}\) is the optimal bandwidth leading to the smallest BECPB for the battery with IDC-based BUF.  
From Equation (22), the derivative of \(E_{\text{bat,M}}\) with respect to \(B\) is  
\[E_{\text{bat,M}}^{\prime}=\frac{\gamma\lambda^{2}B^{2}+2w\lambda\gamma B-w(2 \eta-\gamma w)}{4B^{2}[\eta-\frac{\gamma}{2}(\lambda B+w)]^{2}}, \tag{27}\]  
where \(\lambda=\zeta d^{k}\),\(w=P_{\text{a}}+2P_{\text{d}}\). It can be derived that the equation \(E_{\text{bat,M}}^{\prime}=0\) has two solutions written as  
\[B_{1,M}=\frac{-\gamma w-\sqrt{2\gamma w\eta}}{\gamma\lambda}<0,B_{2,M}=\frac{- \gamma w+\sqrt{2\gamma w\eta}}{\gamma\lambda}>0. \tag{28}\]When \(0<B<B_{2,M}\), \(E^{\prime}_{\text{bat,M}}<0\), \(E_{\text{bat,M}}\) decreases with \(B\). When \(B>B_{2,M}\), \(E^{\prime}_{\text{bat,M}}>0\), \(E_{\text{bat,M}}\) increases with \(B\). As a result, \(B_{2,M}\) is the optimal bandwidth corresponding to the smallest BECPB for the battery with MDC-based BUF.  
We will compare the optimal bandwidth of the two types of batteries. The difference between them can be expressed as  
\[\Delta B=B_{2,M}-B_{2,I} =\frac{-\gamma w+\sqrt{2\gamma w\eta}}{\gamma\lambda}-\frac{- \gamma v+\sqrt{\gamma v\eta}}{\gamma\lambda}\] \[=\frac{-\gamma(w-v)+\sqrt{\gamma\eta}(\sqrt{2w}-\sqrt{\varphi})}{ \gamma\lambda}\] \[\geq\frac{-\gamma(w-v)+\sqrt{\gamma\eta}(\sqrt{w}-\sqrt{v})}{ \gamma\lambda}\] \[=\frac{\sqrt{\gamma}(\sqrt{w}-\sqrt{v})\left[\sqrt{\eta}-\sqrt{ \gamma}(\sqrt{w}+\sqrt{v})\right]}{\gamma\lambda}. \tag{29}\]  
where \(B_{2,M}\) and \(B_{2,I}\) are the optimal bandwidth for battery with MDC-based BUF and IDC-based BUF, respectively. Note that \(w>v\) and \(\sqrt{w}+\sqrt{v}<<1\), thus \(\sqrt{\eta}-\sqrt{\gamma}(\sqrt{w}+\sqrt{v})>0\), \(\Delta B>0\). That is to say, the optimal bandwidth for the battery with MDC-based BUF is always larger.  
As can be shown by Equation (23), the BECPB for the ideal battery decreases with \(B\).