## 4 Energy Model for Cooperative Communication Using LDPC Code  
The total power consumption \(P_{T}\) for a single node consists of two main parts, namely, the power consumption of all the power amplifiers \(P_{PA}\) which is a function of transmission power \(P_{out}\), and the power consumption of all other circuit blocks \(P_{C}\). Thus one can write  
\[P_{T}=P_{PA}+P_{C}. \tag{8}\]  
The power consumption of all the power amplifiers can be calculated using the following equation  
\[P_{PA}=(1+\alpha)P_{out}, \tag{9}\]  
where \(\alpha=(\frac{\xi}{\eta}-1)\), where \(\eta\) is the drain efficiency [24] and \(\xi\) is the peak to average ratio. When the channel only experiences a \(k^{th}\) power path loss with additive white Gaussian noise (AWGN), \(P_{out}\) can be calculated using the link budget relationship as follows. \[P_{out}=\overline{E}_{b}R_{b}\times\frac{(4\pi)^{2}d^{k}}{G_{t}G_{r}\lambda^ {2}}M_{l}N_{f}, \tag{10}\]where \(\overline{E}_{b}\) is the average energy per bit required for a given bit error rate (BER) specification, \(R_{b}\) is the transmission bit rate, \(d\) is the transmission distance, \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gains respectively, \(\lambda\) is the carrier wavelength, \(M_{l}\) is the link margin compensating the hardware process variations and other background noise, \(N_{f}\) is the receiver noise figure defined as \(N_{f}=\frac{N_{r}}{N_{0}}\) where \(N_{r}\) is the power spectral density (PSD) of the total effective noise at the receiver input and \(N_{0}\) is the single-sided thermal noise PSD at the room temperature. The power consumption in the circuit block includes transmitter and receiver power consumption \(P_{ct}\) and \(P_{cr}\), respectively. This power consumption is due to several power blocks such as \(P_{mix}\), \(P_{syn}\), \(P_{filt}\), \(P_{flir}\), \(P_{LNA}\), \(P_{IFA}\), \(P_{DAC}\), and \(P_{ADC}\) which are the power consumption values of the mixer, the frequency synthesizer, the active filters at the transmitter and at the receiver side, the low noise amplifier, the intermediate frequency amplifier, the D/A and A/D converter, respectively. The power consumption block for error correction is not considered as it is same for cooperative case and SISO case. The total energy consumption per bit can be written as  
\[E_{bt}=\frac{(P_{PA}+P_{C})}{R_{b}}, \tag{11}\]  
where \(R_{b}\) is the actual bit rate and can be replaced by \(R_{b}^{eff}=\frac{F-pN_{T}}{F}R_{b}\) when \(pN_{T}\) training symbols are inserted in each block to estimate the channel at the receiving cluster or DGN side. The block size is equal to \(F\) symbols and can be obtained by setting \(F=\lfloor T_{C}R_{S}\rfloor\), where \(R_{S}\) is the symbol rate and \(T_{C}\) is the fading coherence time. The fading coherence time can be estimated from \(T_{C}=\frac{3}{4f_{m}\sqrt{\pi}}\) where the maximum doppler shift \(f_{m}\) is given by \(f_{m}=\frac{v}{\lambda}\) with \(v\) being the velocity and \(\lambda\) being the carrier wavelength [25]. The total energy consumption is estimated by multiplying \(E_{bt}\) by the number of bits \(L\) to be transmitted. Now we develop the mathematical model where we estimate total energy consumption for cooperative communication. The total energy consumption in cooperative case is modeled as  
\[E_{CO} = \sum_{i=1}^{N_{t}-1}L_{i}E_{i}^{t}+E_{da}\sum_{i=1}^{N_{t}}L_{i}+ E_{enc}\sum_{i=1}^{N_{t}}\frac{L_{i}}{r}\gamma_{i} \tag{12}\] \[+(N_{t}-1)E_{i}^{t0}\sum_{i=1}^{N_{t}}\frac{L_{i}}{r}\gamma_{i}\] \[+E_{M}^{l}\sum_{i=1}^{N_{t}}\frac{L_{i}}{r}\gamma_{i}\] \[+\frac{1}{b_{mimo}}\sum_{i=1}^{N_{t}}\sum_{i=1}^{N_{r}}\frac{L_{i }}{r}\gamma_{i}b_{l_{r}}E_{j}^{t},\]  
The energy per bit \(E_{i}^{t}\) is needed to transmit the data from sensors to the cluster head. \(E_{da}\) is the energy dissipation per bit required in the cluster head for data aggregation. It depends on the algorithm complexity and can be expressed as  
\[E_{da}(L)=\left\{