# A Cluster-Based Dual-Adaptive Topology Control Approach in Wireless Sensor Networks
## 4 Performance Evaluation
### Energy Model  
The energy consumption of inter-cluster communications can be measured by the following formula [3]:  
\[E_{x,y}^{bt}=\frac{C_{1}\cdot\gamma(M_{t},M_{r})\cdot(d_{x,y})^{\alpha_{x,y}}+ C_{2}\cdot M_{t}+C_{3}\cdot M_{r}+C_{4}}{R_{b}} \tag{12}\]  
In Equation (12), \(E_{x,y}^{bt}\) is the total energy consumption per bit in an inter-cluster communication between the cluster \(x\)'s MCH/SCH and the cluster \(y\)'s MCH/SCH; \(R_{b}\) is the bit rate; \(\gamma(M_{t}\),\(M_{r})\) is the Signal to Noise Ratio (SNR) threshold at the receiver when \(M_{t}\) and \(M_{r}\) antennas are used for transmission and reception respectively; \(d_{x,y}\) is the transmitter-receiver distance, which is the maximum value of the four distances between the source MCH/SCH and the destination MCH/SCH in a VMIMO operation; \(\alpha_{x,y}\) is path loss exponent (for simplicity, \(\alpha_{x,y}\) = 2 when \(d_{x,y}\) is less than \(d_{\textit{crossover}}\); otherwise, \(\alpha_{x,y}\) = 4); \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are circuit-specific constants shown as follows:  
\[\left\{\begin{array}{l}C_{1}=(1+\delta)\cdot N_{o}\cdot B\cdot N_{f}\cdot G _{o}\cdot M_{l}\\ C_{2}=P_{DAC}+P_{mix}+P_{filt}\\ C_{3}=P_{LNA}+P_{mix}+P_{IFA}+P_{filtr}+P_{ADC}\\ C_{4}=2\cdot P_{syn}\end{array}\right. \tag{13}\]  
In Equation (13), \(\delta\) is a factor that depends on the drain efficiency of the power amplifier and the underlying modulation scheme; \(N_{o}\) is the single-sided thermal noise Power Spectral Density (PSD); \(B\) is the passband bandwidth; \(N_{f}\) is the receiver noise figure (\(N_{f}\) is defined as \(N_{r}/N_{o}\), where \(N_{r}\) is the PSD of the total effective noise at the receiver input); \(G_{o}\) is a constant that depends on the transmitter and receiver antenna gains; \(M_{l}\) is a link margin that compensates for hardware variations and other sources of interference; \(P_{DAC}\), \(P_{mix}\), \(P_{LNA}\), \(P_{IFA}\), \(P_{filtr}\), \(P_{J\!D\!D}\), and \(P_{syn}\) are the power consumption values for the digital-to-analog converter, the mixer, the low noise amplifier, the intermediate frequency amplifier, the active filters at the transmitter and the receiver sides, the analog-to-digital converter, and the frequency synthesizer, respectively.  
For a given inter-cluster distance between a pair of neighboring MCHs/SCHs (e.g., \(d_{x,y}\)), one of the four antenna modes (i.e., SISO, SIMO, MISO, MIMO) is employed based on the principle of minimizing the value of \(E_{x,y}^{bt}\). For relatively small distances, a circuit power is more dominant than a transmission power, a single-antenna mode (e.g., SISO) is more energy-efficient than a multi-antenna mode (e.g., one of SIMO, MISO, MIMO). As the transmitter-receiver distance increases, the tradeoff shifts in favor of multi-antenna modes. The energy consumption per bit in the source MCH/SCH (e.g., that of the cluster \(x\)'s MCH/SCH) is computed by the following formula:  
\[\Delta e_{x,y}^{s}=\frac{C_{1}\cdot\gamma(M_{t},M_{r})\cdot(d_{x,y})^{\alpha_{ x,y}}+C_{2}\cdot M_{t}+0.5\cdot C_{4}}{R_{b}} \tag{14}\]  
The energy consumption per bit in the destination MCH/SCH (e.g., that of the cluster \(y\)'s MCH/SCH) is computed by the following formula:  
\[\Delta e_{x,y}^{d}=\frac{C_{3}\cdot M_{r}+0.5\cdot C_{4}}{R_{b}} \tag{15}\]  
In intra-cluster communications, we also adopt Equations (14) and (15) to compute the energy consumption for sending node and receiving node respectively, where a single-antenna mode is employed, and \(d_{x,y}\) is replaced for the distance between sending node and receiving node.