## 4 Hierarchical Cooperative MIMO Clustering
### Energy Model Based Cluster Sizing  
In this section, we use an underlying communication energy model to determine the number of clusters (and thus, the cluster sizes). Using knowledge of the network under consideration, we find number of clusters required for minimum energy expenditure within the network. In Section 5, we compare the number of clusters based on energy model with the results we achieve using the proposed distributed HMIMO scheme. Let \(d_{u,v}\) be the average inter-cluster distance. Expected distance between cluster/node \(u\) and cluster/node \(v\) located at coordinates \((x_{i},y_{i})\) and \((x_{j},y_{j})\) is given by [14]:  
\[m[d_{u,v}]=\int\int_{R}\sqrt{x^{2}+y^{2}}\frac{1}{4a}dxdy \tag{4}\]  
where \(x=x_{i}-x_{j}\), \(y=y_{i}-y_{j}\) and the length of one side of the grid is \(2a\). We assume distant dependent Friss free-space model (\(d^{2}\) power loss) [15] for per node energy consumption. Thus energy consumption in a sensor node for transmitting one bit of information over a distance \(d\) is:  
\[E_{local}=E_{circuit}+\epsilon_{fs}(d_{toCH}^{*})^{2} \tag{5}\]  
The energy required to transmit one bit from the clusterhead to the cooperating node is also equal to Equation (5). Since the CCS is far from sensors, presumably the energy consumption follows multi-path model. Therefore, assuming \(k\) clusters and \(N/k\) nodes in a cluster, energy per bit consumed in a cluster is given by [15]:  
\[E_{CH}=(\frac{N}{k}-1)E_{circuit}+\frac{N}{k}E_{DA}+E_{circuit}+\epsilon_{mp}( d_{u,v}^{*})^{4} \tag{6}\]  
where \(E_{DA}\) is the energy consumed in data aggregation and \(E_{circuit}\) is [3]:  
\[M_{t}(P_{DAC}+P_{mix}+P_{filt})+2P_{syn}+ \tag{7}\] \[M_{r}(P_{LNA}+P_{mix}+P_{IFA}+P_{filt}+P_{ADC})\]  
and \(\epsilon_{mp}(d_{u,v}^{*})^{4}\) is given by:  
\[(1+\alpha)(\gamma(M_{t},M_{r})N_{o}BN_{f}G_{o}M_{l}(d_{u,v}^{*})^{4}) \tag{8}\]  
where \(\gamma(M_{t},M_{r})\) is the required SNR at the receiver. \(N_{0}\) is the power spectral density, \(B\) is the bandwidth, \(N_{f}\) is the receiver's noise figure, \(G_{0}\) is antenna gain and \(M_{l}\) is the link margin. \(M_{t}\) and \(M_{r}\) is the number of transmitters and receivers respectively. For the case where we have single input and single output at the transmitter and receiver ends, \(M_{t}\) and \(M_{r}\) are both one and \(P_{*}\) are power consumption values. We first find number of hops in a given network setting. Assume a uniform distribution of clusters in a given network. Assuming each side of the square field has \(2a/\sqrt{k}\) clusters, number of hops (h) in the network is approximately given by:  
\[h=\frac{k+\sqrt{k}k}{2} \tag{9}\]  
From Equations (1) and (9), we derive the expression for total energy consumption in the network for \(L\) bits. \[E_{SISO}=(E_{CH}+(\frac{N}{k^{*}}-1)E_{local})L.h \tag{10}\]where \(k^{*}\) from [15] is:  
\[k^{*}=\sqrt{\frac{0.5855N\epsilon_{fs}a^{2}}{\epsilon_{mp}(d_{u,v}^{*})^{4}-E_{ circuit}}} \tag{11}\]  
Equation (11) gives the number of clusters in the network for multihop traditional SISO communication [15] (for minimum energy expenditure). We construct an overlay network on top of these clusters. The bottom layer communicates with the top layer in MIMO mode. Number of clusters in overlay network which minimizes the total energy consumption as compared to when data is forwarded via multihop SISO is given by:  
\[E_{total}(k_{h})<E_{total}(k_{(h-1)}) \tag{12}\]  
where \(h\) indicates the hierarchical level. Energy consumed in MIMO communication (\(E_{MIMO}\)) is the sum of local energy consumption incurred in local communication between the clusterhead and the cooperating node and the energy consumed in the long haul communication. \[E_{local}=E_{circuit}+\epsilon_{fs}(d_{toCH}^{*})^{2} \tag{5}\]0  
From Equation (10), energy consumption in a clustered structure is given by:  
\[E_{local}=E_{circuit}+\epsilon_{fs}(d_{toCH}^{*})^{2} \tag{5}\]1  
For long haul communication clusterhead to clusterhead distance or clusterhead to CCS distance is very large.