# Research on Joint Resource Allocation for Multibeam Satellite Based on Metaheuristic Algorithms
## 2 Problem Formulation
### Link Budget Model  
The carrier-to-noise power ratio of the satellite downlink can be calculated as follows [1; 18]:  
\[\begin{array}{rl}[C/N]&=[P_{i}]+[G_{t}]-[OBO]+[G_{r}]-10\log_{10}\bigl{(}T_ {sys}\bigr{)}\\ &-[L_{FS}]-[L_{O}]+228.6-[BW]\end{array} \tag{6}\]  
where \(P_{i}\) denotes the transmit power of the satellite antenna (assumed to be numerically equal to the power assigned to the beam); \(G_{t}\) and \(G_{r}\) denote the gain of the transmit and receive antennas, respectively; \(OBO\) is the output backoff factor, taken as a fixed value of 5 dB [20]; \(T_{sys}\) represents the receiver system noise temperature, usually at 320 K [1]; \(L_{FS}\) denotes the free-space path loss; \(L_{O}\) denotes the sum of other losses; and \(BW\) is the beam bandwidth.  
We consider four primary interference factors, including carrier-to-adjacent-beam interference (CABI), carrier-to-adjacent-satellite interference (CASI), carrier-to-cross-polarization interferences (CXPI), and carrier-to-third-order intermodulation product interference (CIMI) [20]. Then, taking interference into account, the carrier-to-noise ratio is [27]:  
\[\begin{array}{c}\left[\frac{C}{N+1}\right]^{-1}=\left[\frac{C}{N}\right]^{- 1}+\left[\frac{C}{T}\right]^{-1}=\\ \left[\frac{C}{N}\right]^{-1}+\left[\frac{C}{ABI}\right]^{-1}+\left[\frac{C}{ ASI}\right]^{-1}+\left[\frac{C}{XPI}\right]^{-1}+\left[\frac{C}{XPI}\right]^{-1} +\left[\frac{C}{XPI}\right]^{-1}\end{array} \tag{7}\]  
Figure 1: Four-color frequency reuse pattern. Red and green indicate beams using right-hand circular polarization (RHCP), whereas blue and yellow indicate beams using left-hand circular polarization (LHCP).  
where \(E_{b}\) denotes the bit energy; \(C\) is the carrier power; and \(R_{b}\) represents the data rate (i.e., the bit rate, in bps), satisfying the relation \(R_{b}=1/T_{b}\), where \(T_{b}\) denotes the bit duration and the relationship between the signal bit energy and the carrier power (i.e., \(E_{b}=CT_{b}\)). We assume that the noise bandwidth is equal to the beam bandwidth. Therefore, the density ratio of energy per bit to noise plus interference is calculated as follows:  
\[\frac{E_{b}}{N_{0}+I_{0}}=\frac{CT_{b}}{N_{0}+I_{0}}=(\frac{C}{N_{0}+I_{0}}) \times\frac{B_{N}}{R_{b}B_{N}}=(\frac{C}{N+I})\frac{BW}{R_{b}}. \tag{8}\]  
The signal is modulated with a \(4/8/16/32/64\) element code, with the coding rate taking values from \(1/4\) to \(9/10\). Hence, the ratio of energy per symbol to noise plus interference power spectral density is [26]:  
\[\frac{E_{s}}{N_{0}+I_{0}}=\frac{E_{b}r\log_{2}M}{N_{0}+I_{0}}=(\frac{C}{N+I}) \frac{BW}{R_{b}}r\log_{2}M. \tag{9}\]  
Typically, such a system employs an ACM (adaptive coding and modulation) strategy to maximize spectral efficiency. According to [12], the beam data rate is:  
\[R_{b}=R_{s}\times\Gamma(\frac{E_{s}}{N_{0}})=\frac{BW}{1+\alpha_{r}}\times \Gamma(\frac{E_{s}}{N_{0}}), \tag{10}\]  
where \(\alpha_{r}\) denotes the roll-off factor, \(R_{s}\) denotes the symbol rate, and \(\Gamma\) is the spectral efficiency (in bit/symbol) of the modulation and coding scheme (MODCOD), satisfying the following conditions [13]:  
\[\frac{E_{s}}{N_{0}+I_{0}}\geq\frac{E_{s}}{N_{0}}\bigg{|}_{MODCOD}+\mu, \tag{11}\]  
where \(\mu\) is the desired margin of the link (specified as 1.0 dB), whereas \(\left.E_{s}/\left.N_{0}\right|_{MODCOD}\right.\) denotes the threshold value required for the MODCOD scheme employed by the link to obtain the spectral efficiency (\(\Gamma\)) under ideal channel conditions. Accordingly, to calculate the data rate, we assume a certain MODCOD mode, then use Equations (6)-(10) to calculate \(E_{s}/\left(N_{0}+I_{0}\right)\). Then, we check whether condition (11) is satisfied, thus maximizing the spectral efficiency of the link to the greatest extent possible.