# Shapley's Value as a Resource Optimization Strategy for Digital Radio Transmission over IBOC FM
## 3 Results  
### Description of the Proposed Scenario  
To comprehend the utilization of the Shapley value as a strategy for resource optimization within an IBOC broadcasting scheme in FM, a scenario is outlined involving twelve (12) nodes or digital broadcasting stations conducting IBOC hybrid transmission processes over FM. These transmissions occur within an RF channel that, for this specific case, is assumed to provide a total bandwidth (BW) of 1150 kbps due to conditions affecting the channel's maximum performance. Recorded in Table 6 is each class of traffic per node, where it can be observed that the total bandwidth required is 1387 kbps, exceeding the total bandwidth available in the RF channel. This indicates that the operation mode will be in a state of saturation. To calculate the value of BW, the expression provided by Equation (7) is used:  
\[BW_{T}=\frac{1}{T_{s}}\sum_{k=1}^{N_{sp}}log_{2}\bigg{[}1+\frac{SNR_{k}}{\Gamma }\bigg{]} \tag{7}\]  
\(N_{sp}\): Number of subcarriers. \(T_{s}\): Time of an OFDM symbol (\(T_{s}\) = 2.902 ms) for the particular case of IBOC FM. \(SNR_{k}\): Signal to Noise Ratio (SNR) present on the subcarrier \(k\). \(\Gamma\): This is known as the SNR gap, which represents the loss in SNR incurred by using a specific discrete coding scheme. It is suggested in [26] that the value of \(\Gamma\) can be calculated for practical purposes using Equation (8):  
\[\Gamma=-\frac{1}{1.6}ln\bigg{[}\frac{BER_{obj}}{0.2}\bigg{]} \tag{8}\]  
where the BER target (\(BER_{obj}\)) corresponds to the desired Bit Error Rate value to be sustained, which, in this particular case, is set to \(10^{-6}\). For the proposed scenario, we \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Service Mode**} & \multicolumn{5}{c}{**Approximate Transfer Speed [kbps]**} & \multirow{2}{*}{**Waveform**} \\ \cline{2-2} \cline{4-6}  & **P1** & **P2** & **P3** & **P4** & **PIDS** \\ \hline MP1 & 98 & N/A & N/A & N/A & 1 & Hybrid \\ \hline MP2 & 98 & N/A & 12 & N/A & 1 & Extended hybrid \\ \hline MP3 & 98 & N/A & 25 & N/A & 1 & Extended hybrid \\ \hline MP4 & 98 & N/A & 25 & 25 & 1 & Extended hybrid \\ \hline MP5 & 25 & 74 & 25 & N/A & 1 & All-digital extended hybrid \\ \hline MP6 & 50 & 49 & N/A & N/A & 1 & All-digital extended hybrid \\ \hline \hline \end{tabular} N/A means not applicable.  
\end{table} 
Table 4: Transmission rates of primary logic channels. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Service Mode**} & \multicolumn{5}{c}{**Approximate Transfer Speed [kbps]**} & \multirow{2}{*}{**Waveform**} \\ \cline{2-2} \cline{4-6}  & **S1** & **S2** & **S3** & **S4** & **S5** & **SIDS** \\ \hline MS1 & 0 & 0 & 0 & 98 & 6 & 1 & Hybrid \\ \hline MS2 & 25 & 74 & 25 & 0 & 6 & 1 & Extended hybrid \\ \hline MS3 & 50 & 49 & 0 & 0 & 6 & 1 & Extended hybrid \\ \hline MS4 & 25 & 98 & 25 & 0 & 6 & 1 & Extended hybrid \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Transmission rates of secondary logic channels. consider \(N=12\) nodes comprising the RF system in a saturated state (\(BW_{T}\leq\sum_{i=1}^{N}d_{i}\)), with a total available bit rate \(BW_{T}=E=1150\) kbps. As mentioned above, a bankrupt set will be considered consistent with the saturation state of the RF channel to calculate the transferable utility value for each of the coalitions. Algorithm 1 to perform the calculation of each of the transferable utility values is as follows:  
```
Nj: Number of Players M_Coalitions: Matrix of possible coalitions V_Coalition: Transferable utility value for each coalition
1. Initialize Z as a vector containing numbers from 1 to Nj. 2. Set n_coal to 0. 3. For each coalition size i from 1 to Nj:  a. Increment n_coal by n choose i, where n is Nj. 4. Initialize M_Coalitions as a matrix of zeros with dimensions (n_coal \(\times\) Nj). 5. Initialize c (counter) to 0. 6. For each coalition size i from 1 to Nj:  a. Generate all combinations S of Z taken i at a time. 7. For each combination S:  i. Increment the counter. ii. Initialize Suma_d to 0.  iii.