# Energy Performance of LR-FHSS: Analysis and Evaluation
## 4 Current Consumption Model of an LR-FHSS ED
### Unconfirmed Transmission
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**State Index**} & \multirow{2}{*}{**Description**} & \multicolumn{2}{c}{**Time**} & \multicolumn{2}{c}{**Current**} \\ \cline{3-5}  & & **Parameter** & **Value (ms)** & **Parameter** & **Value (mA)** \\ \hline
1a & Pre-transmission & \(T_{preTx}\) & 2.370 & \(I_{preTx}\) & 3.8 \\
1 & Transmission & \(T_{Tx}\) & Equation (4) & \(I_{Tx}\) & 25.7 \\
1’ & Channel hop & \(T_{Hop}\) & 0.225 & \(I_{Hop}\) & 12.3 \\
1b & Post-Transmission & \(T_{postTx}\) & See Table 6 & \(I_{postTx}\) & 3.7 \\
2 & Wait until Rx1 & \(T_{Rx1 unit}\) & 1000 & \(I_{Rx1 unit}\) & 0.0005 \\
3a & Pre-receive window Rx1 & \(T_{preRx1}\) & 1.300 & \(I_{preRx1}\) & 2.3 \\
3 & Receive window Rx1 & \(T_{Rx1}\) & See Table 6 and Equation (9) & \(I_{Rx1}\) & 5.8 \\
3b & Post-receive window Rx1 & \(T_{postRx1}\) & 0.700 & \(I_{postRx1}\) & 1.2 \\
4 & Wait until Rx2 & \(T_{Rx2 unit}\) & 911.2 & \(I_{Rx2 unit}\) & 0.0005 \\
5a & Pre-receive window Rx2 & \(T_{preRx2}\) & 1.500 & \(I_{preRx2}\) & 1.8 \\
5 & Receive window Rx2 & \(T_{Rx2}\) & 198.4 & \(I_{Rx2}\) & 5.8 \\
5b & Post-reception window Rx2 & \(T_{preRx2}\) & 0.700 & \(I_{postRx2}\) & 1.2 \\
6 & Sleep state & \(T_{Sleep}\) & Equation (2) & \(I_{Sleep}\) & 0.0005 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Characterization of the states of an unconfirmed uplink transmission in LR-FHSS with DR8 (indicated in Figure 6) in terms of their time and current consumption. \begin{table}
\begin{tabular}{c c c} \hline \hline
**CR** & **N** & **M (bytes)** \\ \hline
1/3 & 3 & 2 \\
2/3 & 2 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of N and M depending on the CR used for the payload. the DR for Rx1 is DR1 for uplink frames transmitted with DR8 or DR10, whereas it is DR2 for uplink frames sent with DR9 or DR11. By default, the DR for Rx2 is fixed to DR0. Rx1 will always be opened by the ED regardless of the communication mode (i.e., unconfirmed or confirmed). We calculate \(T_{Rx1}\) using Equation (9). \[T_{Rx1}=N_{symb}\times T_{symb} \tag{9}\]  
Following the LoRa specification [34], Equation (10) models the duration of a symbol, \(T_{symb}\), in the first receive window. \[T_{symb}=\frac{2^{SF}}{BW} \tag{10}\]  
However, we have observed that there is a discrepancy between the calculations using Equation (9) and the values measured in our testbed (cf. Figure 6). Specifically, we measured a shorter value in our scenario, as the device waits for six symbols before closing Rx1. Therefore, we can use Equation (9) to calculate the duration of the receive windows, but by using \(N_{symb}=6\). In Rx2, the ED might be utilizing channel activity detection (CAD), which is a power-saving technique that shortens the duration of the second receive window when no incoming frame is being detected in that window [35]. Then, \(T_{Rx2}\) denotes the duration of Rx2 and is calculated using Equation (11). \[T_{Rx2}=\frac{2^{SF}+32}{BW} \tag{11}\]  
To complete the whole set of experiments, Table 6 depicts the values missing from Table 4, which were computed via the given equations or measured depending on the DR and FRM Payload size. For DR0 and DR5, values were extracted or derived from [20]. We found that \(T_{freqHops}\) accounts for approximately 0.2%, on average, of the total transmission time. We can now utilize \(I_{avg\_unTx}\) to calculate the lifetime of a battery-operated LR-FHSS ED that periodically performs unconfirmed transmissions. This performance parameter is essential, as it gives an approximation of the amount of time that a battery-operated LR-FHSS device may function without requiring the recharging or replacing of its battery. Equation (12) shows how \(T_{lifetime\_unTx}\) depends on the battery capacity, \(C_{battery}\), expressed in mAh, and on \(I_{avg\_unTx}\), expressed in mA.