# Energy Performance of LR-FHSS: Analysis and Evaluation
## 4 Current Consumption Model of an LR-FHSS ED
### Unconfirmed Transmission
 \tag{3}\]  
The total transmission time of a frame, \(T_{Tx}\), depends on the time needed to transmit the physical header replicas, the physical layer payload, and the total time of the frequency channel hops performed during such header and payload transmission, denoted \(T_{header}\), \(T_{payload}\), and \(T_{freqHops}\), respectively, as expressed in Equation (4). \[T_{Tx}=T_{header}+T_{payload}+T_{freqHops} \tag{4}\]  
The duration of the header transmission, \(T_{header}\), can be obtained as defined in Equation (5) [28]. \(N\) indicates the number of times that the header is transmitted, the values of which are listed in Table 5. Note that the CR value refers to the coding rate used for the payload transmission, not for the header transmission. For the latter, CR is always defined as \(CR=1/2\). \[T_{header}=N\times 233.472\ ms \tag{5}\]  
\(T_{payload}\) can be calculated as expressed in Equation (6). \(L_{PHY}\) and \(M\) denote the PHYPayload size and the fragment size, respectively, with both being measured in bytes. The possible values for \(M\) are shown in Table 5. Note that other works based on the expressions listed in Table 108 of the Regional Parameters v1.0.4 specification [28] provide a different equation for \(T_{payload}\) (see Appendix A), which we found is not accurate. We encountered inconsistencies between the measured values and the theoretical calculations predicted in the expressions provided in the current version of the specification [28]. Following discussions with members of the LoRa Alliance, we provide the correct equation for \(T_{payload}\) as Equation (6). We need to take into account that the total amount of bytes to be transmitted in order to send the PHYPayload is \(L_{PHY}\) + 2 + 6/8, since in addition  
Figure 7: Expanded view of a frequency channel hop in an uplink transmission with LR-FHSS. to \(L_{PHY}\), the 2-byte CRC and 6 bits of Trellis termination also need to be sent. FEC uses a convolutional encoding, so to help the decoder, six zeros are pushed in the encoder at the end of the packet. \[T_{payload}=\frac{L_{PHY}+2+\frac{6}{8}}{M}\times 102.4\ ms \tag{6}\]  
\(T_{hop}\) denotes the duration of a frequency channel hop, referred to as state 1' in Table 4. To compute the amount of time that the ED spends hopping between channels, we have to first calculate the total number of said hops for a single uplink transmission, denoted as \(N_{hops}\) and calculated using Equation (7). Again, this depends on the CR that is used in the transmission of the payload. Note that for a given uplink transmission there will be one frequency channel hop after each header transmitted and one channel hop after each fragment transmission (except for the last one). \[N_{hops}=N+\left\lfloor\frac{L_{PHY}+2+\frac{6}{8}}{M}\right\rfloor \tag{7}\]  
Finally, the actual duration of the total frequency channel hopping time for an uplink frame, \(T_{freqHops}\), can be calculated as in Equation (8). \[T_{freqHops}=N_{hops}\times T_{hop} \tag{8}\]  
We next determine \(T_{Rx1}\). After transmission of the uplink frame, the NS can transmit a downlink frame to the ED: either a data frame or an acknowledgment (ACK). The downlink frame is intended to be received in one of the two receive windows. Even if the uplink transmission is performed with LR-FHSS, downlink transmission will use the LoRa modulation. A receive window must be at least as long as the physical layer preamble of the downlink transmission to ensure the ED can detect the incoming downlink frames. The preamble consists of eight symbols for DR0 and DR1 and twelve symbols for the rest of the LoRa DRs (i.e., DR3, DR4, and DR5); the symbols are denoted as \(N_{symb}\). When using LR-FHSS,