# Interference-Robust Transmission in Wireless Sensor Networks
## 4 Proposed Transmission Scheme
For an estimated SNR \(\hat{\gamma}\), it determines the transmission rate by the highest transmission rate \(R_{m}\) that satisfies \(\hat{\gamma}\geq\hat{\gamma}_{m}\), where \(\hat{\gamma}_{m}\) is the minimum SNR for transmission mode \(m\), which can provide the desired packet error rate of \(p_{s}\) with the use of maximum payload size in the absence of interference and can be represented by:  
\[\hat{\gamma}_{m}=b_{m,s}^{-1}\left(1-{p_{s}}^{1/L_{\rm max}}\right). \tag{28}\]  
Here, \(b_{m,s}^{-1}\) denotes the inverse function of \(b_{m,s}\). As described in Section 2, the packet loss can occur due to low SNR and/or packet collision. If the packet loss is mainly due to the low SNR, it may be desirable to decrease the transmission rate \(R\). If it is mainly due to the packet collision, it may be desirable to decrease the transmission time \(T\) to reduce the collision probability. Consider the case that the transmission failure consecutively occurs due to the packet collision, and the transmitter node decreases the transmission rate. Then, the transmission time will be increased, and thus, the packet collision problem may rather be exacerbated. This problem can be alleviated by adjusting the transmission rate and the transmission time together. If the number of consecutive transmission failures reaches a threshold \(N_{\rm fail}\), the transmitter node reduces the transmission rate, while keeping the transmission time \(T\) unchanged. If the transmission rate is adjusted from \(R_{a}\) to \(R_{b}\), it may be desirable to adjust the payload size from \(L_{a}\) to \(L_{b}\) as:  
\[L_{b}=R_{b}T=R_{b}\left(\frac{L_{a}}{R_{a}}\right). \tag{29}\]  
It may be feasible for the transmitter node to adjust the transmission time \(T\) according to the interference condition, while adjusting the transmission rate according to the channel condition. The normalized throughput, defined by \(\tilde{S}=S_{m}/R_{m}\), can be estimated by:  
\[\tilde{S}=\frac{T}{T+\beta^{\prime}}\sum\limits_{i=1}^{W}I\left\{ \begin{array} []{l}\text{ACK packet is received for the $i$-th data packet}\\ \end{array} \right\} \tag{30}\]  
after performance measurement of \(W\)-packet transmission. Note that this metric is not affected by the adjustment of the transmission rate since it is normalized with respect to the transmission rate. After each \(W\)-packet transmission, the transmitter node updates the normalized throughput, say \(\tilde{S}_{\rm new}\). Comparing \(\tilde{S}_{\rm new}\) with a previous one, say \(\tilde{S}_{\rm old}\), it can adjust the transmission time \(T\) to increase the normalized throughput. Let \(\Delta\) be the step size for the adjustment of transmission time and \(I_{T}\left(=\pm 1\right)\) be a parameter indicating whether the transmission time was increased or decreased previously. If \(\tilde{S}_{\rm new}>\eta\tilde{S}_{\rm old}\), where \(\eta\geq 1\), it implies that the transmission time was effectively adjusted. In this case, it may be desirable to keep the adjustment. The transmitter node increases or decreases the transmission time by \(\Delta\) according to \(I_{T}\). If \(\tilde{S}_{\text{old}}>\eta\tilde{S}_{\text{new}}\), it implies that the previous adjustment was not effective, requiring the change of the sign of \(I_{T}\) (i.e., \(I_{T}\leftarrow-I_{T}\)). Then, the transmitter node adjusts the transmission time by \(I_{T}\Delta\). Otherwise, the transmitter node does not adjust the transmission time. It may also be desirable to change the step size in consideration of the difference between \(\tilde{S}_{\text{new}}\) and \(\tilde{S}_{\text{old}}\). If the difference is large, it may be desirable to use a larger step size to quickly adjust the transmission time. Algorithm 2 summarizes the adjustment of the transmission time.