# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## 5 Numerical Results
### Results on Average BER  
In Figure 5, we present the end-to-end average BER for BPSK modulation signals of an 8-hop relay system against the average SNR of each hop over Nakagami-\(m\) fading channels with \(m=3\) in the presence of impulsive noise with various levels of impulsiveness (i.e., from highly impulsive with \(A=5\times 10^{-5}\), moderately impulsive with \(A=5\times 10^{-3}\) to rarely impulsive with \(A=1,10\), the ratio \(\rho\) is fixed as 0.2). The theoretical curves are obtained using the closed-form expressions given by (A5)and (18) and are found to agree well with the simulation results, thus validating our analysis. It can be seen that when the parameter \(A\) is equal to 10, the corresponding curve is extremely close to the result obtained under AWGN assumption, which is in accordance with our statement that when \(A\) is equal to or greater than 10, the MCA channel degenerates to an AWGN channel. It can also be observed from Figure 5 that the performance under impulsive noise is dramatically different from that under the AWGN. Under a highly impulsive noise condition, a three-region performance behavior is observed: the BER curve first decreases almost linearly with increasing SNR, and then remains almost stagnant for some SNR region until finally decreases again after increasing the SNR to a larger value. This phenomenon can be best explained by the envelop distribution analyses of the MCA interferences in [43]. It has been observed that the distributions of the noise envelop \(z\) are divided into three parts, the first corresponds to smaller values of \(z\), in which the Gaussian noise component dominates; the second corresponds to larger values of \(z\), wherein the impulsive noise component dominates; and the third corresponds to intermediate values of \(z\), for which the CDF is virtually constant for these values. The portion of the distribution corresponding to these intermediate values of \(z\) is termed the \"null region\" [43]. Therefore, we can conclude that the first decrease in the BER plot is mainly affected by the Gaussian noise component with small envelopes; and the second decrease is mainly determined by the impulsive noise component while the stagnant flat-region in the BER plot is due to the existence of the \"null region\" in the envelope distribution of the MCA noise. An advantage of our analysis lies in its flexibility for the performance evaluation under various channel conditions. For illustration purposes, Figure 6 shows the BER results of the Rayleigh fading channels in various noise conditions and the purely AWGN channel by setting the appropriate Nakagami-\(m\) and MCA parameters. Figure 5: End-to-end average BER vs. average SNR per link over Nakagami-\(m\) (\(m=3\)) channels with various noise conditions. The effects of the impulsive index \(A\) and the ratio \(\rho\) on the end-to-end BER performance of the multi-hop relay network are investigated in Figure 7. It shows the average BER under various values of \(A\) and \(\rho\) for an 8-hop relay network with average SNR being 25 dB. It is observed that the BER is not a monotonic function of the impulsive index \(A\) with fixed values of \(\rho\). When the channel noise changes from highly impulsive (\(A=10^{-5}\)) to approximately Gaussian (\(A=10\)), the BER first increases for some range of \(A\) before reaching a peak point; then with further increase of \(A\) to around 10, the BER gradually decrease to the value under AWGN channel. The value of \(\rho=\infty\) corresponds to the case of purely AWGN channel, thus the BER value does not depend on \(A\) in this case and is always equal to the value of AWGN case. In Figure 8, we present the end-to-end average BER performance of a relay network against total power consumption under two different power allocation schemes, i.e., the scheme of equal power allocation among all transmitting nodes, and the scheme of balanced power allocation taken into consideration the path loss of each hop.