# Energy Efficiency of a Decode-and-Forward Multiple-Relay Network with Rate Adaptive LDPC Codes +
## 4 Simulation Results and Discussion  
For Monte Carlo simulations, different code lengths of LDPC codes were used but for every length \(n\), 10,000 frames were delivered to the destination with 20 decoding iterations. Five different code rates were used with 4-QAM. A Nakagmi-\(m\) fading factor \(m=3\) and data rate \(C_{R}=1/2\) was used to collect the results, unless otherwise mentioned. We note that wherever mentioned the subscript with Nakagami-\(m\) factor shows whether this is the fading factor on the first or second hop. In Figure 3, a basic BER performance comparison was made where amplify-and-forward relays were studied against decode-and-forward relays. The results clearly demonstrate the supremacy of LDPC coded decode-and-forward relays. AF relaying system had the worst performance because of the well known reason of noise amplification at the relays along with the source signal. We can see that however, uncoded DF relays provided an almost 8 dB SNR gain but the LDPC coded system provided a much larger performance margin against the uncoded system. It is noted that in uncoded DF relaying the relays only perform a hard-decision decoding. In Figure 4, the BER performance for various numbers of relays is shown. As the number of relays increased, a better BER performance was achieved, e.g. for 3 relays a BER of \(10^{-5}\) was gained for a SNR of \(-3.0\) dB approximately, whereas for 7 relays the required SNR to achieve the same BER of \(10^{-5}\), reduced to \(-4.5\) dB. Figure 5 shows that if the amount of transmit power is decreased by a factor of 2 for the same system, the system performance also decreases. This can be observed by comparing Figure 4 and Figure 5.  
It is evident from the two figures that when the transmit power is halved, the target BER of \(10^{-5}\) for 7 relays is achieved at an SNR of about \(-1.5\) dB. Hence, approximately a 3 dB extra SNR will be needed to achieve the target BER when transmit power is decreased by a factor of 2. Figure 6 represents the relationship between the energy efficiency of the system versus the number of cooperative relays. As the number of relays increased, the energy efficiency \(\eta\) dropped very quickly. This is because the total transmit power increased with the increasing number of relays, however, the data rate was not increased at the same pace. In other words, the resultant SNR to attain a certain target BER did not increase as quickly, and consequently, energy efficiency decreased. Nonetheless, this is also clear that when the transmit power was halved but assigned equally assigned,an improvement in the energy efficiency of the system could be seen. Figure 3: A BER (bit error rate) performance comparison between amplify-and-forward (AF), decode-and-forward (DF) coded, and non-coded relay assisted communications, LDPC code length \(n=648\), Nakagami-\(m\) fading, \(m_{1}=m_{2}=3\), 4-QAM (quadrature amplitude modulation), and no. of bits \(=10^{5}\). Figure 4: EPA-I (equal power allocation): BER of system with multiple relays with an equal transmit power, code rate 1/2, codeword length \(n=648\), Nakagami-\(m\) fading, \(m_{1}=m_{2}=3\), and 4-QAM. Figure 5: EPA-I: BER of system with multiple relays with equal transmit power (decreased by a factor of 2), code rate 1/2, codeword length \(n=648\), Nakagami-\(m\) fading \(m_{1}=m_{2}=3\), and 4-QAM. In Figures 7 and 8, we analyze our system for an optimized power allocation strategy with different conditions. In Figure 7, we can observe the three curves, the first (red) curve shows the BER performance of the system when the system adapts to the different code rates but does not do any temporal storage of the data at the relays. In this case, if a relay has to send the data at a lower code rate than the rate with which it has received the data, no bits are buffered at the relay and after sending the required bits according to the new chosen code rate, the remaining bits are disposed off.