# Energy Efficiency of a Decode-and-Forward Multiple-Relay Network with Rate Adaptive LDPC Codes +
## 4 Simulation Results and Discussion
Figure 6: Energy efficiency vs. number of relays for target BER = \(10^{-5}\), code rate 1/2, codeword length \(n=648\), and 4-QAM. Figure 7: BER of multiple relays (three relays) with an equal transmit power (decreased by a factor of 2), codeword length \(n=648\), \(C_{R}\) at source 1/2, and 4-QAM. As an example case, we consider three relays with a LDPC codeword of length \(n=648\). We observe that if the source \(S\) sends data with a code rate of \(3/4\), then the decoded sequence length, i.e., \(k\) will be 486 bits. For the first relay when the relay adapts \(C_{R}\) according to the channel conditions and if it adapts to a new code rate of \(1/2\) then it only sends 324 bits and disposes the remaining 162 information bits. Moreover, a new parity check matrix \(\mathbf{H}_{1}\) with revised parameters, \(k\) and \(n-k\) is defined at the relay. For the relay 2, if the same output code rate as input code rate, i.e., \(C_{R}=3/4\) is chosen then no bits will be disposed off. If at relay three, only 162 information bits are transferred because of worse channel conditions on the link between relay and destination, as \(m_{rd}=1\), hence, the rest of the bits are disposed off at relay three. Therefore, the destination receives the data from the first relay with a code rate of \(1/2\), i.e., \(D\) receives 324 information bits in a frame, from the second relay with a code rate of \(1/4\), the destination receives 162 bits and from the third relay with a code rate of \(3/4\), and \(D\) receives a frame of length 486 bits. Consequently, when all three frames have received from the relays, they are of different lengths. In Figure 9 we have outlined the whole scenario. We observe that by disposing the remaining bits, the system can avoid the overhead of buffering the information at relay(s). Furthermore, there are lower delays in the information transmission. Moreover, the destination combines the only bits that are received in the same time-slot on orthogonal channels from all the relays. Please note, we have used the Nakagami-\(m\) factor as \(m_{1}\) for the first hop, i.e., for the channels between source and multiple relays and it is denoted as \(m_{rd}\) for the links between relays and destination. Moreover, the above example addresses a rather generic scenario and does not necessarily use the same parameters as for the simulation results in Figure 7.  
Figure 8: BER of multiple relays with equal transmit power = \(1/2\), code rate = \(1/2\), code length = 648, and 4-QAM. Furthermore, in Figure 7, the second (blue) curve demonstrates the BER performance of the system when there is no code rate adaptation, the input code rate and output code rate at the relay are the same. The last (green) curve shows the bit-error-rate performance when the system applies the code rate adaptation based on channel conditions and also buffers the remaining bits at the relay. In this case, not only are the bits temporarily stored at the relay but they are also sent using the extra time-slots. As a result the destination receives the complete sequence from each relay in multiple time-slots. Then \(D\) applies the post-detection combining on all the sequences received from relays. If we compare all three curves, we see that at the start the worst performance is shown by the system which applies the code rate adaptation without buffering any information at the relays. Still, at a higher SNR this curve surpasses the performance of the code rate non-adaptive system and shows a very close performance to the system that does a temporal buffering of data at the relays, the reason being that because of the multiple relays, the destination receives multiple copies of the data sequences. If some of the information sequences are of not of the same length, the system can still achieve a good diversity gain. This fact is more evident when the number of relays increases. The advantages of this approach are manifold, i.e.,  
1. No temporal buffering of the data at the relays;
2. Lesser processing at the relays;
3.