All the uplink and downlink channels are assumed to be under uncorrelated Rayleigh fading, but with different variances, namely Σ(u)11 = −10 dB, Σ(u)22 = 0 dB, Σ(u)33 = 10 dB, Σ(u)44 = 20 dB, Σ(d)11 = 20 dB, Σ(d)22 = 10 dB, Σ(d)33 = 0 dB, Σ(d)44 = −10 dB. Similarly, the direct channel is assumed to be hs∼ CN (0, 1). Binary phase-shift
0 5 10 15 20 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 CSNR (dB) SER Direct 2 P S Ortho + OPA, L=2 Nonortho + BF, L=2 Ortho + OPA, L=3 Nonortho + BF, L=3 Ortho + OPA, L=4 Nonortho + BF, L=4
Figure 4.2 SER of the non-orthogonal relaying model with optimal beamform- ing scheme and the orthogonal relaying scheme with power allocation scheme in [1]. The number of relays L = 2, 3, 4.
keying (BPSK) modulation is used at the source. Assume σ2
s = σu2 = σd2 = 1, the
average channel signal-to-noise ratio (CSNR) is simply defined as PS/σ2s = PS/σu2 =
PS/σd2 = PS. The total transmit power at the relays is chosen to be PR= PS.
Fig. 4.2 compares the symbol error rate (SER) performances achieved with the non-orthogonal relaying with optimal beamforming scheme and the orthogonal relay- ing with power allocation scheme in [1]. The number of relays are L = 2, 3, 4. It can be seen from the figure that the performance improvement (from the “equivalent” cod- ing gain) with non-orthogonal relaying is significant, about 3 dB/relay at high SNR. However, since the destination does not know full CSI, the diversity order does not increase with the number of relays. This also agrees with the results of the orthogonal case under Assumption B in [1] and of the non-orthogonal case under Assumptions II and III in [2] where the diversity order is proved to be 2 only. As a reference, the SER obtained by direct transmission with a total transmit power of P = PS + PR
of the direct transmission is significantly worse than that of the relay-assisted trans- mission. It should be pointed out that, in general, the performance improvement of the relay-assisted transmission strongly depends on the channel conditions among the source, relays and destination. The performance curves shown in Fig. 4.2 should be interpreted for the specific channel conditions under consideration.
Next, we compare the non-orthogonal relaying scheme under consideration with the orthogonal relaying scheme considered in [1] under the same CSI assumption. A broadcasting feedback channel accommodating 2 or 6 feedback bits is assumed. To adapt with these limited feedback rates, the quantized version of the beamforming vector (given in Section 4.3.2) and of the power allocation scheme used in [1] (given in Appendix 4.B) are employed. In the procedures, the number of test channel vectors is N = 20, 000 and the termination criterion is chosen as ² = 10−4δ(W(t−1)).
As predicted from Corollary 1, the simulation results in Fig. 4.3 confirm that when infinite-rate (i.e., analog) feedback is available, the beamforming scheme in the non-orthogonal case is superior in terms of the average SNR as compared to the power allocation scheme in the orthogonal case. This is due to the fact that the phases of both uplink and downlink channels can be perfectly matched at each relay, resulting in a constructive superposition (basically addition of signal amplitudes) of all the co-phased signals at the destination. Consequently, the energy of the combined signal is higher than that in the orthogonal case where energies of different signals are accumulated, not their amplitudes. When no feedback is available, it is expected that the non-orthogonal relaying scheme is inferior to the orthogonal scheme. For the case with only 2 feedback bits, performance of the orthogonal scheme is still better. This also reflects the sensitivity of the non-orthogonal relaying scheme to the quality of the phase estimates available at the relays. However, the gap is very small and can be practically removed and even reversed with more feedback bits (e.g., with 6 feedback bits or more in our simulation setup). It is also noted that under the assumed channel model, the performance of the power allocation scheme does not improve much as the number of feedback bits increases.
0 2 4 6 8 10 12 14 16 18 20 −5 0 5 10 15 20 25 CSNR (dB) Average output SNR (dB) ∞ bits feedback 6 bits feedback 2 bits feedback Without feedback Relay Selection Approximate SNR (21) Orthogonal Non−orthogonal
Figure 4.3 Average signal-to-noise ratios of the non-orthogonal relaying scheme with beamforming and the orthogonal relaying scheme with power al- location scheme in [1]. The number of relays L = 4.
Since the average SNRs of the two relaying schemes are not much different, their bandwidth efficiencies mostly depend on the number of channel uses (which is also the number of relays) required in the second phase. As shown in Fig. 4.4, the ergodic capacity of the non-orthogonal scheme is about 2.5 times larger than that of the orthogonal scheme at any SNR region (i.e., approximately (L + 1)/2 times for the cases with 2 or 6 feedback bits). Without feedback information, the non-orthogonal scheme performs a little bit worse at low and medium SNR regions. For a benchmark comparison, the capacity of the model considered in [2] under the assumption of full CSI available at the destination is also plotted. As expected, a decrease in capacity can be clearly observed when less CSI is available at both the transmitters and receiver.
Figs. 4.3 and 4.4 also plot curves (marked with filled circles) to validate the approximate [SNR given in (4.21) and the upper bound of the ergodic capacity given in (4.20), respectively. It is observed that both [SNR and the upper bound of the ergodic capacity are very close to the actual values. Other performance curves in Figs. 4.3
0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 CSNR (dB)
Average capacity (bits/ time slot)
With full CSI [1] ∞ bits feedback 6 bits feedback 2 bits feedback Without feedback Relay selection
Capacity upper bound (20)
Non−orthogonal Orthogonal
Figure 4.4 Ergodic capacity of the non-orthogonal relaying scheme with beam- forming and the orthogonal relaying scheme with power allocation scheme in [1]. The number of relays L = 4.
and 4.4 also show that the relay selection scheme in (4.27) does not perform as well as the beamforming scheme with 2-bit feedback channel. However, it is superior to the power allocation scheme (even with analog feedback case). Thus, relay selection is a suitable solution when one needs to balance between the implementation complexity and performance gain.