In this section, we numerically evaluate the performance of the schemes proposed in Section 2.3, and we compare them with the lower bound from Section 2.2.
In Fig. 2.5(a) we let the power at the helper be fixed at P2 = 1, and N = 1,
and we plot the achievable distortion D with respect to the SNR, given by P1/N . It
can be seen that SLU achieves the lowest distortion among the considered schemes, and the performance of I-HDA and S-VQ, denoted by Dh∗(Ω) and D∗vq(Ω), respectively,
(a) Achievable distortion performance in func- tion of P1for P2= 1 and ρ = 0.3.
(b) Achievable distortion and cut-set bound as a function of P1for P2= 5 and ρ = 0.8.
(c) Achievable distortions and cut-set bound for P = P1= P2= 1 as a function of ρ.
(d) Achievable distortions and cut-set bound for P = P1= P2= 5 as a function of ρ.
Figure 2.5: Upper and lower bounds on the distortion with respect to SNR and corre- lation for the Gaussian one-helper problem.
reduce to D∗u(Ω) in this regime of operation. On the other hand, separate source and channel coding achieves the worst performance among the considered schemes. In Fig. 2.5(b), we let P2 = 5. Contrary to the previous case, while uncoded transmission
still achieves the best distortion for low SNR values, its performance deteriorates as SNR increases, and the pure digital scheme has a better performance in this regime. I-HDA and S-VQ schemes reduce to the pure uncoded performance at low SNR values, while they outperform both digital and pure uncoded transmission schemes at higher SNR values. We note that both I-HDA and S-VQ achieve the same distortion in general, although I-HDA uses successive decoding, and are operationally different in their digital components. In S-VQ, the sources are quantized and are directly mapped to de channel input, and hence, the correlation between the quantization codewords is exploited. On
the other hand, in I-HDA, the sources are quantized and mapped to DPC channel inputs. In this case, the correlation between the DPC codewords is exploited instead. While in this multi-terminal setup both structures achieve the same performance, we believe that in other communication scenarios their performance will be different.
Now, we consider a symmetric power scenario for which P1= P2= P , and plot the
upper and lower bounds on the achievable distortion D with fixed SNR with respect to the source correlation ρ, which quantifies the quality of the helper’s observation. We consider P = 1 in Fig. 2.5(c). Observe that all schemes achieve the lower bound, and are thus optimal, when ρ = 0, which corresponds to the case with independent, hence useless, helper observation. Since the helper is useless, the setup reduces to a Gaus- sian point-to-point channel, for which both separation and ULC are optimal. Uncoded transmission, I-HDA and S-VQ achieve the optimal performance at ρ = 1, i.e., when both users have access to the main source signal, while digital transmission is subopti- mal. In this case, the helper and the main transmitter can fully cooperate by generating correlated channel inputs by exploiting the source correlation, although they still have individual power constraints. However, separation based schemes cannot generate cor- related inputs distributedly, since source and channel coding are done independently. The suboptimality of digital transmission with respect to uncoded transmission for MMSE reconstruction in this setup was proven in [52]. Note that digital transmission is outperformed by the other schemes for any ρ > 0, while I-HDA, S-VQ and uncoded transmission achieve the same distortion. In Fig 2.5(c) we consider the upper and lower bounds for P = 5. In this case, pure digital transmission outperforms analog transmis- sion for low ρ, while analog transmission achieves lower distortions for high correlation values. In general, the gains from separation based schemes are obtained only by the distributed compression of the source, while gains in SLU are obtained only by gener- ating correlated channel inputs that result in beamforming gains. When the correlation is low, higher gains can be obtained from distributed compression, whereas when the correlation is high, distributed beamforming provides higher performance. Nevertheless, I-HDA and S-VQ schemes outperform both pure schemes and achieve lower distortions, since both schemes exploit both types of gains. We observe that for high correlation values, HDA schemes reduce to the performance of uncoded transmission. Note that, as expected, at ρ = 0 and ρ = 1 we have the same optimality results as before.
While we have not been able to prove it analytically, we believe that the uncoded transmission is optimal in those regions where the performance of HDA and uncoded transmission coincide. This is reminiscent of the optimality of uncoded transmission in the MAC setup considered in [13]. However, we note that the optimality conditions in [13] differ from the conditions under which uncoded transmission achieves the lowest distortion for intermediate correlation values.
transmitters, the encoders can generate correlated channel inputs that outperforms the pure digital transmission scheme based on separate source and channel coding, for which the channel inputs are independent. However, while the proposed HDA schemes achieve significant better performance than both the pure analog and pure digital schemes, in general the proposed transmission schemes are far from the derived distortion lower bound. We believe that this mainly stems from the looseness of the proposed lower bound, and tighter lower bounds in this setting is a challenging future research problem.