−5 0 5 10 15 20 10−4
10−3 10−2 10−1 100
1/(Rσ2w) in dB
FER
(a)
R= 1 R= 2 R= 3 R= 5 R= 10
−5 0 5 10 15 20 10−4
10−3 10−2 10−1 100
1/(Rσw2) in dB (b)
LF = 50 LF = 20 LF = 10 LF = 5
Figure 4.9: Performance of frame BP in a system with M = 20, N = 60, pa = 0.2 for different repetition factors R with a frame of length LF = 60 in (a) and for different frame lengths LF over the SNR in a system with
a repetition code of R= 5 in (b).
in this range. Here the likelihood for decoding errors is relatively high and increasing the frame length (which automatically increases the information symbols) increases the likelihood that at least one information symbol is decoded erroneously causing a frame error. At a certain SNR point this behavior changes. Here the interplay between SPE and decoder changes.
Short frames degrade the performance of the SPE which yields activity LLRs of low magnitude. As seen previously in Fig.4.7, these low activity LLRs clip the code symbol LLRs from the multiuser detector to the decoder thereby decreasing the decoder performance. Hence, the decoder can only perform well, if the activity LLRs from the SPE are good enough. However, we see that the performance gain of increasing the frame length is only moderate in the high SNR range, showcasing that short frames already suffice to achieve good activity LLRs at hight SNR.
Phase Transition Diagram
In the following we investigate the performance of the frame BP in a wider set of parameters. To this end, we consider the impact of the spreading sequence length and the node activity probability pa on the FER. This is done by using a modified version of the so-called Donoho-Tanner phase transition diagram [DT09], plotting the tuples of M and pa, where a FER < 10−3 can be achieved. A FER of 10−3 would result in a success probability of 99.9% for direct random access, which is a reasonable value for most
applications. However, especially for ultra reliable M2M, different values have to be considered by simulations. The phase transition diagram is plotted for the noise free detection and is thus only states whether detection is possible in general.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8
1 R = 1
R = 2 R = 3 R = 5
pa
Rel.spreadingseq.lengthM/N (a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
pa
Rel.spreadingseq.lengthM/N (b)
GOMPR= 5 bcSIC R= 5
Figure 4.10: Phase transition diagram of frame BP in the noise free case with different code rates in (a) and phase transition diagram for GOMP and bcSIC with a repetition code of R= 5 in (b). The number of nodes is N = 30 in both cases and the frame length is LF = 30.
Fig. 4.10 (a) plots the phase transition diagram for the frame BP for dif-ferent repetition factors R. Note that the y-axis shows the relative spreading sequence length M/N, as the results are independent of the particular choice for N and solely depend on the ratio. The curves indicate two significant results. First, decreasing the code-rate by increasing the repetition factor indeed requires lower spreading, thereby confirming the results stated above.
Exemplary, with R = 2, the spreading sequence length can be much lower than in the previous case. However, one has to keep in mind that the half rate repetition code has to be involved by looking at the number of resources that is required. The same holds true for the lower rate codes with higher repetition factors respectively. Here the spreading required is even lower, showcasing that the frame BP fully exploits the code dimension in order to decrease the spreading sequence length. Fig. 4.10 (b) shows the phase transition diagrams for two state-of-the-art algorithms in the same setup with repetion factor R = 5. First, the GOMP with a subsequent decoder stage is used. Second, the bcSIC which is a code exploiting non-linear algorithm from [Sch15] is used. Both algorithms exhibit good detection properties and allow for detection in overloaded systems up to a certain pa. However, both
10−4 10−3 10−2 10−1 100 10−4
10−3 10−2 10−1 100
FAR
MDR
(a)
Ω = 10 Ω = 1 Ω = 0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
FAR
MDR
(b)
1/σ2
w = 5dB
1/σ2
w = 0dB
1/σ2
w =−2dB
1/σ2
w =−10dB
Figure 4.11: Pareto frontiers in (a) and ROC in (b) for a Bayes-Risk activity detector based on activity LLRs taken from an frame BP. In a system with M = 20, N = 15, pa = 0.2, R = 1, LF = 30.
schemes cannot exploit the code-space as a means for resources in the same manner as the frame BP does. This shows that the Frame BP is superior to the GOMP and bcSIC.
4.4.1 Pareto Optimality of the Activity LLRs
Compared to the frame LLRs used at the Bayes-Risk detector from Sec-tion 3.4.2, the frame BP yields true frame activity LLRs which capture the statistical process of node (in)activity properly. It is therefore interesting to see, how the activity LLRs of the frame BP perform in terms of being conservative or liberal. We therefore take the frame LLRs at the output of a frame BP and feed them into a Bayes-Risk detector (3.21). The resulting Pareto frontiers and Receiver Operating Characteristics (ROC) are plotted in Fig. 4.11. We see that the activity LLR exhibit a good trade-off between false alarm and missed detection rates. Even with Ω = 1, false alarm and missed detection rates are almost equal. As expected, varying Ω impacts both error rates. However, compared to the frame Bayes-Risk in Section 3.4.2, where a variation on Ω lead to extreme changes in the activity error rates, the impact of Ω is rather moderate. Compared to the Bayes-Risk detector, the calculation of the frame LLRs in the frame BP seems to better capture the underlying probabilistic model of node activity. These results may simplify the application of an ARQ scheme that initiates retransmissions based on the activity LLRs delivered from the frame BP. This question is left for
further research.
4.4.2 Random Sequence vs. Direct Sequence Spread-ing
Throughout this chapter random sequence spreading was assumed such that the composite signature matrix changes each code symbol. This randomness significantly improves the performance of the activity LLR calculation. The reason for this is that each of the LF multiuser detectors observes the same sparsity pattern through another matrix A. Assuming that A does not change means that the extrinsic activity information each multiuser detector generates is nearly the same. Especially, in the noise free region, each multiuser detector observes nearly the same symbol. If the matrixA changes each symbol randomly, each multiuser detector observes a different receive symbol y yielding different extrinsic information. Further, the multiuser interference also changes randomly by applying random sequence spreading.
To demonstrate the impact of random sequence spreading, we consider a setup with N = 60 nodes and a spreading sequence length of M = 20. The frame length is LF = 60 and the repetition factor is R = 5. According to the investigations above, the frame BP yields zero BER at 1/Rσw2 = 12dB.
We compare the evolution of the activity LLRs La within the frame BP over the number of iterations carried out is a system with Nact = 4 active nodes.
Fig. 4.12 (a) shows that in case random sequence spreading, the activity
1 2 3 4 5 6 7 8 9 10
−1,200
−1,000
−800
−600
−400
−200 0 200 400 600 800
Iteration Number Ln
(a) Random seq. spreading
Active Nodes Inactive Nodes
1 2 3 4 5 6 7 8 9 10
−1,200
−1,000
−800
−600
−400
−200 0 200 400 600 800
Iteration Number (b) Direct seq. spreading
Figure 4.12: Evaluation of the activity LLRs within a frame BP over the number of iterations carried out for random sequence spreading in (a) and with direct sequence spreading in (b).
LLRs for active nodes fast turn negative, while the remaining activity LLRs turn positive. Additionally, we see that the evaluation of the activity LLRs is rather stable with increasing number of iterations. This behavior is not true with short spreading yielding to a non changing composite signature matrix. Here all activity LLRs are subject to strong variations, yielding a high uncertainty about the node activity. Consequently, we observe that the frame BP yields huge performance gains with changing signature matrices.