3.4 Activity LLR based Decision Rules
3.4.2 The Bayes-Risk Decision Rule
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Frame Lenght LF
AER
(a)
−5 0 5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
1/σw2 in dB
MDR
(b)
M = 5 M = 10 M = 15 M = 20
Figure 3.3: Frame activity error rate versus frame length LF at a fixed SNR of 15dB on (a) and missed detection rate versus SNR for MAP detection at a frame length of L = 20 for different spreading sequence length M. The number of nodes is N = 20 with pa = 0.2.
and missed detection error reads Rn,l = CFAPr (xn,l = 0)
ZA
pyl(yl|xn,l = 0)dyl
Prob. of false alarm
+
CMDPr (xn,l ∈ A)
ZI
pyl(yl|xn,l ∈ A)dyl
Prob. of missed detection
. (3.19)
ZI and ZA are the regions in the observation space where the detector assigns the observation to the hypotheses HI for inactivity and HA for activity, respectively [VT04]. As shown in appendix A.1 the minimization of the Bayes-Risk for node n can be cast as a Likelihood Ratio Test (LRT)1
log pyl(yl|xn,l = 0) Pr (xn,l = 0)CFA
xn,l∈Apyl(yl|xn,l) Pr (xn,l)CMD
HI
≷
HA
0 log pyl(yl|xn,l = 0) Pr (xn,l = 0)
xn,l∈Apyl(yl|xn,l) Pr (xn,l)
HI
≷
HA
log CMD CFA
Ln,l
HI
≷
HA
log CMD
CFA . (3.20) The right hand side of (3.20) clearly defines a threshold for the activity LLR Ln,l based on the ratio of costs defined as Ω = CCFA
MD. The Bayes-Risk activity decision rule reads
φΩ(Ln,l) =
Sˆxl\n if Ln,l ≥ log Ω1
Sˆxl ∪n if Ln,l < log Ω1 . (3.21) Considering (3.21), the estimated support set SXˆ depends on the particular choice for Ω. Clearly, controlling Ω allows for controlling the activity error rates. Exemplary, decreasing Ω increases the decision threshold, which in turns leads to a higher likelihood that nodes are estimated as active.
Conversely, increasing Ω leads to the opposite results, meaning that it is more likely that nodes are estimated as inactive. Setting Ω = 1 turns the decision threshold to 0 corresponding to the MAP decision rule. In terms of conservative and liberal the parameter Ω allows to seamlessly adjust between both.
1Likelihood ratio tests are commonly carried out over likelihood functions only. In this case the prior is included inside the test which leads to a slightly different test. However, in this thesis, we stick to the term Likelihood Ratio Test
Pareto Optimality
Even though Bayes-Risk detection allows minimizing one activity error rate to arbitrary low values, one has to keep in mind that the other error rate automatically increases. This is not a surprise, since both error rates are coupled. Such characteristic is known as Pareto Optimality [BV07]. The set of false alarm and missed detection rates that can be achieved by changing Ω forms the so-called Pareto frontier.
10−3 10−2 10−1 100 10−3
10−2 10−1 100
FAR
MDR
(a)
Ω = 1 Ω = 0.125 Ω = 0.05
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
TAR
(b)
1/σ2
w = 20 dB
1/σ2
w = 15 dB
1/σ2
w = 10 dB
1/σ2
w = 5 dB
Figure 3.4: Performance of Bayes-Risk activity detection in a system M = 10, N = 20, pa = 0.2. Missed detection rate versus False Alarm rate in (a), and ROC in (b).
Fig. 3.4 (a) exemplary shows the Pareto frontier for Bayes-Risk activity detection. The solid lines denote the Pareto frontiers for different SNR, where higher SNR generally allows for lower activity error rates. Changing Ω at a given SNR simultaneously changes false alarm and missed detection rates, while the performance follows the Pareto frontier. Here we see that decreasing Ω simultaneously decreases the missed detection rate while the false alarm rate increases. Increasing Ω yields the opposite, a increased missed detection ate and a decreased false alarm rate. In summary we have the following connection between Ω and the activity error rates.
• Ω ↑⇒MDR ↑and FAR ↓
• Ω ↓⇒MDR ↓and FAR ↑
Fig. 3.4 (b) shows the corresponding ROC for different values of Ω and different SNRs. The black solid lines connect points for one particular SNR and different Ω. Following the concept of liberal and conservative
detectors, we see that for each SNR Bayes-Risk detection enables adjusting the detector to be liberal or conservative. Instead of following the path for MAP detection, which was shown to be quite conservative, Bayes-Risk detection allows adjusting the activity error rates. Exemplary, both graphs in Fig.3.4 also plot these paths for different choices of Ω, where Ω = 1 corresponds to MAP detection.
We also observe that for a fixed Ω, activity error rates develop rather uncontrolled with changing SNR, i.e., the particular path for a certain Ω does not follow a predictable path and both activity error rates change with changing SNR. Especially in the low SNR range the path for a fixed Ω is bent.
10−4 10−3 10−2 10−1 100 10−4
10−3 10−2 10−1 100
FAR
MDR
(a)
Ω = 1 Ω = 0.125 Ω = 0.05
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
TAR
(b)
M = 20 M = 15 M = 10 M = 5
Figure 3.5: Performance of Bayes-Risk activity detection in a system N = 20, pa = 0.2 for varying spreading sequence length M for a fixed SNR = 15dB. Missed detection rate versus False Alarm rate in (a), and ROC in (b).
The impact of overloading on the performance of Bayes-Risk detection is shown in Fig. 3.5 (a) and (b) where the Pareto frontier and the ROC is plotted for different spreading sequence length M with a fixed SNR of 15 dB.
Overloading the system decreases the performance and the Pareto frontier shifts towards higher activity error rates in this case. This shift is similar as in Fig. 3.4 (a), suggesting that higher overlaoading has a similar effect as decreasing the SNR. This effect can also be observed in the ROC in 3.5(b).
Even though Bayes-Risk detection allows controlling the tendency for the activity error rates, it is still far away from a predictable control. The reason for this is twofold. First, the connection between Ω and activity error rate is not known in closed form and one has to simulate all possible rates and
store them in a look-up table. Second, changing the SNR also changes the connection between Ω and the activity error rates.
Frame-Based Transmissions
Extending the Bayes-Risk concept toward frame-based reads φΩ(Ln,l) =
SXˆ\n if
lLn,l ≥ log Ω1 SXˆ ∪n if
lLn,l < log Ω1 . (3.22) As seen in Section 3.4.1 the bias contained in the frame activity LLRs leads to a high missed detection rate for MAP detection. In the terminology of Risk this means we have a conservative detection. However, Bayes-Risk detection allows counteracting this conservative behavior by properly adapting Ω to enable liberal detection. Considering the Pareto frontier in Fig. 3.6 (a) and the ROC in Fig. 3.6 (b) for a frame length of LF = 20 shows the impact of this effect. The Ω required to achieve certain activity error
10−4 10−3 10−2 10−1 100 10−4
10−3 10−2 10−1 100
FAR
MDR
(a)
Ω = 0.1 Ω = 0.0125
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
TAR
(b)
1/σ2
w= 15 dB
1/σ2
w= 10 dB
1/σ2
w= 5 dB
Figure 3.6: Performance of frame-based Bayes-Risk activity detection in a system M = 10, N = 20, pa = 0.2, LF = 20. Missed detection rate versus false alarm rate in (a), and ROC in (b).
rates has decreased and the point with Ω = 1 is not shown on the Pareto frontier, as the false alarm rate is zero. Comparing the Pareto frontiers for the frame-based detection with symbol-by-symbol detection in Fig. 3.4 (a) shows that the Ω required to control the activity error rates has decreased and much lower values of Ω are required to enable liberal detection. However, even with this conservative impact, we can still trace out the full Pareto
frontier by choosing Ω properly. Comparing the ROC curves in Fig. 3.6 (b) for the frame-based Bayes-Risk to the symbol-by-symbol detection show in Fig. 3.4 (b) shows that we have a SNR gain through the processing of multiple activity LLRs. The ROC curves are shifted to the upper left corner for the frame-based setup. This confirms the results obtained for MAP detection where we have shown that longer frames exhibit SNR gains.