1
Additional illustrations related to the IEEE SPL paper, “Testing the Energy of Random Signals in a Known Subspace: an Optimal Invariant Approach” F.-X. Socheleau, D. Pastor The aim of this short paper is to present additional results, associated to [1], that could not be added in the original paper due to lack of space. More precisely, Figure 1 shows the receiver operating characteristics (ROC) for detector (13) as defined in [1, Sec. III]. ROC curves are obtained for different mismatch-to-maximal interference ratios ν and two values of signal-to-noise ratios (SNR). The SNR is defined as in [2, Eq. (5.20)], that is µ2 T ⊥ z PΦ z. σ2 As expected, the higher the SNR, the higher the detection probability. The performance also strongly depends on the value of ν , which is defined in the paper as ∆
SNR =
∆
ν = τ /(σ 2 ̺),
where ̺ is the maximal interference-to-noise ratio, set to 15 dB in the simulations. As ν increases
(or equivalently as the value of τ increases), the interference gets more energetic in the signal subspace P⊥ z which forces the Φ detector to become more conservative and to increase its detection threshold, leading to a lower detection probability. 1
1
0.9
0.8
Detection Probability
Detection Probability
0.9
0.7 0.6 0.5 0.4
SNR = 10 dB
0.3
ν=0 ν=0.01 ν=0.05 ν=0.25
0.2 0.1 0 −4 10
−3
10
−2
10
Level γ
−1
10
10
0.8
0.7
SNR = 15 dB
0.6
ν=0 ν=0.01 ν=0.05 ν=0.25
0.5
0
(a)
0.4 −4 10
10
−3
10
−2
Level γ
10
−1
0
10
(b)
Fig. 1. ROC curves of detector (13) for various mismatch-to-maximal interference ratios ν. (a) SNR set to 10 dB, (b) SNR set to 15 dB. The maximal interference-to-noise ratio ρ is set to 15 dB.
R EFERENCES [1] F.-X. Socheleau and D. Pastor, “Testing the energy of random signals in a known subspace: an optimal invariant approach,” IEEE Signal Processing Letters, vol. XX, no. X, pp. XX – XX, XX 2014. [2] L. L. Scharf and B. Friedlander, “Matched Subspace Detectors,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 2146 – 2157, 1994.
