Channel Estimation Error in Distributed Detection Systems Hamidreza Ahmadi
Outline Detection Theory Neyman-Pearson Method Classical l Distributed ib t Detection ti Fusion and local sensor rules Channel Aware Distributed Detection Perfect and Estimated CSI in the FC Conclusion
Detection Theory y( (NP Method) Binary Hypotheses Testing Px ( H1) P ( x H 0) H0: x = n H1 :x= A + n Decision Rule: 2 n N(0, σ ) H1 x >τ γ (x) = H0 x <τ False Alarm Probability: P FA = Pr( γ(x) = H1 H0) = Pr(x > τ x = n) P( H0 H1) = 1 PD PH ( 1 H0) = PFA Detection Probability: P D = Pr( γ(x) = H1 H1) = Pr(x > τ x = S+ n)
Detection Theory y( (NP Method) Neyman-Pearson Method: Generally Detection probability and False alarm are changing by changing threshold. Max P D PFA α Likelihood Ratio Test (LRT) : L(x) H1 P(x H1) > = τ P(x H0) < H0 Algorithm: P = P( Λ>τ H 0) =α τ P ( τ) FA D
Classical Distributed Detection Parallel Topology Each Sensor based on its observation,independently, makes its own decision about the Hypotheses and sends it to the fusion center. u =γ (y ) i i i u =γ (u,u,...,u ) 0 0 1 2 N Applicable to Power and Bandwidth limited Networks Performance loss because of accessing to only partial information in the center as compared with centralized Globally y Optimal Sensor and fusion rules
Classical Distributed Detection NP Method in DD: For Fixed global false alarm, what are the Optimum local and fusion rules to Maximize global detection probability. Max P D0 ( γ, γ,..., γ ) 0 1 N Subject to PF0 α Optimal FC Rule: LRT H1 1 N P(U H 1) P(u i H 1) > Λ (U) = = τ P(U H 0) i= 1 P(u H 0) < i H0 Optimal Local Sensor rules: More complicated because of distributed nature. With Conditional independence of sensor observation: LRT H1 P(y i H 1) > Λ (y i) = τ P(y H 0) < i H0 i
Classical Distributed Detection Person by person optimization (PBPO) LRT thresholds at the sensors are coupled with each others and Fusion Center s. Each sensor threshold is optimized assuming fixed decision rules at all other sensors and the FC. This is iteratively done until we reach optima value. It gives a necessary but not sufficient condition for optimality, so several Initialization are necessary. Error Exponents: PBPO is intractable in networks with large number of sensors, So in asymptotic regime, threshold is selected such that gives the best error exponent.( identical thresholds)
Channel Aware Distributed Detection. Sources of uncertainty:noise, fading, Shadowing, interference in Observation and Transmission channel Separation Approach for DD with uncertainty: Communication schemes between sensors and center are separated from the SP algorithms in decision rules.
Channel Aware Distributed Detection Channel Aware Fusion rule: (1) The ultimate goal is not recovering Ui. (2) θ I( θ ; y,...,y ) I( θ ;u ˆ,...,u ˆ ) 1 k 1 k Optimal detector should consider Channel Conditions in the Fusion Rule. (LRT is optimum rule).
Channel Aware Distributed Detection Channel Aware local sensor rules: For globally optimal detection, transmission channel is considered in the local sensors. (Energy efficient) The sensor thresholds are different for different channels. ( Using channel statistics instead of instant CSI)
Channel Aware Distributed Detection Perfect CSI at the fusion center: Example: Independent transmission & observation channels, BPSK (ui= -1 or 1), coherent reception.
Channel Aware Distributed Detection Suboptimal Fusion Rules: High SNR (Chair-Varshney): Low SNR (MRC and EGC): Also can be used when detection indexes of sensors are not known in the FC.
Channel Aware Distributed Detection BPSK with perfect CSI in the FC, using optimal and suboptimal rules
Channel Aware Distributed Detection Estimated CSI in the Fusion center T sensor decisions are packed and sent with a training bit. Assume there is a block fading with coherence time less than the packet length. yk,1 = uthk + nk,1 (u = 1) LMMSE complex channel estimation: E ĥh = E(h y ) =α y, σ = (1 + ) σ 2 b 1 k k k1 k,1 k1 k,1 error 2 n σ = E σ +σ 2 2 2 w b error n t Λ BPSK K * Λ = Re(y h ) Λ 1 ˆ BPSK EGC = Re(y k,tφk ), φk = e K = K k = 1 K 1 ˆ * BPSK MRC k,t k K k 1 jϕ k
Channel Aware Distributed Detection Different Modulation with perfect and imperfect CSI using LRT
Channel Aware Distributed Detection BPSK with optimal and suboptimal fusion rules
Channel Aware Distributed Detection OOK modulation and the impact of Number of sensors on p Pd.
Channel Aware Distributed Detection BFSK, Sensors with different Pds
Conclusion Neyman-Pearson as a detection criterion for maximizing detection probability with the constraint on false alarm rate. Decentralized Detection,, a power and bandwidth efficient detection which uses LRT as an optimal rule in the sensors and FC. Transmission channel aware sensor and fusion rules can improve the detection performance of DD systems in fading channels. In high SNR rules using estimated CSI perform like the ones with In high SNR, rules using estimated CSI perform like the ones with perfect CSI.
Channel Estimation in DD Systems Thank You.