188 Statistical channel parameter estimation 107 N-1 10 02040608 12 0^ 0.2040608 1214 50 Fig.7.3 The empirical average of M.versus the AS with M as a parameter:N=500,=10dB,and M·08=14 (7.36) providespromion of he dependence berwennThfc haorimlies )can be used as an Performance of the NA Estimators 一2Cm tha he MUSIC an188 Statistical channel parameter estimation 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−3 10−2 10−1 100 Emp. CCDF, σφ˜ = 1o Emp. CCDF, σφ˜ = 0.5 o Emp. CCDF, σφ˜ = 2o Emp. CCDF, σφ˜ = 4o Emp. CCDF, σφ˜ = 8o Approx. CCDF z ˆP[ϕ ≥ z] = (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−2 10−1 100 z ˆP[ϕ ≥ z] Emp. CCDFs Approx. CCDF N = 1 N = 3 N = 5 N = 10 N = 50 (b) Fig. 7.2 Empirical (Emp.) and approximate (Approx.) CCDFs of the absolute normalized estimation error of the SS-ML azimuth estimator applied in an SDS scenario (a) with σφ˜ as a parameter, N = 10; (b) with N as a parameter, σφ˜ = 2◦ . 0 1 2 3 4 5 6 7 2 4 6 8 10 12 14 16 18 σφ˜ in [ ◦ ] Mmax = 8 Mmax = 12 Mmax = 16 Empirical approximation (7.36) ˆµMa Fig. 7.3 The empirical average µˆMa of Ma versus the AS σφ˜ with Mmax as a parameter: N = 500, γ = 10dB, and φ¯ = 90◦ . µˆMa = Mmax, while µˆMa = Mmin for large ASs (σφ˜ ≥ 5 ◦ ). In between, µˆMa decreases inversely proportionally to σφ˜. The equation µˆMa · σφ˜ = 14 (7.36) provides a good approximation of the dependence between µˆMa and σφ˜. The fact that µˆMa = Mmin for σφ˜ ≥ 5 ◦ implies that the 1 st-order GAM approximation is not accurate in that range of the AS. Equation (7.36) can be used as an empirical criterion for the preselection of the size of ULA arrays given an AS value and vice-versa. Performance of the NA Estimators Fig. 7.4 depicts the empirical CCDF of the absolute normalized estimation error ϕ of the NA estimators. The CCDF of ϕ of the SS ML azimuth estimator is also reported for comparison purpose. The parameter setting is N = 10, σφ˜ = 2◦ and γ = 25 dB. It can be observed that the SML and the DML estimators perform similarly and better than the MUSIC and SS ML estimators. The MUSIC estimator outperforms the SS-ML estimator. The results show that the NA estimators have significantly lower probability of large estimation errors compared to the conventional SS-ML estimator