32 500 Number of Occurrences 400 300 200 100 0 0 1 Chapter | 1 Wireless Direction-Finding Fundamentals 2 3 4 5 Number of Signals (a) 6 7 8 350 Number of Occurrences 300 250 200 150 100 50 0 0 1 2 5 4 3 Number of Signals (b) 6 7 8 FIGURE 1.19 Histogram of the number of signals resolved by MUSIC (a) and by a beamformer (b) for a random scenario with five emitters using an 8-element uniformly spaced array with K = 100 snapshots, based on 1000 random experiments, -15 SNR 10. a direction-finding scenario with realistic mixtures of emitters with different directions and powers. Results are usually presented for one or a few selected scenarios--most frequently for two equal-power emitters. These results are certainly important and instructive, but they may create unrealistic expectations about the performance of super-resolution methods in real-world deployments. APPENDIX A.1.1 The Array Manifold and Its Derivatives The array manifold a() plays a key role in the performance of direction-finding systems. Here we present the form of the manifolds for linear and circular arrays, and we calculate their first- and second-order derivatives, which are needed to compute the accuracy of direction estimation achieved by different algorithms. For simplicity we assume that the antennas and all the emitters are in the same plane, so that the array manifold can be characterized by azimuth only.