Principles of Modern Radar - Volume 2 1891121537

17.6 Multichannel Processing for Detection Performance Improvement 809can be modeled as additive white Gaussian noise with variance σ 2 n , the resulting DOAestimation accuracy is given by√σ θT (θ T ) ={={ ∣E∣∣ˆθ } T − θ T 2− ∣∣ E {ˆθ }∣T − θ T ∣22 |a|2σ 2 n[‖ṡ θ (θ T )‖ 2 −∣ ṡ H θ (θ T ) s θ (θ T ) ∣ ∣ 2‖s θ (θ T )‖ 2 ]} −1/2(17.41)Using omnidirectional antennas, that is, g i (θ) = g (i = 1, ..., M), the estimation accuracyin (17.41) becomesσ θT (θ T ) =[2 |a|2 g 2 4π ] 2 −1/2f (θσn2 λ 2 T )(17.42)where the dependence on the DOA θ T is embedded in the term f (θ T ) given byf (θ T ) =M∑(x i cos θ T − y i sin θ T ) 2 − 1 [ M 2∑(x i cos θ T − y i sin θ T )](17.43)Mi=1i=1If the following symmetry conditions hold for the planar arrayM∑x i = 0i=1&M∑y i = 0i=1&M∑xi 2 =i=1M∑i=1y 2 i&M∑x i y i = 0 (17.44)i=1equation (17.43) becomesf (θ T ) = 1 M∑ ( x22 i + yi2 ) ∑M M∑= xi 2 = yi 2 = 1 M∑Ri 2 (17.45)2i=1where R i is the Euclidean distance of the i-th element from the origin of the referencesystem. In this case, the estimation accuracy in (17.42) is independent of the DOA θ T .This is a desirable characteristic when the considered application does not require specificdirections to be preferred. Thus, in the following we consider only symmetric array configurations,namely, array configurations for which the symmetry conditions in (17.44)hold.The selection of the most appropriate configuration for passive radar applicationsinvolves the choice of the number M of antenna elements, the identification of its shapeand the determination of the array radius R, defined as the radius of the smallest circlecontaining all the array elements. Considering an array with a relatively small numberof elements, a suitable configuration might be selected on the basis of its pattern characteristics,such as the 3 dB aperture and the PSLR. In fact, a selection based only onthe estimation accuracy in (17.41) would not be appropriate since such expression is significantonly in the absence of ambiguities, namely, when grating lobes do not occur. Aparameter related to the presence of grating lobes, such as the PSLR, has therefore to beconsidered together with the estimation accuracy. Moreover, since this latter parameter isi=1i=1i=1