Radar systems considerations for phased array aperture design using conformal transformations on Riemannian manifolds

Static and rotating phased array radars are analyzed using conformal co-ordinate transformations on Riemann manifolds. By the use of a suitable metric tensor on a Riemann surface we obtain an expression for the parametric distance between two points that is used to calculate the area of surveillance mapped on to the uv-plane. The invariance of the beamwidth in uv-space allows us to determine the total number of beams required for scanning while the total search function time is obtained as an integral representing the volume under a Riemann time-manifold. Co-ordinate transformations are obtained for rotating arrays based on Sanson-Flamteed projections and we derive the corresponding search function times for each radar type. The packing densities of the beams are examined for both static and rotating arrays and we examine the number of radiating elements that are required so that free-space surveillance time is maintained on the basis of the number of array faces in simultaneous use.