GPS World, February 2017

of the components of the electric field and are the phase components of the field is the angular frequency of the carrier and is the wave number Only the real part of Equation 1 is physically relevant with the complex exponential containing information about the phase of the oscillating field Switching from the linear to the circular basis vector set 2 where and are the unit vectors of the RHCP and LHCP components respectively and omitting the explicit time and spatial dependence we can write the normalized electric field as 3 The polarization state of an electromagnetic signal is fully described by and More generally the electric field of any plane wave impinging at the antenna can be expressed in the form 4 Dual Polarized Antenna Array A circular DP antenna features two orthogonal circular polarization output ports meaning each element ideally receives the voltage induced by the RHCP and LHCP field components separately on the two different antenna ports Due to antenna imperfections and the coupling effect part of the received RHCP field is received by the LHCP port and vice versa These undesired voltages are responsible for the emergence of the cross polar components In view of this we characterize the antenna in terms of its response to circularly polarized plane waves and express the electric field using the Jones vector notation in the circular basis as 5 where is the complex total electric field received by the RHCP port is the complex electric field induced at the RHCP port by a purely RHCP electromagnetic wave indicated as a co polar component is the complex crosspolar 68 GPS WORLD WWW GPSWORLD COM FEBRUARY 2017 component of the electric field obtained by exciting the antenna with a purely LHCP electromagnetic wave is the azimuth angle and is the elevation angle of the impinging signal assuming the antenna to be at the origin of the spherical coordinate system Similar statements apply for the total electric field received by the LHCP port and for the copolar and cross polar components If and are the complex voltages induced at the RHCP and LHCP antenna outputs by the signal in Equation 4 respectively the antenna outputs are given by 6 where is the vector parameter carrying the information about the direction of arrival DoA of the incident signal and is the time delay of the incident signal With an M element array of DP sensors we can use and to represent the complex array responses of the DP antenna array 7 where and define the steering vector of the DP antenna array given a signal incident at angle and polarization defined by the Jones vector Problem Formulation The analog signals collected by the antenna array are then passed through the receiver front end where they are amplified filtered and shifted to baseband The resulting complex baseband signal with bandwidth B that is received by an antenna array with M DP sensor elements at polarization port P is 8 where defines the superimposed satellite signal replicas with l 1 identifying the LOS signal and l 2 L the non LOS multipath signals and denotes the superimposed radio frequency interference RFI signals with i ranging from 1 to I Additionally we assume temporally and spatially uncorrelated complex white Gaussian noise can be expressed in terms of the steering vectors given the lth signals incident angle the polarization vector and a complex scalar term