Principles of Modern Radar - Volume 2 1891121537

13.9 Problems 627with the result in (a). Choose the reference angle α such that the x-component of thefield is real.4. An elliptically polarized wave is traveling along the z-direction with the phasor electricfield given by E = ( ˆx2 + ŷ1e jπ/4) e − jkz . Find the tilt angle, ellipticity angle, andsense of rotation of the polarization ellipse. Find a complex unit vector to represent itspolarization state in a linear (x − y) basis. It is required to represent this wave as a sumof two orthogonally polarized elliptical states, E = E 1 + E 2 = (ê 1 E 1 + ê 2 E 2 ) e − jkz .Assume that the field E 1 has tilt angle ψ = 30 ◦ and ellipticity angle τ =−22.5 ◦ .Using the linear-to-elliptical polarization basis transformation in (13.35), calculateE 1 ,E 2 , and determine the complex vectors for the orthogonal elliptical polarizationsstates ê 1 and ê 2 .5. Write the elliptically polarized wave of Problem 4 as the sum of two orthogonalcircularly polarized waves represented by the basis vector [ê r ê r⊥ ] (see (13.30)).6. An elliptically polarized wave is traveling along the z-direction with the phasorelectric field given by E = ( ˆx5 + ŷ8e jπ/8) e − jkz . Find the tilt angle, ellipticity angle,and sense of rotation of the polarization ellipse. Find the Stokes parameters and determinethe location of the polarization state on the Poincaré sphere. Also, determinethe antipodal polarization state and write its Stokes vector.7. If the Stokes parameters of a wave are given by g 0 = 3,g 1 =−2,g 2 = 1,g 3 = 2, findE x and E y , the tilt angle and the ellipticity angle. Show the location of its polarizationstate on the Poincaré sphere.8. Consider a wave with field components given by(E x = 2 cos ωt + π )8(E y = 3 cos ωt + 3π 2).Find the Stokes parameters and their Poincaré representation (13.21) in terms of tiltand ellipticity angles. Show the polarization state location on the Poincaré sphere.9. A partially polarized wave for which d = 0.5, AR = 2,ψ = 0 ◦ , and S = 1 (W/m 2 ),where S is the average power density of the wave, is incident on an antenna for whichAR = 4,ψ = 22.5 ◦ , and the effective aperture A e = 1m 2 . Find the received power.10. (a) A wave for which d = 0.4, AR = 3,ψ = 45 ◦ , and S = 1 (W/m 2 ) is separatelyreceived by six antennas, all of unit effective aperture, with polarization as follows:(i) linear horizontal, (ii) linear vertical, (iii) linear slant (45 ◦ ), (iv) linear slant(135 ◦ ), (v) left circular, and (vi) right circular. In each case, find the receivedpower.(b) Is there a wave to which all the six antennas listed in part (a) respond equally? Ifso, what are the wave parameters (tilt angle and axial ratio)?11. Show that a very thin square plate or disk of area A p has the scattering matrix in linearpolarization basis given byS = C[ 1 00 1], C = 2√ π A pλ