Principles of Modern Radar - Volume 2 1891121537

13.3 Scattering Matrix 605in (13.34) asS = U ∗ S d U † (13.37)Given a transmit polarization state, the scattering matrix elements for co-polarized andcross-polarized returns are obtained from the target geometry and the return signal. KnowingS, one may calculate its eigenvalues and the eigenvectors and determine the transmitpolarization state that produces the maximum return [13, 14].Let the transmit polarization state that yields the maximum backscattered power fromthe target be denoted in terms of the two eigenvectors in (13.35) as e t = a 1 ê 1 + a 2 ê 2 . Thetotal scattered power is given byW s = E s · E ∗ s = Se t · (Se t ) ∗ (13.38)Using (13.34) and the orthogonality of the two eigenvectors, it follows thatW s = ∣ ∣a 1∣ ∣2 ∣ ∣λ 1∣ ∣2 +∣ ∣ a 2∣ ∣ 2 |λ 2 | 2 =( ∣∣a 1∣ ∣2 +∣ ∣ a 2∣ ∣2 ) ∣ ∣ λ 1∣ ∣2 −∣ ∣ a 2∣ ∣2 ( ∣ ∣ λ 1∣ ∣2 −∣ ∣ λ 2∣ ∣2 ) (13.39)Requiring without loss of generality that ∣∣ λ 1∣ ∣2 > |λ2 | 2 , (13.39) states that the backscatteredpower is maximum when a 2 = 0. Therefore, the eigenvector ê 1 in (13.35) correspondsto the transmit polarization state ê 1 (ψ,τ) that maximizes the backscattered power. If thereceive antenna has the same polarization state, then the total received power would bemaximum as well (see (13.26)), hence, ê 1 (ψ,τ) is called Copol max [13, 35–37]. Theangles ψ and τ denote the target orientation angle and the ellipticity angle, respectively,for maximum power. The orthogonal polarization state ê 2 = ê 1 (ψ + π/2, −τ)results in asecondary maximum corresponding to a saddle point of the scattering matrix, denoted asCopol saddle [36, 37]. Equivalently, these states also correspond to two cross-polarizationnulls and are referred to as null polarizations [14, 23]. It can be shown that the correspondingeigenvalues are given by [14]λ 1 = me j2vλ 2 = m tan 2 γ e − j2v (13.40)where m is the maximum amplitude of the received signal and its square indicates anoverall measure of target size or RCS, v is called the target skip angle and is related tothe number of bounces of the signal reflected off the target (−π/4 ≤ v ≤ π/4), and γ iscalled the target polarizability angle (0 ≤ γ ≤ π/4), indicating the target’s ability to polarizeincident unpolarized radiation (0 ◦ for fully polarized return and 45 ◦ for unpolarizedreturn). The angle τ determines the ellipticity of the optimal polarization eigenvector ê 1 orê 2 , and is a measure of target symmetry with respect to right- and left-circular polarizations(0 ◦ for symmetric and 45 ◦ for totally asymmetric). It is also referred to as target helicityangle [14]. Likewise, ψ is called the orientation angle and is a measure of the orientationof the target around the radar line of sight. For roll-symmetric targets, such as a cone, ifthe axis of symmetry can be aligned to be coplanar with a ray from radar to target, thetarget coordinate system may be chosen such that the scattering matrix becomes diagonal(no cross-polarization terms), and the orientation angle can be determined directly fromthe polarization eigenvectors [14]. These five parameters (m,ψ,v,τ,γ ) uniquely definethe scattering matrix (within a constant phase factor; see (13.33)), and provide significantphysical insight into the target scattering process [13, 14].