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the shape of a section of a sphere, can also work well The feed system consists of a coaxial line or waveguide from the receiver and/or transmitter, and a horn or helical driven element at the focal point of the reflector Conventional dish feed is shown in Fig 27-11A Cassegrain dish feed is shown in Fig 27-11B The larger the diameter of the reflector in wavelengths, the greater the gain, the f/b ratio, and the f/s ratio, and the narrower the main lobe A dish antenna must be at least several wavelengths in diameter for proper operation The reflecting element can be sheet metal, a screen, or a wire mesh If a screen or mesh is used, the spacing between the wires must be a small fraction of a wavelength

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1 A matrix B is Positive De nite if and only if x Bx > 0 for all nonzero x 2 A matrix B is Positive Semi-de nite if and only if x Bx 0 for all non-zero x 3 A matrix B is Negative De nite if and only if x Bx < 0 for all non-zero x 4 A matrix B is Negative Semi-de nite if and only if x Bx 0 for all non-zero x When the matrix fails to have any of the above properties, then the matrix is sign inde nite Properties of the eigenvalues and inverse of a positive de nite matrix are discussed further in Exercise B15 Corresponding properties apply to negative de nite matrices, with appropriate changes to the statements

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The singular value decomposition (SVD) of a matrix A Rm n is de ned as A = U V where U Rm m and V Rn n are orthonormal matrices and Rm n is a diagonal matrix The diagonal of is de ned as ( 1 , 2 , , p ) where p = min(m, n) The diagonal elements of are ordered so that 1 > 2 > > p 0 The i are the singular values of the matrix A The rank of A is the smallest integer r such that i = 0 for i > r The rst r columns of U form an orthonormal basis for the range space of A while the last (n r) columns of U span the null space of A The rst r columns of V form an orthonormal basis for the range of A , while the last (n r) columns of V form an orthonormal basis for the null space of A Additional properties of the singular value decomposition and algorithms for its computation can be found in [59]

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Helical A helical antenna is a circularly polarized, high-gain, unidirectional antenna A typical helical antenna is shown in Fig 27-12 The reflector diameter should be at least 08 at the lowest operating frequency The radius of the helix should be approximately 017 at the center of the intended operating frequency range The longitudinal spacing between helix turns should be approximately 025 in the center of the operating frequency range The overall length of the helix should be at least at the lowest operating frequency A helical antenna can provide about 15 dBd forward gain Helical antennas are sometimes used in space communications systems Corner Reflector A corner reflector, employed with a /2 dipole driven element, is illustrated in Fig 27-13 This provides some gain over a /2 dipole by itself The reflector is made of wire mesh, screen, or sheet metal The flare angle of the reflecting element is approximately 90 Corner reflectors are widely used in terrestrial communications at UHF and microwave frequencies Several /2 dipoles can be fed in phase and placed along a common axis with a single, elongated reflector, forming a collinear corner reflector array Horn The horn antenna is shaped like a squared-off trumpet or trombone horn It provides a unidirectional radiation and response pattern, with the favored direction coincident with the opening of the horn The feed line is a waveguide that joins the antenna at the narrowest point (throat) of the horn

Let the matrix A be orthogonal and let A be a computed version of A that is not orthogonal due to numeric errors involved in its computation The main topic of this section is orthogonalization of A The orthogonalized version of A will be denoted by Ao A traditional orthogonalization approach is calculated as Ao = A A A

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