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For A Rn n , the set of scalars i C and (nonzero) vectors xi C n satisfying or ( i I A)xi = 0n Axi = i xi are the eigenvalues and eigenvectors of A We are only interested in nontrivial solutions (ie, solution xi = 0 is not of interest) Nontrivial solutions exist only if ( i I A) is a singular matrix Therefore, the eigenvalues of A are the values of such that | I A| = 0 This yields an n-th order polynomial in If A is a symmetric matrix, then all of its eigenvalues and eigenvectors are real If xi and xj are eigenvectors of symmetric matrix A and their eigenvalues are not equal (ie, i = j ), then the eigenvectors are orthogonal (ie xi xj = 0) A square matrix A is idempotent if and only if AA = A Idempotent matrices are sometimes also called projection matrices Idempotent matrices have the following properties: 1 rank(A) = T r(A); 2 the eigenvalues of A are all either 0 or 1; 3 the multiplicity of 1 as an eigenvalue is the rank(A); 4 A(I A) = (I A)A = 0; and, 5 A , (I A) and (I A ) are idempotent Example B1 Select a state estimation gain vector L such that the error state model for the discrete-time system xk+1 pk = = 08187 00906 00000 00000 10000 10000 xk + xk 00906 00047 fk + L(pk pk )

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At ultrahigh frequencies (UHF) and microwave frequencies, high-gain antennas are reasonable in size because the wavelengths are short

has eigenvalues as 09 01j The actual system is modeled as xk+1 pk = = 08187 00000 00906 10000 00000 10000 xk + xk 00906 00047 fk

The observability is considered in Example 316 and will not be repeated here The error state xk = xk xk has the state space model xk+1 = 08187 L1 00906 (1 L2 ) xk = L xk

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Waveguides In Chap 25, waveguides were briefly mentioned as an option for feeding antenna systems at UHF and microwave frequencies Here s a little more about them A waveguide is a hollow metal pipe having a rectangular or circular cross section The EM field travels down the pipe, provided that the wavelength is short enough (or the cross-sectional dimensions of the pipe are large enough) In order to efficiently propagate an EM field, a rectangular waveguide must have height and width that both measure at least 05 , and preferably more than 07 A circular waveguide should be at least 06 in diameter, and preferably 07 or more

The eigenvalues of the discrete-time, error-state, transition matrix are the roots of the equation 0 = = |zI L | (z 08187) 00906

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The characteristic impedance (Zo) of a waveguide varies with frequency In this sense, it differs from coaxial or parallel-wire lines, whose Zo values are generally independent of the frequency A properly installed and well-maintained waveguide is an exceptional transmission line because dry air has practically no loss, even at UHF and microwave frequencies But it is important that the interior of a waveguide be kept free from dirt, dust, insects, spiderwebs, and condensation Even a small obstruction can seriously degrade the performance and cause the waveguide to become lossy The main limitation of a waveguide, from a practical standpoint, is inflexibility You can t run a waveguide from one point to another in a haphazard fashion, as you can do with coaxial cable Bends or turns in a waveguide present a particular problem, because they must be made gradually Installing a waveguide for a UHF or microwave antenna is a little like putting in the conduit for a new electrical circuit in a home or business It s a significant construction project! Waveguides are impractical for use at frequencies below approximately 300 MHz, because the required cross-sectional dimensions become prohibitively large

L1 (z + L2 1)

= z 2 + (L2 18187)z + (08187 08187L2 + 00906L1 ) To achieve the desired eigenvalues, we must have 18 = L2 18187 082 = 08187 08187L2 + 00906L1

Dish A dish antenna must be correctly shaped and precisely aligned The most efficient shape, especially at the shortest wavelengths, is a paraboloidal reflector However, a spherical reflector, having

The solution is L = [01833, 00187]

For a square matrix B Rn n and x Rn , the scalar mapping x Bx is called a quadratic form It is straightforward using eqn (B3) to show that the quadratic form depends only on the symmetric portion of B: x Bx = x Bs x + x Ba x = x Bs x,

conventional feed (A) and Cassegrain feed (B)

see Exercise B14 Therefore, it is standard practice to assume that B is symmetric When a quadratic form is de ned, the designer is often interested in the sign of the scalar output, x Bx, which depends on the properties of the symmetric matrix B

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