The main purpose of PUMA is to solve the problem<\/p>\n
\\(\\displaystyle\\underset{P\\in \\mathbb{H}^N_R}{\\text{min}}\\underset{\\omega}{\\text{max}}\\frac{P(\\omega)}{R(\\omega)}\\) \u00a0\u00a0 for all \\(\\omega\\) in the passband<\/p>\n
where \\(P,R\\) are positive polynomials of degree at most 2N and \\(\\mathbb{H}^N_R\\) (admissible polynomials) is the set of polynomials of degree at most 2N such that the minimum phase function \\(g_P(\\omega)\\) is admissible with<\/p>\n
\\( \\displaystyle |g_P(\\omega)|^2 = \\left( 1+ \\frac{R(\\omega)}{P(\\omega)}\\right)^{-1} \\)<\/p>\n
Therefore, given a load A with transmission zeros \\(\\alpha_i\\) and reflection coefficient \\(A_{22}\\) at the second port, in order to check if a polynomial \\(P(\\omega)\\) is admissible, we proceed as follows<\/p>\n
\\(\\displaystyle \\Delta(P) = \\frac{1}{j}\\left(\\begin{array}{cccc}\\frac{1-|\\gamma_1|^2}{2j\\text{Im}(\\alpha_1)} & \\frac{1-\\gamma_1\\overline{\\gamma_2}}{\\alpha_1 – \\overline{\\alpha_2}} & \\cdots & \\frac{1-\\gamma_1\\overline{\\gamma_N}}{\\alpha_1 – \\overline{\\alpha_N}}\\\\ \\frac{1-\\gamma_2\\overline{\\gamma_1}}{\\alpha_2 – \\overline{\\alpha_1}} & \\frac{1-|\\gamma_2|^2}{2j\\text{Im}(\\alpha_2)} & \\cdots & \\frac{1-\\gamma_2\\overline{\\gamma_N}}{\\alpha_2 – \\overline{\\alpha_N}} \\\\ \\vdots & \\vdots & \\ddots & \\vdots\\\\ \\frac{1-\\gamma_N \\overline{\\gamma_1}}{\\alpha_N -\\overline{\\alpha_1}} & \\frac{1-\\gamma_N\\overline{\\gamma_2}}{\\alpha_N-\\overline{\\alpha_2}}& \\cdots & \\frac{1-|\\gamma_N|^2}{2j\\text{Im}(\\alpha_N)}\\end{array} \\right)\\)<\/p>\n
with \\(\\gamma_i = A_{22}(\\alpha_i)\/g_P(\\alpha_i)\\).<\/p>\n
Thus we can write the previous problem as<\/p>\n
\\(\\displaystyle\\underset{P}{\\text{min}}\\underset{\\omega}{\\text{max}}\\frac{P(\\omega)}{R(\\omega)}\\)\u00a0\u00a0 s.t\u00a0\u00a0 \\(\\Delta(P)\\succeq 0\\) \u00a0\u00a0 for all \\(\\omega\\) in the passband<\/p>\n
This is a min-max problem that minimises the maximum of a linear criterium over the convex cone of positive polynomials P. However the matrix \\(\\Delta(P)\\) depends not linearly on P what makes the problem not a standard one.<\/p>\n
In order to deal with the min-max problem we introduce the variable \\(L=\\underset{\\omega}{\\text{max}}P(\\omega)\/R(\\omega)\\). This allows us to restate the problem as<\/p>\n\\(\\displaystyle \\underset{P}{\\text{min}} L \\)\n
such that:<\/p>\n
\\(\\displaystyle P(\\omega)\\geq 0\\) \u00a0\u00a0 for all \\(\\omega\\) real<\/p>\n
\\(\\displaystyle \\frac{P(\\omega)}{R(\\omega)}\\leq L\\) \u00a0\u00a0 for all \\(\\omega\\) in the passband<\/p>\n
\\(\\displaystyle \\Delta(P)\\succeq 0\\)<\/p>\n
A polynomial P of degree 2N is non-negative on the real line if and only if there exist a positive semidefinite \\((N+1)x(N+1)\\) matrix \\(\\Theta_0\\) such that<\/p>\n
\\(\\displaystyle \\psi_N(\\omega) \\Theta_0 \\psi_N^T(\\omega) = P(\\omega)\\)<\/p>\n
where \\(\\psi_N(\\omega)\\) is the corresponding basis vector of degree N. For instance, the monomials of degree 0 to N:<\/p>\n
\\(\\displaystyle \\psi_N = [\\omega^N, \\omega^{N-1}, \\cdots, 0]\\)<\/p>\n
This matrix \\(\\Theta_0\\) is called the Gram<\/em> matrix.<\/p>\n From the Gram<\/em> matrix \\(\\Theta_0\\) it is possible to obtain the coefficients of the corresponding polynomial P with the linear operation<\/p>\n \\(\\displaystyle P = Z\\cdot \\text{vec}\\Theta_0 \\)<\/p>\n where Z is a matrix of size \\((2N-1)x(N^2+N)\/2\\) and \\(\\text{vec}\\Theta_0 \\) is the column vector with the coefficients in the lower triangle of \\(\\Theta_0\\).<\/p>\n The polynomials \\(Q(\\omega)\\) of degree 2N that are positive on an interval \\([a,b]\\) are parametrised as<\/p>\n \\(\\displaystyle Q(\\omega) = F(\\omega) – (\\omega-a)(\\omega-b) G(\\omega)\\)<\/p>\n where \\(F(\\omega)\\) and \\(G(\\omega)\\) are polynomials positive on the real axis of degree 2N and 2N-2 respectively.<\/p>\n Therefore, using the previous theorem, polynomials F,G are positive if and only if there exist positive definite matrices \\(\\Theta_1\\) of size \\((N+1)x(N+1)\\) and \\(\\Theta_2\\) of size \\(NxN\\) such that<\/p>\n \\(\\displaystyle \\psi_N(\\omega) \\Theta_1\\psi_N^T(\\omega) = F(\\omega)\\)<\/p>\n \\(\\displaystyle \\psi_{N-1}(\\omega) \\Theta_2 \\psi_{N-1}^T(\\omega) = G(\\omega)\\)<\/p>\n As in the previous case, the coefficients of the polynomial Q depend linearly on the matrices \\(\\Theta_1, \\Theta_2\\)<\/p>\n \\(\\displaystyle Q = \\Omega \\cdot \\left[\\begin{array}{c} \\text{vec}\\Theta_1\\\\\\text{vec}\\Theta_2\\end{array} \\right]\\)<\/p>\n with \\(\\Omega\\) a matrix of size\u00a0 \\((2N-1)xN^2\\)<\/p>\n Using the previous parametrisation of positive polynomials we can use the positive definite Gram matrices \\(\\Theta_0,\\Theta_1, \\Theta_2\\) to ensure that \\(P(\\omega)\\) is a positive polynomial and also that \\(R(\\omega)\\cdot L – P(\\omega)(\\omega)\\) is positive in the passband.<\/p>\n Thus we re-parametrice the problem in function of the Gram<\/em> matrices<\/p>\n \\(\\displaystyle \\begin{array}{l} P\\rightarrow P(\\Theta_0) \\\\ \\Delta(P) \\rightarrow \\Delta(\\Theta_0) \\\\ Q=RL-P \\rightarrow Q(\\Theta_1,\\Theta_2)\\end{array}\\)<\/p>\n Note that by doing that it is necessary to impose that the matrix \\(\\Theta_0\\) and the matrices \\(\\Theta_1,\\Theta_2\\) represent the same polynomial P. We do that by imposing the linear equalities \\(Q=R\\cdot L – P \\) or equivalently \\(-RL+P+Q = 0 \\). Using the above relations we obtain the linear system<\/p>\n \\(\\displaystyle [-R,Z,\\Omega] \\cdot \\left[\\begin{array}{c}L\\\\ \\text{vec}\\Theta_0 \\\\ \\text{vec}\\Theta_1\\\\\\text{vec}\\Theta_2\\end{array} \\right] = 0 \\)<\/p>\n Finally the problem remains<\/p>\n \\(\\displaystyle \\underset{\\Theta_0,\\Theta_1,\\Theta_2,L}{min}(L)\\)\u00a0\u00a0\u00a0 subject to \\(\\Theta_0\\succeq 0\\), \\(\\Theta_1\\succeq 0\\), \\(\\Theta_2\\succeq 0\\), \\(\\Delta(\\Theta_0)\\succeq 0\\)\u00a0\u00a0 and \u00a0 \\(\\displaystyle [-R,Z,\\Omega] \\cdot \\left[\\begin{array}{c}L\\\\ \\text{vec}\\Theta_0 \\\\ \\text{vec}\\Theta_1\\\\\\text{vec}\\Theta_2\\end{array} \\right] = 0 \\)<\/p>\n This is a standard non linear semi-definite program under linear equality constrains that is solved by means of the matlab toolbox NonSDP<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":" The main purpose of PUMA is to solve the problem \u00a0\u00a0 for all in the passband where are positive polynomials of degree at most 2N and (admissible polynomials) is the set of polynomials of degree at most 2N such that the minimum phase function is admissible with Therefore, given a\u2026<\/p>\nPositive polynomials on an interval<\/h2>\n
Semi-definite program<\/h2>\n