Contents
The main purpose of PUMA is to solve the problem
\(\displaystyle\underset{P\in \mathbb{H}^N_R}{\text{min}}\underset{\omega}{\text{max}}\frac{P(\omega)}{R(\omega)}\) for all \(\omega\) in the passband
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
\( \displaystyle |g_P(\omega)|^2 = \left( 1+ \frac{R(\omega)}{P(\omega)}\right)^{-1} \)
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
- Compute the stable function \(g_P(\omega)\) as the minimum phase factor of \((1+R/P)^{-1}\)
- Check for the existence of a schur function \(b(\omega)\) such that \(b(\alpha_i) = A_{22}(\alpha_i)/g_P(\alpha_i)\). This is done by testing the positive definiteness of the matrix
\(\displaystyle \Delta(P) = \frac{1}{j}\left(\begin{array}{cccc}\frac{1-|\gamma_1|^2}{2j\text{Im}(\alpha_1)} & \frac{1-\gamma_1\overline{\gamma_2}}{\alpha_1 – \overline{\alpha_2}} & \cdots & \frac{1-\gamma_1\overline{\gamma_N}}{\alpha_1 – \overline{\alpha_N}}\\ \frac{1-\gamma_2\overline{\gamma_1}}{\alpha_2 – \overline{\alpha_1}} & \frac{1-|\gamma_2|^2}{2j\text{Im}(\alpha_2)} & \cdots & \frac{1-\gamma_2\overline{\gamma_N}}{\alpha_2 – \overline{\alpha_N}} \\ \vdots & \vdots & \ddots & \vdots\\ \frac{1-\gamma_N \overline{\gamma_1}}{\alpha_N -\overline{\alpha_1}} & \frac{1-\gamma_N\overline{\gamma_2}}{\alpha_N-\overline{\alpha_2}}& \cdots & \frac{1-|\gamma_N|^2}{2j\text{Im}(\alpha_N)}\end{array} \right)\)
with \(\gamma_i = A_{22}(\alpha_i)/g_P(\alpha_i)\).
Thus we can write the previous problem as
\(\displaystyle\underset{P}{\text{min}}\underset{\omega}{\text{max}}\frac{P(\omega)}{R(\omega)}\) s.t \(\Delta(P)\succeq 0\) for all \(\omega\) in the passband
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.
Solving a min-max problem
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
\(\displaystyle \underset{P}{\text{min}} L \)such that:
\(\displaystyle P(\omega)\geq 0\) for all \(\omega\) real
\(\displaystyle \frac{P(\omega)}{R(\omega)}\leq L\) for all \(\omega\) in the passband
\(\displaystyle \Delta(P)\succeq 0\)
Positive polynomials
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
\(\displaystyle \psi_N(\omega) \Theta_0 \psi_N^T(\omega) = P(\omega)\)
where \(\psi_N(\omega)\) is the corresponding basis vector of degree N. For instance, the monomials of degree 0 to N:
\(\displaystyle \psi_N = [\omega^N, \omega^{N-1}, \cdots, 0]\)
This matrix \(\Theta_0\) is called the Gram matrix.
From the Gram matrix \(\Theta_0\) it is possible to obtain the coefficients of the corresponding polynomial P with the linear operation
\(\displaystyle P = Z\cdot \text{vec}\Theta_0 \)
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 \(\Theta_0\).
Positive polynomials on an interval
The polynomials \(Q(\omega)\) of degree 2N that are positive on an interval \([a,b]\) are parametrised as
\(\displaystyle Q(\omega) = F(\omega) – (\omega-a)(\omega-b) G(\omega)\)
where \(F(\omega)\) and \(G(\omega)\) are polynomials positive on the real axis of degree 2N and 2N-2 respectively.
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
\(\displaystyle \psi_N(\omega) \Theta_1\psi_N^T(\omega) = F(\omega)\)
\(\displaystyle \psi_{N-1}(\omega) \Theta_2 \psi_{N-1}^T(\omega) = G(\omega)\)
As in the previous case, the coefficients of the polynomial Q depend linearly on the matrices \(\Theta_1, \Theta_2\)
\(\displaystyle Q = \Omega \cdot \left[\begin{array}{c} \text{vec}\Theta_1\\\text{vec}\Theta_2\end{array} \right]\)
with \(\Omega\) a matrix of size \((2N-1)xN^2\)
Semi-definite program
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.
Thus we re-parametrice the problem in function of the Gram matrices
\(\displaystyle \begin{array}{l} P\rightarrow P(\Theta_0) \\ \Delta(P) \rightarrow \Delta(\Theta_0) \\ Q=RL-P \rightarrow Q(\Theta_1,\Theta_2)\end{array}\)
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
\(\displaystyle [-R,Z,\Omega] \cdot \left[\begin{array}{c}L\\ \text{vec}\Theta_0 \\ \text{vec}\Theta_1\\\text{vec}\Theta_2\end{array} \right] = 0 \)
Finally the problem remains
\(\displaystyle \underset{\Theta_0,\Theta_1,\Theta_2,L}{min}(L)\) subject to \(\Theta_0\succeq 0\), \(\Theta_1\succeq 0\), \(\Theta_2\succeq 0\), \(\Delta(\Theta_0)\succeq 0\) and \(\displaystyle [-R,Z,\Omega] \cdot \left[\begin{array}{c}L\\ \text{vec}\Theta_0 \\ \text{vec}\Theta_1\\\text{vec}\Theta_2\end{array} \right] = 0 \)
This is a standard non linear semi-definite program under linear equality constrains that is solved by means of the matlab toolbox NonSDP.