We deal now with the problem of rational matching with prescribed transmission zeros:

\(\displaystyle\underset{S_{22}\in \mathbb{F}^N_R}{\text{min}}\underset{\omega}{\text{max}}\left|S_{22}(\omega)\right|\)    for all \(\omega\) in the passband

We aim to minimise the maximum in the passband of the reflection parameter of the global system. The reflection of the global system is parametrised in a rational form as \(S_{22}=\frac{p}{q}\) with  \(qq^* – pp^* = R \) where \(p,q\) are polynomials of degree at most N and \(R\) is a positive polynomial having the prescribed transmission zeros as roots.

Also note that the global system is composed of a matching filter chained to the prescribed load. Therefore we need to ensure that such system can indeed be obtained as a passive stable filter cascaded with the load, namely the reflection of the system at the second port \(S_{22}\) must be feasible (after specifying the transmission zeros it corresponds to the set of responses \(S_{22}\in \mathbb{F}^N_R\)).

For a load A with transmission zeros \(\alpha_i\) and reflection \(A_{22}\) at the second port, the feasible responses are those that satisfies:

\(\displaystyle S_{22}(\alpha_i) = A_{22}(\alpha_i)\)

Thus the previous problem is an optimisation among all rational responses \(S_{22}\) of degree N satisfying the previous set of interpolation conditions. However we do not know  how to solve this problem optimally yet. Instead we can consider a relaxed version of the set of feasible responses.

Admissible responses

We define a minimum phase (with no zeros in the analicity domain) schur function \(g(\omega)\)  as admissible if and only if there exist another function \(S_{22}\) that is feasible (\(S_{22}\in \mathbb{F}\)) and has better (smaller) modulus everywhere in the frequency axis \(|S_{22}(\omega)|\leq |g(\omega)|\).

We denote by \(\mathbb{G}\) the set of admissible responses. Similarly we define \(\mathbb{G^N_R}\) as the set of admissible responses parametrised in a rational form \(S_{22}=\frac{p}{q}\) with  \(qq^* – pp^* = R \) where \(p,q\) are polynomials of degree at most N.

Characterisation of admissible responses

Suppose \(g\in \mathbb{G}\) (admissible), then there exist a function \(f\in \mathbb{F}\) (feasible) such that for all \(\omega\in\mathbb{R}\), \(|f(\omega)|\leq |g(\omega)|\). Since \(f\) is feasible it satisfies the interpolation conditions \(f(\alpha_i)=A_{22}(\alpha_i)\).

In that case there exist a schur function \(b(\omega) = \frac{f(\omega)}{g(\omega)}\) satisfying the interpolation conditions \(b(\alpha_i)={A_{22}(\alpha_i)}/{g(\alpha_i)}\).

Thus we state: A minimum phase schur function \( g(\omega)\) is admissible for a load A with transmission zeros \(\alpha_i\) and reflection \(A_{22}\) at the second port, if and only if there exist a schur function \(b(\omega)\) such that

\(\displaystyle b(\alpha_i)=\frac{A_{22}(\alpha_i)}{g(\alpha_i)}\)

Admissible polynomials

As before, we also parametrise the admissible functions in the belevitch form as \(g=\frac{p}{q}\) with  \(qq^* – pp^* = R\). Note that the modulus square of \(g\) can be express only in function of the positive polynomials \(P = pp^*\).

\(\displaystyle |g(\omega)|^2=G_P(\omega) = \frac{P(\omega)}{P(\omega)+R(\omega)}\)    for all \(\omega\) real

being \(g_P\) the minimum phase spectral factor of  \(G_P\).

Given the positive polynomial \(R\) of degree at most 2N, we call admissible polynomials the set of positive polynomials \(P\) of degree  at most 2N such that  \(g_{P}\) is admissible. We represent this set by \(\mathbb{H^N_R}\)

Convexity

Theorem: the set \(\mathbb{H^N_R}\) is a convex set.

Proof: We proof that if  \(g_{P_1}\) and \(g_{P_2}\) are admissible (there exist \(f_1,f_2\) feasible such that \(|f_1(\omega)|\leq |g_{P_1}(\omega)|\), \(|f_2(\omega)|\leq |g_{P_2}(\omega)|\), \(\forall \omega\in \mathbb{R}\)), then \(g_{P_3}\) with \(P_3 = \lambda P_1 + (1-\lambda)P_2\),   \(0\leq \lambda \leq 1\) is admissible as well (there exist \(f_3\) feasible such that for all \(\omega\in \mathbb{R}\), \(|f_3(\omega)|\leq |g_{P_3}(\omega)|\)).

The check for feasibility of \(f_3\) is straightforward, \(f_1,f_2\) satisfies  \(f_1(\alpha_i) = A_{22}(\alpha_i)\), \(f_2(\alpha_i)=A_{22}(\alpha_i)\), then \(f_3 = \lambda f_1 + (1-\lambda) f_2\) satisfies those interpolation conditions as well.

Now we need to show that \(|f_3(\omega)|^2\leq G_{P_3}(\omega) \), \(\forall \omega\in\mathbb{R}\).

\( |g_P|^2\) is a concave function in P (its second derivative is negative):

\(\displaystyle \frac{d}{dP}|g_P|^2 = \frac{R}{(P+R)^2}\)                    \(\displaystyle \frac{d^2}{d^2P}|g_P|^2 = -\frac{2R}{(P+R)^3}\)

Therefore (calling \(g_i=|g_{P_i}|\)):

\( \displaystyle G_{P_3} \geq \underset{A}{\underbrace{\lambda g_{1}^2 + (1-\lambda) g_{2}^2}}\)

Also by applying the triangle inequality

\( \displaystyle |f_3|^2=|\lambda f_1 + (1-\lambda) f_2|^2 \leq \left( \lambda |f_1| + (1-\lambda) |f_2| \right)^2 \leq \underset{B}{\underbrace{\left( \lambda g_{1} + (1-\lambda) g_{2} \right)^2}} \)

It can be easily verified  that \(A(\omega)\geq B(\omega)\) for all \(\omega\) real. In particular, computing \(A(\omega)-B(\omega)\)

\(\displaystyle A(\omega)-B(\omega) = 2(\lambda – \lambda^2)\left(g_{1}(\omega) – g_{2}(\omega) \right)^2\geq 0\)

We obtain \(|f_3(\omega)|^2\leq B\omega) \leq A(\omega) \leq G_{P_3}(\omega)\). Thus the polynomial \(P_3\) is admissible.

Relaxed matching problem

Now we can modify the matching problem to seek the best among all admissible responses instead of the feasible ones:

\(\displaystyle\underset{S_{22}\in \mathbb{G}^N_R}{\text{min}}\underset{\omega}{\text{max}}\left|S_{22}(\omega)\right|\)    for all \(\omega\) in the passband

This time, we are minimising the maximum reflection in the passband among all admissible responses (not feasible) that are parametrised in the belevitch form. With the rational parametrisation

\(\displaystyle |g(\omega)|^2 = \frac{P(\omega)}{P(\omega)+R(\omega)} = \left(1+\frac{R(\omega)}{P(\omega)} \right)^{-1}\)

we can state the matching problem over the set of admissible polynomials as:

\(\displaystyle\underset{P\in \mathbb{H}^N_R}{\text{min}}\underset{\omega}{\text{max}}\frac{P(\omega)}{R(\omega)}\)    for all \(\omega\) in the passband

Thanks to the convexity of the set \(\mathbb{H}^N_R\), the convexity of the previous problem can be proved, and therefore the optimality of the solution is guaranteed.

 

Link to the original matching problem

Consider now the original problem where the reflection of the global system has been parametrised in the rational form as well

• Original problem

\(\displaystyle L_{opt} = \underset{S_{22}\in \mathbb{F}^N_R}{\text{min}}\underset{\omega}{\text{max}}|S_{22}(\omega)|\)    for all \(\omega\) in the passband

• Relaxed problem

\(\displaystyle L_{rel} = \underset{S_{22}\in \mathbb{G}^N_R}{\text{min}}\underset{\omega}{\text{max}}|S_{22}(\omega)|\)    for all \(\omega\) in the passband

Note that every function \(S_{22}\in \mathbb{F}^N_R\) can be expressed as a blascke product times a minimum phase function (being this one admissible)

\(\displaystyle S_{22}(\omega)= b(\omega) g(\omega) \)

where \(h(\omega)\) is admissible and \(|S_{22}(\omega)|=|g(\omega)|\) for all \(\omega\) real. Therefore for  each feasible function \(S_{22}\) there exist an admissible function \(g\) that provides the same criterium in previous problem. Thus by solving the relaxed problem we obtain a lower bound for the optimal level in the original problem

\(\displaystyle L_{rel}\leq L_{opt}\)

Note the original problem is not convex, once a local minimum is found, it can not be guaranteed to be the global one. However we can use the lower bound \(L_{rel}\) to determine how far from the optimum the obtained solution is.