Contents
We deal now with the problem of rational matching with prescribed transmission zeros:
for all 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 with where are polynomials of degree at most N and 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 must be feasible (after specifying the transmission zeros it corresponds to the set of responses ).
For a load A with transmission zeros and reflection at the second port, the feasible responses are those that satisfies:
Thus the previous problem is an optimisation among all rational responses 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 as admissible if and only if there exist another function that is feasible () and has better (smaller) modulus everywhere in the frequency axis .
We denote by the set of admissible responses. Similarly we define as the set of admissible responses parametrised in a rational form with where are polynomials of degree at most N.
Characterisation of admissible responses
Suppose (admissible), then there exist a function (feasible) such that for all , . Since is feasible it satisfies the interpolation conditions .
In that case there exist a schur function satisfying the interpolation conditions .
Thus we state: A minimum phase schur function is admissible for a load A with transmission zeros and reflection at the second port, if and only if there exist a schur function such that
Admissible polynomials
As before, we also parametrise the admissible functions in the belevitch form as with . Note that the modulus square of can be express only in function of the positive polynomials .
for all real
being the minimum phase spectral factor of .
Given the positive polynomial of degree at most 2N, we call admissible polynomials the set of positive polynomials of degree at most 2N such that is admissible. We represent this set by
Convexity
Theorem: the set is a convex set.
Proof: We proof that if and are admissible (there exist feasible such that , , ), then with , is admissible as well (there exist feasible such that for all , ).
The check for feasibility of is straightforward, satisfies , , then satisfies those interpolation conditions as well.
Now we need to show that , .
is a concave function in P (its second derivative is negative):
Therefore (calling ):
Also by applying the triangle inequality
It can be easily verified that for all real. In particular, computing
We obtain . Thus the polynomial 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:
for all 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
we can state the matching problem over the set of admissible polynomials as:
for all in the passband
Thanks to the convexity of the set , 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
for all in the passband
- Relaxed problem
for all in the passband
Note that every function can be expressed as a blascke product times a minimum phase function (being this one admissible)
where is admissible and for all real. Therefore for each feasible function there exist an admissible function 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
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 to determine how far from the optimum the obtained solution is.