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The broadband matching problem aims to minimise within a specified frequency interval I the reflected power from the system () composed of a matching filter () cascaded with load (). Note that the load is represented here as a two port device.
Statement of the general matching problem
The problem is stated in its general form as the minimisation over the filter of the maximum in the band of over the frequency variable . If the filter is parametrised in function of its reflection parameter , the only constraint imposed is that the reflection must remain passive and stable. Therefore is a stable function with on the whole frequency axis.
Chaining operation
The scattering parameter can be computed by the chain operation ():
where stand for the transmission parameters of the matching filter.
For a lossless device, the scattering matrix is unitary, this means (). Using this property we can express:
This expression correspond to the pseudohyperbolic distance between the function and .
Optimisation over the Schur functions
The problem is stated then as the minimisation of the previous pseudohyperbolic distance over the functions that are passive and stable.
The functions that are passive () and stable (analytic) in a domain are called schur functions. We denote the set of such functions by .
Thus the aim is to solve:
for all in the passband.
Infinite dimension solution
If the only restriction of the function is being stable and passible (without limitation on the degree), the previous problem can be solved optimally. This is possible because the set is a convex set.
The optimal solution to the general problem was already obtained by Helton in [ref]. He computed, for a given load the function that provides a reflection lowest possible absolute value within the passband. This reflection level, being obtained without limitation on the degree of , represents a lower bound on the reflection level that can be obtained for . We can see an example of the Helton bound for a particular load in the following picture, along with the function of infinite degree providing the optimal reflection level.
However, if the function is constrained to be rational with a low degree, this bound will not be accurate. Symilarly, the optimal rational function with a given low degree will differ from the one obtained with no degree limitation.
Rational matching problem
If we now restrict the function only to be rational, i.e. with and polynomials of degree at most N, the previous problem becomes too difficult to solve. Intead we use a parametrisation allowing for the transmission zeros to be prescribed as well.
Belevitch representation
We use the belevitch form to parametrise the global system:
with a uni-modular constant, and polynomials of degree at most N satisfying with . Note that q is a stable polynomial (analytic in the stability domain). Additionally, the polynomial is fix by the user.
This parametrisation is customary in classical filter design where the belevitch form is used to parametrise lossless devices. Similarly the transmission zeros are often set to given positions in the complex plane, usually with the purpose of increasing the out-of-band selectivity or to have some control on the group delay in the passband.
Feasible responses
Now given the chaining operation introduced before:
we define the set of feasible functions () as the image of the schur functions () under the chaining application:
In other words, a function is feasible for a given load if and only if there exists a Schur function such that .
Rational matching with prescribed transmission zeros
With S being parametrised in the belevitch form as before, we denote by the set admissible functions satisfying (the set of rational responses where the transmission zeros are prescribed).
With those definitions, the rational matching problem can be stated as the minimisation in the passband of the reflection of the global system knowing that there must exist a function rational of given degree that chained to the load provides the desired reflection:
for all in the passband.
Note that the problem is stated here on instead of . However, it is not relevant since for lossless systems and real, .
The Fano/Youla characterisation
Having defined the set of feasible responses as the reflection parameters that can be obtained by connecting a passive stable device to a prescribe load, we need a charactesation for it. In other words, what requisites must be satisfied by a reflection to be feasible?, of course we know that the requisite is that there must exist a passive stable filter F such that the cascade of that filter with the load shows the reflection at the second port. However thats too abstract.
A better characterisation can be obtained from the chaining equation
We look at the transmission zeros of the load (the points inside the stability domain of the complex plane for which . At those points, for any passibe stable filter F the right term in the previous equation vanishes since the denominator can not be zero inside the stability domain. Therefore we obtain the interpolation condition
This is a necessary condition for a function to be feasible. In addition, as the main result of Fano/Youla work, it was shown that this condition is indeed necessary and sufficient.
(this stand for the case of simple transmission zeros of the load inside the complex plane, if the load has either transmission zeros of higher multiplicity or transmission zeros on the frequency axis, the previous condition has to be modified including equalities on the derivatives as well).
In conclussion, if a response satisfied the previous conditions for a given load A then it exist a passive stable filter F that chained with the load provides the reflection at the second port (the function is feasible).
Thus this interpolation conditions represent a characterisation of the feasible responses .
Nevanlinna-Pick interpolation
The last tool required to understand how puma works is the Nevanlinna-Pick interplation theorem, it states:
Given N points in the lower half of the complex plane (points with negative imaginary part), and N complex values with , there exist a schur function (passive stable in the lower half plane) such that if and only if the followig matrix is positive definite.
This matrix is called the Pick matrix.
Nevanlinna parametrisation
Nevanlinna also provided a parametrisation of all solutions to the interpolation problem with (schur functions). In the general case, when the pick matrix is positive definite (with N interpolation conditions)
where are polynomials of degree at most N and is a schur function.
Remark: in our context, this expression is actually equivalent to the chaininig operation where the polynomials are in fact the polynomials involved in the belevitch model of the load .
In the particular case where the pick matrix is positive semi-definite and singular, there is a unique solution to the interpolation problem in the form of a blaschke product :
where M stand for the rank of the pick matrix, belong to the analicity domain and with real.