Transfer function

Transfer function \(h\) is expression of the state of the system in terms of its state at the precedent step of evolution.

In the signal processing systems, the transfer function determines the output signal in terms of the input one.

In the linear optics, the transfer can be expressed as linear operator, that describes the transform of light inside some system.

In TORI, the term usually refers to the Transfer equation, the Abel equation; the transfer function is assumed to be given function that appear in these equations. The solutions of these equations are called superfunctions and Abel functions. For a given transfer function, the additional restrictions should be applied to make the superfunction and the Abel function unique .

Transfer equation and Abel equation
In mathematics, the transfer function is generic name for a given function that appears in the transfer equation and in the Abel equation.

If \(h\) is declared as transfer function, then, the superfunction \(F\) is solution of the transfer equation
 * \((1) ~ ~ ~ ~ ~ F(z\!+\!1)=h(F(z))\)

and the Abel function \(G\) is solution of the Abel equation
 * \((2) ~ ~ ~ ~ ~ G(h(z))=G(z)+1\)

Transfer function and the Abel function
It is convenient to consider such Abel function \(G\) that is inverse function of the  superfunction \(F~\); \(~H(G(z))=z\) Then, the Transfer function can be expressed as follows:
 * \((3) ~ ~ ~ ~ ~ h(z)=F(1+G(z))\)

Such an expression can be used to define the \(c\)th iterate of the transfer function
 * \((4) ~ ~ ~ ~ ~ h^c(z)=F(c+G(z))\)

for various \(c\). In particular, for integer values of \(c\), \(h^{-1}\) is inverse function of \(h\), while \(h^{0}\) is identify function, \(h^{1}\!=\!h~\); \(~h^{2}(z)\!=\!h(h(z))~\) and so on.

Once the superfunction \(F\) and the Abelfunction \(G=F^{-1}\) are specified, the definition (4) of the \(c\)th iteration of the transfer function can be used also for fractional, irrational and even for complex values of \(c\).

Table of superfunctions and Abel functions
In principle any holomorphic function can be declared as a superfunction \(F\), and the inverse function \(G=F^{-1}\) can be declared as the Abel function. Then function \(h\) by equation (3) is corresponding transfer function. This gives simple way to construct the table of superfunctions and Abel functions.

Constant as transfer function
Consider \(F(z)=C=\)const as superfunction. The inverse function \(G\) cannot be defined at any complex domain, but the transfer function that is always constant can be considered as the transfer function; \(h(z)=C\).

This is example of a transfer function identical to its superfunction.

Unity increment as the transfer function
Consider the addition of a constant \(b\); let \(F(z)=b+z\) be superfunction. Then, the Abel function is \(G(z)=z-b\). The corresponding transfer function
 * \((5) ~ ~ ~ ~ ~ h(x)= b+ (1+(z-b))=z+1 = +\!+(z)\)

is just unity increment.

In particular, expression (5) is valid at \(b=1\); this is another example when the transfer function identical to its superfunction.

Addition as transfer function
Consider the multilication by a constant \(b\); let \(F(z)=b z\) be superfunction. Then, at \(b\!\ne\! 0\), the Abel function is \(G(z)=z/b\).

The corresponding transfer function
 * \((6) ~ ~ ~ ~ ~ h(x)= b*(1+z/b)=z+b\)

In such a way, the addition of constant \(b\) is superfunction with respect to the unity increment and transfer function with respect to multiplicaiton by a constant \(b\).

multiplication as transfer function
Consider the exponentiation, let \(F(z)=\exp_b(z)\) be superfunction. Then, at \(b\!\ne\! 0\), \(b\!\ne\! 1\), the Abel function is \(G(z)=\log_b(z)\). The corresponding transfer function
 * \((7) ~ ~ ~ ~ ~ h(x)= \exp_b(1+\log_b(z))= b z\)

In such a way, the multiplication by constant \(b\) is superfunction with respect to the addition of constant \(b\) and transfer function with respect to exponentiation to base \(b\).

Exponentiation as transfer function
Consider the tetration, let \(F(z)=\mathrm{tet}_b(z)\). Then, at least for \(b\!>\!1\), the Abel function is arctetration, \(G(z)=\mathrm{ate}_b(z)\).

The corresponding transfer function
 * \((8) ~ ~ ~ ~ ~ h(x)= \mathrm{tet}_b(1+\mathrm{ate}_b(z))=b^z\)

So, the exponentiation is superfunction with respect to to multipliction and transfer function with respect to tetration

In the similar way, tetration is superfunction with respect to exponential and transfer function with respect to pentation, and so on for other the ackermann functions.

Another superfunction for a linear function
Consider the linear superfunction, let \(F(z)=a\!+\!b z\). Then, for \(b\!\ne\! 0\), the Abel function \(G(z)=z/b\! -\! a\). Then, the corresponding tansfer function
 * \((9) ~ ~ ~ ~ ~ h(x)= a+ b \times (z/b-a) = z+ (1\!-\!b) a\)

which looks preffy similar to equation (6); it is the simple case when the different superfunctions correspond to the same transfer function.

Linear fraction as superfunction
Consider
 * \((10) \displaystyle ~ ~ ~ ~ ~ F(z)=\frac{a+b z}{p+q z}~ ~, ~ ~\) where \(a,b,p,q\) are constants (for example, some complex numbers).

Then
 * \((11) \displaystyle ~ ~ ~ ~ ~ G(z)=F^{-1}(z)=\frac{a- p z}{-b+q z}\)

and the corresponding transfer function
 * \((12) \displaystyle ~ ~ ~ ~ ~ f(z)=F(1+G(z))=\frac{b^2+(bp-bq-aq)z}{bp+bq-qa - q^2 z}\)

is also the linear fraction function.

Denoting
 * \((13) \displaystyle ~ ~ ~ ~ ~ A=-b^2\)
 * \((14) \displaystyle ~ ~ ~ ~ ~ B=-bp+bq+aq\)
 * \((15) \displaystyle ~ ~ ~ ~ ~ P=-bp-bq+qa\)
 * \((16) \displaystyle ~ ~ ~ ~ ~ Q=q^2\)

the transfer function can be written in the simple form
 * \((17) \displaystyle ~ ~ ~ ~ ~ f(z)=\frac{A+Bz}{P+Qz}\)

The corresponding superfunction \(F\) and the Abel function \(G\) are expressed with (10) and (11).

Consider the case, that \(A\), \(B\), \(P\), \(Q\) are given. They determine the transfer function \(f\). Then, if the solution \(a\), \(b\), \(p\), \(q\) of equations (13),(14),(15),(16) exists, then, \(F\) by (10) and \(G\) by (11) are corresponding superfunction and the Abel function.

More complicated transfer functions
In principle any pair of biholomorphic \(F\) and \(G=F^{-1}\) with wide enough range of holomorphism can be declared as superfunciton and Abel function; then \(f(z)=F(1+G(z))\) appears as the corresponding transfer function. In such a way, the table of superfunctions can be constructed.

In general the transfer function can be defined on the the set of vectors or set of functions. In this case, the transfer function is called also functional. (Such a name may assyme that the Functional is considered as something superior to just a function of single variable). Such a case my be important for the signal processing .

The transfer function determines the transformation of a function (that is interpreted as a signal), as it passes through some filter. In principle, the signal may pass through the filter several times (or to pass through the set of identical filters). If the filter is interpreted as a piece of a system with translational symmetry, then the superfunction may express the evolution of the signal inside the filter. For the multidimensional transfer function, the definition of the Abel function cannot be applied as is; it should be generalizad.