contrary, a system of memory corresponds to the input values at times other than the
current time. In many physical systems, memory is directly related to the storage of energy.
 
Though there are many properties of system, we only focus on the first four
properties for our use. For a LTI system,  
output=Input * Impulse response of the system (system function).  
Considering that the circuit to be a system, then the voltage and current can be the input
and output.  For example,
If the input is current I, then the output will be voltage V.  V(s) = I(s)*Z(s)
If the input is voltage V, then the output will be current I.  I(s) = V(s) *Y(s)
1.3.2   Positive Real function
Z and Y parameters are the response of the system. If the system is passive and
reciprocal, then Z and Y parameters belong to a class of functions called positive real
function (abbreviated p.r.)[9]. A function of complex frequency variable s is positive and
real if it has the following general properties:
•  It is real for real values of s
•   It has a positive real part for values of s with a positive real part
It is clear that to ensure the realizability of linear, passive networks, the driving-point
impedances and admittances of the network are p.r. functions. Also, all the detailed
properties of these network functions are derivable from their p.r. character alone.  There
are 3 necessary and sufficient conditions for any function to be p.r. function:
(1) The real part of the Zin should be positive in the right half s plane.
(2) All the poles and zeros should not be in the right half of the plane.
(3) If the system has poles in the imaginary axis, then it must have real
positive residue.
The function cannot be p.r. unless its reciprocal is p.r.[9]. This is an important property
which we will use in Brune’s synthesis.  1.4.  Network synthesis procedure
“Circuit Analysis” determines characteristics of given circuits. “Network Synthesis”
is the inverse. It determines  circuits with given (desired) characteristics. For linear time-
invariant RLC circuits, Kirchhoff’s laws led to simultaneous linear differential equations in
voltages and/or currents, with constant coefficients [10].  
A notable example is Foster. However, he only focused on lossless network. And
then later, Caure made some research on the network and developed the Caure I and Caure
II circuit. In 1931, then, O. Brune finished the network synthesis with rational function. For
briefer references to more people who did lots contributions to the development of network
synthesis but not mentioned in this paper, see [11]
In this project, we will only investigate Foster I and Brune’s network synthesis.
1.4.1   General Foster equivalent circuit
First, the frequency-dependent element characteristic data X(s) of a circuit element is
given like this:                   (1)
In the equation, the sampled frequency is s=j2ߨf.
There are 2 types of Foster Circuit: Foster I (Impedance Type) and Foster II
(Admittance Type).  
In Foster I, the impedance function is expanded into the following equations:
The first and second terms correspond to a series connection of an inductor L and a
resistor R. The third term is series connections of parallel C-G circuits, and the last term is
series connections of L-G-C-R parallel resonant circuits [12].  Figure 3: Foster I type circuit
In Foster II, the admittance function is expanded into the following equations:
The first and second terms correspond to a parallel connection of a capacitor C and a
conductor G. The third term is parallel connections of series L-R circuits, and the last term Investigation of Lumped Element Equivalent Circuit for Distributed Microwave Circuit(3):http://www.youerw.com/yingyu/lunwen_1765.html