## 2 Theory and Models
### Linear Passive Elements and Causality  
If an element of an electric circuit is linear, the interrelation between the voltage, \(v(t)\), across it and the current, \(i(t)\), through it can be established using the model of the linear time invariant (LTI) system. Such a system can be completely described in a time domain by means of its impulse response (IR). Impedance impulse response, \(z(t)\), describes the LTI system whose input is the current and output is the voltage:  
\[v(t)=z(t)*i(t)=\int\limits_{-\infty}^{+\infty}z(\theta)i(t-\theta)d\theta, \tag{48}\]  
where \(*\) denotes the convolution expanded after the second equals sign.  
If the voltage is considered to be the system input and, consequently, the current is its output, the LTI system models admittance with impulse response \(y(t)\):  
\[i(t)=y(t)*v(t). \tag{49}\]If the LTI system under consideration belongs to the class of stable systems [47], it can be equivalently described by frequency response (FR), which is the Fourier transform of the system impulse response. Thus, the link between the voltage (3) and current (4) spectra can be established in its multiplicative form as:  
\[V(f)=Z(f)I(f),\ I(f)=Y(f)V(f), \tag{50}\]  
where \(Z(f)\) and \(Y(f)\) are the FR of the impedance and admittance, respectively.  
The relation (50), together with the properties of multiplying a generalized function by an ordinary one [44], gives the immediate relation between the coefficient of the complex form Fourier series for the voltage (3) and current (4):  
\[M_{v}^{(kF)}=Z(kF)M_{i}^{(kF)},\ M_{i}^{(kF)}=Y(kF)M_{v}^{(kF)}, \tag{51}\]  
As soon as FRs are known, the link between CSCD (21) and the equivalent characteristic of the voltage and current can be established in explicit form:  
\[\mathcal{S}_{vi}(\alpha,f)=Z\Big{(}f+\frac{\alpha}{2}\Big{)}\mathcal{S}_{i}( \alpha,f)=Y^{*}\Big{(}f-\frac{\alpha}{2}\Big{)}\mathcal{S}_{v}(\alpha,f), \tag{52}\]  
where \(\mathcal{S}_{v}(\alpha,f)\) and \(\mathcal{S}_{i}(\alpha,f)\) are the spectral correlation densities (SCD) of the voltage and current, respectively.  
Relation (52) allows writing the immediate representation of the CSCF at cyclic frequency, \(\alpha\), via the SCF of a voltage and current, denoted, respectively, as \(\mathcal{S}_{v}^{(\alpha)}(f)\) and \(\mathcal{S}_{i}^{(\alpha)}(f)\):  
\[\mathcal{S}_{vi}^{(\alpha)}(f)=Z\Big{(}f+\frac{\alpha}{2}\Big{)}\mathcal{S}_{ i}^{(\alpha)}(f)=Y^{*}\Big{(}f-\frac{\alpha}{2}\Big{)}\mathcal{S}_{v}^{( \alpha)}(f). \tag{53}\]  
In turn, the components of CSCD, i.e., the weights of Dirac delta functions, can be expressed in the closed form:  
\[\mathcal{S}_{vi}^{(m,k)}=Z\bigg{(}\bigg{[}k+\frac{m+((m))_{2}}{2}\bigg{]}F \bigg{)}\mathcal{S}_{i}^{(m,k)}=Y^{*}\bigg{(}\bigg{[}k-\frac{m-((m))_{2}}{2} \bigg{]}F\bigg{)}\mathcal{S}_{v}^{(m,k)}, \tag{54}\]  
where \(\mathcal{S}_{v}^{(m,k)}\) and \(\mathcal{S}_{i}^{(m,k)}\) can be evaluated in accordance with (22), assuming that both of their multipliers are chosen from the same signal representation.  
In particular, at zero cyclic frequency \(\alpha=0\), (54) gets reduced to:  
\[\mathcal{S}_{vi}^{(0,k)}=Z(kF)\Big{|}M_{i}^{(kF)}\Big{|}^{2}=\Big{|}M_{v}^{( kF)}\Big{|}^{2}Y^{*}(kF). \tag{55}\]  
Furthermore, the average power components distributed over frequency are:  
\[P_{r}^{(kF)}=\Re\mathcal{S}_{vi}^{(0,k)}=R(kF)\Big{|}M_{i}^{(kF)}\Big{|}^{2}= G(kF)\Big{|}M_{v}^{(kF)}\Big{|}^{2}, \tag{56}\]  
where \(R(f)=\Re\epsilon Z(f)\) and \(G(f)=\Re\epsilon Y(f)\) represent, respectively, the element resistance and conductance as the frequency dependent function.