**3. Hybrid time-frequency techniques for computing the solution of MPDAEs**

In this section, we will finally discuss the hybrid time-frequency numerical techniques that can be used to evaluate the solution of MPDAEs describing the operation of nonlinear electronic radio frequency circuits running in an aperiodic slow time scale and a periodic fast time scale. Section 3.1 addresses an efficient technique often referred to as *multitime envelope transient harmonic balance* (multitime ETHB). Then, Section 3.2 presents an advanced parti‐ tioned time-frequency technique, which is an improved version of multitime ETHB and has demonstrated to be even more efficient than this technique.

#### **3.1. Multitime envelope transient harmonic balance**

*2.5.2. Multirate partial differential algebraic equations' systems*

*multirate partial differential algebraic equations'* (MPDAE) system [11]:

*E C tt tt*

The mathematical relation between (1) and (15) establishes that if *b*

to the periodicity of the problem in the *tC* dimension we will have

*2.5.3. Initial and boundary conditions for envelope-modulated regimes*

following initial periodic-boundary value problem

¶ ¶

¶ ¶

*t t*

*E C*

*E C E C*

*tt tt*

^

*t t*

¶ ¶

(15), i.e., *x*(*t*) may be retrieved from its bivariate form *x*

¶ ¶

(15), then the univariate forms *b*(*t*)= *b*

60 Modeling and Simulation in Engineering Sciences

Let us consider a general nonlinear RF circuit described by the differential algebraic equations' system of (1), and let us suppose that this circuit is driven by the envelope-modulated signal of (12). Considering the above stated, we are able to reformulate the excitation *b*(*t*) and the state variables *x*(*t*) vectors as bidimensional entities, in which *t* is replaced by *tE* for the slowly varying parts (the envelope time scale) and by *tC* for the fast-varying parts (the RF carrier time scale). This bidimensional formulation converts the DAE system of (1) into the following

<sup>µ</sup> <sup>µ</sup> <sup>µ</sup> \$ ( ( , )) ( ( , )) ( ( , )) ( , ). *E C E C*

(*t*, *t*) and *x*(*t*)= *x*

solutions of (1) are available on diagonal lines *tE* =*t*, *tC* =*t*, along the bivariate solutions of

Consequently, if one wants to obtain the univariate solution in the generic [0,*tFinal*] interval, due

on the rectangular domain 0, *tFinal* × 0, *TC* , where *t* mod *TC* represents the remainder of division of *t* by *TC*. The main advantage of this MPDAE approach is that it can result in significant improvements in simulation speed when compared to DAE-based alternatives.

Dynamical behavior of RF circuits driven by stimuli of the form of (12) can be described by the MPDAE system of (15) together with a set of initial and periodic boundary conditions. In fact, bivariate forms of the circuits' state variables can be achieved by computing the solution of the

( ( )) ( ( )) ( ( )) ( )

+ +=

ˆ ˆ , , <sup>ˆ</sup> <sup>ˆ</sup> , ,,

( ) ( ) ( ) ( )

*t t t tT*

*C C E EC*

ˆ ˆˆ 0, , ,0 , ,

= =

*qx qx fx b*

*xi xx*

+ +=

*EC EC*

^

<sup>µ</sup> ( ) ( , mod ) *<sup>C</sup> x x t tt T* = (16)

*EC EC*

*tt tt*

(*tE*, *tC*) and *x*

^(*tE*, *tC*), by simply setting *tE* = *tC* = *t*.

^(*t*, *<sup>t</sup>*) satisfy (1) [11]. Therefore, univariate

^(*tE*, *tC*) satisfy

(17)

*tt tt*

*qx qx fx b* (15)

Let us consider the initial-boundary value problem of (17) and let us define a semi-discretiza‐ tion of the rectangular domain 0, *tFinal* × 0, *TC* in the *tE* slow time dimension described by the following general non uniform grid

$$0 = \mathbf{t}\_{E,0} < \mathbf{t}\_{E,1} < \dots < \mathbf{t}\_{E,i-1} < \mathbf{t}\_{E,i} < \dots < \mathbf{t}\_{E,K\_E} = \mathbf{t}\_{\text{Final}}, \quad \mathbf{h}\_{E,i} = \mathbf{t}\_{E,i} - \mathbf{t}\_{E,i-1}, \tag{18}$$

in which *KE* represents the total number of steps in *tE* and *hE*,*<sup>i</sup>* denotes the grid size at each time step *i*. If we replace the derivatives of the MPDAE in *tE* with a finite-differences approximation (e.g., a backward differentiation formula, the modified trapezoidal rule, etc.), then we obtain for each slow time instant *tE*,*<sup>i</sup>* , from *i* = 1 to *i* = *KE*, a periodic boundary value problem in *tC*. For simplicity, and clarity, let us suppose that the Backward Euler rule is used. In such a case, we obtain

$$\begin{aligned} \frac{q\left(\hat{\mathfrak{x}}\_{i}\left(t\_{\mathcal{C}}\right)\right) - q\left(\hat{\mathfrak{x}}\_{i-1}\left(t\_{\mathcal{C}}\right)\right)}{h\_{\mathcal{E},i}} + \frac{dq\left(\hat{\mathfrak{x}}\_{i}\left(t\_{\mathcal{C}}\right)\right)}{dt\_{\mathcal{C}}} + \mathcal{J}\left(\hat{\mathfrak{x}}\_{i}\left(t\_{\mathcal{C}}\right)\right) = \hat{\mathfrak{b}}\left(t\_{\mathcal{E},i}, t\_{\mathcal{C}}\right), \end{aligned} \tag{19}$$
 
$$\hat{\mathfrak{x}}\_{i}\left(0\right) = \hat{\mathfrak{x}}\_{i}\left(T\_{\mathcal{C}}\right),$$

where *x* ^ *i* (*tC*) is an approximation to the exact solution **x** ^*(tE,i,tC)*. Thus, once *<sup>x</sup>* ^ *<sup>i</sup>*−1(*tC*) is computed, the solution on the next slow time instant, *x* ^ *i* (*tC*), is evaluated by solving (19). Consequently, it is straightforward to conclude that we have to solve a set of *KE* periodic boundary value problems if we want to obtain the solution *x* ^(*tE*, *tC*) in the entire 0, *tFinal* <sup>×</sup> 0, *TC* domain. With multitime ETHB, each one of these periodic boundary value problems is solved using the harmonic balance method. For each slow time instant *tE*,*<sup>i</sup>* , the resultant HB system is the *n*(2*K* + 1) algebraic equation system defined by

$$\frac{\mathbf{Q}(\hat{\mathbf{X}}(t\_{E,i})) - \mathbf{Q}(\hat{\mathbf{X}}(t\_{E,i-1}))}{h\_{E,i}} + j\boldsymbol{\Omega}\mathbf{Q}(\hat{\mathbf{X}}(t\_{E,i})) + \mathbf{F}(\hat{\mathbf{X}}(t\_{E,i})) = \hat{\mathbf{B}}(t\_{E,i}),\tag{20}$$

where **B ^** (*tE*,*<sup>i</sup>* ) and **X ^** (*tE*,*<sup>i</sup>* ) are the vectors containing the Fourier coefficients of the excitation sources and of the solution (the state variables), respectively, at *tE* = *tE*,*<sup>i</sup>* . **F**(•) and **Q**(•) are unknown functions that can be computed by evaluating *f*(·) and *q*(·) in the time domain and then calculating their Fourier coefficients. Ω is the diagonal matrix (6), and the **X ^** (*tE*,*<sup>i</sup>* ) vector can be described as

$$\hat{\mathbf{X}}(t\_{E,i}) = \left[\hat{\mathbf{X}}\_1(t\_{E,i})^T, \hat{\mathbf{X}}\_2(t\_{E,i})^T, \dots, \hat{\mathbf{X}}\_n(t\_{E,i})^T\right]^r,\tag{21}$$

where each one of the state variable frequency components, **X ^** *<sup>v</sup>*(*tE*,*<sup>i</sup>* ), *v* = 1,…,*n*, is a (2*K*+1)×1 vector defined as

$$\hat{\mathbf{X}}\_{\mathbf{v}}(t\_{E,i}) = \left[X\_{\mathbf{v},-K}(t\_{E,i}), \dots, X\_{\mathbf{v},0}(t\_{E,i}), \dots, X\_{\mathbf{v},K}(t\_{E,i})\right]^{\mathrm{r}}.\tag{22}$$

The HB system of (20) can be rewritten as

$$\mathbf{H}(\hat{\mathbf{X}}(t\_{\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon},i}})) = \frac{\mathbf{Q}(\hat{\mathbf{X}}(t\_{\boldsymbol{\varepsilon},i})) - \mathbf{Q}(\hat{\mathbf{X}}(t\_{\boldsymbol{\varepsilon},i-1}))}{h\_{\boldsymbol{\varepsilon},i}} + j\boldsymbol{\Omega}\mathbf{Q}(\hat{\mathbf{X}}(t\_{\boldsymbol{\varepsilon},i})) + \mathbf{F}(\hat{\mathbf{X}}(t\_{\boldsymbol{\varepsilon},i})) - \hat{\mathbf{B}}(t\_{\boldsymbol{\varepsilon},i}) = \mathbf{0},\tag{23}$$

or simply as

$$\mathbf{H}(\hat{\mathbf{X}}(t\_{E,i}))=\mathbf{0},\tag{24}$$

in which **H**(**X ^** (*tE*,*<sup>i</sup>* )) is the error function at *tE* = *tE*,*<sup>i</sup>* . In general, the nonlinear algebraic system of (24) is iteratively solved using Newton's method

Hybrid Time-Frequency Numerical Simulation of Electronic Radio Frequency Systems http://dx.doi.org/10.5772/64152 63

$$d\mathbf{H}(\hat{\mathbf{X}}^{[r]}(t\_{E,i})) + \frac{d\mathbf{H}(\hat{\mathbf{X}}(t\_{E,i}))}{d\hat{\mathbf{X}}(t\_{E,i})}|\_{\hat{\mathbf{X}}(t\_{E,i}) = \hat{\mathbf{X}}^{[r]}(t\_{E,i})} \left[\hat{\mathbf{X}}^{[r+1]}(t\_{E,i}) - \hat{\mathbf{X}}^{[r]}(t\_{E,i})\right] = 0,\tag{25}$$

which requires that we have to solve a linear system of *n*(2*K* + 1) equations at each iteration *r* to compute the new estimate **X ^** *<sup>r</sup>*+1 (*tE*,*<sup>i</sup>* ). Consecutive Newton iterations will be computed until a desired accuracy is achieved, i.e., until **H**(**X ^** (*tE*,*<sup>i</sup>* )) <*δ*, where *δ* is the allowed residual size.

The system of (25) requires the computation of the Jacobian matrix **J**(**X ^** (*tE*,*<sup>i</sup>* )), i.e., the derivative of the vector **H**(**X ^** (*tE*,*<sup>i</sup>* )), with respect to the vector **X ^** (*tE*,*<sup>i</sup>* ),

1, 1, 1 , 1 , 2 , , 2, 2, 2 , , , 1 2 , , , , 1 , ˆˆ ˆ ( ( )) ( ( )) ( ( )) ˆˆ ˆ () () () ˆˆ ˆ ( ( )) ( ( )) ( ( )) <sup>ˆ</sup> ( ( )) <sup>ˆ</sup> ˆˆ ˆ ( ( )) () ( ) ( ) <sup>ˆ</sup> ( ) ˆ ( ( )) ˆ ( ) *E i E i E i E i E i n Ei E i E i E i E i E i Ei E i n Ei E i n Ei E i tt t tt t tt t d t t tt t d t t t* ¶¶ ¶ ¶¶ ¶ ¶¶ ¶ = = ¶¶ ¶ ¶ ¶ **HX HX HX XX X HX HX HX H X J X XX X X H X X** L L L LLL , , 2 , , ˆ ˆ ( ( )) ( ( )) ˆ ˆ () () *n Ei n Ei E i n Ei t t t t* é ù ê ú ¶ ¶ ¶ ¶ ë û **HX HX X X** <sup>L</sup> (26)

This matrix has a block structure, consisting of *n*×*n* square submatrices (blocks), each one with dimension (2*K* + 1). The general block of row *m* and column *l* can be expressed as

$$\frac{d\mathbf{H}\_{\boldsymbol{m}}(\hat{\mathbf{X}}(t\_{E,\boldsymbol{\iota}}))}{d\hat{\mathbf{X}}\_{\boldsymbol{l}}(t\_{E,\boldsymbol{\iota}})} = \frac{1}{h\_{\boldsymbol{\varepsilon},\boldsymbol{\iota}}} \frac{d\mathbf{Q}\_{\boldsymbol{m}}(\hat{\mathbf{X}}(t\_{E,\boldsymbol{\iota}}))}{d\hat{\mathbf{X}}\_{\boldsymbol{l}}(t\_{E,\boldsymbol{\iota}})} + j\boldsymbol{\Omega} \frac{d\mathbf{Q}\_{\boldsymbol{m}}(\hat{\mathbf{X}}(t\_{E,\boldsymbol{\iota}}))}{d\hat{\mathbf{X}}\_{\boldsymbol{l}}(t\_{E,\boldsymbol{\iota}})} + \frac{d\mathbf{F}\_{\boldsymbol{m}}(\hat{\mathbf{X}}(t\_{E,\boldsymbol{\iota}}))}{d\hat{\mathbf{X}}\_{\boldsymbol{l}}(t\_{E,\boldsymbol{\iota}})}.\tag{27}$$

#### **3.2. Partitioned time-frequency technique**

where *x* ^ *i*

where **B ^** (*tE*,*<sup>i</sup>*

can be described as

vector defined as

or simply as

in which **H**(**X**

**^** (*tE*,*<sup>i</sup>*

(*tC*) is an approximation to the exact solution **x**

harmonic balance method. For each slow time instant *tE*,*<sup>i</sup>*

, , 1

,

*E i t t* ^ *i*

ˆ ˆ ( ( )) ( ( )) ˆ ˆˆ ( ( )) ( ( )) ( ), *E i E i*

sources and of the solution (the state variables), respectively, at *tE* = *tE*,*<sup>i</sup>*

where each one of the state variable frequency components, **X**

, , 1

)) is the error function at *tE* = *tE*,*<sup>i</sup>*

(24) is iteratively solved using Newton's method

,

*E i t t*

then calculating their Fourier coefficients. Ω is the diagonal matrix (6), and the **X**

, 1, 2, , ˆ ˆˆ ˆ ( ) [ ( ), ( ), , ( )], *<sup>T</sup> <sup>T</sup> T T*

, ,, ,0, , , <sup>ˆ</sup> ( ) [ ( ), , ( ), , ( )] . *<sup>T</sup>*

, , ,,

*<sup>t</sup> jt t t <sup>h</sup>*

ˆ ˆ ( ( )) ( ( )) <sup>ˆ</sup> ˆ ˆˆ ( ( )) ( ( )) ( ( )) ( ) 0, *E i E i E i E i Ei Ei*


*jt t t <sup>h</sup>*

unknown functions that can be computed by evaluating *f*(·) and *q*(·) in the time domain and

it is straightforward to conclude that we have to solve a set of *KE* periodic boundary value

multitime ETHB, each one of these periodic boundary value problems is solved using the

the solution on the next slow time instant, *x*

62 Modeling and Simulation in Engineering Sciences

problems if we want to obtain the solution *x*

+ 1) algebraic equation system defined by

The HB system of (20) can be rewritten as

) and **X ^** (*tE*,*<sup>i</sup>* ^*(tE,i,tC)*. Thus, once *<sup>x</sup>*

, ,,

*E i Ei Ei*

) are the vectors containing the Fourier coefficients of the excitation

*E i Ei Ei n Ei* **X XX X** *t tt t* = ¼ (21)

*v Ei v K Ei v Ei vK Ei t X t Xt X t* **X** = ¼¼ - (22)

**^** *<sup>v</sup>*(*tE*,*<sup>i</sup>*

, <sup>ˆ</sup> ( ( )) 0, *E i* **H X** *<sup>t</sup>* <sup>=</sup> (24)

. In general, the nonlinear algebraic system of


^

, the resultant HB system is the *n*(2*K*

(*tC*), is evaluated by solving (19). Consequently,

^(*tE*, *tC*) in the entire 0, *tFinal* <sup>×</sup> 0, *TC* domain. With

*<sup>i</sup>*−1(*tC*) is computed,

. **F**(•) and **Q**(•) are

) vector

**^** (*tE*,*<sup>i</sup>*

), *v* = 1,…,*n*, is a (2*K*+1)×1

Although multitime ETHB can take advantage of the signals' time rate disparity, it does not take into account the circuit's heterogeneities, i.e., it uses the same numerical algorithm to compute all the circuit's state variables. Thus, if the circuit evidences some heterogeneity (e.g., modern wireless architectures combining RF, baseband analog circuitry, and digital components in the same circuit), this tool cannot benefit from such a feature. This lack of ability to perform some distinction between nodes or blocks within the circuit had already been identified by Rizzoli et al. [17] and is the main limitation of multitime ETHB. To cope with this deficiency, the partitioned time-frequency technique separates the circuit's state variables (node voltages and branch currents) into fast (*active*) and slowly varying (*latent*) subsets. That implies the MPDAE system of (15) to be first considered as coupled active-latent MPDAE subsystems

$$\begin{aligned} \frac{\partial \mathbf{q}\_{A}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right)}{\partial t\_{\varepsilon}} + \frac{\partial \mathbf{q}\_{A}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right)}{\partial t\_{\varepsilon}} + \mathbf{f}\_{A}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right) - \hat{\mathbf{b}}\left(t\_{\varepsilon},t\_{\varepsilon}\right)}{\partial t\_{\varepsilon}} \\ \frac{\partial \mathbf{q}\_{L}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right)}{\partial t\_{\varepsilon}} + \frac{\partial \mathbf{q}\_{L}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right)}{\partial t\_{\varepsilon}} + \mathbf{f}\_{L}\left(\hat{\mathbf{x}}\_{A}\left(t\_{\varepsilon},t\_{\varepsilon}\right),\hat{\mathbf{x}}\_{L}\left(t\_{\varepsilon},t\_{\varepsilon}\right)\right) = \hat{\mathbf{b}}\left(t\_{\varepsilon},t\_{\varepsilon}\right) \end{aligned} \tag{28}$$

with

$$\mathbf{x}(t) = \hat{\mathbf{x}}(t, t) = \begin{bmatrix} \mathbf{x}\_A(t) = \hat{\mathbf{x}}\_A(t, t) \\ \mathbf{x}\_L(t) = \hat{\mathbf{x}}\_L(t, t) \end{bmatrix}, \quad \mathbf{x}\_A(t) \in \mathbb{R}^{n\_A}, \quad \mathbf{x}\_L(t) \in \mathbb{R}^{n\_L}, \quad n\_A + n\_L = n,\tag{29}$$

where *xA*(*t*) and *xL*(*t*) are the vectors containing, respectively, the fast-varying and the slowly varying state variables. As we will see, with this partition stratagem, fast-varying state variables can be computed with multitime ETHB, while slowly varying ones are being evaluated with a unidimensional time-step integration scheme. This tactic also allows the moderate nonlinearities to be treated in the frequency domain, while severe nonlinearities are appropriately evaluated in the time domain [16].

With the purpose of providing an elucidatory explanation of the partitioned time-frequency technique, let us consider a typical wireless system, composed of RF and baseband blocks. In such a case, the state variables in the RF block can be described as fast carrier envelope modulated waveforms defined as

$$\mathbf{x}\_A(t) = \mathbf{x}\_{A,\nu}(t) = \sum\_{k=-K}^{K} X\_{k,\nu}(t) e^{j k 2 \pi f\_c t}, \quad \nu = 1, \dots, n\_A \tag{30}$$

while state variables in the baseband block can be seen as slowly varying aperiodic functions of the form

$$\mathbf{x}\_L(t) = \mathbf{x}\_{L,\boldsymbol{\nu}}(t) = \boldsymbol{\psi}\_{\boldsymbol{\nu}}(t), \quad \boldsymbol{\nu} = \mathbf{l}, \ldots, n\_L. \tag{31}$$

In (30), *Xk*,*<sup>v</sup>*(*t*) represents the Fourier coefficients of *xA*,*<sup>v</sup>*(*t*), which are slowly varying in the baseband time scale, and *fC* is the high-frequency carrier. As stated above, signals of the form *xA*,*<sup>v</sup>*(*t*) will be denoted as active, whereas signals of the form *xL*,*<sup>v</sup>*(*t*) will be designated as latent. The latency (slowness) of *xL*,*<sup>v</sup>*(*t*) indicates that these variables belong to a circuit block where there are no fluctuations dictated by the fast carrier. Thus, it is straightforward to conclude that all of the *xL*,*<sup>v</sup>*(*t*) can be efficiently represented with much less sample points than any of the *xA*,*<sup>v</sup>*(*t*). Moreover, since the *xL*,*<sup>v</sup>*(*t*) state variables do not evidence any periodicity, they cannot be evaluated in the frequency domain. In contrast, if the number of harmonics *K* is not too large, the fast carrier oscillation components of *xA*,*<sup>v</sup>*(*t*) can be efficiently computed with harmonic balance. Taking the above into account we can easily conclude that distinct numer‐ ical strategies will be required if we want to simulate, in an efficient way, circuits having such signal format disparities.

In the following we provide a brief theoretical description of the partitioned time-frequency technique fundamentals. For that, let us now consider the bivariate forms of *xA*,*<sup>v</sup>*(*t*) and *xL*,*<sup>v</sup>*(*t*), denoted by *x* ^ *<sup>A</sup>*,*v*(*tE*, *tC*) and *x* ^ *<sup>L</sup>* ,*v*(*tE*, *tC*), and defined as

$$
\hat{\mathfrak{X}}\_{A,\mathbf{v}}(t\_E, t\_{\mathcal{C}}) = \sum\_{k=-K}^{K} X\_{k,\mathbf{v}}(t\_E) \mathbf{e}^{\not p \cdot 2\pi f\_{\mathcal{C}} t\_{\mathcal{C}}} \tag{32}
$$

and

implies the MPDAE system of (15) to be first considered as coupled active-latent MPDAE

( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( )

ˆˆ ˆˆ ,, , ,, , <sup>ˆ</sup> ˆ ˆ ,, , ,

*qx x qx x fx x b*

*qx x qx x fx x b*

+ +=

*AA L*

*LA L*

*A L A L*

ë û <sup>=</sup> ¡ ¡ (29)

*EC EC EC*

(28)

*tt tt tt*

*EC EC EC*

*tt tt tt*

+ +=

( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( )

ˆˆ ˆˆ ,, , ,, , <sup>ˆ</sup> ˆ ˆ ,, , ,

() (, ) <sup>ˆ</sup> () (, ) <sup>ˆ</sup> , () , () , , () (, ) <sup>ˆ</sup> *A L A A n n*

é ù <sup>=</sup> = = ê ú Î Î +=

where *xA*(*t*) and *xL*(*t*) are the vectors containing, respectively, the fast-varying and the slowly varying state variables. As we will see, with this partition stratagem, fast-varying state variables can be computed with multitime ETHB, while slowly varying ones are being evaluated with a unidimensional time-step integration scheme. This tactic also allows the moderate nonlinearities to be treated in the frequency domain, while severe nonlinearities are

With the purpose of providing an elucidatory explanation of the partitioned time-frequency technique, let us consider a typical wireless system, composed of RF and baseband blocks. In such a case, the state variables in the RF block can be described as fast carrier envelope

2

while state variables in the baseband block can be seen as slowly varying aperiodic functions

In (30), *Xk*,*<sup>v</sup>*(*t*) represents the Fourier coefficients of *xA*,*<sup>v</sup>*(*t*), which are slowly varying in the baseband time scale, and *fC* is the high-frequency carrier. As stated above, signals of the form *xA*,*<sup>v</sup>*(*t*) will be denoted as active, whereas signals of the form *xL*,*<sup>v</sup>*(*t*) will be designated as latent. The latency (slowness) of *xL*,*<sup>v</sup>*(*t*) indicates that these variables belong to a circuit block where there are no fluctuations dictated by the fast carrier. Thus, it is straightforward to conclude that all of the *xL*,*<sup>v</sup>*(*t*) can be efficiently represented with much less sample points than any of

*jk f t*

p

= = å = ¼ (30)

(31)

, , ( ) ( ) ( ) , 1, , *<sup>C</sup>*

, ( ) ( ) ( ), 1, , . *L Lv v <sup>L</sup> xt x t t v n* = = =¼ y

*x t x t X te v n*

*A Av k v A*

*K*

*k K*

=-

*xt xtt xt xt n n n*

subsystems

with

¶ ¶

64 Modeling and Simulation in Engineering Sciences

¶ ¶

¶ ¶

*AA L AA L*

*LA L LA L*

*t t tt tt tt tt*

*E C EC EC EC EC*

*E C*

*EC EC EC EC*

*L L x t x tt*

appropriately evaluated in the time domain [16].

modulated waveforms defined as

of the form

*x t x tt*

*tt tt tt tt*

¶ ¶

*t t*

$$
\hat{\mathfrak{X}}\_{L,\mathbf{v}}(t\_E, t\_C) = \psi\_{\mathbf{v}}(t\_E), \tag{33}
$$

where *tE* and *tC* are, respectively, the slow envelope time dimension and the fast carrier time dimension. As can be seen, since the *x* ^ *<sup>L</sup>* ,*v*(*tE*, *tC*) state variables have no dependence on *tc*, they have no fluctuations in the fast time axis. The reason is that they belong to a circuit block where there are no carrier frequency oscillations. As a result, for each slow time instant *tE*,*<sup>i</sup>* defined on the grid of (18), each of the *x* ^ *<sup>L</sup>* ,*v*(*tE*,*<sup>i</sup>* , *tC*) is merely a constant signal that can be simply represented by the *k* = 0 component. Therefore, there is no necessity to perform the conversion between time and frequency domains for *x* ^ *<sup>L</sup>* ,*v*(*tE*,*<sup>i</sup>* , *tC*), which means that these state variables can be processed in a purely time-domain scheme. In contrast, for each slow time instant *tEi*, each of the *x* ^ *<sup>A</sup>*,*v*(*tE*,*<sup>i</sup>* , *tC*) is a waveform that has to be represented as a Fourier series adopting a convenient harmonic truncation at some order *k* = −*K*,…,*K*, i.e., each of the *x* ^ *<sup>A</sup>*,*v*(*tE*,*<sup>i</sup>* , *tC*) is a waveform that requires a total of 2*K* + 1 harmonic components for a convenient frequency domain representation. In summary, while active state variables have to be represented by a set of 2*K* + 1 Fourier coefficients arranged in (2*K* + 1)×1 vectors of the form

$$\hat{\mathbf{X}}\_{A,\boldsymbol{\nu}}(t\_{E,i}) = \left[X\_{A,\boldsymbol{\nu},-K}(t\_{E,i}), \dots, X\_{A,\boldsymbol{\nu},0}(t\_{E,l}), \dots, X\_{A,\boldsymbol{\nu},K}(t\_{E,l})\right]^T, \quad \boldsymbol{\nu} = \mathbf{l}, \dots, \mathbf{n}\_A,\tag{34}$$

latent state variables can be represented as 1×1 scalar quantities, i.e., they can be simply represented as

$$
\hat{\mathbf{X}}\_{L,\boldsymbol{\nu}}(t\_{E,l}) = X\_{L,\boldsymbol{\nu},0}(t\_{E,l}), \quad \boldsymbol{\nu} = \mathbf{l}, \ldots, n\_L. \tag{35}
$$

By considering this, we can easily deduce that the size of the **X ^** (*tE*,*<sup>i</sup>* ) vector defined by (21) will be significantly decreased, as well as the total number of equations in the HB system of (23). Furthermore, another crucial aspect is that we are no longer forced to carry out the conversion between time and frequency domains for the latent state variables expressed in the form of (35), as well as for the components of **H**(**X ^** (*tE*,*<sup>i</sup>* )) corresponding to latent blocks of the circuit. Given that the *k* = 0 order Fourier coefficient *Xv*,0(*tE*,*<sup>i</sup>* ) is exactly the same as the constant *tC* time value *x* ^ *<sup>v</sup>*(*tE*,*<sup>i</sup>* ), components of the HB system of (23) that have no dependence on active state variables will not be required for any direct or inverse Fourier transformation operations.

Considerable Jacobian **J**(**X ^** (*tE*,*<sup>i</sup>* )) matrix size reductions will also be achieved with this technique. Indeed, by considering the latency of state variables in some parts of the circuit, some blocks of the Jacobian matrix (26) are simply reduced to 1×1 scalar elements. These scalar elements contain the dc sensitivity of **H**(**X ^** (*tE*,*<sup>i</sup>* )) to the latent components of **X ^** (*tE*,*<sup>i</sup>* ).

With the state variable **X ^** (*tE*,*<sup>i</sup>* ) and the error function **H**(**X ^** (*tE*,*<sup>i</sup>* )) vector size reductions, as also the resulting Jacobian **J**(**X ^** (*tE*,*<sup>i</sup>* )) matrix size reduction, it is possible to avoid dealing with large linear systems in the iterations of (25). Thus, a less computationally expensive Newton-Raphson iterative solver is required to solve (23). In conclusion, partitioning the circuit into active and latent sub-circuits (blocks) can lead us to significant computation and memory savings when computing the solution.
