**2.1 Frequency coefficient extraction**

Let us denote *i*(*t*) the transient current which is considered also as the excitation of the under test electronic structure. The sampled data corresponding to this test signal is supposed discretized from the starting time *tmin* to the stop time *tmax* with time step equal to *∆t*. In this case, the number *n* of time-dependent samples is logically, equal to:

$$m = \text{int}\left(\frac{t\_{\text{max}} - t\_{\text{min}}}{\Delta t}\right) \tag{1}$$

with int(*x*) expresses the lowest integer number greater than the real *x*. Accordingly, via the fast Fourier transform (fft), the equivalent frequency-dependent spectrum of *i*(*tk*) (with *tk* = *k.∆t* and *k* = {1…*n*}) can be determined. The frequency data emanated by this mathematical transform are generally as a complex number denoted by *I*( ) () *f ff k k* = *tit* [ ]. Therefore, the

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 7

frequency-dependent magnetic- or H-field expressed by 0 <sup>0</sup> *H xyz f* (,, , ) which is recorded at the point 0 *M*(,, ) *xyz* chosen arbitrarily, into a time-dependent data denoted by 0 *Hxyz t* ( , , ,) by using the ifft-operation. These points 0 *M*(,, ) *xyz* belong in the X-Y plane positioned at *z* = *z*0. In this case, the frequency range considered varying from *fmin* to *fmax* and the frequency step *fstep* of 0 00 *H f H xyz f* () (,, , ) = must be well-synchronized with that of the excitation signal frequency coefficients *ck*. Under this condition, the time-dependent data desired <sup>0</sup> *Ht Hxyz t* () ( , , ,) = which is generated by the specific excitation signal *i*(*t*) can be calculated with the inverse fast Fourier transform (ifft) of the convolution product between

with ℜ*e x*{ } represents the real part of the complex number *x*. Fig. 2 depicts the flow work highlighting the different operations to be fulfilled for the achievement of the transient EMfield computation-method proposed. This method enables to provide the time-dependent

Fig. 2. Flow work illustrating the transient H-field radiation computation-method proposed knowing the temporal range, *tmin* and *tmax* step *Δt* of the excitation signal *i*(*tk*) and also the frequency-dependent H-field 0 *H f* ( ) (here the under bar indicates the complex variables)

*Ht ei* ( ) [ ( ). ( )] = ℜ { *fft c f H*<sup>0</sup> *f* } . (4)

<sup>0</sup> *cf If I* ( ) ( )/ = and 0 *H f* ( ) written as:

magnitude of this signal spectrum from DC to a certain frequency in function of the initial time-step sampled data can be extracted as depicted in Fig. 2. Consequently, by denoting *I*<sup>0</sup> the sinusoidal current magnitude for generating the magnetic field spectrum 0 *H f* ( ) , the following complex coefficients of the input current in function of frequency as illustrated in Fig. 1:

Fig. 1. Extraction of the frequency coefficients from the excitation signal spectrum

It corresponds to the discrete data at each sample of frequency *fk* = *k*.*fstep* for *k* = {1…*n*} having a step-frequency given by:

$$f\_{step} = \frac{1}{t\_{\text{max}} - t\_{\text{min}}}.\tag{3}$$

We underline that this method requires a frequency range [*fmin*, *fmax*] whose the lowest frequency value *fmin* of *I f* ( ) -data is equal to the step frequency *fstep*. It means that the spectrum value can be extrapolated linearly to generate the excitation signal steady-state component at *f* = 0. In practice, it does not change the calculation results because according to the signal processing theory, the DC-component of transient waves with ultra-short time duration at very low frequencies is usually negligible. The upper frequency *fmax* should correspond to the frequency bandwidth containing 95-% of the excitation signal considered spectrum energy.

#### **2.2 Routine process of the computation-method proposed**

The computation-method under study is mainly consisted of two different steps. The first step is focused on the time-domain characterization of the excitation current considered *i*(*tk*) in the specific interval range varying from *tmin* to *tmax* with step *∆t*. In this first step, the complex frequency-coefficients *<sup>k</sup> c* should be extracted through the fft-operation as explained in previous subsection. The following second step is the conversion of the 6 Fourier Transform Applications

magnitude of this signal spectrum from DC to a certain frequency in function of the initial time-step sampled data can be extracted as depicted in Fig. 2. Consequently, by denoting *I*<sup>0</sup> the sinusoidal current magnitude for generating the magnetic field spectrum 0 *H f* ( ) , the following complex coefficients of the input current in function of frequency as illustrated in

> 0 ( ) *step*

<sup>⋅</sup> <sup>=</sup> . (2)

*step <sup>f</sup> t t* <sup>=</sup> <sup>−</sup> . (3)

*I*

*Ik f*

*k*

Fig. 1. Extraction of the frequency coefficients from the excitation signal spectrum

**2.2 Routine process of the computation-method proposed** 

It corresponds to the discrete data at each sample of frequency *fk* = *k*.*fstep* for *k* = {1…*n*} having

We underline that this method requires a frequency range [*fmin*, *fmax*] whose the lowest frequency value *fmin* of *I f* ( ) -data is equal to the step frequency *fstep*. It means that the spectrum value can be extrapolated linearly to generate the excitation signal steady-state component at *f* = 0. In practice, it does not change the calculation results because according to the signal processing theory, the DC-component of transient waves with ultra-short time duration at very low frequencies is usually negligible. The upper frequency *fmax* should correspond to the frequency bandwidth containing 95-% of the excitation signal considered

The computation-method under study is mainly consisted of two different steps. The first step is focused on the time-domain characterization of the excitation current considered *i*(*tk*) in the specific interval range varying from *tmin* to *tmax* with step *∆t*. In this first step, the complex frequency-coefficients *<sup>k</sup> c* should be extracted through the fft-operation as explained in previous subsection. The following second step is the conversion of the

max min 1

*c*

Fig. 1:

a step-frequency given by:

spectrum energy.

frequency-dependent magnetic- or H-field expressed by 0 <sup>0</sup> *H xyz f* (,, , ) which is recorded at the point 0 *M*(,, ) *xyz* chosen arbitrarily, into a time-dependent data denoted by 0 *Hxyz t* ( , , ,) by using the ifft-operation. These points 0 *M*(,, ) *xyz* belong in the X-Y plane positioned at *z* = *z*0. In this case, the frequency range considered varying from *fmin* to *fmax* and the frequency step *fstep* of 0 00 *H f H xyz f* () (,, , ) = must be well-synchronized with that of the excitation signal frequency coefficients *ck*. Under this condition, the time-dependent data desired <sup>0</sup> *Ht Hxyz t* () ( , , ,) = which is generated by the specific excitation signal *i*(*t*) can be calculated with the inverse fast Fourier transform (ifft) of the convolution product between <sup>0</sup> *cf If I* ( ) ( )/ = and 0 *H f* ( ) written as:

$$H(t) = \Re \left\{ \langle \mathcal{J}f[\underline{\mathfrak{C}}(f).\underline{H}\_0(f)] \rangle \right\}. \tag{4}$$

with ℜ*e x*{ } represents the real part of the complex number *x*. Fig. 2 depicts the flow work highlighting the different operations to be fulfilled for the achievement of the transient EMfield computation-method proposed. This method enables to provide the time-dependent

Fig. 2. Flow work illustrating the transient H-field radiation computation-method proposed knowing the temporal range, *tmin* and *tmax* step *Δt* of the excitation signal *i*(*tk*) and also the frequency-dependent H-field 0 *H f* ( ) (here the under bar indicates the complex variables)

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 9

Note that in the present study, the elementary dipoles are supposed ideal, thus, there is no coupling between each other. By reason of linearity, the total EM-field at any point *M*(*x*,*y*,*z*)

8

1 (,,) (,,) *Mk k Hxyz H xyz* =

After implementation of the computation algorithm introduced in Fig. 2 in Matlab program,

Fig. 4. Configuration of the radiating source considered which is comprised of magnetic

In this subsection, comparison results between the transient EM-field maps radiated by the elementary dipoles displayed in Fig. 4 from the direct calculation and from the method

In order to highlight the influence of the form and the transient variation of the disturbing currents in the electronic structure, the considered short-duration pulse excitation current *i*(*t*) is assumed as a bi-exponential signal analytically defined in appendix B. One points out that in order to take into account the truncation effect between *tmin* and *tmax*, the considered sampling data from *i*(*t*) should be multiplied by a specific time gate. So that accordingly,

negligible if the assumed time step is well-accurate. The numerical application was made by taking the current amplitude *IM* = 1A and the time-constants *τ*1 = *τ*2/2 = 2 ns. So, from the analytical relation expressed in (B-4), we have *ω*95% ≈ 3.07 Grad.s-1. Fig. 5 displays the

The time interval range of signal test was defined from *tmin* = 0 ns to *tmax* = 20 ns with step *∆t* = 0.2 ns. One can see that this baseband signal presents a frequency bandwidth *fmax* of about 2 GHz, where belongs more than 95-% of the spectrum signal energy. The data

= generates the frequency-coefficient values of *i*(*t*) according to the

should be assumed as a sine cardinal. But here, this effect can be

we obtain the EM-calculation results presented in the next subsection.

<sup>=</sup> ∑ JJG JJG . (5)

is consequently the sum of each dipole contribution:

dipoles placed in the horizontal plane *xy*

**3.2.1 Description of the excitation signal** 

ω

transient plot of this current excitation.

relation expressed in (2) as described earlier in subsection 2.2.

**3.2 Validation results** 

proposed are presented.

each component *I*( )

calculated *I fft i t* ( ) [ ( )] ω

H-field here denoted as *H*(*t*) according to the arbitrary form of the excitation and also knowing the frequency-dependent H-field data in the frequency range starting from the lowest value to the upper frequency limit equal to the inverse of the time-step *∆t* of the discrete data *x*(*tk*).
