*2.1.2 Near-field vector data sampling*

The complexity and size of an EM simulation are governed by the detail and size of the structure and surrounding space, and the frequency band of interest. These simulating features affect the number of mesh cells and the time step of the EM solving. To achieve high spatial resolution for the analysis, the excitation signal BW must be adequate, and it can be larger than the operational BW of the antenna. To guarantee convergence condition, the time step can be much smaller than the time step corresponding to the excitation signal Nyquist BW. Therefore, it does not need to sample all the computed near-field data in the simulating time. The duration of the EM solving depends on the volume of the simulation space, degree of EM energy


#### **Table 1.**

*EM solving and sampling parameters of a simulation example.*

**Figure 1.** *Reconstructed structure and mapping of the Vivaldi edge points to the smooth polynomial curve.*

stored in the structure, the length of the excitation signal and accuracy requirement of the analysis, especially for low-frequency or long-delay-time components.

An example of EM solving and sampling parameters for a Vivaldi antenna structure is presented in **Table 1**. EM field vector data are sampled in the simulating space and time. Each vector is represented by three scalar components. With singleprecision floating point format, each scalar component uses 32 bits in size. The total size of the sampled EM near-field data can reach hundreds of gigabytes.

The Vivaldi edges are the main parts of this traveling-wave antenna structure, and a significant proportion of EM energy concentrates along these edges. Space, time and frequency distributions of the EM fields along these edges contain the most important information about the EM energy propagation; therefore, accurate sampling of field data in these regions is essential.

The sampling is based on fitting a nine-order polynomial smooth curve on the stepped structure of the reconstructed structure from the meshed structure, as shown **Figure 1**. As expected, this curve represents the Vivaldi curve of the input structure, at which there is a transition in material property from conductor to dielectric/vacuum. Therefore, there is also a corresponding transition in the field vectors at the edge. To preserve this attribute in the sampled data, the EM field vectors at the edge points of the reconstructed structure are mapped to the nearest points on the smooth curve, and these represent the field vectors on the Vivaldi edge of the input structure. The tangential and normal vector components of the EM field vectors can also be divided from the sampled vectors as shown in **Figure 1**. A three-dimensional interpolation from the field vector data at mesh cell vertices

*Near-Field Propagation Analysis for Traveling-Wave Antennas DOI: http://dx.doi.org/10.5772/intechopen.100856*

around the mapped point is also a method for the sampling. However, because of the dependence of interpolated result on distance to the mapped point and the difference between field vectors inside the conducting patch and field vectors at the edge of the patch (especially in the vector direction), there can be a significant error in the interpolation result if a mapped point closes to a mesh cell vertex inside the conducting patch.

The field vector data for the lateral and end edges of the conducting patches are sampled by extracting the vector data directly on mesh cell vertices corresponding to these edges. The field vector data at positions in dielectric/vacuum volumes are sampled by interpolated data from the nearest mesh cell vertices.

#### **2.2 Time- and frequency-domain-based analysis methods**

#### *2.2.1 Impulse response analysis*

With the time-domain-based method (TDbM), the analysis is started by an EM simulation with an impulse excitation signal (e.g., Gaussian signal) covering a certain analysis BW. The observed signals are directly extracted from the EM-simulating data. The EM responses on the structure can be analyzed directly in the time domain or can be transformed to the frequency domain by the discrete Fourier transform (DFT). An ineffectiveness of the analysis method is that the EM solver must be rerun whenever there is a change in the analysis BW, and the frequency-domainbased method (FDbM) is a solution proposed to improve the analysis performance.

By the FDbM, the EM solver was only run once with an excitation signal covering a wide enough BW of all analysis sub-bandwidth segments. Then, transformations to the frequency domain for the excitation signal and the observed signals are implemented based on DFT. To compensate the unequal in magnitude and phase of the frequency components of the excitation signal, an equalization is implemented for the frequency components of the observed signals.

Then, the compensated observed signals are transformed into the time domain by inverse DFT. Thus, when the excitation signal is processed and transformed to the time domain, all of its frequency components are equal in magnitude and phase; it is a full Nyquist BW impulse signal. The compensated observed signals in time domain are the responses of this impulse. However, in practical, the compensation can gain noise levels excessively for low-energy-level frequency components for an observed signal with a certain signal-to-noise ratio (SNR), especially at the upper end of the simulating band. Therefore, the responses with full Nyquist BW cannot be achieved in practice. Rectangular, Gaussian and Kaiser windows are used to limit the analysis sub-bandwidth segments in this work.

A frequency-modulated continuous wave (FMCW) is chosen for the excitation signal. This linear chirp sweeps from 0 to 180 GHz in 0.5 ns and covering a BW of 0– 210 GHz. **Figure 2** shows this excitation signal and the impulse signals corresponding to the different windows in the time and frequency domains. The impulses and responses in **Figure 3** are the result of process using a Gaussian window in a low-pass analysis band of 0–64 GHz and a band-pass analysis band of 20–90 GHz (20 dB BW is standard) in the time and frequency domains. The response signals are curve normal vector component of the E-field at points along a Vivaldi edge located a distance *l* from the excitation source as shown by the smooth curve in **Figure 1**.

The differences between the TDbM and the FDbM results are also examined. For TDbM, the EM simulation uses a Gaussian impulse excitation signal with a BW of 0–64 GHz in time domain. For FDbM, the EM simulation uses an FMCW excitation signal with a BW of 0–210 GHz in time domain, and a Gaussian window

**Figure 2.**

*The FMCW excitation signal and FDbM impulse signals with different windows in the (a) time and (b) frequency domains.*

**Figure 3.**

*The FDbM impulse signal and responses in the (a) time and (b) frequency domains at the points along the Vivaldi edge.*

with a low-pass BW of 0–64 GHz is used in the analysis frequency limitation. The impulse signal pair, the response pairs, and their errors are shown in **Figure 4**. These comparisons demonstrate a good agreement between the TDbM and FDbM.

## *2.2.2 MUSIC algorithm for ToA estimation*

As a well-known super-resolution algorithm, MUSIC is often used for signal analysis [15, 16], especially in cases of the overlap of many signals and/or limitation in analysis BW. The MUSIC algorithm is applied in this work for ToA or time delay estimation of dominant components/clusters in the observed signals.

The signal applied at the antenna excitation port is *x t*ð Þ. Because of multiple overlapping EM energy flows propagating along different paths in the structure, the observed signal at an arbitrary position in the simulating space is considered as a

**Figure 4.** *The impulse and responses pairs with (a) TDbM and FDbM and (b) the corresponding errors.*

superposition of different versions from the excitation source. This can be presented by a convolution of *x t*ð Þ and the impulse response of the propagation channel *h t*ð Þ:

$$\mathbf{s}(t) = \mathbf{x}(t) \* h(t). \tag{1}$$

In general, *h t*ð Þ can be continuous, but in the case of limited excitation signal BW and limits in space/time resolution of the EM simulation, it can be approximated by *N* discrete components (*N* can be large):

$$h(t) \approx \sum\_{n=0}^{N-1} a\_n \delta(t - \tau\_n),\tag{2}$$

where *δ* is a Dirac function, and *an* and *τ<sup>n</sup>* are the amplitude and delay time, respectively, of the *n*-th component.

The channel impulse response in the frequency domain is

$$H(f) \approx \sum\_{n=0}^{N-1} a\_n e^{-j2\pi f \tau\_n}.\tag{3}$$

Considering noise (as white noise from EM solver errors) and with a sufficient SNR of the signals in the analysis band, an estimation of *H f* ð Þ can be implemented as

$$
\hat{H}(f) = \frac{\hat{\mathbf{S}}(f)}{\mathbf{X}(f)} = H(f) + w'(f), \tag{4}
$$

where *X f* ð Þ is the Fourier transform of *x t*ð Þ, ^ *S f* ð Þ is the Fourier transform of *s t*ð Þ with the noise and *w*<sup>0</sup> ð Þ*f* is considered as the corresponding white noise in the frequency domain.

Thus,

$$\hat{H}(f) = \sum\_{n=0}^{N-1} a\_n e^{-j2\pi f \tau\_n} + w(f),\tag{5}$$

where *w f* ð Þ is the total noise including the error of approximation in Eq. (3). In Eq. (5), *H f* ^ ð Þ can be considered as a harmonic model [15, 17], and *<sup>τ</sup><sup>n</sup>* are parameters needed to be estimated.

The MUSIC algorithm is applied to solve this problem in this work. *H f* ^ ð Þ is sampled with *Ns* samples and a frequency step of Δ*f* in the frequency domain:

$$\hat{H}[k] = \sum\_{n=0}^{N-1} a\_n e^{-j2\pi(k\Delta f)\mathbf{r}\_n} + w[k], k = \mathbf{0}, \mathbf{1}, \dots, N\_s - \mathbf{1}$$

$$w[k] = w(k\Delta f). \tag{6}$$

The matrix form of *H k* ^ ½ � is

$$
\hat{\mathbf{H}} = \mathbf{V}\mathbf{a} + \mathbf{w},\tag{7}
$$

where

$$\hat{\mathbf{H}} = \left[\hat{H}[\mathbf{0}]\hat{H}[\mathbf{1}]\dots\hat{H}[\mathbf{N}\_{\circ}-\mathbf{1}]\right]^{\mathrm{T}}$$

$$\mathbf{V} = \left[\mathbf{v}(\tau\_{0})\mathbf{v}(\tau\_{1})\dots\mathbf{v}(\tau\_{N-1})\right]$$

$$\mathbf{v}(\tau\_{n}) = \left[\mathbf{1}\ e^{-j2\pi((1)\Delta f)\tau\_{n}}\dots e^{-j2\pi((N\_{\circ}-1)\Delta f)\tau\_{n}}\right]^{\mathrm{T}}$$

$$\mathbf{a} = \left[a\_{0}a\_{1}\dots a\_{N-1}\right]^{\mathrm{T}}$$

$$\mathbf{w} = \left[w[\mathbf{0}]w[\mathbf{1}]\dots w[\mathbf{N}\_{\circ}-\mathbf{1}]\right]^{\mathrm{T}},\tag{8}$$

T is a transpose of a vector.

The analysis band limitation is implemented by a rectangular window:

$$\hat{H}[k] = u[k] \left( \sum\_{n=0}^{N-1} a\_n e^{-j2\pi (k\Delta f)\tau\_n} + \omega \begin{bmatrix} k \end{bmatrix} \right), k = 0, 1, \ldots, N\_s - 1$$
 
$$u[k] = \begin{cases} 1 & \text{if } F\_{cL} \le k\Delta f \le F\_{cH} \\ 0 & \text{otherwise}, \end{cases} \tag{9}$$

where *FcL* and *FcH* are the lower and upper frequency bounds of the analysis band, respectively.

A Toeplitz data matrix **Z** is established from **H**^ array with a dimension factor *D* chosen as *D* ≥ *N*0, where *N*<sup>0</sup> ≤ *N* is the size of the signal subspace chosen in the analysis:

$$\mathbf{Z} = \begin{bmatrix} \hat{H}[D] & \dots & \hat{H}[\mathbf{0}] \\ \vdots & \ddots & \vdots \\ \hat{H}[\mathbf{N}\_{i}-\mathbf{1}-D] & \dots & \hat{H}[\mathbf{D}] \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \\ \hat{H}[\mathbf{N}\_{i}-\mathbf{1}] & \dots & \hat{H}[\mathbf{N}\_{i}-\mathbf{1}-D] \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \\ \hat{H}^{\ast}[D] & \dots & \hat{H}^{\ast}[\mathbf{N}\_{i}-\mathbf{1}-D] \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \\ \hat{H}^{\ast}[\mathbf{N}\_{i}-\mathbf{1}-D] & \dots & \hat{H}^{\ast}[\mathbf{N}\_{i}-\mathbf{1}] \end{bmatrix} \tag{10}$$

where H is a Hermitian matrix transpose.

*Near-Field Propagation Analysis for Traveling-Wave Antennas DOI: http://dx.doi.org/10.5772/intechopen.100856*

A covariance matrix **R** is calculated based on this **Z** matrix:

$$\mathbf{R} = \mathbf{Z}\mathbf{Z}^{\dagger}.\tag{11}$$

With the **R** covariance matrix, eigen decomposition is implemented and the signal and noise subspaces are separated [18]:

$$\mathbf{R} = \sum\_{i=1}^{D+1} \lambda\_i \mathbf{u}\_i \mathbf{u}\_i^{\mathcal{H}},\tag{12}$$

where *λ<sup>i</sup>* and **ψ***<sup>i</sup>* are eigenvalues and eigenvectors, respectively. The ToA parameters can be estimated based on the peak positions of the MUSIC spectrum:

$$P(\mathbf{r}) = \frac{1}{\left\| \Phi^{\mathcal{H}} \mathbf{e}(\mathbf{r}) \right\|^2},\tag{13}$$

where **ϕ** ¼ **ψ***<sup>N</sup>*0þ<sup>1</sup>**ψ***<sup>N</sup>*0þ<sup>2</sup> … **ψ***<sup>D</sup>*þ<sup>1</sup> � � spans the noise subspace, and **<sup>e</sup>**ð Þ¼ *<sup>τ</sup>* 1 *e <sup>j</sup>*2*π*ð Þ<sup>1</sup> *<sup>τ</sup>=Ns* … *e <sup>j</sup>*2*π*ð Þ *<sup>D</sup> <sup>τ</sup>=Ns* � �<sup>T</sup> is a steering vector [19].

This MUSIC algorithm is applied to directly analyze the observed signals (curve normal components of the E-field vectors at the points a distance *l* along an edge of a slot line structure as presented in **Figure 5(b)**). The analysis is implemented in the cases of *N*<sup>0</sup> ¼ 1 and *N*<sup>0</sup> ¼ 500.

• Analysis of expected single component *N***<sup>0</sup>** ¼ **1**

**Figure 5.** *The structural models with excitation ports of (a) a Vivaldi antenna and (b) a slot line.*

In the case of *N*<sup>0</sup> ¼ 1, only one component in the signal subspace is expected. This corresponds to the most dominant component or cluster of the observed signal, and other components are considered as noise. The peak of the MUSIC spectrum is expected to indicate the ToA or time delay of this component or cluster.

MUSIC parameters are set as *N*<sup>0</sup> ¼ 1; *D* ¼ 35; *Ns* ¼ 1, 250; and *dt* ¼ 1 ps in this analysis. The analysis is implemented in different bands of *FcL* and *FcH*. The response signals with the same bands limited by the Gaussian windows are calculated based on FDbM as the reference signals. **Figures 6** and **7** show the results of estimated ToAs and propagation velocities based on the response signals and the MUSIC spectra.

**Figure 6** shows that the FDbM response signal shape changes with increasing *l*. The signal part to the left of the peak is getting wider than the part to the right of the peak, which tends to increase in negative potential. This denotes a left-shifting tendency in the FDbM response peaks versus distance *l*. Thus, the estimated ToAs based on the FDbM response peaks tend to shrink with distance *l*, while the peak positions of the MUSIC spectra increase regularly with *l*. In **Figure 7**, the feature can

#### **Figure 6.**

*The MUSIC spectra and FDbM impulse response signals at the points along the slot line edge with analysis bands of (a) 0–60, (b) 0–90, (c) 20–60, and (d) 20–90 GHz.*

#### **Figure 7.**

*The estimated propagation velocities based on the FDbM response signal peaks and the MUSIC spectra in different analysis bands along the slot line edge.*

be seen more clearly with the estimated propagation velocities. The estimated velocities based on the response peaks are faster than those based on MUSIC, and the difference increases with an increase in *l*, with an exception in the lowfrequency band case (0–30 GHz), where the two methods show similar results. Furthermore, the estimated results based on MUSIC show reasonably constant velocities, independent of *l*, especially in wideband cases.

• Analysis of multi-components *N*<sup>0</sup> ¼ 500

The analysis is observed at a point *l* ¼ 30 *mm* on the slot line edge with a parameter set of *N*<sup>0</sup> ¼ 500; *D* ¼ 738; *Ns* ¼ 1, 250; and *dt* ¼ 1 ps and an analysis band of 0–90 GHz. The dimension factor *D* equals 738 to ensure that a full slide over the samples generates the **Z** matrix with a size of 738 � 1, 024, and matrix columns contain all of the frequency components of the analysis band without redundancy. This ensures to capture adequately signal frequency information. In **Figure 8**, besides the MUSIC spectra, FDbM impulse response signals with a Gaussian window and a rectangular window to limit frequency in the same analysis band of 0–90 GHz are also calculated and presented.

With the larger *N*<sup>0</sup> in this analysis, the MUSIC spectrum shows synchronicity with the FDbM impulse response signals (limited by rectangular window); as seen in **Figure 8**, the spectrum is also eliminated at nulls of the response signal. This shows that accuracy of the estimation of the dominant component ToAs can be reduced at proximity of the zero crossings in the impulse response signals. In the case of *N*<sup>0</sup> ¼ 1, this effect cannot be observed. Another observable feature in this analysis is that the variability in the MUSIC spectrum magnitude is not following the trend of the component or cluster magnitude of the corresponding response signal. Thus, the magnitude estimation for the separate components or clusters needs additional information other than the MUSIC spectrum.
