# A New Conformal Map for Polynomial Chaos Applied to Direction-of-Arrival Estimation via UCA Root-MUSIC

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. gPC Approximation

#### 2.2. Conformally Mapped gPC

#### 2.3. Tanh Map

#### 2.4. Setup

## 3. Results

## 4. Discussion

#### 4.1. Comparison of Classic and Mapped gPC

#### 4.2. Consequences for the UCA Root-MUSIC Algorithm

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The ${E}_{1.6}$ Bernstein ellipse. Its size $\rho $ is equal to the sum of its semiminor and semimajor axes—in this case, 1.6. Its foci $-1$ and 1 are shown by the black dots.

**Scheme 1.**An overview of the conformally mapped gPC algorithm. A new random variable $\tilde{x}$ is defined by applying a conformal map $x=g\left(\tilde{x}\right)$ on the random variable x. An adapted quadrature rule $({\tilde{x}}_{i},{\tilde{w}}_{i})$ is constructed for the new weight function $\tilde{w}\left(\tilde{x}\right)$ via the modified moments ${m}_{k}$, along with its own orthogonal polynomials $\left\{{\tilde{\Phi}}_{k}\right\}$. Generalized Polynomial Chaos (gPC) is applied to the composite function $\left(\right)$ and a mapped gPC approximation of f is found after performing the inverse conformal map $\tilde{x}={g}^{-1}\left(x\right)$.

**Figure 2.**The ${E}_{1.6}$ Bernstein ellipse (dashed black line) and its image (solid blue line) under the Ellipse-to-Strip map (

**a**), the KTE map (

**b**), the Sausage map (

**c**) and the Ellipse-to-Slit map with slits along the imaginary axis (

**d**).

**Figure 3.**The tanh map for real $\tilde{x}$ at different values of $\kappa $, together with the trivial map $g\left(\tilde{x}\right)=\tilde{x}$.

**Figure 4.**The Bernstein ellipse with size $\rho =1.6$ (singularity located at $p=\pm 1.1125$) in the x-space (

**a**) and the corresponding Bernstein ellipses ${E}_{{\tilde{\rho}}_{\mathrm{re}}}$ and ${E}_{{\tilde{\rho}}_{\mathrm{im}}}$ when using $\kappa =1.05{\kappa}_{\mathrm{eq}}$ (

**b**), $\kappa =0.95{\kappa}_{\mathrm{eq}}$ (

**c**) and $\kappa ={\kappa}_{\mathrm{eq}}$ (

**d**). The singularities are depicted with hollow dots.

**Figure 5.**The different regions, denoting the relative sizes of both Bernstein ellipses, in the $\left(\right|p|,\kappa )$-space.

**Figure 6.**Starting with a Bernstein ellipse ${E}_{1.6}$ in the x-space, the Bernstein ellipses in the $\tilde{x}$-space with sizes ${\tilde{\rho}}_{\mathrm{eq}}=2.72$, $0.95{\tilde{\rho}}_{\mathrm{eq}}=2.59$, $0.9{\tilde{\rho}}_{\mathrm{eq}}=2.45$, $0.85{\tilde{\rho}}_{\mathrm{eq}}=2.31$ and $0.8{\tilde{\rho}}_{\mathrm{eq}}=2.18$ are shown (

**b**). The corresponding regions $g({E}_{\tilde{\rho}};{\kappa}_{\mathrm{eq}})$ in which analytic continuability of f is assumed are shown in (

**a**). The original Bernstein ellipse ${E}_{1.6}$ is shown with a dashed line as a reference and the singularities are depicted with hollow dots.

**Figure 7.**The size of ${E}_{\tilde{\rho}}$ as a function of the size of ${E}_{\rho}$ when using the tanh map. ${\tilde{\rho}}_{\mathrm{eq}}$ corresponds to the maximally achievable value of $\tilde{\rho}$.

**Figure 10.**The absolute and relative errors on $\mu $ (

**a**,

**c**) and $\sigma $ (

**b**,

**d**) with regard to their reference value when applying the $\mathrm{Beta}(x;1,1)$ distribution. The precision floor is shown by a dashed line.

**Figure 11.**The absolute and relative errors on $\mu $ (

**a**,

**c**) and $\sigma $ (

**b**,

**d**) with regard to their reference value when applying the $\mathrm{Beta}(x;2,2)$ distribution. The precision floor is shown by a dashed line.

**Figure 12.**The absolute and relative errors on $\mu $ (

**a**,

**c**) and $\sigma $ (

**b**,

**d**) with regard to their reference value when applying the $\mathrm{Beta}(x;3,3)$ distribution. The precision floor is shown by a dashed line.

**Figure 13.**(

**a**) The approximation of f using classic and mapped gPC and (

**b**) the error on the approximation with regard to the reference curve, with $w\left(x\right)=\mathrm{Beta}(x;2,2)$ and $N=15$. The reference curve was constructed by sampling the full simulation.

**Figure 14.**(

**a**) The empirical CDFs of $\widehat{\varphi}$ when sampling the classic and mapped gPC expansion with the full simulation as a reference, each constructed with LHS and ${10}^{6}$ samples, with $w\left(x\right)=\mathrm{Beta}(x;2,2)$ and $N=15$. (

**b**) The error of the classic and mapped gPC CDFs in comparison to the reference CDF.

**Table 1.**The reference values of $\mu $ and $\sigma $. Computed with MC using LHS and ${10}^{6}$ samples.

${\mathit{\mu}}_{\mathbf{ref}}$ | ${\mathit{\sigma}}_{\mathbf{ref}}$ | |
---|---|---|

$\mathrm{Beta}(x;1,1)$ | 49.510102 | 2.527614 |

$\mathrm{Beta}(x;2,2)$ | 50.046675 | 1.347420 |

$\mathrm{Beta}(x;3,3)$ | 50.163825 | 0.956752 |

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**MDPI and ACS Style**

Van Brandt, S.; Verhaevert, J.; Van Hecke, T.; Rogier, H.
A New Conformal Map for Polynomial Chaos Applied to Direction-of-Arrival Estimation via UCA Root-MUSIC. *Sensors* **2022**, *22*, 5229.
https://doi.org/10.3390/s22145229

**AMA Style**

Van Brandt S, Verhaevert J, Van Hecke T, Rogier H.
A New Conformal Map for Polynomial Chaos Applied to Direction-of-Arrival Estimation via UCA Root-MUSIC. *Sensors*. 2022; 22(14):5229.
https://doi.org/10.3390/s22145229

**Chicago/Turabian Style**

Van Brandt, Seppe, Jo Verhaevert, Tanja Van Hecke, and Hendrik Rogier.
2022. "A New Conformal Map for Polynomial Chaos Applied to Direction-of-Arrival Estimation via UCA Root-MUSIC" *Sensors* 22, no. 14: 5229.
https://doi.org/10.3390/s22145229