The Fractional Lower Order Moments Based ESPRIT Algorithm for Noncircular Signals in Impulsive Noise Environments

Abstract

By applying the covariation statistics, this paper generalizes the non-circularity to the scenario of complex isotropic symmetric alpha-stable ($S\alpha S$) signals. The matrices based on fractional lower order moments for the extended sensor array outputs, which have been shown to have similar forms to the covariance matrix for the Gaussian distributed signals and noise, are formulated for the case of complex $S\alpha S$ signals and noise. Therefore, the similarity transformation based estimating signal parameter via rotational invariance techniques can be applied to these matrices resulting to the improved direction of arrival estimates for noncircular signals in the presence of impulsive noise environments. The robustness of our proposed algorithm, especially for low generalized signal to noise ratio situations, and for quite highly impulsive noise environments is demonstrated by Monte-Carlo simulations.

Appendices

Appendix 1: Proof for Lemma 1

For a complex noncircular $S\alpha S$ signal $z$, we can describe $z$ as $z = z_{0} e^{j\varphi }$, where $z_{0}$ and $\varphi$ represent the amplitude and the phase of $z$, respectively. That is, $z_{0}$ is the zero-phase version of $z$. From the properties of covariation, we have

$$ \begin{aligned} \left[ {z,z^{ * } } \right]_{\alpha } &= \left[ {z_{0} e^{j\varphi } ,z_{0} e^{ - j\varphi } } \right]_{\alpha } \hfill \\ \quad \quad \quad \, &= e^{j\varphi } \left( {e^{ - j\varphi } } \right)^{ * } \left| {e^{ - j\varphi } } \right|^{\alpha - 2} \left[ {z_{0} ,z_{0} } \right]_{\alpha } \hfill \\ \quad \quad \quad \, &= e^{j \cdot 2\varphi } \left[ {z_{0} ,z_{0} } \right]_{\alpha } \hfill \\ \end{aligned} $$

(34)

and

$$ \begin{aligned} \left[ {z,z} \right]_{\alpha } &= \left[ {z_{0} e^{j\varphi } ,z_{0} e^{j\varphi } } \right]_{\alpha } \hfill \\ \quad \quad \quad \, &= e^{j\varphi } \left( {e^{j\varphi } } \right)^{ * } \left| {e^{j\varphi } } \right|^{\alpha - 2} \left[ {z_{0} ,z_{0} } \right]_{\alpha } \hfill \\ \quad \quad \quad \, &= \left[ {z_{0} ,z_{0} } \right]_{\alpha } \hfill \\ \end{aligned} $$

(35)

Based on the definitions of the non-circularity rate and the non-circularity phase in Eq. (14), for the noncircular $S\alpha S$ signal with non-circularity rate $\rho = 1$, combining (34) and (35), we can get $\varphi = {\phi \mathord{\left/ {\vphantom {\phi 2}} \right. \kern-0pt} 2}$, where $\phi$ is the non-circularity phase of $z$. Therefore, the formula (15) can be proved.

Appendix 2: A Brief Interpretation of the Proof for Eq. (19)

As shown in (18), ${\mathbf{C}}_{\text{nc}}$ is a $2M \times 2M$ matrix which is defined as the covariation matrix of the extended sensor array output ${\mathbf{y}}(t)$. Partitioning ${\mathbf{C}}_{\text{nc}}$ as

$$ {\mathbf{C}}_{\text{nc}} = \left[ \begin{aligned} {\mathbf{C}}^{1} \quad {\mathbf{C}}^{2} \hfill \\ {\mathbf{C}}^{3} \quad {\mathbf{C}}^{4} \hfill \\ \end{aligned} \right] $$

(36)

we can obtain that the corresponding $(i,j){\text{th}}$ entries of these four sub-matrices ${\mathbf{C}}^{1}$,${\mathbf{C}}^{2}$, ${\mathbf{C}}^{3}$, and ${\mathbf{C}}^{4}$, take the values $[x_{i} (t),x_{j} (t)]_{\alpha }$, $[x_{i} (t),x_{j}^{ * } (t)]_{\alpha }$, $[x_{i}^{ * } (t),x_{j} (t)]_{\alpha }$ and $[x_{i}^{ * } (t),x_{j}^{ * } (t)]_{\alpha }$, respectively.

For the $(i,j)\,{\text{th}}$ entry $[x_{i} (t),x_{j} (t)]_{\alpha }$ of the first sub-matrix ${\mathbf{C}}^{1}$, using the independence assumption between the signal ${\mathbf{s}}(t)$ and the noise ${\mathbf{n}}(t)$ and by the properties of covariation, we have that

$$ \begin{aligned} C_{ij}^{1} &= \left[ {x_{i} (t),x_{j} (t)} \right]_{\alpha } \hfill \\ \quad \,\, &= \left[ {{\mathbf{A}}_{i} (\theta ){\mathbf{s}}(t) + n_{i} (t),{\mathbf{A}}_{j} (\theta ){\mathbf{s}}(t) + n_{j} (t)} \right]_{\alpha } \hfill \\ \quad \,\, &= \left[ {{\mathbf{A}}_{i} (\theta ){\mathbf{s}}(t),{\mathbf{A}}_{j} (\theta ){\mathbf{s}}(t)} \right]_{\alpha } + \left[ {n_{i} (t),n_{j} (t)} \right]_{\alpha } \hfill \\ \end{aligned} $$

(37)

where ${\mathbf{A}}_{i} (\theta ) = [e^{{ - j(i - 1)\psi_{1} }} ,e^{{ - j(i - 1)\psi_{2} }} , \ldots ,e^{{ - j(i - 1)\psi_{P} }} ]\quad i = 1,2, \ldots ,M$.

From Lemma 1, we can describe the noncircular signal vector ${\mathbf{s}}(t)$ as

$$ {\mathbf{s}}(t) = {\varvec{\Phi}}^{{\frac{1}{2}}} {\mathbf{s}}_{0} (t) $$

(38)

where ${\mathbf{s}}_{0} (t) = [s_{01} (t),s_{02} (t), \ldots ,s_{0P} (t)]^{T}$, in which the $\left\{ {s_{0i} (t)} \right\}_{i = 1}^{P}$ are the corresponding zero phase versions of the signals $\left\{ {s_{i} (t)} \right\}_{i = 1}^{P}$, and ${\varvec{\Phi}}^{{\frac{1}{2}}} = {\text{diag}}\left\{ {e^{{{{j\phi_{1} } \mathord{\left/ {\vphantom {{j\phi_{1} } 2}} \right. \kern-0pt} 2}}} ,e^{{{{j\phi_{2} } \mathord{\left/ {\vphantom {{j\phi_{2} } 2}} \right. \kern-0pt} 2}}} , \ldots ,e^{{{{j\phi_{P} } \mathord{\left/ {\vphantom {{j\phi_{P} } 2}} \right. \kern-0pt} 2}}} } \right\}$.

Therefore, the first item of (37) can be further expressed as

$$ \left[ {{\mathbf{A}}_{i} (\theta ){\mathbf{s}}(t),{\mathbf{A}}_{j} (\theta ){\mathbf{s}}(t)} \right]_{\alpha } = \left[ {\sum\limits_{k = 1}^{P} {a_{i} (\theta_{k} )e^{{j\phi_{k} /2}} s_{0k} (t),} \sum\limits_{l = 1}^{P} {a_{j} (\theta_{l} )e^{{j\phi_{l} /2}} s_{0l} } (t)} \right]_{\alpha } $$

(39)

Using the independent assumption of the signals, and the properties P1 and P2 for covariation, we have that

$$ \begin{aligned} \left[ {{\mathbf{A}}_{i} (\theta ){\mathbf{s}}(t),{\mathbf{A}}_{j} (\theta ){\mathbf{s}}(t)} \right]_{\alpha } \; &= \sum\limits_{k = 1}^{P} {a_{i} (\theta_{k} )e^{{j\phi_{k} /2}} (a_{j} (\theta_{k} )e^{{j\phi_{k} /2}} )^{\langle \alpha - 1\rangle } } \left[ {s_{0k} (t),s_{0k} (t)} \right]_{\alpha } \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\; &= \sum\limits_{k = 1}^{P} {a_{i} (\theta_{k} )a_{j}^{ * } (\theta_{k} )\gamma_{0k} } \hfill \\ \end{aligned} $$

(40)

in which $\gamma_{0k} = \left[ {s_{0k} (t),s_{0k} (t)} \right]_{\alpha } \quad k = 1,2, \ldots P$.

Also, due to the noise assumption it holds that

$$ \left[ {n_{i} (t),n_{j} (t)} \right]_{\alpha } = \kappa_{n} \delta_{i,j} $$

(41)

where $\delta_{i,j}$ is the kronecker delta function and $\kappa_{n} = \left[ {n_{i} (t),n_{i} (t)} \right]_{\alpha }$.

Combining (40) with (41), we have

$$ \begin{aligned} C_{ij}^{1} &= \left[ {x_{i} (t),x_{j} (t)} \right]_{\alpha } \hfill \\ \quad \, &= \,\sum\limits_{k = 1}^{P} {a_{i} (\theta_{k} )a_{j}^{ * } (\theta_{k} )\gamma_{0k} } + \kappa_{n} \delta_{i,j} \hfill \\ \end{aligned} $$

(42)

The analysis for $(i,j){\text{th}}$ entries of the remaining three sub-matrices ${\mathbf{C}}^{2}$,${\mathbf{C}}^{3}$, and ${\mathbf{C}}^{4}$, can be obtained similarly, and therefore is omitted here. Interested readers can refer to Ref. [20] for details.

Thus, incorporating $C_{ij}^{2}$, $C_{ij}^{3}$, and $C_{ij}^{4}$ with $C_{ij}^{1}$, we have ${\mathbf{C}}_{\text{nc}}$ expressed in (19).