Novel Direction Findings for Cyclostationary Signals in Impulsive Noise Environments

Abstract

We consider the problem of direction-of-arrival (DOA) estimation for cyclostationary signals in impulsive noise modeled as a complex symmetric α-stable (SαS) process. Since the DOA estimation based on second-order cyclic statistics degrades seriously in an α-stable distribution noise environment, we define a novel pth-order cyclic correlation by fusing the fractional lower order statistics and second-order cyclic correlation. After briefly introducing the statistical characteristics of pth-order cyclic correlation and building the extended data model, we first propose a novel extended pth-order cyclic MUSIC algorithm (EX-POC-MUSIC) by exploiting both pth-order cyclic correlation and pth-order cyclic conjugate correlation. The algorithm allows us to select desired signals and to ignore interference in the communication system. Second, in order to increase the resolution capabilities and the noise robustness significantly, an improved EX-POC-MUSIC algorithm called the extended pth-order cyclic Root-MUSIC (EX-POC-RMUSIC) algorithm is also presented. This algorithm has all the merits of the EX-POC-MUSIC algorithm, and it is also a fast DOA estimation algorithm because it avoids spatial spectrum searching. Under some conditions, both proposed algorithms are able to handle more sources than the number of sensors. Simulation results strongly verify the effectiveness of the two algorithms.

Appendix: Proof of Theorem 1

Note that C _ik is a complex number. We can show that the real part $\operatorname{Re}\{C_{ik}\}$ and imaginary part $\operatorname{Im} \{C_{ik}\}$ are bounded respectively with 0<p<α/2,1<α<2.

(41)

Recalling the fact that for any complex number Y=Y ₁+jY ₂, we have

$$ \operatorname{Re}(Y) = Y_{1}\le |Y_{1}|\le \sqrt{Y_{1}^{2}+Y_{2}^{2}} = |Y| \\ $$

(42)

so

(43)

Substituting (1) into (43) for x _i (t) and x _k (t) and using the conditional expression, we have

(44)

It is easy to see that if $\sum_{m=1}^{K_{\alpha}} A_{im} s_{m}(t)$ and $\sum_{r=1}^{K_{\alpha}} A_{kr} s_{r}(t)$ are constant, then $X_{1}= \sum_{m=1}^{K_{\alpha}} A_{im} s_{m}(t) + n_{i}(t)$, $X_{2}= \sum_{r=1}^{K_{\alpha}} A_{kr} s_{r}(t) + n_{k}(t)$ are jointly SαS. The works [6, 9, 18] show that if X ₁ and X ₂ are jointly SαS, then

$$ E\bigl\{ |X_{1}|^{p_{1}} |X_{2}|^{p_{2}}\bigr\} < \infty,\quad p_{1} + p_{2} < \alpha $$

(45)

We can always select constant G so that (46) is satisfied for all possible values of $s_{1}(t),\ldots, s_{K_{\alpha}}(t)$:

(46)

Hence,

$$ E\bigl\{ \bigl| x_{i}(t) \bigr|^{p} \bigl| x_{k}(t) \bigr|^{p} \bigr\} < E_{s_{1}} E_{s_{2}/s_{1}} \cdots E_{s_{K_{\alpha}}/s_{1},\ldots, s_{K_{\alpha} - 1}} \{ G \} = G < \infty $$

(47)

Similarly, from (42), we can get $\operatorname{Re}(Y) = Y_{1} \ge - |Y_{1}|\ge - \sqrt{Y_{1}^{2} + Y_{2}^{2}} = - |Y|$; hence,

$$ \operatorname{Re} \{ C_{ik} \} \ge - E \bigl\{ |X_{1}|^{p_{1}} |X_{2}|^{p_{2}} \bigr\} > - G > - \infty $$

(48)

Combining (47) with (48), we obtain

$$ \bigl| \operatorname{Re} \{ C_{ik} \} \bigr| < G < \infty $$

(49)

The proof of the imaginary part $\operatorname{Im}\{C_{ik}\}$ of C _ik is similar to that of $\operatorname{Re}\{C_{ik}\}$. Thus, the proof of Theorem 1 is complete.