Robust Symbol Rate Estimation of PSK Signals Under the Cyclostationary Framework

Abstract

In many communication channels the noise is non-Gaussian and usually exhibits impulsive characteristics. The symbol rate estimation problem of phase-shift keying (PSK) signals in this kind of impulsive noise environment is addressed. Since the performance of the cyclic statistics-based symbol rate estimation methods, which are very effective in Gaussian noise, may deteriorate significantly in the presence of impulsive noise, a robust method under the cyclostationary framework is proposed. Specifically, a robust form of cyclic autocorrelation based on the M-estimate concept is developed. Simulation results show that the proposed robust technique offers a significant performance gain over the cyclic autocorrelation method in impulsive noise.

Appendices

Appendix A: The Computational Complexity

Here we provide only a rough analysis of the attendant increase in computational complexity for the robust cyclic autocorrelation calculation in contrast to the conventional one.

Equation (11) can be rewritten as

$$ \begin{aligned} & R_{R}^{ ( i )} ( \varepsilon,\tau ) = \frac{\sum_{t = 0}^{T - 1} e^{ ( i - 1 )} ( t,\tau,\varepsilon )R ( t,\tau )e^{ - j2\pi \varepsilon t}}{\sum_{t = 0}^{T - 1} e^{ ( i - 1 )} ( t,\tau,\varepsilon )}, \\ & e^{ ( i - 1 )} ( t,\tau,\varepsilon ) = \bigl \vert R ( t,\tau )e^{ - j2\pi \varepsilon t} - R_{R}^{ ( i - 1 )} ( \varepsilon,\tau ) \bigr \vert ^{ - 1}. \end{aligned} $$

(13)

For a fixed value of τ, the complex multiplication operation numbers of different items for the calculation of $R_{R}^{ ( i )} ( \varepsilon,\tau )$ are given in Table 1. Assume that C multiplication operations⁷ are needed for calculating the conventional cyclic autocorrelation R(ε,τ), which means that the initialization in the iterative procedure of R _R (ε,τ) needs C operations. Thus for K iterations, the iterative procedure needs C+K(2T ³+T ²) multiplication operations.

Table 1 Multiplication operation numbers of different items for the calculation of $R_{R}^{ ( i )} ( \varepsilon,\tau )$

In summary, there is an attendant increase in computational complexity (roughly K(2T ³+T ²) multiplication operations) for the robust cyclic autocorrelation calculation, compared with the conventional method.

Appendix B: Influence Function Analysis

The richest quantitative robustness information is provided by the influence function (IF) and derived quantities; therefore, the IF is an important concept in robust signal processing. The IF of an estimator $\hat{\theta}$ at a nominal distribution F is given by

$$ \mathrm{IF} ( x;\hat{\theta},F ) = \lim_{t \to 0} \frac{\hat{\theta} ( ( 1 - t )F + t\Delta_{x} ) - \hat{\theta} ( F )}{t}, $$

(14)

where Δ _x is the point-mass probability on x and t the fraction of contamination. The two most important norms of the IF are the sup-norm $\gamma^{*}(\gamma^{*} = \sup_{x}\vert \mathrm{IF} ( x;\hat{\theta},F ) \vert )$ over x as the central local robustness measure, and the L ₂-norm with respect to F, namely the well-known asymptotic variance of the estimator $V ( \hat{\theta},F ) (V ( \hat{\theta},F ) = \int \mathrm{IF} ( x;\hat{\theta},F )\, dF ( x ))$ as the basic efficiency measure.

The score function is defined as φ(x)=F′(x). For the loss function F(x)=|x| the score function $\varphi ( x ) = x / \vert x \vert = \operatorname{sign} ( x )$ is odd and bounded, which means that φ is B-robust [9] and thus implies that the IF has a finite γ ^∗. On the other hand, in [14] it was determined that in a heavy-tailed noise environment the loss function F(x)=|x| can minimize the maximum asymptotic variance⁸ $V ( \hat{\theta},F )$. Accordingly, F(x)=|x| appears as an optimal residual function for problem (6), and the corresponding solution of the optimization problem (6), i.e., R _R (ε,τ), is robust and reliable in impulsive noise.