Performance Predictions for Parameter Estimators That Minimize Cost-Functions Using Wirtinger Calculus With Application to CM Blind Equalization

We present calculations of the first-order bias and first-order variance of parameters which are estimated batch-wise from the global minimum of a cost-function of complex-valued signals embedded in zero-mean Gaussian noise. The derivation involves the calculation of the multidimensional Taylor series of the non-analytic cost-function up to third order using the elegant Wirtinger calculus. Whereas closed-form expressions for the variance can be obtained straightforwardly from a second-order Taylor series, and have been presented in various other contexts, an exact expression for the bias cannot be derived, in general. In this paper, we propose approximate expressions for the first-order bias and confirm them in a comparison of results from extensive analytical calculations with results from Monte Carlo (MC) simulations for the statistical efficiency of a batch-processing blind equalizer using the constant-modulus (CM) criterion. We study the equalization of independent and identically distributed (i.i.d.) random symbols and obtain asymptotic (for large batch size) expressions for the averages of the bias and variance over zero-mean random (real-valued) signals of binary phase shift keying (BPSK), and (complex-valued) signals of M-ary PSK modulation (M > 2). Finally, we compare the statistical efficiency of the CM estimator with the one of the maximum likelihood (ML) blind estimation of the path parameters and equalized symbols with CM constraint.