We study the performance of systems for signal parameters estimation, which are based on the linear minimum mean square error (LMMSE) filtering. Such an estimation technique is widely used in practical scenarios, specifically wireless sensor networks and MIMO communications. We model the estimation system through sums and products of random matrices, involving a d-fold Vandermonde matrix (d>=1) with entries uniformly distributed on the complex unit circle. Then, we derive the mean square error (MSE) of the estimated signal by resorting to asymptotic analysis and by applying the eta-transform operator. We describe how our results can be exploited for the study of practical systems, and we show the agreement existing between the asymptotic results derived through our analysis and the simulation results obtained for small values of d.