Adaptive signal subspace digital receivers for communication in time-varying noise
We develop a general three-stage Moving Average Matched Filter (MAMF) receiver system for digital communications in an environment where the noise conditions are unknown a priori and change constantly and significantly with time. The MAMF is a subset of the class of matched filters which are optimal with respect to enhancing the signal energy relative to the noise power in order to improve discrimination between signals at the receiver. In a time-varying noise environment, a fixed signal cannot be designed and used for transmission which will provide optimal performance at the receiver under all noise conditions. Designing a signal for optimality in a particular noise environment will typically lead to a deteriorated performance in another noise environment relative to a signal which is chosen for the new environment. This deterioration in performance can be so severe that the signal-to-noise ratio (SNR) from the input to the output of the filter is degraded. Ideally, to achieve performance which is more nearly optimal under all noise conditions, the transmitted signal should change or adapt in response to variations in the noise environment.
For practical reasons, it is desirable to concentrate all adaptivity in the receiver rather than the transmitter. Typically, a MAMF receiver consists of two stages - a filtering stage and a detection stage. We develop the general design expressions for a three-stage MAMF receiver in which the additional stage is a linear pre-filter placed before the filtering and detection stages.
Obviously, if the MAMF is optimal for a given noise condition, any operation performed on the received signal plus noise prior to filtering will potentially reduce performance at that given noise condition by some amount. We accept this performance loss in favor of a pre-filtering operation which can effectively manipulate the transmitted signal upon arrival at the receiver and provide more robust performance in the time-varying noise environment.
Specifically, we compare a pre-filter consisting of a unity gain with a prefilter that linearly combines k M x 1 partitions of the transmitted signal vector (i.e. transmitted signal vector of length N = k x M). Proper design of the transmitted signals can ensure that the partitions are linearly independent. In this case, we can view the transmitted signal as representing a k-dimensional subspace of the original M-dimensional signal space. By linearly combining these partitions at the receiver we can achieve any vector within this subspace. We show that we can select these partitions such that the resulting signal vector represents an optimum signal subspace for k noise environments. This is contrasted with the fixed 1-dimensional subspace of the original N-dimensional signal subspace when the pre-filter is a constant gain.
The two MAMF receivers are compared by measuring the signal-to-noise ratio improvement (SNRI) of the filters. The SNRI is defined as the output signal-to-noise ratio (OSNR) measured at the output of the filtering stage over the input signal-to-noise ratio (ISNR) measured at the input to the pre-filtering stage. We demonstrate through simulation that the signal subspace version can be more robust with respect to deviation from the absolute maximum SNRI achievable by either system.
Using maximum likelihood techniques, we derive an optimal detector for an arbitrary bank of L linear pre-filter and MAMF sections. This is shown to outperform a detection scheme that has been derived for use solely in an optimal binary communication scenario.