DOA parameter estimation with 1-bit quantization bounds, methods and the exponential replacement

While 1-bit analog-to-digital conversion (ADC) allows to significantly reduce the analog complexity of wireless receive systems, using the exact likelihood function of the hard-limiting system model in order to obtain efficient algorithms in the digital domain can make 1-bit signal processing challenging. If the signal model before the quantizer consists of correlated Gaussian random variables, the tail probability for a multivariate Gaussian distribution with N dimensions (general orthant probability) is required in order to formulate the likelihood function of the quantizer output. As a closed-form expression for the general orthant probability is an open mathematical problem, formulation of efficient processing methods for correlated and quantized data and an analytical performance assessment have, despite their high practical relevance, only found limited attention in the literature on quantized estimation theory. Here we review the approach of replacing the original system model by an equivalent distribution within the exponential family. For 1-bit signal processing, this allows to circumvent calculation of the general orthant probability and gives access to a conservative approximation of the receive likelihood. For the application of blind direction-of-arrival (DOA) parameter estimation with an array of K sensors, each performing 1-bit quantization, we demonstrate how the exponential replacement enables to formulate a pessimistic version of the Cramér-Rao lower bound (CRLB) and to derive an asymptotically achieving conservative maximum-likelihood estimator (CMLE). The 1-bit DOA performance analysis based on the pessimistic CRLB points out that a low-complexity radio front-end design with 1-bit ADC is in particular suitable for blind wireless DOA estimation with a large number of array elements operating in the medium SNR regime.