  Obtaining digital representations of multivariate continuous-time (CT)
signals is a challenge encountered in many signal processing systems. In
practice, these signals are often acquired to extract some underlying
information, i.e., for a specific task. Employing conventional task-agnostic
analog-to-digital converters (ADCs), typically designed to minimize the mean
squared error (MSE) in reconstructing the CT input signal, can be costly and
energy-inefficient in such cases. In this work, we study task-based ADCs, which
are designed to obtain a digital representation of a multivariate CT input
process with the goal of recovering an underlying statistically related
parameter vector, referred to as \emph{the task}. The proposed system employs
analog filtering, uniform sampling, and scalar uniform quantization of the
input process before recovering the task vector using a linear digital recovery
filter. We optimize the analog and digital filters and derive closed-form
expressions for the achievable MSE in recovering the task vector from a set of
analog signals when utilizing ADCs with a fixed sampling rate and amplitude
resolution. Based on our derivation, we provide guidelines for designing
practical acquisition systems subject to a constraint on the bit rate. Our
analysis proves that the intuitive approaches of either recovering the task
vector solely in digital or designing the analog filter to estimate the task
vector are inferior to the proposed joint design. We then consider the recovery
of a set of matched filter outputs under a rate budget. We numerically verify
our theoretical observations and demonstrate that task-based ADCs substantially
outperform analog matched filtering as well as applying the matched filter
solely in the digital domain. [...]
