  Energy-efficient signal processing systems require estimation methods
operating on data collected with low-complexity devices. Using
analog-to-digital converters (ADC) with $1$-bit amplitude resolution has been
identified as a possible option in order to obtain low power consumption. The
$1$-bit performance loss, in comparison to an ideal receiver with $\infty$-bit
ADC, is well-established and moderate for low SNR applications ($2/\pi$ or
$-1.96$ dB). Recently it has been shown that for parameter estimation with
state-space models the $1$-bit performance loss with Bayesian filtering can be
significantly smaller ($\sqrt{2/\pi}$ or $-0.98$ dB). Here we extend the
analysis to Bayesian smoothing where additional measurements are used to
reconstruct the current state of the system parameter. Our results show that a
$1$-bit receiver performing smoothing is able to outperform an ideal
$\infty$-bit system carrying out filtering by the cost of an additional
processing delay $\Delta$.
