  In this work, the problem of signal parameter estimation from measurements
acquired by a low-complexity analog-to-digital converter (ADC) with $1$-bit
output resolution and an unknown quantization threshold is considered.
Single-comparator ADCs are energy-efficient and can be operated at ultra-high
sampling rates. For analysis of such systems, a fixed and known quantization
threshold is usually assumed. In the symmetric case, i.e., zero hard-limiting
offset, it is known that in the low signal-to-noise ratio (SNR) regime the
signal processing performance degrades moderately by ${2}/{\pi}$ ($-1.96$ dB)
when comparing to an ideal $\infty$-bit converter. Due to hardware
imperfections, low-complexity $1$-bit ADCs will in practice exhibit an unknown
threshold different from zero. Therefore, we study the accuracy which can be
obtained with receive data processed by a hard-limiter with unknown
quantization level by using asymptotically optimal channel estimation
algorithms. To characterize the estimation performance of these nonlinear
algorithms, we employ analytic error expressions for different setups while
modeling the offset as a nuisance parameter. In the low SNR regime, we
establish the necessary condition for a vanishing loss due to missing offset
knowledge at the receiver. As an application, we consider the estimation of
single-input single-output wireless channels with inter-symbol interference and
validate our analysis by comparing the analytic and experimental performance of
the studied estimation algorithms. Finally, we comment on the extension to
multiple-input multiple-output channel models.
