  We investigate quantum error correction using continuous parity measurements
to correct bit-flip errors with the three-qubit code. Continuous monitoring of
errors brings the benefit of a continuous stream of information, which
facilitates passive error tracking in real time. It reduces overhead from the
standard gate-based approach that periodically entangles and measures
additional ancilla qubits. However, the noisy analog signals from continuous
parity measurements mandate more complicated signal processing to interpret
syndromes accurately. We analyze the performance of several practical filtering
methods for continuous error correction and demonstrate that they are viable
alternatives to the standard ancilla-based approach. As an optimal filter, we
discuss an unnormalized (linear) Bayesian filter, with improved computational
efficiency compared to the related Wonham filter introduced by Mabuchi [New J.
Phys. 11, 105044 (2009)]. We compare this optimal continuous filter to two
practical variations of the simplest periodic boxcar-averaging-and-thresholding
filter, targeting real-time hardware implementations with low-latency
circuitry. As variations, we introduce a non-Markovian ``half-boxcar'' filter
and a Markovian filter with a second adjustable threshold; these filters
eliminate the dominant source of error in the boxcar filter, and compare
favorably to the optimal filter. For each filter, we derive analytic results
for the decay in average fidelity and verify them with numerical simulations.
