  Departing from traditional communication theory where decoding algorithms are
assumed to perform without error, a system where noise perturbs both
computational devices and communication channels is considered here. This paper
studies limits in processing noisy signals with noisy circuits by investigating
the effect of noise on standard iterative decoders for low-density parity-check
codes. Concentration of decoding performance around its average is shown to
hold when noise is introduced into message-passing and local computation.
Density evolution equations for simple faulty iterative decoders are derived.
In one model, computing nonlinear estimation thresholds shows that performance
degrades smoothly as decoder noise increases, but arbitrarily small probability
of error is not achievable. Probability of error may be driven to zero in
another system model; the decoding threshold again decreases smoothly with
decoder noise. As an application of the methods developed, an achievability
result for reliable memory systems constructed from unreliable components is
provided.
