  The measurement of the escape time of a Josephson junction might be used to
detect the presence of a sinusoidal signal embedded in noise when standard
signal processing tools can be prohibitive. We show that the prescriptions for
the experimental set-up and some physical behaviors depend on the detection
strategy. More specifically, by exploiting the sample mean of escape times to
perform detection, two resonant regions are identified. At low frequencies
there is a stochastic resonance/activation phenomenon, while near the plasma
frequency a geometric resonance appears. The naive sample mean detector is
outperformed, in terms of error probability, by the optimal likelihood ratio
test. The latter exhibits only geometric resonance, showing monotonically
increasing performance as the bias current approaches the junction critical
current. In this regime the escape times are vanishingly small and therefore
performance are essentially limited by measurement electronics. The behavior of
the likelihood ratio and sample mean detector for different values of incoming
signal to noise ratio are discussed, and a relationship with the error
probability is found. The likelihood ratio test based detectors could be
employed also to estimate unknown parameters in the applied input signal. As a
prototypical example we study the phase estimation problem of a sinusoidal
current, that is accomplished by using the filter bank approach. Finally we
show that for a physically feasible detector the performances are found to be
very close to the Cramer- Rao theoretical bound. Applications might be found
for example in some astronomical detection problems or to analyze weak signals
in the sub-terahertz range.
