Performance of the biased square-law sequential detector in the absence of signal

G. E. Albert's general sequential test for making one of r distinct decisions about a distribution function
F(x_{i}|x_{i-1})
by observing a sequence of
x_{i}
's is presented, and his results, which give the performance of this test as the solutions of integral equations, are stated. This test is more general than the usual sequential probability ratio test of Wald, and includes nonoptimum, as well as Wald's optimum, sequential tests. Albert's equations are used to treat the incoherent detection of a sine wave in Gaussian noise by a biased square-law detector. This is the detector which uses samples
y_{i}
of the envelope of the received waveform to calculate the sums
x_{i} = k \sum_{j=1}^{i} (yj^{2} -
bias
), i = 1, 2, \cdots
, and sequentially compares the
x_{i}
to two thresholds until an
x_{i}
is found which is less than the lower threshold
B
, or greater than the upper threshold
A
. Then if
x_{i} \leq B
, it is decided that the signal is not present (dismissal), and if
A \leq x_{i}
it is decided that the signal is present (alarm). Though this is not the optimum detector for a sine wave in Gaussian noise (unless the SNR is very small), it is a convenient test to implement and is of considerable interest in its own right. For
y_{i}
having the Rayleigh probability distribution
f(y) = y \exp\{-y^{2}/2\}
, i.e., for the received waveform consisting of Gaussian noise alone, {\em exact} solutions are obtained for the probability of (false) alarm and for the average test duration. These results are unique in that they are valid without restrictions on either the design input SNR or on the thresholds A and B. Curves of the probability of false alarm vs. the upper threshold A, and of the average test duration vs. the expected input SNR are presented, and are compared to similar results of Wald.