  The probability density function (pdf) of the received signal of an ambient
backscatter communication system is derived, assuming that on-off keying (OOK)
is performed at the tag, and that the ambient radio frequency (RF) signal is
white Gaussian. The pdf of the received signal is then utilized to design two
different types of non-coherent detectors. The first detector directly uses the
received signal to perform a hypothesis test. The second detector first
estimates the channel based on the observed signal and then performs the
hypothesis test. Test statistics and optimal decision threshold of the
detectors are derived. The energy detector is shown to be an approximation of
the second detector. For cases where the reader is able to avoid or cancel the
direct interference from the RF source (e.g., through successive interference
cancellation), a third detector is given as a special case of the first
detector. Numerical results show that both the first and the second detectors
have the same bit error rate (BER) performance, making the second detector
preferable over the first detector due to its computational simplicity.
