One of the most important objectives in processing a stream of
data is to determine and detect the presence of any invariant or
quasi-invariant “features” in that data stream. These features are
often initially unknown and must be “learned” from the observations.
One of the simplest features of this form is a finite length signal
which occurs repetitively, but not necessarily periodically with time,
and has a waveshape that remains invariant or varies only slowly with
time.
In this discussion, we assume that the data stream has been
pre-processed, perhaps by a detector or discriminator, so as to exhibit
this type of repetitive (but unknown) waveshape or signal structure.
The observed signal, however, is perturbed by additive noise or other
disturbances. It is desired to separate the quasi-invariance of the
data from the truly random environment. The repetitive waveform may
represent, for example, the transmission of an unknown sonar or radar,
a pulse-position modulated noise-like waveform, or a repeated code word.
The problem of concern is to estimate the signal waveshape and to
determine the time of each signal occurrence. We limit this discussion
to the situation where only a single repetitive waveform is present
and the signal sample values are binary. The observed waveform is
assumed to be received at low signal-to-noise ratio so that a single
observation of the signal (even if one knew precisely the arrival time)
is not sufficient to provide a good estimate of the signal waveshape.
The occurrence time of each signal is assumed to be random.