  With the advent of massive data outputs at a regular rate, admittedly, signal
processing technology plays an increasingly key role. Nowadays, signals are not
merely restricted to physical sources, they have been extended to digital
sources as well.
  Under the general assumption of discrete statistical signal sources, we
propose a practical problem of sampling incomplete noisy signals for which we
do not know a priori and the sample size is bounded. We approach this sampling
problem by Shannon's channel coding theorem. We use an extremal binary channel
with high probability of transmission error, which is rare in communication
theory. Our main result demonstrates that it is the large Walsh coefficient(s)
that characterize(s) discrete statistical signals, regardless of the signal
sources. Note that this is a known fact in specific application domains such as
images. By the connection of Shannon's theorem, we establish the necessary and
sufficient condition for our generic sampling problem for the first time.
Finally, we discuss the cryptographic significance of sparse Walsh transform.
