  In the digital world, signals are discrete and finite. The Fourier
representation of discrete and finite signals is FT convolution of the finite
sampling function and the continuous signal. Conventionally, finite sampling is
treated as a segment of infinite sampling. Though this approach perfectly
solves the difference between finite and infinite sampling, it has caused much
trouble for signal processing. Mathematically, there is a kind of sampling
between finite and infinite sampling, and we name this kind of sampling as
half-infinite sampling. Theoretically, finite sampling can also be treated as a
segment of half-infinite sampling. Because we can derive the Fourier
representation of discrete and finite signals from half-infinite sampling, the
FTs of several half-infinite samplings are studied. The results show that the
FT of half-infinite sampling is more concise than that of infinite sampling. A
numerical experiment verified the theoretical derivations successfully, during
which step sampling was proposed. Besides, several interesting equations were
built.
