  In this paper we focus our attention on a particle that follows a
unidirectional quantum walk, an alternative version of the nowadays widespread
discrete-time quantum walk on a line. Here the walker at each time step can
either remain in place or move in a fixed direction, e.g., rightward or upward.
While both formulations are essentially equivalent, the present approach leads
to consider Discrete Fourier Transforms, which eventually results in obtaining
explicit expressions for the wave functions in terms of finite sums, and allows
the use of efficient algorithms based on the Fast Fourier Transform. The wave
functions here obtained govern the probability of finding the particle at any
given location, but determine as well the exit-time probability of the walker
from a fixed interval, which is also analyzed.
