  We set up Wigner distributions for $N$ state quantum systems following a
Dirac inspired approach. In contrast to much of the work on this case,
requiring a $2N\times 2N$ phase space, particularly when $N$ is even, our
approach is uniformly based on an $N\times N $ phase space grid and thereby
avoids the necessity of having to invoke a `quadrupled' phase space and hence
the attendant redundance. Both $N$ odd and even cases are analysed in detail
and it is found that there are striking differences between the two. While the
$N$ odd case permits full implementation of the marginals property, the even
case does so only in a restricted sense. This has the consequence that in the
even case one is led to several equally good definitions of the Wigner
distributions as opposed to the odd case where the choice turns out to be
unique.
