  In this paper we consider the problem of constructing $T$-optimal
discriminating designs for Fourier regression models. We provide explicit
solutions of the optimal design problem for discriminating between two Fourier
regression models, which differ by at most three trigonometric functions. In
general, the $T$-optimal discriminating design depends in a complicated way on
the parameters of the larger model, and for special configurations of the
parameters $T$-optimal discriminating designs can be found analytically.
Moreover, we also study this dependence in the remaining cases by calculating
the optimal designs numerically. In particular, it is demonstrated that $D$-
and $D_s$-optimal designs have rather low efficiencies with respect to the
$T$-optimality criterion.
