  The conventional non-coherent differential detection of continuous phase
modulation (CPM) is quite robust to channel impairments such as phase and
Doppler shifts. Its implementation is on top of that simple. It consists in
multiplying the received baseband signal by its conjugate and delayed version
of one symbol period. However it suffers from a signal-to-noise ratio gap
compared to the optimum coherent detection. In this paper, we improve the error
rate performance of the conventional differential detection by using a delay
higher than one symbol period. We derive the trellis description as well as the
branch and cumulative metric that take into account a delay of $K$ symbol
periods. We then derive an optimization criterion of $K$ based on the minimum
Euclidean distance between two differential signals. We finally determine
optimized delays for some popular CPM formats whose values are confirmed by
error rate simulations.
