On worst case interference for M-ary signaling with correlation receivers

Specification of the worst case additive interference for
M
-ary signaling and receivers using predecision correlation is obtained. The interference is constrained by average power and peak amplitude upper bounds on its Fourier transform. In addition, no statistical description is assumed. The performance measure is a weighted
(\pm 1)
sum of the correlations between the received corrupted signal Fourier transform
R_{m}(f)
and each member of the signal set Fourier transforms
\{S_{m}(f)\}
. This measure also depends on the out-of-sync time shift
t
and the signal present probability
p
. Functional analysis concepts are utilized to state and solve the resulting generalized Lagrange multiplier problem in
L_{2}
. The worst case interference function
W^{0}(f)
is shown to have an average power equal to the average power upper bound and is independent of
p
, but dependent on
t
. A key parameter may need numerical techniques for evaluation. This power level is achieved with
W^{0}
as a clipped multiple of
| \sum S_{m}|
that follows the phase of
\sum S_{m}(f)
. The clipping level is the peak amplitude upper bound. The worst case performance is explicitly expressed in terms of
W^{0}
. Upper and lower bounds are also derived with respect to arbitrary signal sets. Additional bounds are obtained for interference functions independent of the signal set magnitudes or phases. As a recommendation to minimize the worst case performance, the communicator should transmit with a high average probability per signal
pM^{-1}
for a given fixed {\em sum}
\sum S_{m}(f)
and
\tilde{S}_{m}(f)S_{n}(f)=0, m \neq n
, all
m,n
. However, to weaken the effectiveness of some sub-worst case interference, compared with the worst case. the communicator should transmit with low average probability per signal
pM^{-1}
and a phase of
\sum S_{m}(f)
close to
\pi
. Further investigation of good signal sets is warranted.