PAR-constrained sequences are widely used in communication systems and radars
due to various practical needs; specifically, sequences are required to be
unimodular or of low peak-to-average power ratio (PAR). For unimodular sequence
design, plenty of efforts have been devoted to obtaining good correlation
properties. Regarding channel estimation, however, sequences of such properties
do not necessarily help produce optimal estimates. Tailored unimodular
sequences for the specific criterion concerned are desirable especially when
the prior knowledge of the channel is taken into account as well. In this
paper, we formulate the problem of optimal unimodular sequence design for
minimum mean square error estimation of the channel impulse response and
conditional mutual information maximization, respectively. Efficient algorithms
based on the majorization-minimization framework are proposed for both problems
with guaranteed convergence. As the unimodular constraint is a special case of
the low PAR constraint, optimal sequences of low PAR are also considered.
Numerical examples are provided to show the performance of the proposed
training sequences, with the efficiency of the derived algorithms demonstrated
