  This work studies the impact of time- synchronization in molecular timing
(MT) channels by analyzing three different modulation techniques. The first
requires transmitter-receiver synchronization and is based on modulating
information on the release timing of information particles. The other two are
asynchronous and are based on modulating information on the relative time
between two consecutive releases of information particles using
indistinguishable or distinguishable particles. All modulation schemes result
in a system that relate the transmitted and the received signals through an
additive noise, which follows a stable distribution. As the common notion of
the variance of a signal is not suitable for defining the power of stable
distributed signals (due to infinite variance), we derive an expression for the
geometric power of a large class of stable distributions, and then use this
result to characterize the geometric signal-to-noise ratio (G-SNR) for each of
the modulation techniques. In addition, for binary communication, we derive the
optimal detection rules for each modulation technique. Numerical evaluations
indicate that the bit error rate (BER) is constant for a given G-SNR, and the
performance gain obtained by using synchronized communication is significant.
Yet, it is also shown that by using two distinguishable particles per bit
instead of one, the BER of the asynchronous technique can approach that of the
synchronous one.
