  The diversity multiplexing tradeoff of a general two-hop asynchronous
cooperative network is examined for various relaying protocols such as
non-orthogonal selection decode-and-forward (NSDF), orthogonal selection
decode-and-forward (OSDF), non-orthogonal amplify-and-forward (NAF), and
orthogonal amplify-and-forward (OAF). The transmitter nodes are assumed to send
pulse amplitude modulation (PAM) signals asynchronously, in which information
symbols are linearly modulated by a shaping waveform to be sent to the
destination. We consider two different cases with respect to the length of the
shaping waveforms in the time domain. In the theoretical case where the shaping
waveforms with infinite time support are used, it is shown that asynchronism
does not affect the DMT performance of the system and the same DMT as that of
the corresponding synchronous network is obtained for all the aforementioned
protocols. In the practical case where finite length shaping waveforms are
used, it is shown that better diversity gains can be achieved at the expense of
bandwidth expansion. In the decode-and-forward (DF) type protocols, the
asynchronous network provides better diversity gains than those of the
corresponding synchronous network throughout the range of the multiplexing
gain. In the amplify-and-forward (AF) type protocols, the asynchronous network
provides the same DMT as that of the corresponding synchronous counterpart
under the OAF protocol; however, a better diversity gain is achieved under the
NAF protocol throughout the range of the multiplexing gain. In particular, in
the single relay asynchronous network, the NAF protocol provides the same DMT
as that of the 2 {\times} 1 multiple-input single-output (MISO) channel.
