  In this paper, we complement Verd\'{u}'s work on spectral efficiency in the
wideband regime by investigating the fundamental tradeoff between rate and
bandwidth when a constraint is imposed on the error exponent. Specifically, we
consider both AWGN and Rayleigh-fading channels. For the AWGN channel model,
the optimal values of $R_z(0)$ and $\dot{R_z}(0)$ are calculated, where
$R_z(1/B)$ is the maximum rate at which information can be transmitted over a
channel with bandwidth $B/2$ when the error-exponent is constrained to be
greater than or equal to $z.$ Based on this calculation, we say that a sequence
of input distributions is near optimal if both $R_z(0)$ and $\dot{R_z}(0)$ are
achieved. We show that QPSK, a widely-used signaling scheme, is near-optimal
within a large class of input distributions for the AWGN channel. Similar
results are also established for a fading channel where full CSI is available
at the receiver.
