  In this work, we consider a discrete-time stationary Rayleigh flat-fading
channel with unknown channel state information at transmitter and receiver. The
law of the channel is presumed to be known to the receiver. In addition, we
assume the power spectral density (PSD) of the fading process to be compactly
supported. For i.i.d. zero-mean proper Gaussian input distributions, we
investigate the achievable rate. One of the main contributions is the
derivation of two new upper bounds on the achievable rate with zero-mean proper
Gaussian input symbols. The first one holds only for the special case of a
rectangular PSD and depends on the SNR and the spread of the PSD. Together with
a lower bound on the achievable rate, which is achievable with i.i.d. zero-mean
proper Gaussian input symbols, we have found a set of bounds which is tight in
the sense that their difference is bounded. Furthermore, we show that the high
SNR slope is characterized by a pre-log of 1-2f_d, where f_d is the normalized
maximum Doppler frequency. This pre-log is equal to the high SNR pre-log of the
peak power constrained capacity. Furthermore, we derive an alternative upper
bound on the achievable rate with i.i.d. input symbols which is based on the
one-step channel prediction error variance. The novelty lies in the fact that
this bound is not restricted to peak power constrained input symbols like known
bounds, e.g. in [1]. Therefore, the derived upper bound can also be used to
evaluate the achievable rate with i.i.d. proper Gaussian input symbols. We
compare the derived bounds on the achievable rate with i.i.d. zero-mean proper
Gaussian input symbols with bounds on the peak power constrained capacity given
in [1-3]. Finally, we compare the achievable rate with i.i.d. zero-mean proper
Gaussian input symbols with the achievable rate using synchronized detection in
combination with a solely pilot based channel estimation.
