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In this work, a discrete-time stationary Rayleigh flat-fading channel with unknown channel state information at transmitter and receiver side is studied. The law of the channel is presumed to be known to the receiver. For independent identically distributed (i.i.d.) zero-mean proper Gaussian input distributions, the achievable rate is investigated. The main contribution of this paper is the derivation of two new upper bounds on the achievable rate with Gaussian input symbols. One of these bounds is based on the one-step channel prediction error variance but is not restricted to peak power constrained input symbols like known bounds. Moreover, it is shown that Gaussian inputs yield the same pre-log as the peak power constrained capacity. The derived bounds are compared with a known lower bound on the capacity given by Deng and Haimovich and with bounds on the peak power constrained capacity given by Sethuraman et al.. Finally, the achievable rate with i.i.d. Gaussian input symbols is compared to the achievable rate using a coherent detection in combination with a solely pilot-based channel estimation.