In a MIMO system, if the receiver has perfect CSI, i.e. knows that channel matrix $\boldsymbol{H}$, then the achievable rate is: $$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\boldsymbol{H}^H\right|$$ where $P$ is the transmit power and $N_R,N_T$ the number of receive and transmit antennas respectively. What happens if I don't know the channel, but estimate it? Will it simply be $$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\hat{\boldsymbol{H}}^H\right|$$ where $\hat{\boldsymbol{H}}$ is the channel estimate?

I'm interested specifically in LMMSE estimation (and Gaussian channel).

In a MIMO system, if the receiver has perfect CSI, i.e. knows that channel matrix $\boldsymbol{H}$, then the achievable rate is: $$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\boldsymbol{H}^H\right|\tag1$$ where $P$ is the transmit power and $N_R,N_T$ the number of receive and transmit antennas respectively. What happens if I don't know the channel, but estimate it? Will it simply be $$ R = \log \left|\boldsymbol{I}+\frac{P}{N_RN_T}\boldsymbol{H}\hat{\boldsymbol{H}}^H\right|\tag2$$ where $\hat{\boldsymbol{H}}$ is the channel estimate?

I'm interested specifically in LMMSE estimation (and Gaussian channel).