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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).

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  • $\begingroup$ Can't be your equation $(2)$, because in that case, you'd just pick $\hat{\boldsymbol{H}}$ to maximize the $R$, not to estimate the channel as good as possible; for example $\hat{\boldsymbol H}= 10^{20}\cdot \mathbf I$ would probably be a pretty good pick. $\endgroup$
    – Marcus Müller
    Commented Nov 20, 2019 at 13:33
  • $\begingroup$ @MarcusMüller I'll let OP answer but I believe the question was if a channel matrix estimate has been calculated, is it a good approach to just plug that estimate in to equation 1 to get an achievable rate estimate? Or, is there a better way to go about estimating achievable rate $\endgroup$
    – Engineer
    Commented Nov 20, 2019 at 13:46
  • $\begingroup$ @MarcusMüller Your estimate of the matrix is pretty good though :) $\endgroup$
    – Engineer
    Commented Nov 20, 2019 at 13:47

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I found the following relation which answers my question

$$R=\log\det \left(\boldsymbol{C}_{MMSE}^{-1}\right)$$

where $\boldsymbol{C}_{MMSE}$ is the covariance matrix of the error in the data $\boldsymbol{e}=\boldsymbol{x}-\hat{\boldsymbol{x}}$

I found the answer in the paper "An Iteratively Weighted MMSE Approach to Distributed Sum-Utility Maximization for a MIMO Interfering Broadcast Channel" but it is said to be well known.

Therefore, we need to assume some method for estimation the data using the estimated channel, for example, match-filter or MMSE estimator, and then calculate its error (as a function of the real and estimated channels).

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