# Achievable rate for MIMO channel with estimated channel matrix

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).

• 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. – Marcus Müller Nov 20 '19 at 13:33
• @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 – Engineer Nov 20 '19 at 13:46
• @MarcusMüller Your estimate of the matrix is pretty good though :) – Engineer Nov 20 '19 at 13:47

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).