MMSE and sparse recovery methods both aim at improving the classical LS channel estimation by injecting prior information. On the one hand, the prior information is learnt over time for MMSE (it is assumed that the channel is a correlated gaussian with a covariance matrix learnt from previous channel estimates). On the other hand, the prior information is based on physics for sparse recovery methods (it is assumed that only a few paths contribute significantly to the channel so that provided the plane wave assumption holds, it can be expressed as a sparse linear combination of steering vectors).
I wonder if the two methods have already been thoroughly compared on realistic channels.
These are estimation methods that minimize the estimation error under different prior information.
MMSE estimation uses the exact channel distribution to find the estimator that minimizes the MSE.
If we don't know the exact channel distribution, but know that the channel is sparse in some domain, then compressive sensing methods can be used to find the best estimate that satisfy that condition.
Since the estimators build on different prior information, a comparison will be unfair. The MMSE estimator is better since it knows more about the channel.
Thanks for the answer.
These methods are indeed based on different prior informations.
My question was precisely about comparisons between the different kinds or prior info.
The distribution information may be better but it requires time to be estimated, and thus using MMSE requires the channel distribution to be constant over large periods of time.
On the other hand, the sparsity of the channel is a prior information based on the physics of propagation and is thus usable immediately.
In a practical context and a given environment (provided channels are indeed relatively sparse), I wonder how many channel estimations are required to form empirical covariance matrices accurate enough so that the MMSE beats sparse recovery methods. Also, I wonder if in some environment with high mobility, the MMSE would not be applicable because the channel distribution varies too fast. In that case, sparse recovery methods should always be better, no matter the numbr of channel estimations used to form empirical covariances.
This is a good point from a practical perspective. Then it isn't the MMSE estimator that you are interested in, but an approximate MMSE estimator based on sample covariance matrices. I haven't seen such a comparison.
I wrote a paper a few years ago called "Massive MIMO with Imperfect Channel Covariance Information". We compared approximate MMSE with LS, but not any sparsity methods and only for Gaussian distributed channels.
I would guess that the estimation of the sample covariance matrix also becomes easier the more sparsity exists. The most important thing is that you have a number of samples equal to the number of dominant eigendirections, so that you get a good sense of the span of the low-rank subspace where the channel will exist. Then you can project the received pilot signals onto that subspace and apply any estimation method (approximate MMSE if you can trust the eigenvalues in the subspace, otherwise something more LS-like).
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