This sounds like a blind channel estimation problem. Blind channel estimation is used such as in emerging massive MIMO systems where pilot contamination can otherwise limit the advantage of adding additional transmitters.

A very simple example of blind channel estimation is decision directed least squares using the least squares technique that I describe at this post How determine the delay in my signal practically , with an estimate of the transmit signal based on hard decisions at the receiver. This technique works well in higher SNR conditions when the uncorrected error rate is still reasonably low (actual numbers would depend on the actual conditions but I would guess that channels for error rates on the order of $10^{-2}$ to $10^{-3}$ could still be determined and with that those error rates significantly improved based on decisions alone for the estimated tx signal).

For much more details on blind channel estimation, see this paper and linked references by Xiaotian Li and others that describes statistical methods such as the Signal Subspace Method which is widely used in MIMO and OFDM. This is a good choice when there are a large number of received symbols but the author goes into other deterministic methods based on the least squares methods such as I have linked that would be more appropriate for a smaller number of samples such as with to the OP’s question. The link to the paper is here:

 https://www.hindawi.com/journals/jam/2014/146207/

This sounds like a blind channel estimation problem. Blind channel estimation is used such as in emerging massive MIMO systems where pilot contamination can otherwise limit the advantage of adding additional transmitters.

A very simple example of blind channel estimation is decision directed least squares using the least squares technique that I describe at this post How determine the delay in my signal practically , with an estimate of the transmit signal based on hard decisions at the receiver. This technique works well in higher SNR conditions when the uncorrected error rate is still reasonably low (actual numbers would depend on the actual conditions but I would guess that channels for error rates on the order of $10^{-2}$ to $10^{-3}$ could still be determined and with that those error rates significantly improved based on decisions alone for the estimated tx signal).

For much more details on blind channel estimation, see this paper and linked references by Xiaotian Li and others that describes statistical methods such as the Signal Subspace Method which is widely used in MIMO and OFDM. This is a good choice when there are a large number of received symbols but the author goes into other deterministic methods based on the least squares methods such as I have linked that would be more appropriate for a smaller number of samples such as with to the OP’s question. The paper is:  Xiaotian Li - Blind Channel Estimation Based on Multilevel Lloyd-Max Iteration for Nonconstant Modulus Constellations.

This sounds like a blind channel estimation problem. Blind channel estimation is used such as in emerging massive MIMO systems where pilot contamination can otherwise limit the advantage of adding additional transmitters.

A very simple example of blind channel estimation is decision directed least squares using the least squares technique that I describe at this post How determine the delay in my signal practically , with an estimate of the transmit signal based on hard decisions at the receiver. This technique works well in higher SNR conditions when the uncorrected error rate is still reasonably low (actual numbers would depend on the actual conditions but I would guess that channels for error rates on the order of $10^{-2}$ to $10^{-3}$ could still be determined and with that those error rates significantly improved based on decisions alone for the estimated tx signal.

For much more details on blind channel estimation, see this paper and linked references by Xiaotian Li and others that describes statistical methods such as the Signal Subspace Method which is widely used in MIMO and OFDM. This is a good choice when there are a large number of received symbols but the author goes into other deterministic methods based on the least squares methods such as I have linked that would be more appropriate for a smaller number of samples such as with to the OP’s question. The link to the paper is here:

https://www.hindawi.com/journals/jam/2014/146207/

This sounds like a blind channel estimation problem. Blind channel estimation is used such as in emerging massive MIMO systems where pilot contamination can otherwise limit the advantage of adding additional transmitters.

A very simple example of blind channel estimation is decision directed least squares using the least squares technique that I describe at this post How determine the delay in my signal practically , with an estimate of the transmit signal based on hard decisions at the receiver. This technique works well in higher SNR conditions when the uncorrected error rate is still reasonably low (actual numbers would depend on the actual conditions but I would guess that channels for error rates on the order of $10^{-2}$ to $10^{-3}$ could still be determined and with that those error rates significantly improved based on decisions alone for the estimated tx signal).

For much more details on blind channel estimation, see this paper and linked references by Xiaotian Li and others that describes statistical methods such as the Signal Subspace Method which is widely used in MIMO and OFDM. This is a good choice when there are a large number of received symbols but the author goes into other deterministic methods based on the least squares methods such as I have linked that would be more appropriate for a smaller number of samples such as with to the OP’s question. The link to the paper is here:

https://www.hindawi.com/journals/jam/2014/146207/