# Capacity and channel estimation algorithms: explanation of results

I study a spectral efficiency of a system with a precoding scheme with different channel schemes.

Algorithm A for channel estimation gives a smaller estimation error instead of Algorithm B.

I have computed the precoding schemes using the resulted estimated from Alg. A and Alg.B. As a result, the capacity with precoding for Algorithm A is better ( bigger) than for Algorithm B.

• How can I explain the results?
• Can I say that the reason is in estimation error? ( estimate of Alg A is better)
• Probably yes. But could you please make clear what you mean by "the capacity" and how you computed it? Jul 6, 2020 at 8:51
• @AlexTP I Am using the same expression as it was discussed in link, link Jul 6, 2020 at 13:58
• Be careful, channel capacity and data error are different criteria. It seems that the expression in your links is the capacity with CSI at receiver, i.e. channel is known at receiver. I would be skeptical to ask what you injected to the expression in your links to compute the capacity. Specifically, how did you use the channel estimation for the expression which assumes channel is known? Jul 6, 2020 at 15:10
• @AlexTP i have used the estimates for calculation F( precoding matrices) Jul 7, 2020 at 10:48
• @AlexTP and H is the perfect CSI Jul 7, 2020 at 10:56

The channel capacity expression you cited is the one with CSI at receiver that means the receiver know perfectly the realization of fading channel, denoted $$\mathbf{H}$$, but not the realization of additive white Gaussian noise. In that case, the channel capacity is the max (or sup) data rate at which there exists a receiver that achieves decoding error probability smaller than any infinitesimal $$\epsilon > 0$$, and can be computed using the expression in your links (with adding expected value operator depending assumptions) see chapter 8.