# A question of channel capacity in precoding case

I’ve a general question about precoding.

The channel capacity after precoding is $$\log_2(\det(I+F’H’HF))$$ where $$\det$$ is the determinant, $$I$$ the identity matrix, $$F$$ the precoding matrix, $$H$$ the channel matrix and $$’$$ the complex conjugate transposition.

If $$F$$ is a square unitary matrix, then

\begin{align}\log_2(\det(I+F’H’HF))&=\log_2(\det(F’(I+H’H)F)) \\ &=\log_2(\det(F’)\det(I+H’H)\det(F))) \\ &=\log_2(\det(I+H’H)), \end{align}

which is unchanged from the no precoding case. Have I made any mistake?

If I haven’t, what is the benefit of precoding in such case?

Your equation is correct, i.e. $$\log\det(I+F^*H^*HF)=\log\det(I+H^*H)$$.
In reality, uniform power allocation is generally suboptimal. And the capacity of a MIMO channel ($$\mathbf y=H\mathbf x+\mathbf n$$) can be much higher, given by
$$C=\max_{R_{\mathbf x}} \log\det(I+HR_{\mathbf x}H^*)$$ where $$R_{\mathbf x}$$ is the autocorrelation of $$\mathbf x$$, and is generally subject to a total transmit power constraint, e.g. $$\text{tr}(R_{\mathbf x})\le P$$. With EVD, $$R_{\mathbf x}=UDU^*$$, we see $$U$$ is the unitary precoding matrix and $$D$$ represents power allocation across spatial streams.
From this, it's clear that the unitary precoding matrix ($$U$$) matters, when $$D$$ is not identity (i.e. non-uniform power allocation). Different $$U$$ can give drastically different channel capacity.