A question of channel capacity in precoding case

Question

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?

Answer 1

Your equation is correct, i.e. $\log\det(I+F^*H^*HF)=\log\det(I+H^*H)$.

However, the conclusion that unitary precoding doesn't change the capacity of a MIMO channel is not true, in general. It is, however, true, when we're constrained to uniform power allocation across spatial streams, as your equation above implies.

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.