MMSE is approximately zero-forcing at high SNR

Question

In https://stanford.edu/~dntse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter8.pdf page 360, it is stated that for a flat fading MIMO channel with receiver CSI only: at high SNR, the $\mathbf{K}_{z_k}^{-\frac{1}{2}}$ operation reduces to the projection of $\mathbf{y}$ onto the subspace orthogonal to that spanned by $\mathbf{h}_1, \ldots, \mathbf{h}_{k-1}, \mathbf{h}_{k+1}, \ldots, \mathbf{h}_{n_t}$ and the linear MMSE receiver reduces to the decorrelator.

where $\mathbf{K}_{z_k}:=N_0 \mathbf{I}_{n_r}+\sum_{i \neq k}^{n_t} P_i \mathbf{h}_i \mathbf{h}_i^*$ is the covariance of the noise and interference for the kth stream.

I am confused on how exactly the square root of the covariance is the projection onto the orthogonal subspace? This doesn't seem obvious to me.

Answer 1

Before I answer the question we need to understand the concepts involved:

1. $\mathbf{K}_{z_k}$ is the covariance matrix of the noise plus interference for the kth data stream (excluding the contribution from the kth transmitter). It includes thermal noise and the inter-stream interference that is present in the system.

2. $\mathbf{K}_{z_k}^{-\frac{1}{2}}$ is essentially a whitening operation that serves to decorrelate the received vector $\mathbf{y}$ by transforming the noise-plus-interference into a white noise with identity covariance. This is because taking the inverse square root of a covariance matrix and then applying it to a random vector will whiten that vector.

The referenced document suggests that at high SNR, the operation of $\mathbf{K}_{z_k}^{-\frac{1}{2}}$ can be interpreted as a projection onto a subspace orthogonal to the space spanned by the interference vectors. This doesn't mean that the square root of the covariance matrix is literally the projection, but rather that the effect it has on the received signal vector $\mathbf{y}$ at high SNR is equivalent to projecting $\mathbf{y}$ onto the orthogonal subspace.

This equivalence arises because at high SNR, the components of $\mathbf{K}_{z_k}$ due to noise ($N_0 \mathbf{I}_{n_r}$) become negligible compared to the components due to interference $\left(\sum_{i \neq k}^{n_t} P_i \mathbf{h}_i \mathbf{h}_i^*\right)$. Thus, the inverse square root primarily serves to undo the correlation introduced by the interference terms, which is similar to projecting onto the orthogonal complement of the interference subspace.

Essentially, at high SNR, the dominant effect is to remove the interference, and since the interference vectors are the $\mathbf{h}_i$ for $i \neq k$, the result of the whitening operation is analogous to projecting onto the space orthogonal to these vectors. The decorrelator refers to a receiver that completely removes the inter-stream interference by projecting the received signal onto the orthogonal complement of the interference subspace. Hence, in the high SNR limit, the linear MMSE receiver, which is optimal in the sense of minimizing the mean squared error, converges to the decorrelator.

In practice, however, the equivalence might not be perfect, and the actual implementation would still involve computing $\mathbf{K}_{z_k}^{-\frac{1}{2}}$ rather than an explicit projection, especially at moderate SNRs where the noise cannot be neglected.