1. System model
Wireless system model:
\[\mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{n}, \qquad \mathbf{H} \in \mathbb{C}^{N_r \times N_t}, \quad \mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})\]The MMSE solution is
\[\hat{\mathbf{x}} = \mathbf{W} \mathbf{y}, \qquad \mathbf{W} = (\mathbf{H}^H \mathbf{H} + \sigma^2 \mathbf{I})^{-1} \mathbf{H}^H = \mathbf{A}^{-1} \mathbf{H}^H\]where \(\mathbf{A} = \mathbf{H}^H \mathbf{H} + \sigma^2 \mathbf{I}\).
2. QR-based MMSE with Extended Channel Matrix
The extended channel and received signal are
\[\bar{\mathbf{H}} = \begin{bmatrix} \mathbf{H} \\ \sigma \mathbf{I}_{N_t} \end{bmatrix}, \qquad \bar{\mathbf{y}} = \begin{bmatrix} \mathbf{y} \\ \mathbf{0} \end{bmatrix}\]The QR decomposition of \(\bar{\mathbf{H}}\) is
\[\bar{\mathbf{H}} = \mathbf{Q} \mathbf{R}, \qquad \mathbf{Q} = \begin{bmatrix} \mathbf{Q}_1 \\ \mathbf{Q}_2 \end{bmatrix}\]where \(\mathbf{Q}_1 \in \mathbb{C}^{N_r \times N_t}\) is the signal (channel) block, \(\mathbf{Q}_2 \in \mathbb{C}^{N_t \times N_t}\) the noise (regularization) block, and \(\mathbf{R} \in \mathbb{C}^{N_t \times N_t}\) is upper triangular.
Therefore, the inverse of \(\mathbf{R}\) comes for free from the noise block.
\[\mathbf{H} = \mathbf{Q}_1 \mathbf{R}\] \[\sigma \mathbf{I} = \mathbf{Q}_2 \mathbf{R} \rightarrow \mathbf{R}^{-1} = \frac{1}{\sigma}\,\mathbf{Q}_2\]The MMSE weight matrix is built from the two blocks \(\mathbf{Q}_1\) and \(\mathbf{Q}_2\) by
\[\mathbf{W} = \mathbf{A}^{-1} \mathbf{H}^H = \mathbf{R}^{-1} \mathbf{R}^{-H} \mathbf{R}^H \mathbf{Q}_1^H = \mathbf{R}^{-1} \mathbf{Q}_1^H = \frac{1}{\sigma} \mathbf{Q}_2 \mathbf{Q}_1^H\]2.1 Bias
The MMSE estimate is biased, which causes the estimated value to be smaller than the true value.
Bias from the extended QR.
\[\mathbf{y}_r = \mathbf{Q}_1^H \mathbf{y} = \mathbf{Q}_1^H \mathbf{H} \mathbf{x} + \mathbf{Q}_1^H \mathbf{n} = \mathbf{Q}_1^H \mathbf{Q}_1 \mathbf{R} \mathbf{x} + \mathbf{Q}_1^H \mathbf{n} = (\mathbf{I} - \mathbf{Q}_2^H \mathbf{Q}_2) \mathbf{R} \mathbf{x} + \mathbf{Q}_1^H \mathbf{n}\]Back-substitution \(\hat{\mathbf{x}} = \mathbf{R}^{-1} \mathbf{y}_r\) treats \(\mathbf{R}\) as the channel, whereas the signal is in fact multiplied by \((\mathbf{I} - \mathbf{Q}_2^H \mathbf{Q}_2) \mathbf{R}\).
Bias from the weight matrix. The weight matrix obeys
\[\mathbf{W} \mathbf{H} = \mathbf{A}^{-1} \mathbf{H}^H \mathbf{H} = \mathbf{A}^{-1} (\mathbf{A} - \sigma^2 \mathbf{I}) = \mathbf{I} - \sigma^2 \mathbf{A}^{-1}\]so the estimate splits into a signal part and a noise part,
\[\hat{\mathbf{x}} = \mathbf{W} \mathbf{y} = (\mathbf{I} - \sigma^2 \mathbf{A}^{-1}) \mathbf{x} + \mathbf{W} \mathbf{n}\]The diagonal elements of \((\mathbf{I} - \sigma^2 \mathbf{A}^{-1})\) are all strictly less than 1, since \(\sigma^2 \mathbf{A}^{-1}\) is positive definite.
The two biases are the same:
\[\mathbf{R}^{-1} (\mathbf{I} - \mathbf{Q}_2^H \mathbf{Q}_2) \mathbf{R} = \mathbf{I} - \mathbf{R}^{-1} \mathbf{Q}_2^H \mathbf{Q}_2 \mathbf{R} = \mathbf{I} - \sigma^2 \mathbf{R}^{-1} \mathbf{R}^{-H} = \mathbf{I} - \sigma^2 \mathbf{A}^{-1} = \mathbf{W} \mathbf{H}\]3. QR-based MMSE with \(\mathbf{H}\)
The QR decomposition is
\[\mathbf{H} = \mathbf{Q} \mathbf{R}, \qquad \mathbf{Q}^H \mathbf{Q} = \mathbf{I}_{N_t}\]where \(\mathbf{Q} \in \mathbb{C}^{N_r \times N_t}\) and upper triangular \(\mathbf{R} \in \mathbb{C}^{N_t \times N_t}\).
The estimated signal is
\[\hat{\mathbf{x}} = (\mathbf{R}^H \mathbf{R} + \sigma^2 \mathbf{I})^{-1} \mathbf{R}^H \mathbf{y}_r\]where \(\mathbf{y}_r = \mathbf{Q}^H \mathbf{y}\) is the rotated received signal.
This approach is the basis of nonlinear receivers, but it has no advantages for MMSE.
4. Signal-to-Noise Ratio (SNR)
\[\mathrm{SNR}_k = \frac{1}{\sigma^2 (\mathbf{A}^{-1})_{kk}} - 1\]With the extended QR decomposition, we have
\[(\mathbf{A}^{-1})_{kk} = \lVert (\mathbf{R}^{-1})_{k,:} \rVert^2\] \[(\mathbf{A}^{-1})_{kk} = \frac{1}{\sigma^2} \lVert (\mathbf{Q}_2)_{k,:} \rVert^2\]Therefore,
\[\mathrm{SNR}_k = \frac{1}{\lVert (\mathbf{Q}_2)_{k,:} \rVert^2} - 1\]