MMSE-based Channel Estimation

Regarding channel estimation, minimum mean square error (MMSE) is a popular algorithm widely utilized in a variety of applications. This short document is just written to reveal the rationale between the algorithm.

Without loss of generality, we take an OFDM system for instance. The received signals on pilot, a.k.a. reference signal (RS) resource elements (RE) can be written as \(\mathbf{y}_p = \mathbf{X}_p \mathbf{h}_{p} + \mathbf{n}_p\), where

• \(\mathbf{X}_p \in \mathbb{C}^{P \times P}\) is a diagonal matrix with pilot symbols as its diagonal elements, \(P \in \mathbb{Z}^+\) the number of pilot in the pattern.
• \(\mathbf{h}_p \in \mathbb{C}^P\) is the channel frequency response (CFR) on the pilot REs.
• \(\mathbf{n}_p \in \mathbb{C}^P \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I}_P)\) is the additive Gaussian white noise.

The CFR of the target RE can be denoted by \(\tilde{\mathbf{h}} = \mathbf{G} \mathbf{y}_p\), where \(\mathbf{h}\) is the target CFR and \(\mathbf{G}\) is the MMSE channel estimator.

Then, the mean square error (MSE) can be obtained as

\begin{align} e &= \mathrm{E}\left[\|\tilde{\mathbf{h}} - \mathbf{h}\|_2^2\right] \nonumber \\ &= \mathrm{E}\left[\|\mathbf{G}\mathbf{y}_p - \mathbf{h}\|_2^2\right] \label{eq:mse} \end{align}

Just as its name indicates, MMSE algorithm aims to minimize the MSE. To this end, we let \(\cfrac{\partial e}{\partial \mathbf{G}} = \mathrm{E}\left[(\mathbf{G}\mathbf{y}_p - \mathbf{h}) \mathbf{y}_p^H\right] = \mathbf{0}\). Then, we can get the MMSE channel estimator

\begin{align} \mathbf{G} &= \mathrm{E}\left[\mathbf{h}\mathbf{y}_p^H\right] \cdot \mathrm{E}^{-1}\left[\mathbf{y}_p\mathbf{y}_p^H\right] \nonumber \\ &= \mathrm{E}\left[\mathbf{h}\mathbf{h}_p^H \right] \cdot \left(\mathrm{E}\left[\mathbf{h}_p\mathbf{h}_p^H \right] + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \cdot \mathbf{X}_p^{-1} \nonumber \\ &= \mathbf{R}_{h,p} \left(\mathbf{R}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \mathbf{X}_p^{-1}, \label{eq:mmse} \end{align}

where

• \(\gamma\) is the SINR on pilot REs.
• \(\mathbf{R}_{h,p} \triangleq \mathrm{E}\left[\mathbf{h}\mathbf{h}_p^H \right]\) is the cross-correlation between the CFR on target REs and pilot REs.
• \(\mathbf{R}_p\triangleq \mathrm{E}\left[\mathbf{h}_p\mathbf{h}_p^H \right]\) is the auto-correlation of the CFR on pilot REs.

Clearly, either cross-correlation or auto-correlation in \eqref{eq:mmse} depends on the ideal statistics of the channel, e.g., power delay profile (PDP) for frequency-domain correlation computation, and Doppler power spectrum for time-domain correlation computation. However, the ideal statistics is not available in practice. In stead, some simple statistics are assumed in realistic systems, e.g., uniform PDP within cyclic prefix (CP) length, U-shape Doppler power spectrum. In this case, the MMSE channel estimator in \eqref{eq:mmse} becomes

\begin{align} \tilde{\mathbf{G}} = \tilde{\mathbf{R}}_{h,p} \left(\tilde{\mathbf{R}}_p + \frac{1}{\gamma}\mathbf{I}_P\right)^{-1}\mathbf{X}_p^{-1}, \label{eq:mmse_nonideal} \end{align}

where \(\tilde{\mathbf{R}}_{h,p}\) and \(\tilde{\mathbf{R}}_p\) are respectively the counterparts of \(\mathbf{R}_{h,p}\) and \(\mathbf{R}_p\) conditioned on non-ideal channel statistics.

Then, the MSE in \eqref{eq:mse} can be written as

\begin{align} e(\tilde{\mathbf{G}}) = \mathrm{Tr}\left[\mathbf{R}_h + \tilde{\mathbf{R}}_{h,p}\left(\tilde{\mathbf{R}}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \left(\mathbf{R}_p + \frac{1}{\gamma}\mathbf{I}_P \right) \left(\tilde{\mathbf{R}}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \tilde{\mathbf{R}}_{h,p}^H - \tilde{\mathbf{R}}_{h,p}\left(\tilde{\mathbf{R}}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \mathbf{R}_{h,p}^H - \mathbf{R}_{h,p}\left(\tilde{\mathbf{R}}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1} \tilde{\mathbf{R}}_{h,p}^H\right]. \label{eq:mse_nonideal} \end{align}

Particularly, for ideal channel statistics, the MSE further becomes

\begin{align*} e(\mathbf{G}) = \mathrm{Tr}\left[\mathbf{R}_h - \mathbf{R}_{h,p}\left(\mathbf{R}_p + \frac{1}{\gamma}\mathbf{I}_P \right)^{-1}\mathbf{R}_{h,p}^H\right]. \end{align*}