# Technical note for the FS-MMSE-IC receiver

Titouan Petitpied, R Tajan, G. Ferré, Pascal Chevalier, S Traverso

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Titouan Petitpied, R Tajan, G. Ferré, Pascal Chevalier, S Traverso. Technical note for the FS-MMSE-IC receiver. 2019. hal-02063288

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Distributed under a Creative Commons Attribution 4.0 International License
# Technical note for the FS-MMSE-IC receiver

T. Petitpied$^{T}$†, R. Tajan$^{T}$, G. Ferré$^{T}$, P. Chevalier†‡, S. Traverso†

$^{T}$IMS, Univ. Bordeaux, Bordeaux INP, CNRS (UMR 5218), F-33405 Talence (FRANCE)

† THALES, 4 avenue des Louvresses, 92230 Gennevilliers (FRANCE)

‡ CEDRIC laboratory – CONSERVATOIRE NATIONAL DES ARTS ET MÉTIERS
292 rue Saint-Martin, 75141 Paris Cedex 3 (FRANCE)

Email: forename.name@{$^{T}$ims-bordeaux.fr, †thalesgroup.com, ‡cnam.fr}

February 2019

## 1 Definitions

For any scalars or vectors $x$ and $y$ with convenient sizes, we define:

$$ egin{cases} x_n = e_n^\dagger x \\ \mathbf{R}_{xy} = \mathbb{E}_{x,y}[xy^\dagger] \\ \mathbf{R}_x = \mathbf{R}_{xx} \end{cases} $$

where $e_n$ is the $(n+1)^{th}$ element of the canonic base. The operator $\mathbb{E}_{x,y}[xy^\dagger]$ represents the statistical average with respect to $x$ and $y$.

For any matrix $A$, we define $A^*$ the conjugate matrix of $A$, $A^T$ its transpose matrix and $A^\dagger = (A^T)^*$.

## 2 Context & Problem formulation

We suppose having the following expressions:

$$ egin{cases} \mathbf{r} = \mathbf{H}\mathbf{U}\mathbf{s} + \mathbf{w} \\ \mathcal{I}(\mathbf{f}, p) = \mathbf{f}^\dagger\mathbf{r} + p \\ \mathcal{J}(\mathbf{f}, p) = \mathbb{E}_{\mathcal{I}(\mathbf{f}, p)} |\mathcal{I}(\mathbf{f}, p) - s_n|^2 \end{cases} $$

where:

* $\mathbf{f}$: a $(2N \times 1)$ vector,

* $p$: a scalar,

* $\mathbf{H}$: a $(2N \times 2N_s)$ deterministic convolution matrix,

* $\mathbf{U}$: a $(2N_s \times N_s)$ up-sampling matrix,

* $\mathbf{D} = \mathbf{U}^\dagger$: a down-sampling matrix,

* $\mathbf{s}$: a $(N_s \times 1)$ probabilistic vector such that $\mathbb{E}[\mathbf{s}] = \mathbf{s}^d$ and $\mathbf{R}_{s_n, s_m} = (v_n^d + |s_n^d|^2)\delta(n-m) \implies \mathbb{E}[(\mathbf{s} - \mathbf{s}^d)(\mathbf{s} - \mathbf{s}^d)^\dagger] = \mathbf{V}^d = \text{diag}(\mathbf{v}^d)$,

* $\mathbf{w}$: a $(2N \times 1)$ probabilistic vector such that $\mathbb{E}[\mathbf{w}] = \mathbf{0}_N$ and $\mathbf{R}_{\mathbf{w}} = \sigma_w^2 \mathbf{I}_N$,
Typically, the vector **r** represents a band-limited received signal after the convolution operation of the shaping filter and the channel **H**, corrupted by a centered circular white noise **w** with average power $\sigma_w^2$. Moreover, this holds if and only if the shaping filter bandwidth is bounded by $\frac{2}{\tau}$. Hence, **r** represents a Fractionally-Spaced (FS) system model where $\mathcal{I}(\mathbf{f}, p)$ is the estimator of **s** with Mean Square Error (MSE) $\mathcal{J}(\mathbf{f}, p)$. Furthermore, considering the prior information ($\mathbf{s}^d, \mathbf{V}^d$) on **s** leads to an Interference-Cancellation (IC) structure.

**Problem formulation:** We search for $\tilde{s}_n^e = \mathcal{I}(\mathbf{f}_n, p_n)$, $\tilde{v}_n^e = \mathcal{J}(\mathbf{f}_n, p_n)$ such that:

$$ (\mathbf{f}_n, p_n) = \underset{(\mathbf{f}, p)}{\operatorname{argmin}} \mathcal{J}(\mathbf{f}, p) $$

Consequently, $\tilde{v}_n^e$ represents the Minimum MSE of the considered system model (FS-MMSE-IC) and with estimator $\tilde{s}_n^e$ of $s_n$.

# 3 Problem resolution

## 3.1 Expression of the MSE minimizers

$$ \begin{aligned} \mathcal{J}(\mathbf{f}, p) &= \mathbb{E}_{\mathcal{I}(\mathbf{f}, p)} |\mathcal{I}(\mathbf{f}, p) - s_n|^2 \\ &= \mathbf{f}^\dagger \mathbf{R}_r \mathbf{f} + \mathbf{f}^\dagger \mathbb{E}[\mathbf{r}] p^* - \mathbf{f}^\dagger \mathbf{R}_{rs_n} + p \mathbb{E}[\mathbf{r}^\dagger] \mathbf{f} + |p|^2 - p \mathbb{E}[s_n^*] - \mathbf{R}_{s_n} \mathbf{r} \mathbf{f} - \mathbb{E}[s_n] p^* + \mathbf{R}_{s_n} \end{aligned} $$

Minimizing $\mathcal{J}(\mathbf{f}, p)$ leads us to nullify its the first partial derivatives.

$$ \begin{aligned} \frac{\partial \mathcal{J}(\mathbf{f}, p)}{\partial \mathbf{f}} &= 0 \iff 2\mathbf{R}_r \mathbf{f}_n + 2\mathbb{E}[\mathbf{r}] p_n^* - 2\mathbf{R}_{rs_n} = 0 \\ \iff \mathbf{f}_n &= \mathbf{R}_r^{-1} (\mathbf{R}_{rs_n} - \mathbb{E}[\mathbf{r}] p_n^*) \end{aligned} $$

$$ \begin{aligned} \frac{\partial \mathcal{J}(\mathbf{f}, p)}{\partial p^*} &= 0 \iff 2\mathbb{E}[\mathbf{r}^\dagger] \mathbf{f}_n + 2p_n^* - 2\mathbb{E}[s_n^*] = 0 \\ \iff p_n &= \mathbb{E}[s_n] - \mathbf{f}_n^\dagger \mathbb{E}[\mathbf{r}] \end{aligned} $$

We check that such an extremum corresponds to a minimum :

$$ \left\{ \begin{aligned} \frac{\partial^2 \mathcal{J}(\mathbf{f}, p)}{\partial \mathbf{f}^2} &= 2\mathbf{R}_r \ge 0 \\ \frac{\partial^2 \mathcal{J}(\mathbf{f}, p)}{\partial p^* \partial p^*} &= 2 \ge 0 \end{aligned} \right. $$

## 3.2 Computation of the MSE minimizers

Let us first compute :

$$ \mathbb{E}[s_n] = s_n^d $$

$$ \mathbb{E}[\mathbf{r}] = \mathbf{H}\mathbf{U}\mathbf{s}^d $$

$$ \mathbf{R}_r = \mathbf{H}\mathbf{U}\mathbf{R}_s\mathbf{D}\mathbf{H}^\dagger + \sigma_w^2 \mathbf{I}_N $$

$$ \mathbf{R}_{rs_n} = \mathbf{H}\mathbf{U}\mathbf{R}_{ss_n} $$
Then, the MSE minimizers can be computed:

$$
\begin{align*}
\mathbf{f}_n^\dagger &= (\mathbf{R}_{s_n\mathbf{r}} - p_n\mathbb{E}[\mathbf{r}^\dagger])\mathbf{R}_\mathbf{r}^{-1\dagger} \\
&= (\mathbf{R}_{s_n\mathbf{r}} - \mathbb{E}[s_n]\mathbb{E}[\mathbf{r}^\dagger] + \mathbf{f}_n^\dagger|\mathbb{E}[\mathbf{r}]|^2)\mathbf{R}_\mathbf{r}^{-1\dagger} \\
&= (\mathbf{R}_{s_n\mathbf{r}} - \mathbb{E}[s_n]\mathbb{E}[\mathbf{r}^\dagger])\mathbf{R}_\mathbf{r}^{-1\dagger}(\mathbf{I}_N - |\mathbb{E}[\mathbf{r}]|^2\mathbf{R}_\mathbf{r}^{-1\dagger})^{-1} \\
&= (\mathbf{R}_{s_n\mathbf{r}} - \mathbb{E}[s_n]\mathbb{E}[\mathbf{r}^\dagger]) (\mathbf{R}_\mathbf{r}^{-1\dagger} - |\mathbb{E}[\mathbf{r}]|^2)^{-1} \\
&= (\mathbf{R}_{s_n\mathbf{s}} - \mathbb{E}[s_n]\mathbb{E}[\mathbf{s}^\dagger])\mathbf{D}\mathbf{H}^\dagger (\mathbf{H}\mathbf{U}(\mathbf{R}_\mathbf{s} - \mathbb{E}[\mathbf{s}]^2)\mathbf{D}\mathbf{H}^\dagger + \sigma_w^2\mathbf{I}_{2N})^{-1} \\
&= v_n^d\mathbf{e}_n^\dagger\mathbf{D}\mathbf{H}^\dagger (\mathbf{H}\mathbf{U}\mathbf{V}^d\mathbf{D}\mathbf{H}^\dagger + \sigma_w^2\mathbf{I}_{2N})^{-1}
\end{align*}
$$

Using twice the Woodbury identity on any $(K_1 \times K_2)$ matrix $A$ and $(K_2 \times K_1)$ matrix $B$:

$$
\begin{align*}
\mathbf{A}(\mathbf{B}\mathbf{A} + \mathbf{I}_{K_2})^{-1} &= \mathbf{A}(\mathbf{B}\mathbf{I}_{K_1}\mathbf{A} + \mathbf{I}_{K_2})^{-1} \\
&= \mathbf{A}(\mathbf{I}_{K_2} - \mathbf{B}(\mathbf{I}_{K_1} + \mathbf{A}\mathbf{B})^{-1}\mathbf{A}) \\
&= (\mathbf{I}_{K_1} - \mathbf{A}\mathbf{B}(\mathbf{I}_{K_1} + \mathbf{A}\mathbf{B})^{-1})\mathbf{A} \\
&= (\mathbf{A}\mathbf{B}\mathbf{I}_{K_1} + \mathbf{I}_{K_1})^{-1}\mathbf{A} \\
&= (\mathbf{A}\mathbf{B} + \mathbf{I}_{K_1})^{-1}\mathbf{A}
\end{align*}
$$

We identify $\begin{cases} \mathbf{A} = \sigma_w^{-2}\mathbf{D}\mathbf{H}^\dagger \\ \mathbf{B} = \mathbf{H}\mathbf{U}\mathbf{V}^d \end{cases}$ leading to:

$$
\begin{align*}
\mathbf{f}_n^\dagger &= v_n^d \mathbf{e}_n^\dagger \mathbf{D} \mathbf{H}^\dagger (\mathbf{H} \mathbf{U} \mathbf{V}^d \mathbf{D} \mathbf{H}^\dagger + \sigma_w^2 \mathbf{I}_{2N})^{-1} \\
&= v_n^d \mathbf{e}_n^\dagger (\mathbf{D} \mathbf{H}^\dagger \mathbf{H} \mathbf{U} \mathbf{V}^d + \sigma_w^2 \mathbf{I}_N)^{-1} \mathbf{D} \mathbf{H}^\dagger \\
&= v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{D} \mathbf{H}^\dagger
\end{align*}
$$

where $\mathbf{\Sigma} = \mathbf{G}\mathbf{V}^d + \sigma_w^2\mathbf{I}_N$ is the MMSE equalization matrix and $\mathbf{G} = \mathbf{D}\mathbf{H}^\dagger\mathbf{H}\mathbf{U}$ is also called *global filter* and gathers the up-sampling, channel and matched filtering and down-sampling operations.

$$
\begin{align*}
p_n &= \mathbb{E}[s_n] - \mathbf{f}_n^\dagger \mathbb{E}[\mathbf{r}] \\
    &= s_n^d - v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} s_n
\end{align*}
$$

### 3.3 Computation of the FS-MMSE-IC and its estimator

The FS-MMSE-IC estimator is given by:

$$
\begin{align*}
\tilde{s}_n^e &= \mathbf{f}_n^\dagger \mathbf{r} + p_n \\
&= s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{D}\mathbf{H}^\dagger \mathbf{r} - \mathbf{G}\mathbf{s}^d) \\
&= s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d)
\end{align*}
$$

where $\mathbf{y} = \mathbf{D}\mathbf{H}^\dagger\mathbf{r}$ represents the received signal after matched filtering and down-sampling.
The FS-MMSE-IC $\tilde{v}_n^e$ of the estimator $\tilde{s}_n^e$ can be computed by first deriving:

$$
\begin{align*}
\mathbf{R}_{\tilde{s}_n^e} &= |s_n^d|^2 + 2(s_n^d)^* v_n^d \mathbf{e}_n^\dagger \operatorname{Re}(\Sigma^{-1} \mathbb{E}_y[(\mathbf{y} - \mathbf{G}\mathbf{s}^d)]) + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} \mathbb{E}_y[(\mathbf{y} - \mathbf{G}\mathbf{s}^d)(\mathbf{y} - \mathbf{G}\mathbf{s}^d)^\dagger] \Sigma^{-1\dagger} \mathbf{e}_n \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} \mathbb{E}_{s,w} |\mathbf{G}(\mathbf{s} - \mathbf{s}^d) + \mathbf{D}\mathbf{H}^\dagger\mathbf{w}|^2 \Sigma^{-1\dagger} \mathbf{e}_n \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} \mathbf{D}\mathbf{H}^\dagger \mathbb{E}_{s,w} |\mathbf{H}\mathbf{U}(\mathbf{s} - \mathbf{s}^d) + \mathbf{w}|^2 \Sigma^{-1\dagger} \mathbf{e}_n \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} \mathbf{D}\mathbf{H}^\dagger (\mathbf{H}\mathbf{U}\mathbf{V}^d \mathbf{D}\mathbf{H}^\dagger + \sigma_w^2 \mathbf{I}_{2N}) \mathbf{H}\mathbf{U}\Sigma^{-1\dagger} \mathbf{e}_n \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} (\mathbf{D}\mathbf{H}^\dagger\mathbf{H}\mathbf{U}\mathbf{V}^d + \sigma_w^2 \mathbf{I}_N) \mathbf{D}\mathbf{H}^\dagger\mathbf{H}\mathbf{U}\Sigma^{-1\dagger} \mathbf{e}_n \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \Sigma^{-1} \mathbf{G}\mathbf{e}_n
\end{align*}
$$

$$
\begin{align*}
\mathbf{R}_{\tilde{s}_n^e, s_n} &= \mathbb{E}_y[(s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1}(\mathbf{y} - \mathbf{G}\mathbf{s}^d)) s_n^*] \\
&= |s_n^d|^2 + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbb{E}_s[(\mathbf{s} - \mathbf{s}^d) s_n^*] \\
&= |s_n^d|^2 + (v_n^d)^2 \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbf{e}_n \\
&= \mathbf{R}_{\tilde{s}_n^e}
\end{align*}
$$

$$
\mathbf{R}_{s_n} = |s_n^d|^2 + v_n^d
$$

This leads to the FS-MMSE-IC $\tilde{v}_n^e$ expression:

$$
\begin{align*}
\tilde{v}_n^e &= \mathbb{E}_{\tilde{s}_n^e} |\tilde{s}_n^e - s_n|^2 \\
&= \mathbf{R}_{\tilde{s}_n^e} - \mathbf{R}_{\tilde{s}_n^e, s_n} - \mathbf{R}_{s_n, \tilde{s}_n^e} + \mathbf{R}_{s_n} \\
&= \mathbf{R}_{s_n} - \mathbf{R}_{s_n, \tilde{s}_n^e} \\
&= |s_n^d|^2 + v_n^d - |s_n^d|^2 - (v_n^d)^2 \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbf{e}_n \\
&= v_n^d (1 - v_n^d c_n)
\end{align*}
$$

where $c_n = \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbf{e}_n$.

3.4 Unbiased FS-MMSE-IC and its estimator

We can note that the estimator $\tilde{s}_n^e$ of $s_n$ has a residual bias :

$$
\begin{align*}
\mathbb{E}[s_n^e | \mathbb{E}[s_n] = \mu] &= \mathbb{E}_{s \sim \{n\}, w} [s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d) | \mathbb{E}[s_n] = \mu] \\
&= s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbb{E}_{s \sim \{n\}} [\mathbf{s} - \mathbf{s}^d | \mathbb{E}[s_n] = \mu] \\
&= s_n^d + (\mu - s_n^d) v_n^d c_n \\
&= v_n^d c_n \mu + s_n^d (1 - v_n^d c_n)
\end{align*}
$$

where the vector $\mathbf{s}_{\sim\{n\}}$ represents all the components of $\mathbf{s}$ but $s_n$.

Removing the additive term $s_n^d(1 - v_n^d c_n)$ and dividing the multiplicative term $v_n^d c_n$ leads to:

$$
\begin{align*}
s_n^e &= (v_n^d c_n)^{-1} (\tilde{s}_n^e - s_n^d (1 - v_n^d c_n)) \\
&= s_n^d + c_n^{-1} \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d)
\end{align*}
$$
We compute the new statistical matrices:

$$ 
\begin{aligned}
\mathbf{R}_{s_n^e} &= |s_n^d|^2 + c_n^{-2} \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbb{E}_y [(\mathbf{y} - \mathbf{G}\mathbf{s}^d)(\mathbf{y} - \mathbf{G}\mathbf{s}^d)^\dagger] \mathbf{\Sigma}^{-1} \mathbf{e}_n \\
&= |s_n^d|^2 + c_n^{-1}
\end{aligned}
$$

$$ 
\begin{aligned}
\mathbf{R}_{s_n^e, s_n} &= \mathbb{E}_y [(s_n^d + c_n^{-1} \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d)) s_n^*] \\
&= |s_n^d|^2 + c_n^{-1} \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbb{E}_s [(\mathbf{s} - \mathbf{s}^d) s_n^*] \\
&= |s_n^d|^2 + v_n^d
\end{aligned}
$$

$$ \mathbf{R}_{s_n} = |s_n^d|^2 + v_n^d $$

And finally we find the unbiased FS-MMSE-IC:

$$ 
\begin{aligned}
v_n^e &= \mathbb{E}_{s_n^e} |s_n^e - s_n|^2 \\
&= \mathbf{R}_{s_n^e} - \mathbf{R}_{s_n^e, s_n} - \mathbf{R}_{s_n, s_n^e} + \mathbf{R}_{s_n} \\
&= |s_n^d|^2 + c_n^{-1} - |s_n^d|^2 - v_n^d - |s_n^d|^2 - v_n^d + |s_n^d|^2 + v_n^d \\
&= c_n^{-1} - v_n^d
\end{aligned}
$$

# 4 Conclusion

The biased FS-MMSE-IC receiver is given by:

$$ 
\begin{cases} 
\tilde{s}_n^e = s_n^d + v_n^d \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d) \\ 
\tilde{v}_n^e = v_n^d (1 - v_n^d c_n) 
\end{cases}
$$

where $c_n = \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} \mathbf{G} \mathbf{e}_n$ is a scalar, $\mathbf{y} = \mathbf{D}\mathbf{H}^\dagger \mathbf{r}$ is the receive signal after matched filtering and down-sampling, $\mathbf{G} = \mathbf{D}\mathbf{H}^\dagger \mathbf{H}\mathbf{U}$ is the global effect including up-sampling, channel and matched filtering and down-sampling, $\mathbf{\Sigma} = \mathbf{G}\mathbf{V}^d + \sigma_w^2 \mathbf{I}_N$ is the MMSE equalization matrix.

Moreover, the unbiased FS-MMSE-IC receiver is given by:

$$ 
\begin{cases} 
s_n^e = s_n^d + c_n^{-1} \mathbf{e}_n^\dagger \mathbf{\Sigma}^{-1} (\mathbf{y} - \mathbf{G}\mathbf{s}^d) \\ 
v_n^e = c_n^{-1} - v_n^d 
\end{cases}
$$

Consequently, the FS-MMSE-IC receiver leads to a FS matched filter followed by a symbol time equalizer.