# Baseband Detection

## General Theory for Baseband MAP Detection
- Let $\mathcal{S}_{l}=\left\{s_{l,1},s_{l,2},\cdots,s_{l,M}\right\}\subset\mathbb{C}^{N}$ be a signal constellation with respect to the baseband orthonormal basis functions $\left\{\phi_{l,1}(t):i=1,2,\cdots,N \right\}$ (Dimension N)
- The baseband equivalent signals are
$$
s_{l,m}(t)=\sum^{N}_{i=1}s_{l,m}\phi_{l,i}(t)
$$
where $s_{l,m}=\left[s_{l,m,1} \cdots s_{l,m,N}\right]$
- The corresponding bandpass signals are
$$
s_{m}(t) = \Re\left\{s_{l,m}(t)e^{i2\pi f_{c}t}\right\}
$$
Note
$$
\mathcal{E}_{m}=\left\Vert s_{m}(t)\right\Vert^{2}=\frac{1}{2}\left\Vert s_{l,m}(t)\right\Vert^{2}=\frac{1}{2}\left\Vert s_{l,m}\right\Vert^{2}=\frac{1}{2}\mathcal{E}_{l,m}
$$

Given $s_{m}(t)$ was transmitted, the bandpass received signal is
$$
r(t) = s_{m}(t)+n(t)
$$
Let $r_{l}(t)$ be the lowpass equivalent signal
$$
r_{l}(t)=\left[r(t)+i\hat{r}(t)\right]e^{-i2\pi f_{c}t}=s_{l,m}(t)+n_{l}(t)
$$
Set
$$
r_{l,i} = \left\langle r_{l}(t),\phi_{l,i}(t) \right\rangle=\int r_{l}(t)\phi^{*}_{l,i}(t)dt\\
n_{l,i} = \left\langle n_{l}(t),\phi_{l,i}(t) \right\rangle=\int n_{l}(t)\phi^{*}_{l,i}(t)dt
$$
Hence we have
$$
r_{l}=s_{l,m}+n_{l}
$$
Recall that
$$
S_{n,l}(f) = 4S_{n}(f+f_{c})=2N_{0}
$$
It follows that $n_{l}\sim \mathbb{C}\mathcal{N}(0,2N_{0}\mathbf{I}_{N})$ and
$$
f(n_{l})= \frac{1}{\left(2\pi N_{0}\right)^{N}}\exp\left(-\frac{\left\Vert n_{l}\right\Vert^{2}}{2N_{0}}\right)
$$

:::success
#### Baseband equivalent MAP detection
$$
\begin{aligned}
\hat{s}_{l,m}
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{\ln P_{m}-\frac{\left\Vert r_{l}-s_{l,m}\right\Vert^{2}}{2N_{0}}\right\}\\
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{N_{0}\ln P_{m}+\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}-\frac{1}{2}\left\Vert s_{l,m}\right\Vert^{2}\right\}
\end{aligned}
$$
:::

:::info
#### Proof
$$
\begin{aligned}
\hat{s}_{l,m}
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{Pr\left\{s_{l,m}\vert r_{l}\right\}\right\}\\
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{ Pr\left\{r_{l}\vert s_{l,m} \right\}\underbrace{Pr\left\{s_{l,m}\right\}}_{P_{m}}\right\}\\
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{\ln P_{m}-\frac{\left\Vert r_{l}-s_{l,m}\right\Vert^{2}}{2N_{0}}\right\}\\
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{\ln P_{m}-\frac{\left\Vert r_{l}\right\Vert^{2}+\left\Vert s_{l,m}\right\Vert^{2}-\overbrace{\left(r_{l}^{\dagger}s_{l,m}+s_{l,m}^{\dagger}r_{l}\right)}^{2\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}}}{2N_{0}}\right\}\\
&=\arg \max_{s_{l,m}\in\mathcal{S}_{l}}\left\{N_{0}\ln P_{m}+\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}-\frac{1}{2}\left\Vert s_{l,m}\right\Vert^{2}\right\}
\end{aligned}
$$
:::

:::warning
#### Compare Bandpass and Baseband MAP Decision Rule (Equivalent)
- Bandpass MAP rule
$$
\begin{aligned}
\hat{m}
=\arg \max_{m}\left\{\frac{N_{0}}{2}\ln P_{m}+r^{\top}s_{m}-\frac{1}{2}\mathcal{E}_{m}\right\}
\end{aligned}
$$
- Baseband MAP rule
$$
\begin{aligned}
\hat{m}
=\arg \max_{m}\left\{N_{0}\ln P_{m}+\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}-\frac{1}{2}\mathcal{E}_{l,m}\right\}
\end{aligned}
$$
where $\mathcal{E}_{m}=\frac{1}{2}\mathcal{E}_{l,m}$ and $r^{\top}s_{m}=\frac{1}{2}\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}$
$$
\begin{aligned}
\hat{m}
&=\arg \max_{m}\left\{\frac{N_{0}}{2}\ln P_{m}+r^{\top}s_{m}-\frac{1}{2}\mathcal{E}_{m}\right\}\\
&=\arg \max_{m}\left\{\frac{N_{0}}{2}\ln P_{m}+\frac{1}{2}\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}-\frac{1}{4}\mathcal{E}_{l,m}\right\}\\
&=\arg \max_{m}\left\{N_{0}\ln P_{m}+\Re\left\{r_{l}^{\dagger}s_{l,m}\right\}-\frac{1}{2}\mathcal{E}_{l,m}\right\}
\end{aligned}
$$
:::
###### tags: `Digital Communication` `Book`