Demodulation and Optimal detection in Baseband

Implementation of passband receiver using low-pass equivalent signals

In the previous topic, we dealt with general Detection theory, initially and later applied it to passband Signals. In this section, we will analyse the idea of Demodulation and Detection in Baseband. When we discussed the structure of the optimal detector we have considered projecting r(t) onto sm(t) or $\phi_i(t)$.These are passband operations. The math presents us with a relation between the inner product of two signals in passband and the lowpass equivalent signals in baseband. With this idea we can implement the receiver in baseband. Mathematically we see that,

(1)
\begin{align} < r(t),s_m(t)> = \frac{1}{2} \Re \{<r_l(t),s_{ml}(t)>\} \end{align}
(2)
\begin{align} r_l(t)=\left( r(t)\cos(2\pi f_0t) + \hat{r}(t)\sin(2\pi f_0t)\right) + j \left( \hat{r}(t)\cos(2\pi f_0t) - r(t)\sin(2\pi f_0t) \right) \end{align}

The received signal in baseband, assuming Kth signal was sent is given by,

(3)
\begin{equation} r_l(t)=s_{kl}(t) + n_{l}(t). \end{equation}

We will assume that r(t) and n(t) are passed through a bandpass filter of bandwidth W Hz in passband and then converted to baseband. So n_l(t) is a random process with PSD,

(4)
\begin{eqnarray} S_x_i(f)= S_x_q(f) = \left \{ \begin{array}{ll} N_0 & |f|<W\\ 0 & \mbox{otherwise}.\end{array} \right. \end{eqnarray}

From a example in Proakis(page 81), we can conclude that,

(5)
\begin{eqnarray} S_x_l(f) = \left \{ \begin{array}{ll} 2N_0 & |f|<W\\ 0 & \mbox{otherwise}.\end{array} \right. \end{eqnarray}

Now let,

(6)
\begin{eqnarray} Z_m &=& <r_l(t), s_{ml}(t)> \\ &=& <s_{kl}(t), s_{ml}(t)> + <n_l(t),s_{ml}(t)> \end{eqnarray}

Notice that $<s_{kl}(t),s_{ml}(t)>$ is real when k=m. So it is enough to look at Re(Zm)

(7)
\begin{align} \Re (Z_m) = \Re <s_{kl}(t), s_{ml}(t)> + \Re <n_l(t),s_{ml}(t)> \end{align}

Say $\rho_{km}$ is the correlation between the passband signals sk(t) and sm(t), Ekand Em be the passband energies then,

(8)
\begin{align} \Re (Z_m) = 2\sqrt{E_k} \sqrt{E_m} \rho_{km} + n, \end{align}

where n is a Gaussian random variable with zero mean and variance equal to 2EmN0.


Base-band Receiver using First Principles

In this case we project received signal onto the basis functions of the low-pass signals i.e we project rl(t) onto $\phi_l(t)$ where $\phi_l(t)$ is the ith basis function for the signal set { s1l , ……, sMl}
let,

(9)
\begin{align} Z_i = <r_l(t), \phi_{li}(t)> \end{align}

we note that Zi's are complex numbers.The signal component is $s_{kl}(t),\phi_{li}(t)$ and the noise component is $<n_{kl}(t),\phi_{li}(t)> = n$.

(10)
\begin{eqnarray} \Re \{n\} &=& <n_{lI}(t),\phi_{liI}(t)> - <n_{lQ}(t),\phi_{liQ}(t)> \\ \Im \{n\} &=& <n_{lI}(t),\phi_{liQ}(t)> + <n_{lQ}(t),\phi_{liI}(t)> \\ \end{eqnarray}

since $\phi_{li}(t)$ is of unit energy,$\Re \{n\} \sim N(0,N_0) \mbox{ \& } \Im \{n\} \sim N(0,N_0).$
Example:
Consider FSK modulation scheme at the transmitter , wherein signal set is given by $s_i(t)= \sqrt{\frac{2}{T}}\cos(2\pi(f_0 + (i-1)\Delta f)t) \mbox{rect}(t- \frac{T}{2})$

The lowpass equivalent signals are

(11)
\begin{align} s_{li}(t)=\sqrt{\frac{2}{T}} e^{j(i-1)\Delta f t} \mbox{rect}(t- \frac{T}{2}) \end{align}

Suppose we choose sli(t) to be orthogonal in baseband, i.e $\Delta f =\frac{1}{T}$. We arrive at this condition by imposing orthogonality condition on the baseband signals. Mathematically, <sli(t),slm(t)>=0. Further in that case $\phi_{li}(t)=\frac{1}{\sqrt{2}}s_{li}(t)$

1(2).png
(12)
\begin{align} f_{z|s_{k}}(z)=\frac{1}{\pi^N \mbox{det}(C_z)} \exp[-(\underline{z}-\underline{s}_k)^H C_z^{-1} (\underline{z}-\underline{s}_k)] \end{align}

Where$s_k=[0,\ldots,\sqrt{2E},\ldots,0]^T$ and $C_z = \mbox{diag}(2N_0,2N_0,2N_0,\ldots,2N_0)$

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