Stealth Communications

In some situations, it is advantageous to hide the presence of a communication system. The following signaling scheme accomplishes that objective.

The binary communications system under consideration uses the following signals to transmit equally likely messages over an additive, white Gaussian noise channel (spectral height $\frac{N_0}{2}$)

$$\begin{array}{cccc} s_0(t) & = & 0 & \mbox{for $0 \leq t \l... ... s_1(t) & = & N_t^s & \mbox{for $0 \leq t \leq T$,} \end{array}$$

where $N_t^s$ is a zero-mean, white Gaussian noise process with autocorrelation function $R_s(\tau)=P_s\delta(\tau)$.

Assume first that the following receiver front-end is being used, where $\phi(t)$ denotes an arbitrary signal of unit energy, i.e., $\Vert\phi(t)\Vert^2=\int_0^T \phi^2(t)\,dt=1$.

1. Compute the conditional distribution of the random variable $R$ for each of the two signals $s_0(t)$ and $s_1(t)$.
2. Assume now that the conditional distributions for the random variable $R$ are given by
   
   $$\begin{array}{cccc} H_0: & R & \sim & N(0,\frac{N_0}{2})\\ H_1: & R & \sim & N(0,\frac{N_0}{2}+P_s) \end{array}$$
   
   
   Find the likelihood-ratio test that minimizes the probability of error. Simplify as much as possible.
3. Find the probability of error for your test.
4. Consider now the following receiver front-end
   
   
   

   The functions $\phi_0(t)$ and $\phi_1(t)$ are orthonormal. Find the joint distribution of random variables $R_0$ and $R_1$.

5. For the receiver front-end above, devise the minimum probability of error test for determining which of the two signals $s_0(t)$ and $s_1(t)$ was transmitted. You do not have to compute the probability of error for your test, but you must provide a convincing argument if the probability of error will be larger or smaller than with the first receiver front-end.
6. What is the optimum receiver for this problem?