1.6 Homework 7

Consider an On-Off Keying system where $s_1(t)=A\cos (2\pi f_{c}t+)$ for $0\le t\le T$ and $s_2(t)=0$ for $0\le t\le T$ . The channel is ideal AWGN with zero mean and spectral height $\frac{N_0}{2}$ .

• Assume $$ is known at the receiver. What is the average probability of bit-error using anoptimum receiver?
• Assume that we estimate the receiver phase to be $\hat{}$ and that $\hat{}\neq$ . Analyze the performance of the matched filter with the wrong phase,that is, examine $\stackrel{}{P_e}$ as a function of the phase error.
• When does noncoherent become preferable? (You can find an expression for the $\stackrel{}{P_e}$ of noncoherent receivers for OOK in your textbook.) That is, how big should the phase error be before you would switch tononcoherent?

Proakis and Salehi , Problems 9.4 and 9.14

A coherent phase-shift keyed system operating over an AWGN channel with two sided power spectral density $\frac{N_0}{2}$ uses $s_0(t)=A{p}_T(t)\cos ({}_{c}t+{}_0)$ and $s_1(t)=A{p}_T(t)\cos ({}_{c}t+{}_1)$ where $\forall i, i\in \{0, 1\}\colon \left|{}_i\right|\le \frac{\pi }{3}$ , are constants and that $f_{c}T=\mathrm{integer}$ with ${}_c=2\pi f_c$ .

• Suppose ${}_0$ and ${}_1$ are known constants and that the optimum receiver uses filters matched to $s_0(t)$ and $s_1(t)$ . What are the values of $P_{e0}$ and $P_{e1}$ ?
• Suppose ${}_0$ and ${}_1$ are unknown constants and that the receiver filters are matched to ${\hat{s}}_0(t)=A{p}_T(t)\cos ({}_{c}t)$ and ${\hat{s}}_1(t)=A{p}_T(t)\cos ({}_{c}t+\pi )$ and the threshold is zero.
  Use a correlation receiver structure.
  What are $P_{e0}$ and $P_{e1}$ now? What are the minimum values of $P_{e0}$ and $P_{e1}$ (as a function of ${}_0$ and ${}_1$ )?