# Degradation of BPSK system using matched filter with a timing error

Given a BPSK system with: \begin{align*} s_1(t) &= Au_T(t) \\ s_2(t) &= -Au_T(t) \\ \end{align*} where $$u_T(t)=1$$ for $$0, and zero otherwise. Given $$A > 0$$, and Gaussian white noise of power spectral density $$\frac{N_0}{2}$$.

1. Design a matched filter for this system.
2. Draw to scale the output of the filter due to signal only.
3. Assuming the system has a timing error of $$\varepsilon$$ with respect to the optimal sampling instant, find an expression for the probability of error.

My attempt:

I have solve the first part and find out $$h(t) = \frac{1}{\sqrt{T}}$$. However I have no idea how to do the rest.

• More than you ever wanted to know about matched filters can be found in my answer to the question "Understanding the matched filter", and the graphs at the end of the answer can help in answering 2. and, with a little bit of thought, 3. as well. Dec 20, 2020 at 17:25
• @dilipsarwate Part 2 is fine for me now. Can you please give some additional hint on part 3? Dec 20, 2020 at 17:43
• The graphs shown at the end my answer cited above show that there are two possibilities to be considered, If the next bit is the same as the current bit, then the MF output at the delayed sampling instant is the same as the peak output at the correct sampling instant and so the error probability is unchanged. If the next bit is different from the current bit, then the MF output at the delayed sampling instant is "smaller" than the output at the correct sampling instant and so the error probability is larger. The average error probability is the weighted average of the two numbers thus found. Dec 20, 2020 at 21:16
• I think I understand now, thank you @DilipSarwate Dec 21, 2020 at 9:55