### **Answer to Question 8**

For $P(f)$ being a raised-cosine spectrum over an ideal channel ($H_C(f)=1$), we have:
$$
H_T(f)H_R(f) = P(f)
$$

To minimize $P_E$ in AWGN, the matched filter condition requires:
$$
|H_T(f)| = |H_R(f)|
$$

Substituting this into the first equation:
$$
|H_T(f)|^2 = P(f)
$$

Thus, the optimum design is:
$$
|H_T(f)| = |H_R(f)| = \sqrt{P(f)}
$$