Topic: Signals and Systems
Consider a system with $x(t)$ as input and $y(t)$ as output. The signal $x(t)$ is split into two branches: the first branch passes through a filter A and is multiplied by $j$, and the second branch is direct. The sum of these two branches passes through a filter B to produce $y(t)$. To obtain an output $Y(f)$ which contains only the positive frequency components of $X(f)$ shifted towards the origin by $f_c$, which combination of filters A and B is appropriate?
- A. A is a signum function $\text{sgn}(f)$ and B is an ideal low-pass filter
- B. A is a signum function $\text{sgn}(f)$ and B is an ideal high-pass filter
- C. A is an ideal low-pass filter and B is a signum function $\text{sgn}(f)$
- D. A is an ideal high-pass filter and B is a signum function $\text{sgn}(f)$
Answer: A. A is a signum function $\text{sgn}(f)$ and B is an ideal low-pass filter
Explanation: For the first branch, passing the signal through a signum function $\text{sgn}(f)$ and multiplying by $j$ effectively acts as a Hilbert transform. Adding this to the original signal mathematically cancels out the negative frequency components while doubling the positive frequency components. Afterwards, a Low-Pass Filter (LPF) applied as block B can isolate the desired down-converted baseband signal.