Topic: Signals and Systems
A signal $x(t)=\text{sinc}(\alpha t)$, where $\alpha$ is a real constant, is the input to a Linear Time Invariant system whose impulse response is $h(t)=\text{sinc}(\beta t)$, where $\beta$ is a real constant. If $\min(\alpha, \beta)$ denotes the minimum of $\alpha$ and $\beta$ and similarly, $\max(\alpha, \beta)$ denotes the maximum of $\alpha$ and $\beta$, and K is a constant, which one of the following statements is true about the output of the system (here, $\text{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$).
- A. It will be of the form $K \text{ sinc}(\gamma t)$ where $\gamma=\min(\alpha,\beta)$
- B. It will be of the form $K \text{ sinc}(\gamma t)$ where $\gamma=\max(\alpha,\beta)$
- C. It will be of the form $K \text{ sinc}(\alpha t)$
- D. It cannot be a sinc type of signal
Answer: A. It will be of the form $K \text{ sinc}(\gamma t)$ where $\gamma=\min(\alpha,\beta)$
Explanation: $x(t) = \text{sinc}(\alpha t) \longleftrightarrow X(\omega)$ is a rectangular pulse of width $\alpha$.
$h(t) = \text{sinc}(\beta t) \longleftrightarrow H(\omega)$ is a rectangular pulse of width $\beta$.
$Y(\omega) = X(\omega)H(\omega)$. The multiplication of two rectangular pulses of different widths in the frequency domain results in a rectangular pulse with a width equal to the minimum of the two widths. Therefore, the time-domain output is a sinc function with the parameter $\min(\alpha, \beta)$.