Q21GATE 2008MCQ

A signal $x(t)=sinc(\alpha t)$ where $\alpha$ is a real constant $(sinc(x)=\frac{sin (\pi x)}{\pi x})$ is the input to a Linear Time Invariant system whose impulse response $h(t)=sinc(\beta t)$ , 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 ?

🚀 Solve in Practice Mode

📖 Explanation

\begin{aligned} x(t)&=\sin c (\alpha t)=Sa[\pi \alpha t]\\ h(t)&=\sin c (\beta t)=Sa[\pi \beta t]\\ x(t)&=Sa[\pi \alpha t] \end{aligned}

\begin{aligned} y(t)&=h(t)*x(t)\\ Y(j\omega )&=X(j\omega )H(j\omega ) \end{aligned}

\begin{aligned} y(t)&=\frac{1}{\beta }Sa( \pi \alpha t)\;\;\; if \; \alpha \lt \beta \\ y(t)&=\frac{1}{\beta } \sin c(\alpha t)\;\;\; if \; \alpha \lt \beta \\ y(t)&=\frac{1}{\alpha }\sin c( \beta t)\;\;\; if \; \alpha \gt \beta \\ \end{aligned}

So, output is of the form $k \sin c (\gamma t)$ where, $\gamma =min(\alpha ,\beta )$