Question

A singal x(t) = $\frac{5sin5t}{\pi t}$ is applied as input to a system with h(t) = $10\frac{sin25t}{\pi t}$, if y(t) is output, then

• A. y(0) = $\frac{250}{\pi }$
• B. y(5) = $\frac{10}{\pi }sin\left(25\right)$
• C. y(1) = $\frac{50sin5}{\pi }$
• D. y(3) = $\frac{50}{\pi }sin\left(15\right)$

Answer 1

Answer: (A), (B), (C)

y(1) = $\frac{50sin5}{\pi }$

Given that x(t) is $\frac{5sin5t}{\pi t}$ and h(t) is $10\frac{sin25t}{\pi t}$


so, $X\left(\omega \right)=5rect\left(\omega /10\right)$

$H\left(\omega \right)=10rect\left(\omega /50\right)$
since
$y\left(t\right)=h\left(t\right)\ast x\left(t\right)$

and $Y\left(\omega \right)=H\left(\omega \right).X\left(\omega \right)$
$=5rect\left(\omega /10\right).10rect\left(\omega /50\right)$

$y\left(\omega \right)=50rect\left(\omega /10\right)$

$y\left(t\right)=50\frac{sin\left(5t\right)}{\pi t}$
and
$y\left(0\right)=\frac{250}{\pi }$

$y\left(1\right)=\frac{50}{\pi }sin5$

$y\left(3\right)=\frac{50}{3\pi }sin15$

$y\left(5\right)=\frac{50}{5\pi }sin25=\frac{10}{\pi }sin25$