Question

Suppose for input x(t) a linear time-invariant system with impulse response h(t) produces output y(t), so that x(t) * h(t) = y(t). Further, if |x(t)| * |h(t)| = z(t), which of the following statements is true?

• A.
  $\text{For some but not all t}\in \text{(-}\infty \text{,}\infty \text{), z(t)}\ge \text{y(t)}$
• B.
  $\text{For all t}\in \text{(-}\infty \text{,}\infty \text{), z(t)}\ge \text{y(t)}$
• C.
  $\text{For some but not all t}\in (-\infty \text{,}\infty )\text{, z(t)}\le \text{y(t)}$
• D.
  $\text{For all t}\in \text{(-}\infty \text{,}\infty \text{), z(t)}\le \text{y(t)}$

Answer 1

The correct option is B

$\text{For all t}\in \text{(-}\infty \text{,}\infty \text{), z(t)}\ge \text{y(t)}$
$\text{Since, y(t) = x(t) + h(t)}$
$\text{and z(t) = |x(t)| * |h(t)|}$

Case-1:

then, y(t) and z(t)

Case-2:

then, y(t) = z(t)

Thus, z(t) $\ge$ y(t) , for all 't'.