The GATE Grind

GATE 2014 EE (EE1) – Question 19

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple choice

x(t) is nonzero only for $T_x<t<T_x^\prime$, and y(t) only for $T_y<t<T_y^\prime$. Let z(t) = x(t) * y(t). Which statement is true?

  1. z(t) can be nonzero over an unbounded interval.
  2. $z(t)\ne0$ for $t<T_x+T_y$.
  3. z(t) is zero outside $T_x+T_y<t<T_x^\prime+T_y^\prime$.
  4. $z(t)\ne0$ for $t>T_x^\prime+T_y^\prime$.

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Correct answer: (C) z(t) is zero outside $T_x+T_y<t<T_x^\prime+T_y^\prime$.

Explanation

For a nonzero integrand x(τ)y(t−τ), both Tx<τ<Tx′and Ty<t−τ<Ty′must hold. Such overlap is impossible unless Tx+Ty<t<Tx′+Ty′. Therefore **z(t)is zero outside that interval,C**. The statement supplies a support bound without guaranteeing nonzero values everywhere inside.