The GATE Grind

GATE 2017 EE – Question 13

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

Let $z(t) = x(t) * y(t)$, where "$*$" denotes convolution. Let $c$ be a positive real-valued constant. Choose the correct expression for $z(ct)$.

  1. $c \cdot x(ct) * y(ct)$
  2. $x(ct) * y(ct)$
  3. $c \cdot x(t) * y(ct)$
  4. $c \cdot x(ct) * y(t)$

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Correct answer: (A) $c \cdot x(ct) * y(ct)$

Explanation

Write $z(ct) = \int x(\tau)y(ct - \tau)\,d\tau$ and substitute $\tau = c u$, so $d\tau = c\,du$. Then $z(ct) = c\int x(cu)y(c(t - u))\,du = c\,[x(ct) * y(ct)]$.