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GATE 2023 BM – Question 49

Signals and Systems · LTI Systems and Convolution · 2 marks · Multiple select

Let x(t)=1 for 0≤t≤1 and zero otherwise, and y(t)=x*x. Which statements are correct?

  1. y(t)=0 for all t<0
  2. y(t)=0 for all t>1
  3. y(t)=0 for all t>3
  4. $\int_{0.1}^{0.75}y\prime(t)\,dt\ne0$

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Correct answer: (A) y(t)=0 for all t<0; (C) y(t)=0 for all t>3; (D) $\int_{0.1}^{0.75}y\prime(t)\,dt\ne0$

Explanation

Convolution of the two unit rectangles gives y=t on [0,1], y=2−t on [1,2], and zero elsewhere. Thus A is true, B is false because the triangle continues to two, and C is true. On .1 to .75 the derivative is one, so the integral is .65, nonzero. Correct A,C,D.

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