Convolution MCQ

1. 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 · x(ct) ∗ y(ct)
2. x(ct) ∗ y(ct)
3. c · x(t) ∗ y(ct)
4. c · x(ct) ∗ y(t)

2. x(t) is non-zero only for T_x < t < T’_x, and similarly, y(t) is non-zero only for T_y < t < T_y‘. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is true?

1. z(t) can be non-zero over an unbounded interval.
2. z(t) is non-zero for t < T_x + T_y
3. z(t) is zero outside of T_x + T_y < T’_x + T’_y
4. z(t) is non-zero for t > T’_x > T’_y

3. Two systems with impulse responses h₁(t) and h₂(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

1. product of h₁(t) and h₂(t)
2. sum of h₁(t) and h₂(t)
3. convolution of h₁(t) and h₂(t)
4. subtraction of  h₂(t) from h₁(t)

4. Let y[n] denotes the convolution of h[n] and g[n], where h[n] = (1/2)ⁿ u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1] equals

1. 0
2. 1/2
3. 1
4. 3/2

5. Given two continous time signals x(t) = e^-t and y(t) = e^-2t which exist for t > 0, the convolution z(t) = x(t) ∗ y(t) is

1. e^-t – e^-2t
2. e^-3t
3. e^+t
4. e^-t + e^-2t

6. The convolution of the functions f₁(t) = e^-2t u(t) and f₂(t) = e^t u(t) is equal to

1. $-\frac{1}{3}e^{-2t}+\frac{1}{3}e^{t}$
2. $\frac{1}{3}e^{-2t}+\frac{1}{3}e^{t}$
3. $-\frac{1}{3}e^{-2t}-\frac{1}{3}e^{t}$
4. $-\frac{1}{3}e^{t}+\frac{1}{3}e^{-2t}$