Advanced Practice: Signals and Systems GATE-Level Quiz Quiz

Challenge your understanding of Signals and Systems with these practice questions focused on advanced concepts like convolution, system properties, Fourier analysis, Laplace transforms, and signal sampling. This quiz is designed to help students prepare for competitive exams by strengthening core knowledge and problem-solving skills in Signals and Systems.

1. Convolution Integral Application

   If x(t) = e^{-t}u(t) and h(t) = u(t), where u(t) is the unit step function, what is the result of the convolution y(t) = x(t) * h(t) for t ≥ 0?

   1. B) e^{-t}u(t)
   2. C) (1 + e^{-t})u(t)
   3. A) (1 - e^{-t})u(t)
   4. D) tu(t)

   Explanation: The convolution of x(t) = e^{-t}u(t) and h(t) = u(t) results in y(t) = ∫₀^t e^{-τ} dτ = (1 - e^{-t})u(t). Option A is correct as it reflects this property. Option B is just the input signal and not the convolution. Option C is incorrect as it adds instead of subtracts. Option D gives the convolution of two unit step functions, not e^{-t}u(t) with u(t).

2. System Linearity Test

   Given the system y[n] = 2x[n] - 3, determine whether the system is linear or not.

   1. B) Nonlinear due to the constant term
   2. A) Linear because superposition holds
   3. C) Linear as only scaling is used
   4. D) Nonlinear because it involves x[n] squared

   Explanation: The system y[n] = 2x[n] - 3 is nonlinear because the presence of the constant term -3 violates the superposition property required for linearity. Option A is incorrect since superposition does not hold here. Option C ignores the constant, only considering scaling. Option D is wrong as there is no squaring of x[n].

3. Properties of the Fourier Transform

   Which property of the continuous-time Fourier Transform states that a time shift t₀ in x(t) results in a phase shift in X(jω)?

   1. C) Time shifting
   2. D) Duality
   3. B) Linearity
   4. A) Frequency shifting

   Explanation: The time-shifting property states that if x(t) undergoes a delay by t₀, its Fourier Transform X(jω) gets multiplied by e^{-jωt₀}. Option C correctly identifies this property. Option A refers to frequency shifting, which is a different property. Linearity (Option B) and duality (Option D) have no direct relation to time shifts.

4. Laplace Transform Region of Convergence

   For the causal signal x(t) = e^{2t}u(-t), what is the region of convergence (ROC) of its Laplace Transform?

   1. B) Re(s) u003C 2
   2. A) Re(s) u003E 2
   3. D) Re(s) u003E 0
   4. C) Re(s) u003C 0

   Explanation: The signal e^{2t}u(-t) is non-causal and right-sided, leading to an ROC where Re(s) u003C 2. Option B captures this correctly. Option A is the ROC for a causal right-sided exponential. Option C and Option D are not correct for this signal since they do not account for the correct exponential growth or side.

5. Sampling and Aliasing Scenario

   If a continuous-time signal x(t) = sin(600πt) is sampled at a rate of 400 Hz, what frequency will be observed in the reconstructed signal due to aliasing?

   1. A) 300 Hz
   2. B) 200 Hz
   3. C) 100 Hz
   4. D) 400 Hz

   Explanation: The given signal has a frequency of 300 Hz but when sampled at 400 Hz, frequencies above half the sampling rate (200 Hz) are aliased. The observed frequency will be |400 - 300| = 100 Hz, so Option C is correct. Option A gives the original frequency without considering aliasing. Option B mistakenly selects the Nyquist frequency. Option D is the sampling rate, not the aliased frequency.