BEE Signals and Control Systems 2 — Questions and Answers
Question 1: Which property of the Fourier Transform states that a time shift in a signal results in a phase shift in its frequency domain representation?
- Duality property
- Time-shifting property (Correct answer)
- Modulation property
- Parseval's theorem
Correct answer: Time-shifting property
The time-shifting property states that if x(t) ↔ X(jω), then x(t−t₀) ↔ X(jω)e^(−jωt₀), introducing a linear phase shift.
Question 2: A system has the transfer function H(s) = 10/(s² + 4s + 10). What is the damping ratio ζ?
- 0.2
- 0.4
- 0.632 (Correct answer)
- 2.0
Correct answer: 0.632
Comparing with the standard form ωn²/(s² + 2ζωns + ωn²), ωn = √10 ≈ 3.162 and 2ζωn = 4, so ζ = 4/(2×3.162) ≈ 0.632.
Question 3: The z-transform of the unit step sequence u[n] is:
- z/(z−1), |z| > 1 (Correct answer)
- 1/(z−1), |z| > 1
- z/(z+1), |z| > 1
- 1/z, |z| > 0
Correct answer: z/(z−1), |z| > 1
The z-transform of u[n] = Σ z^(−n) for n≥0 sums to z/(z−1) for |z| > 1.
Question 4: In a PID controller, which component is added primarily to reduce steady-state error to zero for step inputs?
- Proportional term
- Derivative term
- Integral term (Correct answer)
- Feed-forward term
Correct answer: Integral term
The integral term accumulates past error over time, forcing the steady-state error to zero by continuously correcting any residual offset.
Question 5: Which of the following is NOT a property of a Linear Time-Invariant (LTI) system?
- Superposition applies
- The impulse response fully characterizes the system
- Output depends on future inputs (Correct answer)
- Convolution can be used to find the output
Correct answer: Output depends on future inputs
LTI systems are causal in most practical applications, meaning their output depends only on present and past inputs, not future inputs.
Question 6: A discrete-time signal x[n] = cos(0.5πn) has a fundamental period of:
- 2 samples
- 4 samples (Correct answer)
- 8 samples
- It is not periodic
Correct answer: 4 samples
The fundamental period N satisfies 0.5π × N = 2πk, giving N = 4k; the smallest positive integer is N = 4 samples.
Question 7: The gain margin of a system is defined as the gain increase (in dB) needed to bring the system to the verge of instability at:
- The gain crossover frequency
- The phase crossover frequency (Correct answer)
- The resonant frequency
- The bandwidth frequency
Correct answer: The phase crossover frequency
Gain margin is measured at the phase crossover frequency where the open-loop phase shift is −180°, indicating how much gain can be added before instability.
BEE Signals and Control Systems 2
Which property of the Fourier Transform states that a time shift in a signal results in a phase shift in its frequency domain representation?