BEE Signals and Control Systems 3 — Questions and Answers
Question 1: What does the Region of Convergence (ROC) of the z-transform determine about a signal?
- The signal's energy content
- Whether the signal is causal, anti-causal, or two-sided (Correct answer)
- The signal's frequency spectrum
- The signal's autocorrelation
Correct answer: Whether the signal is causal, anti-causal, or two-sided
The ROC determines the nature of the corresponding time-domain signal: a right-half ROC (|z| > r) indicates a causal signal, while a left-half ROC indicates an anti-causal signal.
Question 2: For a second-order underdamped system, the peak overshoot Mp in the step response is determined by:
- Natural frequency ωn only
- Damping ratio ζ only (Correct answer)
- Both ωn and ζ
- Neither ωn nor ζ
Correct answer: Damping ratio ζ only
Peak overshoot Mp = exp(−πζ/√(1−ζ²)) depends solely on the damping ratio ζ, not on the natural frequency.
Question 3: In a Bode magnitude plot, a simple pole at s = −a contributes a slope of how many dB/decade beyond the corner frequency?
- +20 dB/decade
- −20 dB/decade (Correct answer)
- +40 dB/decade
- −40 dB/decade
Correct answer: −20 dB/decade
A single real pole introduces a −20 dB/decade roll-off slope in the magnitude plot for frequencies well above the pole's corner frequency.
Question 4: The convolution of two rectangular pulses of duration T each produces a signal of what shape?
- Rectangular pulse of duration T
- Rectangular pulse of duration 2T
- Triangular pulse of duration 2T (Correct answer)
- Sinc function
Correct answer: Triangular pulse of duration 2T
Convolving two equal-width rectangular pulses yields a triangular pulse whose total duration is twice the original pulse width.
Question 5: The Nyquist sampling theorem states that to perfectly reconstruct a band-limited signal with maximum frequency fmax, the sampling frequency must be:
- At least fmax
- At least 2fmax (Correct answer)
- Exactly 2fmax
- At most 2fmax
Correct answer: At least 2fmax
The Nyquist theorem requires a sampling rate of at least 2fmax (the Nyquist rate) to avoid aliasing and enable perfect reconstruction.
Question 6: Which root locus rule states that the number of branches going to infinity equals the number of open-loop poles minus the number of open-loop zeros?
- Angle of asymptotes rule
- Number of asymptotes rule (Correct answer)
- Centroid rule
- Breakaway point rule
Correct answer: Number of asymptotes rule
The number of asymptotes (branches going to infinity) equals P − Z, where P is the number of poles and Z is the number of finite zeros.
Question 7: An energy signal x(t) has finite energy E. Its Parseval's relation states that E equals:
- ∫|X(f)|df from −∞ to ∞
- ∫|X(f)|²df from −∞ to ∞ (Correct answer)
- ∫|x(t)|dt from −∞ to ∞
- max|X(f)|²
Correct answer: ∫|X(f)|²df from −∞ to ∞
Parseval's theorem equates the total energy computed in the time domain to the integral of the squared magnitude spectrum: E = ∫|X(f)|²df.
BEE Signals and Control Systems 3
What does the Region of Convergence (ROC) of the z-transform determine about a signal?