Signals and Control Systems Flashcards

Which property of the Fourier Transform states that a time shift in a signal results in a phase shift in its frequency domain representation?

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.

A system has the transfer function H(s) = 10/(s² + 4s + 10). What is the damping ratio ζ?

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.

The z-transform of the unit step sequence u[n] is:

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.

In a PID controller, which component is added primarily to reduce steady-state error to zero for step inputs?

Answer: Integral term

The integral term accumulates past error over time, forcing the steady-state error to zero by continuously correcting any residual offset.

Which of the following is NOT a property of a Linear Time-Invariant (LTI) system?

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.

A discrete-time signal x[n] = cos(0.5πn) has a fundamental period of:

Answer: 4 samples

The fundamental period N satisfies 0.5π × N = 2πk, giving N = 4k; the smallest positive integer is N = 4 samples.

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:

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.