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Quiz: System Properties and Analysis

Test your understanding of fundamental system properties and analysis techniques.

1. What mathematical condition defines a linear system?

  1. \(\mathcal{T}[ax_1(t) + bx_2(t)] = a\mathcal{T}[x_1(t)] + b\mathcal{T}[x_2(t)]\) for any inputs and constants
  2. \(\mathcal{T}[x(t - t_0)] = y(t - t_0)\) for all time shifts
  3. \(y(t_0)\) depends only on \(x(t)\) for \(t \leq t_0\)
  4. \(|x(t)| < M_x\) implies \(|y(t)| < M_y\)
Show Answer

The correct answer is A. A system is linear if it satisfies the superposition principle: \(\mathcal{T}[ax_1(t) + bx_2(t)] = a\mathcal{T}[x_1(t)] + b\mathcal{T}[x_2(t)]\) for any inputs \(x_1(t)\) and \(x_2(t)\) and any constants \(a\) and \(b\). This encompasses both additivity and homogeneity properties. Option B describes time-invariance, option C describes causality, and option D describes BIBO stability.

Concept Tested: Linear Systems

See: Linear Systems

2. For a time-invariant system, if \(y(t) = \mathcal{T}[x(t)]\), what must be true?

  1. The system parameters change linearly with time
  2. The system response depends on when the input is applied
  3. \(\mathcal{T}[x(t - t_0)] = y(t - t_0)\) for all signals and time shifts
  4. The system has no memory of past inputs
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The correct answer is C. A time-invariant system satisfies \(\mathcal{T}[x(t - t_0)] = y(t - t_0)\) for all signals \(x(t)\) and all time shifts \(t_0\). This means a time shift in the input produces an equivalent time shift in the output with no change in the waveform. Time-invariant systems have constant parameters that do not change over time, ensuring consistent behavior regardless of when an input is applied.

Concept Tested: Time-Invariant Systems

See: Time-Invariant Systems

3. What is the BIBO stability condition for an LTI system in terms of its impulse response \(h(t)\)?

  1. \(h(t) = 0\) for \(t < 0\)
  2. \(h(t)\) must be a finite-duration signal
  3. \(\int_{-\infty}^{\infty} |h(t)| dt < \infty\)
  4. \(h(t)\) must be a periodic function
Show Answer

The correct answer is C. For LTI systems, BIBO stability requires that the impulse response be absolutely integrable: \(\int_{-\infty}^{\infty} |h(t)| dt < \infty\). This ensures that every bounded input produces a bounded output. Option A describes causality (not stability), option B is too restrictive (stable systems can have infinite-duration impulse responses if they decay sufficiently), and option D is incorrect as impulse responses are generally not periodic.

Concept Tested: Stability, Impulse Response

See: Stability

4. Which of the following best describes a causal system?

  1. A system whose impulse response \(h(t) = 0\) for \(t < 0\)
  2. A system that can predict future input values
  3. A system with zero phase shift at all frequencies
  4. A system that produces outputs before inputs are applied
Show Answer

The correct answer is A. A causal system has an impulse response that satisfies \(h(t) = 0\) for \(t < 0\), meaning the output at any time depends only on present and past input values, never on future inputs. All physical systems that operate in real-time must be causal, as they cannot respond to inputs that have not yet occurred.

Concept Tested: Causality, Impulse Response

See: Causality

5. What is the relationship between the step response \(s(t)\) and the impulse response \(h(t)\) for an LTI system?

  1. \(s(t) = \frac{d}{dt}h(t)\)
  2. \(s(t) = \int_{-\infty}^{t} h(\tau) d\tau\)
  3. \(s(t) = h(t) * u(t)\), which is the same as option B
  4. Both B and C are correct
Show Answer

The correct answer is D. The step response is the integral of the impulse response: \(s(t) = \int_{-\infty}^{t} h(\tau) d\tau\). This can also be expressed as the convolution of the impulse response with the unit step: \(s(t) = h(t) * u(t)\). Both formulations are equivalent and correct. The step response reveals important characteristics including rise time, settling time, overshoot, and steady-state error.

Concept Tested: Step Response, Impulse Response

See: Step Response

  1. \(H(f)\) is the Laplace transform of \(h(t)\)
  2. \(H(f)\) is the derivative of \(h(t)\)
  3. \(H(f)\) is the Fourier transform of \(h(t)\): \(H(f) = \int_{-\infty}^{\infty} h(t)e^{-j2\pi ft} dt\)
  4. \(H(f)\) is the time-reversed version of \(h(t)\)
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The correct answer is C. The frequency response is the Fourier transform of the impulse response: \(H(f) = \int_{-\infty}^{\infty} h(t)e^{-j2\pi ft} dt\). The magnitude \(|H(f)|\) shows how the system amplifies or attenuates different frequencies, while the phase \(\angle H(f)\) shows how the system delays sinusoidal components. Note that option A describes the transfer function \(H(s)\), which is the Laplace transform.

Concept Tested: Frequency Response, Impulse Response

See: Frequency Response

7. What is the impulse response of a memoryless system?

  1. \(h(t) = K\delta(t)\) where \(K\) is a constant gain
  2. \(h(t) = u(t)\) (unit step function)
  3. \(h(t) = e^{-at}u(t)\) (exponential decay)
  4. \(h(t)\) extends over infinite time
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The correct answer is A. A memoryless system produces outputs that depend only on the current input value, with no dependence on past or future inputs. The impulse response is \(h(t) = K\delta(t)\) where \(K\) is a constant gain. Options B and C describe systems with memory, as their impulse responses extend over time beyond \(t = 0\).

Concept Tested: Memoryless Systems, Impulse Response

See: Memoryless Systems

8. Given two LTI systems connected in series (cascade) with transfer functions \(H_1(s)\) and \(H_2(s)\), what is the overall transfer function?

  1. \(H_{total}(s) = H_1(s) + H_2(s)\)
  2. \(H_{total}(s) = H_1(s) \cdot H_2(s)\)
  3. \(H_{total}(s) = \frac{H_1(s)}{1 + H_1(s)H_2(s)}\)
  4. \(H_{total}(s) = H_1(s) - H_2(s)\)
Show Answer

The correct answer is B. For LTI systems in cascade (series), the overall transfer function is the product of individual transfer functions: \(H_{total}(s) = H_1(s) \cdot H_2(s)\). In the time domain, this corresponds to the convolution of impulse responses. Option A describes parallel systems, and option C describes a feedback configuration.

Concept Tested: System Interconnections, Transfer Function

See: System Interconnections

9. For a negative feedback system with forward path \(G(s)\) and feedback path \(H(s)\), what is the closed-loop transfer function?

  1. \(T(s) = G(s) + H(s)\)
  2. \(T(s) = G(s) \cdot H(s)\)
  3. \(T(s) = \frac{G(s)}{1 - G(s)H(s)}\)
  4. \(T(s) = \frac{G(s)}{1 + G(s)H(s)}\)
Show Answer

The correct answer is D. The closed-loop transfer function for a negative feedback system is \(T(s) = \frac{G(s)}{1 + G(s)H(s)}\). The denominator \(1 + G(s)H(s)\) is called the characteristic equation, and its roots (poles of the closed-loop system) determine stability. Negative feedback provides stability, disturbance rejection, and reduced sensitivity to parameter variations.

Concept Tested: Feedback Systems, Transfer Function

See: Feedback Systems

10. An LTI system has impulse response \(h(t) = e^{at}u(t)\) where \(a > 0\). What can you conclude about this system's stability?

  1. The system is stable because it has an exponential impulse response
  2. The system is stable because the impulse response is causal
  3. The system is unstable because \(\int_{-\infty}^{\infty} |h(t)| dt = \int_{0}^{\infty} e^{at} dt\) diverges when \(a > 0\)
  4. Stability cannot be determined without knowing the input signal
Show Answer

The correct answer is C. The system is unstable because with \(a > 0\), the integral \(\int_{-\infty}^{\infty} |h(t)| dt = \int_{0}^{\infty} e^{at} dt\) diverges (approaches infinity). The BIBO stability condition for LTI systems requires the impulse response to be absolutely integrable. A growing exponential violates this condition, meaning bounded inputs can produce unbounded outputs. Note that causality (option B) does not guarantee stability.

Concept Tested: Stability, Impulse Response

See: Stability, Impulse Response