Linear Time-Invariant (LTI) Systems: Impulse Response Quiz Quiz

This quiz explores core principles of Linear Time-Invariant (LTI) systems and the role of impulse response in system analysis. Evaluate your understanding of concepts such as convolution, causality, stability, and the relationship between input, output, and system responses in LTI systems.

1. Impulse Response Uniqueness

   In a Linear Time-Invariant (LTI) system, which statement best describes the relationship between the system's impulse response and its complete behavior?

   1. The impulse response fully determines the system's output for any input.
   2. The impulse response does not affect system behavior for zero inputs.
   3. The impulse response only applies to sinusoidal inputs.
   4. Different LTI systems can have the same unique impulse response but different overall behaviors.

   Explanation: For LTI systems, the impulse response provides a complete characterization; any output can be computed using convolution with the input. The second option is incorrect because the impulse response is not restricted to sinusoidal inputs. The third option is wrong since two LTI systems cannot have different behaviors if their impulse responses are identical. The last option is misleading because a zero input produces a zero output, but that does not diminish the role of the impulse response.

2. Impulse Response and Convolution

   When an LTI system with impulse response h(t) receives an input x(t), which mathematical operation must be used to determine the output y(t)?

   1. Division
   2. Multiplication
   3. Correlation
   4. Convolution

   Explanation: The output of an LTI system is given by the convolution of the input with the system's impulse response. Multiplication and division do not generally yield the correct output for arbitrary inputs and impulse responses. Correlation is a related operation but is not typically used to compute the output of LTI systems in this context.

3. Impulse Response and Causality

   Which condition must the impulse response h(t) of a continuous-time LTI system satisfy for the system to be causal?

   1. h(t) = 0 for t u003C 0
   2. h(t) = 1 for all t
   3. h(t) is an even function
   4. h(t) = 0 for t u003E 0

   Explanation: A causal LTI system cannot respond before an input is applied, so its impulse response must be zero for all times t less than zero. The second option is invalid, since the impulse response is not necessarily constant. The third option corresponds to an anti-causal system, which does not describe causality. Being an even function does not guarantee causality; it refers instead to the function's symmetry.

4. Impulse Response and System Stability

   For a continuous-time LTI system to be BIBO (Bounded Input, Bounded Output) stable, what property should its impulse response h(t) have?

   1. h(t) must be nonzero only at t = 0.
   2. The integral of |h(t)| must converge.
   3. h(t) must be differentiable everywhere.
   4. The area under h(t) must be infinite.

   Explanation: A system is BIBO stable if the integral of the absolute value of its impulse response is finite. If the area under h(t) is infinite, the system could amplify bounded inputs into unbounded outputs. The impulse response being nonzero only at t = 0 describes an impulse system, not necessarily a stable one. Differentiability is not required for stability; it merely concerns the function’s smoothness.

5. Discrete-Time Impulse Response Example

   If the impulse response of a discrete-time LTI system is h[n] = (0.5)^n u[n], where u[n] is the unit step function, which statement about the system is correct?

   1. The impulse response is not defined for negative n.
   2. The system is stable because the impulse response is absolutely summable.
   3. The system is non-causal because h[n] ≠ 0 for n u003C 0.
   4. The system is unstable because the impulse response grows with n.

   Explanation: For the given h[n], the sum of |0.5^n| from n=0 to infinity converges, ensuring absolute summability and thus BIBO stability. The impulse response decays for increasing n, not grows, making the second option wrong. The third is incorrect because u[n] equals zero for n u003C 0, making the system causal. The last option is not accurate, as h[n] is simply zero for negative n, not undefined.