Causality and Stability in Systems: Test Your Concepts Quiz

Explore the core principles of causality and stability in systems through this concise quiz designed to reinforce your understanding of fundamental concepts, including system response, signal behavior, and related theoretical distinctions. Gain clarity on how and why systems behave with respect to input signals, initial conditions, and temporal relationships—perfect for students and enthusiasts in signals and systems analysis.

1. Identifying Causal Systems

   Which of the following statements correctly describes a causal discrete-time system, given input x[n] and output y[n]?

   1. The output y[n] at any time n can depend exclusively on values of x[k] for k u003C 0, regardless of n.
   2. The output y[n] at any time n depends on x[k] for k u003E n.
   3. The output y[n] at any time n depends only on x[n] and x[k] for k u003C n.
   4. The output y[n] at any time n is only determined by x[0].

   Explanation: A causal discrete-time system is one in which the output at any time n depends only on the current and past input values (x[n] and x[k] for k u003C n), never on future inputs. The second option describes a non-causal system since it uses future values. The third option is not general, as output is rarely solely dependent on x[0]. The last option ignores the importance of n, incorrectly suggesting the output relies only on inputs before time zero.

2. Defining System Stability

   Which condition must be satisfied for a system to be considered BIBO (Bounded-Input Bounded-Output) stable?

   1. The output always equals the input.
   2. Unbounded outputs may occur with bounded inputs.
   3. Every bounded input produces a bounded output.
   4. The output depends only on the present input.

   Explanation: A system is BIBO stable if, for every bounded input, the output remains bounded for all time. The second option refers to an identity system but not necessarily stability. The third option explicitly contradicts BIBO stability by allowing unbounded outputs. The fourth option relates to memoryless systems, not directly to stability.

3. Non-Causal System Example

   Given the system y(t) = x(t) + 2x(t + 1), what property does this system exhibit?

   1. It is time-invariant but not linear.
   2. It is non-causal because the output depends on a future input.
   3. It is unstable for all bounded inputs.
   4. It is causal because only current and past inputs are involved.

   Explanation: This system is non-causal since y(t) depends on x(t + 1), a future value of the input. The second option incorrectly assigns additional properties not evident from the definition. The third option incorrectly claims causality, ignoring the dependence on future values. The fourth option is unsupported without information about input or system behavior with respect to stability.

4. Impulse Response and Stability

   For a linear time-invariant (LTI) system with impulse response h[n], under what condition is the system BIBO stable?

   1. h[n] is always zero for negative n.
   2. h[n] alternates between positive and negative values.
   3. h[n] equals zero for all n.
   4. The sum of the absolute values of h[n] over all n is finite.

   Explanation: For an LTI discrete-time system, BIBO stability requires the impulse response to be absolutely summable, meaning the sum of |h[n]| for all n is finite. Option two refers to causality but not stability. The third option suggests a zero system, which is trivially stable but does not define the general case. The fourth option about alternation does not guarantee bounded outputs.

5. Memory in Systems

   Consider a system where the output y(t) = x(t) + x(t - 1). What type of system is this with respect to memory?

   1. It is a memoryless system because only the current input is used.
   2. It is a system with memory, as the output depends on past input values.
   3. It is a non-causal memoryless system.
   4. It is a system without input dependence.

   Explanation: Since the output depends on both the current input x(t) and the past input x(t - 1), the system possesses memory. The second option is incorrect, because dependence on x(t - 1) introduces memory. The third option misclassifies the system as memoryless and non-causal; however, it uses past inputs and is causal. The fourth option is incorrect, as the output clearly depends on the input signal.