Final answer:
The characteristics of the given systems vary: System a is non-linear, time-invariant, memoryless, and causal but not BIBO stable. System b is linear, time-invariant, has memory, and is causal and potentially BIBO stable. System c is time-varying, memoryless, non-causal, and not BIBO stable.
Step-by-step explanation:
The student's question revolves around determining the characteristics of given systems such as linearity, time-invariance, memoryless property, causality, BIBO (Bounded Input Bounded Output) stability, and invertibility. Let's analyze each system one by one.
- a. y[n] = -x[n]⁴: This system is non-linear due to the input being raised to the power of four. It is time-invariant as there is no explicit time dependence that would change if we shifted the input in time. the system is memoryless because the output at time n depends only on the input at the same time n. it is causal as the output only depends on the current and past input values. this system is not BIBO stable because a bounded input could lead to an unbounded output due to the power of four nonlinearity. Invertibility cannot be determined without additional context or constraints as multiple inputs can lead to the same output.
- b. y(t) = ∫ᵗ⁻¹∞ x(τ)dτ: This system is linear as integration is a linear operation. It is time-invariant if the limits of integration change with the shift in the input signal. It has memory as the output depends on past values of the input from time t-1 to infinity. it is causal if we assume the system starts operating at t-1. It is BIBO stable if the input is bounded and absolutely integrable over time. Invertibility depends on the nature of the input signal.
- c. y(t) = x(1 - 5t): This system is time-varying as the input is scaled by a changing factor with time. It is memoryless because the output at any time t depends only on the input at that same time t. It is non-causal because the output depends on future values of the input due to the term 1-5t, which could be positive for t < 1/5. the system is not BIBO stable since for t close to 1/5, small changes in input can result in large changes in output. invertibility can be argued if a unique input can be obtained for a given output, but the time-varying nature makes this unlikely.