Let's analyze each system to determine whether it is linear, time-invariant, and causal:
a. y[n] = x[n]cos(0.2πn)
1) Linearity: This system is not linear because it contains a nonlinear operation, the cosine function. When applying a linear combination of inputs, the output will involve the cosine of the linear combination, which does not satisfy the superposition property.
2) Time-invariance: This system is time-invariant because the cosine function does not depend on the specific time instance n. Shifting the input x[n] will result in a corresponding shift in the output y[n].
3) Causality: This system is causal because the output y[n] only depends on the current and past values of the input x[n]. It does not rely on future values.
b. y[n] = x[n] - x[n-1]
1) Linearity: This system is linear because it satisfies the superposition property. Applying a linear combination of inputs will result in a linear combination of outputs.
2) Time-invariance: This system is time-invariant because shifting the input x[n] by a delay will cause the output y[n] to shift by the same delay.
3) Causality: This system is causal because the output y[n] only depends on the current and past values of the input x[n]. It does not depend on future values.
c. y[n] = |x[n]|
1) Linearity: This system is not linear because it does not satisfy the superposition property. Taking the absolute value of a linear combination of inputs does not result in the same linear combination of outputs.
2) Time-invariance: This system is time-invariant because shifting the input x[n] by a delay will result in the same delay in the output y[n].
3) Causality: This system is causal because the output y[n] only depends on the current and past values of the input x[n]. It does not depend on future values.
d. y[n] = Ax[n] + B (A and B are nonzero constants)
1) Linearity: This system is linear because it satisfies the superposition property. Applying a linear combination of inputs will result in a linear combination of outputs.
2) Time-invariance: This system is time-invariant because it does not depend on the specific time instance n. Shifting the input x[n] will not affect the output y[n].
3) Causality: This system is causal because the output y[n] only depends on the current and past values of the input x[n]. It does not depend on future values.
In summary:
a. Not linear, Time-invariant, Causal
b. Linear, Time-invariant, Causal
c. Not linear, Time-invariant, Causal
d. Linear, Time-invariant, Causal