Final answer:
The discrete-time system y[n] = x[n]x[n-1] + 5n²x[n-3] is not linear because it does not satisfy the property of superposition, it is not time-invariant due to the n² term that depends on the specific time index, but it is causal because the output depends only on present and past inputs.
Step-by-step explanation:
We are discussing the properties of a discrete-time system represented by the equation y[n] = x[n]x[n-1] + 5n²x[n-3]. Specifically, we will analyze its linearity, time-invariance, and causality.
Linearity
A system is linear if it satisfies two properties: additivity and homogeneity (scalability). To test this, we see if the system satisfies superposition. Let's take two inputs x1[n] and x2[n] and scalar constants a and b. If y[n] is the output corresponding to ax1[n] + bx2[n], we must have:
a·y1[n] + b·y2[n] = a·(x1[n]x1[n-1] + 5n²x1[n-3]) + b·(x2[n]x2[n-1] + 5n²x2[n-3])
However, the left side would actually be:
(ax1[n] + bx2[n])(ax1[n-1] + bx2[n-1]) + 5n²(ax1[n-3] + bx2[n-3])
This shows that the system does not satisfy superposition due to the multiplication of x[n] with x[n-1], hence, it is not linear.
Time-Invariance
A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. Mathematically, if x[n] → y[n], then x[n-k] → y[n-k] for any integer k. Shifting the input by k units gives:
x[n-k]x[n-k-1] + 5(n-k)²x[n-k-3]
Comparing this with y[n-k], the term 5(n-k)² introduces a time variance, which is dependent on n. Hence, the system is not time-invariant.
Causality
A system is causal if the output at any time depends only on the present and past input values. Since y[n] depends on x[n], x[n-1], and x[n-3], the output does not depend on future values. Therefore, the system is causal.