216k views
1 vote
Discuss linearity, time-invariance and causality of the following system. y[n]=x[n]x[n−1]+5n²x[n−3]

1 Answer

4 votes

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.

User Tunaranch
by
7.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories