Linearity check for discrete-time systems: Which of the following systems is linear under superposition?

Introduction / Context:Determining linearity is fundamental in signals and systems. A linear system obeys the principles of superposition and homogeneity: the response to a_1 x_1[n] + a_2 x_2[n] must equal a_1 y_1[n] + a_2 y_2[n].

Given Data / Assumptions:

• We test three candidate input–output relations.
• Operations include addition, time shift, multiplication, and squaring.
• Zero initial conditions are assumed (standard linearity test context).

Concept / Approach:

Linearity is preserved by addition and time-shifting but not by multiplication or nonlinear operations like squaring. Therefore we can test each definition against linearity axioms.

Step-by-Step Solution:

Verification / Alternative check:

Apply a_1 x_1 + a_2 x_2 to (b): y = (a_1 x_1 + a_2 x_2) + (a_1 x_1[n−10] + a_2 x_2[n−10]) = a_1 (x_1 + x_1[n−10]) + a_2 (x_2 + x_2[n−10]) = a_1 y_1 + a_2 y_2, confirming linearity.

Why Other Options Are Wrong:

• (a) Nonlinear due to multiplication of input terms.
• (c) Nonlinear due to squaring.
• (d) Incorrect because neither (a) nor (c) is linear.
• (e) Incorrect; (b) is linear.

Common Pitfalls:

• Assuming any time shift or addition is nonlinear—it is linear.
• Overlooking that multiplication of inputs (even with delays) breaks linearity.

Final Answer:

y[n] = x[n] + x[n − 10]