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:
System (a): y[n] = x[n] × x[n − 1]. Multiplication of input samples violates superposition; scaling an input by a factor does not scale the output linearly. Hence nonlinear.System (b): y[n] = x[n] + x[n − 10]. This is a linear time-invariant operator (sum of identity and a time shift). It satisfies both additivity and homogeneity.System (c): y[n] = (x[n])^2. Squaring is nonlinear; scaling the input by α scales the output by α^2, violating homogeneity.
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]
Discussion & Comments