Linearity of Discrete-Time Systems: A Practical Test with Examples

Testing the Linearity of Discrete-Time Systems with Examples

This article explores the concept of linearity in discrete-time (DT) systems. We'll analyze two distinct systems and determine whether they exhibit linear behavior.

Let's consider two DT systems, A and B, defined by the following input/output relationships:

• System A: y[n] = 2^(x[n])* System B: y[n] = nx[n]

Our goal is to determine if Systems A and B are linear. To do this, we'll utilize two fundamental properties of linear systems: superposition and scaling. We'll use two input signals for testing: x1[n] = 0.8n and x2[n] = cos[n], both evaluated for 0 ≤ n ≤ 5.

1. Superposition Principle

A system is considered linear if the output response to a sum of input signals is equivalent to the sum of the individual responses to each input signal applied separately.

Analyzing System A for Superposition:

• Input x1[n] = 0.8n: The output is y1[n] = 2^(0.8n)* Input x2[n] = cos[n]: The output is y2[n] = 2^(cos[n])

To test for superposition, we need to check if the output for the combined input (x1[n] + x2[n]) is equal to y1[n] + y2[n]. In this case, it's not immediately clear, and further analysis (potentially numerical) would be needed to confirm or deny linearity based on superposition.

Analyzing System B for Superposition:

• Input x1[n] = 0.8n: The output is y1[n] = n * (0.8n) = 0.8n^2* Input x2[n] = cos[n]: The output is y2[n] = n * cos[n]

For System B, if we apply the combined input (x1[n] + x2[n]), the output is y[n] = n * (0.8n + cos[n]). This is indeed equal to y1[n] + y2[n], suggesting System B might satisfy the superposition principle.

2. Scaling Property

A system exhibits linearity if scaling the input signal by a constant factor results in the output signal being scaled by the same factor.

Analyzing System B for Scaling:

We'll focus on System B and the input x1[n] = 0.8n, which yields the output y1[n] = n * (0.8n).

If we scale the input by a constant, for example, 2, the new input becomes 2 * x1[n] = 1.6n. This produces the output y[n] = n * (1.6n) = 1.6n^2, which is exactly 2 * y1[n].

This demonstrates that System B adheres to the scaling property.

Important Note: While this analysis provides a strong indication, rigorously proving linearity requires a more formal mathematical approach and consideration of all possible input signals. Software tools like MATLAB or Python with numerical libraries can be helpful for evaluating and visualizing system responses for a range of inputs.

This article offered a practical approach to understanding and testing the linearity of discrete-time systems using illustrative examples.