# Show that the system is linear?

In summary, the conversation discusses the linearity of a system with the given equation y(n) = x(n) + 0.8y(n-1). The attempt at a solution involves finding the impulse response and using it to prove linearity, but the issue of assuming linearity is raised. Another idea is proposed to consider the input and previous output values.

## Homework Statement

Is the following system linear:
$$y(n) = x(n) + 0.8y(n-1)$$

## The Attempt at a Solution

I'm really stuck on this one.

My idea was to assume that the system is casual, find the impulse response and then use the fact that: $$y(n)=x(n)*h(n)$$. However, if I assume that the output is the convolution of the impulse response and the input, I have ALREADY ASSUMED (and not proven) that the system is linear.

So I am lost.

I can't wrap my brain around the fact that the output depends on the input and the previous output.

Another idea I had is that if I show that I input a sinusoid and I get a scaled and phase shifted sinusoid, then the system is LTI.

Any help?

What if, assuming y(0) = x(0), you wrote:

y(n) = x(n) + 0.8y(n-1) = x(n) + 0.8x(n-1)... + (0.8)^n x(0),

would that help?

## 1. What does it mean for a system to be linear?

A linear system is one that follows the principle of superposition, meaning that the output is directly proportional to the input. This means that if we double the input, the output will also double. Additionally, the system must also satisfy the properties of homogeneity and additivity to be considered linear.

## 2. How do you show that a system is linear?

To show that a system is linear, we can use the principle of superposition. This involves breaking down the input into smaller parts and analyzing the output of each part separately. If the output of each part is directly proportional to its corresponding input, then the system is linear.

## 3. Can a nonlinear system exhibit linearity?

No, a nonlinear system cannot exhibit linearity. A nonlinear system does not follow the principle of superposition and therefore cannot satisfy the properties of homogeneity and additivity required for linearity.

## 4. Why is it important to determine if a system is linear?

Determining if a system is linear is important because it allows us to make predictions about the system's behavior and performance. If we know that a system is linear, we can use mathematical tools and techniques to analyze and understand its behavior, which can help us in designing and optimizing the system.

## 5. Can a system be linear in one domain but nonlinear in another?

Yes, a system can be linear in one domain but nonlinear in another. For example, a system may be linear in the time domain but nonlinear in the frequency domain. This is because different domains may require different mathematical descriptions and principles to analyze the system's behavior.

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