# How is this system nonlinear

Suppose we have a system S and the relationship between $$x(t)$$ and $$y(t)$$ is $$y(t) = x(t)+1$$

However despite the fact the system is made of linear operators the system is not linear.How is this possible?

Can we have a system made of nonlinear operation but the system becomes linear?

• Adding a constant is not a linear map. It's affine. Your specific system is easily seen to be non-linear as it maps 0 to something other than 0. Commented Sep 8, 2022 at 22:15
• What is a linear map , please explain. Commented Sep 8, 2022 at 22:17
• en.wikipedia.org/wiki/Linear_map Commented Sep 9, 2022 at 7:15
• Sorry, but I'll be repeating my previous comment: you are not going to learn this stuff by asking one basic question at a time. Maybe something like allsignalprocessing.com/lessons/… ? Commented Sep 9, 2022 at 12:11

## 3 Answers

This system may cause a hallucination because of its similarity to the algebraic equation $$y = x + 1$$ which is a linear one indeed as stated in the wikipedia link to System_of_linear_equations

However, signal processing defines linearity according to system theory which defines linearity as an input-output relationship (a.k.a. mapping) possesed by a system $$T\{\}$$ so that

$$T\{ a ~x_1(t) + b ~x_2(t) \} = a ~T\{ x_1(t) \} + b ~T\{ x_2(t) \} \tag{1}$$

is satisfied for every input $$x_1,x_2$$ and every complex constants $$a,b$$.

One can easily verify that Eq.1 is not satisfied by a system whose input output relation is: $$y(t) = x(t) + 1$$, simply because of the fixed-constant $$1$$ present at the output for any input, even for $$x(t) = 0$$ for all $$t$$.

Note that this system is also called incrementally linear, which can be very easily linearized, so that another system defined as $$y_2(t) = y(t) -1 = x(t)$$ will be linear.

• "... also called incrementally linear ..." or "affine" -- I'm pretty sure they mean the same thing. Commented Sep 9, 2022 at 0:26
• @TimWescott affine if you prefer a mathematical jargon, incrementally linear if signal processing... I prefer the latter, as the affine may refer to different mathematical constructs than system input-output relations. Commented Sep 9, 2022 at 0:43
• It doesn't matter if you call it an algebraic equation or not, $y=x+1$ is not linear. Its graph in a cartesian coordinate system is a line, but that's not the same thing as the equation being linear. Commented Sep 9, 2022 at 7:13
• @Jazzmaniac $$y - x = 1 \\ y + x = 2$$ is a set of linear equations. Commented Sep 9, 2022 at 15:03

A linear system means that the system's output should scale with the input, and the system's output should combine given two inputs. Look at the scaling property, you scale the input and this results in an output:

$$y_1(t)=\alpha x(t)+1$$

and now we need to test is this equal to just scaling the output by $$\alpha$$? No, it is not because if you simply scaled the output the result is:

$$\alpha y(t)=\alpha (x(t)+1)=\alpha x(t)+\alpha$$

Since $$y_1(t) \neq \alpha y(t)$$, this is enough to say the system is non-linear and you can do another similar experiment to show the summing of two inputs does not give you the sum of the two outputs. Here is the result of combining two inputs:

$$y'(t)=(x_1(t)+x_2(t))+1$$

and here is the result of combining two outputs:

$$y_1(t)+y_2(t)=x_1(t)+1+x_2(t)+1=x_1(t)+x_2(t)+2$$

Since $$y_1(t)+y_2(t) \neq y'(t)$$, the system does not have the combining property either.

Paraphrasing Fat32’s answer: in signal processing, and other linear systems contexts, a linear system is one that satisfies superposition.

One component of superposition is homogeneity: for the system

$$y = f(x)$$

we must have that

$$f(0) = 0$$

Screenshot from linked page:
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# How is this system nonlinear

Suppose we have a system S and the relationship between $$x(t)$$ and $$y(t)$$ is $$y(t) = x(t)+1$$

However despite the fact the system is made of linear operators the system is not linear.How is this possible?

Can we have a system made of nonlinear operation but the system becomes linear?

• Adding a constant is not a linear map. It's affine. Your specific system is easily seen to be non-linear as it maps 0 to something other than 0. Commented Sep 8, 2022 at 22:15
• What is a linear map , please explain. Commented Sep 8, 2022 at 22:17
• en.wikipedia.org/wiki/Linear_map Commented Sep 9, 2022 at 7:15
• Sorry, but I'll be repeating my previous comment: you are not going to learn this stuff by asking one basic question at a time. Maybe something like allsignalprocessing.com/lessons/… ? Commented Sep 9, 2022 at 12:11

## 3 Answers

This system may cause a hallucination because of its similarity to the algebraic equation $$y = x + 1$$ which is a linear one indeed as stated in the wikipedia link to System_of_linear_equations

However, signal processing defines linearity according to system theory which defines linearity as an input-output relationship (a.k.a. mapping) possesed by a system $$T\{\}$$ so that

$$T\{ a ~x_1(t) + b ~x_2(t) \} = a ~T\{ x_1(t) \} + b ~T\{ x_2(t) \} \tag{1}$$

is satisfied for every input $$x_1,x_2$$ and every complex constants $$a,b$$.

One can easily verify that Eq.1 is not satisfied by a system whose input output relation is: $$y(t) = x(t) + 1$$, simply because of the fixed-constant $$1$$ present at the output for any input, even for $$x(t) = 0$$ for all $$t$$.

Note that this system is also called incrementally linear, which can be very easily linearized, so that another system defined as $$y_2(t) = y(t) -1 = x(t)$$ will be linear.

• "... also called incrementally linear ..." or "affine" -- I'm pretty sure they mean the same thing. Commented Sep 9, 2022 at 0:26
• @TimWescott affine if you prefer a mathematical jargon, incrementally linear if signal processing... I prefer the latter, as the affine may refer to different mathematical constructs than system input-output relations. Commented Sep 9, 2022 at 0:43
• It doesn't matter if you call it an algebraic equation or not, $y=x+1$ is not linear. Its graph in a cartesian coordinate system is a line, but that's not the same thing as the equation being linear. Commented Sep 9, 2022 at 7:13
• @Jazzmaniac $$y - x = 1 \\ y + x = 2$$ is a set of linear equations. Commented Sep 9, 2022 at 15:03

A linear system means that the system's output should scale with the input, and the system's output should combine given two inputs. Look at the scaling property, you scale the input and this results in an output:

$$y_1(t)=\alpha x(t)+1$$

and now we need to test is this equal to just scaling the output by $$\alpha$$? No, it is not because if you simply scaled the output the result is:

$$\alpha y(t)=\alpha (x(t)+1)=\alpha x(t)+\alpha$$

Since $$y_1(t) \neq \alpha y(t)$$, this is enough to say the system is non-linear and you can do another similar experiment to show the summing of two inputs does not give you the sum of the two outputs. Here is the result of combining two inputs:

$$y'(t)=(x_1(t)+x_2(t))+1$$

and here is the result of combining two outputs:

$$y_1(t)+y_2(t)=x_1(t)+1+x_2(t)+1=x_1(t)+x_2(t)+2$$

Since $$y_1(t)+y_2(t) \neq y'(t)$$, the system does not have the combining property either.

Paraphrasing Fat32’s answer: in signal processing, and other linear systems contexts, a linear system is one that satisfies superposition.

One component of superposition is homogeneity: for the system

$$y = f(x)$$

we must have that

$$f(0) = 0$$

Screenshot from linked page:
user
The content structure of the passage is a discussion about the linearity of a system in the context of signal processing, with a specific example of a system that is not linear despite using linear operators. It includes a question about the nature of the system and several answers explaining why the system is nonlinear.

TYPE: NO_QUESTION

However, upon closer inspection, the passage does contain a question in the form of "How is this possible?" and "Can we have a system made of nonlinear operation but the system becomes linear?".

Revised analysis:
TYPE: MULTIPLE_QUESTIONS

The passage contains multiple questions:
1. "How is this possible?" - This question is asking for an explanation of why the system is nonlinear despite using linear operators.
2. "Can we have a system made of nonlinear operation but the system becomes linear?" - This question is asking if it is possible for a system composed of nonlinear operations to become linear.

No answers to these specific questions are provided in the text, but the comments and answers do provide explanations and additional context.