Hi,

I thought i saw a place in the thread where you had a linear equation that you didnt think was linear so i thought i would throw out some definitions that might help.

For example, the capacitor:
dv/dt=i/C

or when charging:
dv=i*dt/C

would be linear in 'i', so with a constant current we'd get a ramping voltage.

On the other hand:
dv=(E-Vc(t))/RC

where E is a DC voltage source and Vc(t) the changing capacitor voltage, that would come out looking exponential:
Vc(t)=E*(1-e^(-t/RC))

But anyway, the idea of using:
(x2-x1)/(y2-y1)=K

for any *TWO* points, tells us that we do need at least two inputs and two outputs to check for linearity. Of course that may not be enough either though, because really this equation has to be satisfied for ANY two points on the curve, not just one particular set. There may be a time when we can get one particular set that satisfies this, but then picking a new, different set shows non linearity. So the more points tested the better, at least for a truly general test that is. For example for the equation described by two conditions:
y=x {0<=x<=5}
y=x+1 {5<x<10}

for any two points inside the interval from 0 to 5 inclusive would show complete linearity, while for anything between 5 and 10 exclusive would show non linearity.

If you already knew this that's fine, but you also need to apply it

If the system is already known to be linear you can use the impulse response.

 

Hi,

I thought i saw a place in the thread where you had a linear equation that you didnt think was linear so i thought i would throw out some definitions that might help.

Are you referring to y(x) = A * dy(x)/dx ?

And that it is a linear function in A?

I did not say that this equation is not linear. I said that the function is not linear.
The equation mapping one function from another function is indeed linear.

 

Consider the function:
y(x) = A*x^2

Then this function is linear in A and not linear in x. But what is a function? A function is something that takes in some argument. When we say that a function is linear in A, then we automatically think of the function:
y(A) = A*x^2 and definitely not y(x) = A*x^2

Or, to see more clearly, changing x with A and A with x:
y(x) = x*A^2

Where A^2 is some constant (you must choose one argument) => for A^2->A for simplicity, we have y(x) = x*A = Ax

This is similar for:
y(x) = A*dy(x)/dx
which is y(A) = A*dy/dx = A*constant or changing the notation, we have y(x) = x*constant

If he meant for y(x) = A*dy/dx to be a linear function, then this is equivalent to y(A) = Ax. And he said that it is not the same thing. But it is the same thing if you choose A to be your argument.
If, on the other hand, you choose x to be your argument, then this is not a function! This is a linear equation where the unknown is a function y(x), and the result is y(x) = c*e^x/A.
That is an exponential, not a linear function.

We could also say that the equation is a function in x, because its solution is a function, but it is an exponential function. We can say that because this equation maps a function to another function. And one of the functions was chosen to be 0 and the other one is y(x).

Either way, he got it wrong. If the argument is A, then the form is y=ax. If the argument is x, then the function is exponential, not linear.

If there is still confusion between functions and equations, or between signals and systems, I recommend this video:

The general form of a linear function is y(x) = ax. That is, in the engineering sense. In math, the general form is y(x)=ax+b


PS: I don't want to sound mean, I'm sure he knows a lot of stuff.

 

Hi,

Well sometimes you have to read into it, take the true meaning from the context.
For example, suppose i say to you:
"The sky is blue."

What do you think of immediately? (Be honest or just think to yourself)

If we were standing outside, you'd think, "Oh, he must mean the sky above is blue, which it is".
On the other hand, if we were standing inside a museum, i could very well be talking about the sky in a painting on the wall.

In either case you would know what 'sky' i was talking about because we where in a place where you could draw a conclusion based on background information that although was not specifically stated is apparent.

So if an author was talking about a linear equation, they might write:
"Here is a linear equation: u=A*x^3"

and that might seem odd but if we believe the the author then we have to take it as linear in A. If we dont believe the author then we would think they made a mistake.
So i guess we have the right to question it, but once the result is confirmed then we are forced to believe the author and realize that it must be linear in A.

The word "Linear" has so many interpretations that i hesitate to get involved in these discussions anymore.
A simple example:
H=(s^2+s+1)/(5*s^3+2*s^2+s+3)

is often considered linear, yet on inspection we dont see anything immediate that would tell us this is a linear system.

 

Well, I don't believe that math should be subjective like the example with the sky is, but only objective. Math is math. Math is not poetry.

Regarding the different interpretations of the word "linear" depending on the context, well, this is a problem created by humans for humans due to their lack of creativity. We should give different names for different things even if they have meanings very close one to another. I really believe that this is a problem with how information is communicated today. When it comes to symbols, math has been made very rigorous in the last hundreds of years and this is good, but when it comes to words, it is less rigorous. I firmly believe that the words are very important too. And the way they are placed in a story is also important. Words can add so much to the understanding if they are chosen in a wise way. You could make math seem more easy and in the same time don't change how rigorous it is. Maybe in the next hundreds of years.

One example that I can think about is how linear in math is different than linear in engineering. Linear in math means y=ax+b. In engineering (but also in linear algebra) is y=ax. And that is because of the superposition principle. Why not name m-linear and e-linear. I don't know.

About the linear equation. I'm okay with an author giving me y=A*x^3 as a linear equation. Maybe it is a little unusual because the author should stick to the convention used by everyone else, but I'm okay with that if he specifies what he is talking about (of course, maybe that is the first time I see an equation and I will not understand what he is saying about the little note and his own convention, and then when I will read another book, I will be confused because this other author does not say anything about his convention).

Or, maybe, it could be the case that he talked a whole chapter about the equation y=A*x^3 being non-linear in x, and then the next chapter he talks about the same equation being linear in A. So, of course, he will not change the notation. He will just state that the equation is linear in A.

But I am not talking about linear equations. I am talking about linear functions. If an author gives me f(x)=A*x^3 and says that this function is linear in A, then he is wrong. The function f(A) is linear in A. The function f(x) is totally different than f(A). These are two different functions and we can see that by plotting their graphs. f(x) is not linear in A.
Having the equation y=A*x^3, I can define two functions: one linear y(A), and one non-linear y(x). But also A(y), or x(A), etc. These are all different functions.
And because you can have a lot of different functions from a simple equation with some variables, then the way you specify f() is very important.

I admit that some author could talk a whole chapter about the function f(x). And then the next chapter he could say that this function is linear in A. It could happen. It would be something wrong, but it could easily happen. I say that it is something wrong because a function is a relation, a dependence of one variable to another variable. Either you say "function" or "variable1-variable2" is the same thing. And if you say that "y-x" is linear in A, is nonsensical.

 

To what I believe to be the main question:

Or if a system is linear, then it is linear, no matter what input I apply to it? Is the linearity of the system independent of the input?

I would say yes, a linear time invariant system is linear regardless of the input. This can be proven mathematically. But in this case you've defined the system as linear, not just tested it with some cases. If your "black box" is really a transfer function or ODE that describes an LTI system, then it's not really a black box. It's an LTI system.

If your "black box" is really completely unknown, you might see what looks like an LTI system response for some input, but I don't see how you could ever prove it. A person could certainly create a black box system that behaves like an LTI system with some inputs and not others.

Edit...I just noticed a couple more posts while I was typing this one -- need to pay more attention to the entire thread... Seems like there is a serious disconnect in the discussion between linear functions, equations, systems...I assume we are talking about linear time invariant systems that satisfy scaling and superposition for any combination of inputs. They are defined by transfer functions which have polynomials of varying order. The transfer functions are certainly not straight lines that go through the origin on a graph.

 

To what I believe to be the main question:



I would say yes, a linear time invariant system is linear regardless of the input. This can be proven mathematically. But in this case you've defined the system as linear, not just tested it with some cases. If your "black box" is really a transfer function or ODE that describes an LTI system, then it's not really a black box. It's an LTI system.

If your "black box" is really completely unknown, you might see what looks like an LTI system response for some input, but I don't see how you could ever prove it. A person could certainly create a black box system that behaves like an LTI system with some inputs and not others.

Ok. So for a black box, I can never know if it is a real LTI or not. I understood this.

But, in the same time, I can never choose the output of a black box (that I want to believe is an LTI) in response to the unit step function to have the form: something + constant. If I want to model the output response and in the same time I want the system to be LTI, then this constant must be 0. If I must model that constant too (because it might be some noise) then I model the whole circuit as a black box (which I want to believe is an LTI) and an adder that adds the noise separately (this if I am okay with the noise being an additive noise source). I could also view this whole thing as a bigger black box but this black box could never be an LTI because of the noise. The smaller black box, on the other hand, could be a potential LTI.

Edit...I just noticed a couple more posts while I was typing this one -- need to pay more attention to the entire thread... Seems like there is a serious disconnect in the discussion between linear functions, equations, systems...I assume we are talking about linear time invariant systems that satisfy scaling and superposition for any combination of inputs. They are defined by transfer functions which have polynomials of varying order. The transfer functions are certainly not straight lines that go through the origin on a graph.

Yes. LTI that satisfy the superposition principle.
But I am not talking about the transfer function being straight lines or not. Their linearity is different than the linearity of the LTI. Again, the same word is used but they have different uses in this context. We can easily see this because, as you said, we can have a linear system whose transfer function is not linear, but whose ODE is linear. And this because the transfer function comes from the ODE where the degree of the polynomials is decided by the derivative of the input and output, i.e. if its first-order ODE, second-order ODE, etc.
I am referring to the linearity of Linear TIs.