I have a feeling that if you can do that, you might get a Nobel Prize or something similar.And the definition is about the two properties?
I am not satisfied with this definition. I understand that it works but I want another one.
What if we say that the input x(t) = x*f(t), where x is the physical variable like voltage, and f(t) is a varying function. For example: x(t)=Vcos(ft), where x = V, f(t) = cos(ft).I have a feeling that if you can do that, you might get a Nobel Prize or something similar.
I am still surprised you have not opened your textbook.
You are definitely missing the entire point. Do the math like I suggested.V1 = A*u(t)
V2 = B*u(t)^2
I do not get it where you are trying to go with this.
The first system is linear and we can verify this by checking the scaling property and additivity property. Because it is defined like it is defined, then for every input, the output will have the form A*Vin.
The second system is not linear.
But I am talking about something else. I do not know a priori how the system is defined. I just have a system, a black box, to which I input the unit step function and I observe the response. And I see that the response is such that the system is linear. Can I conclude based on this that the system is linear for any input? In other words, is the linearity of the system independent on what input I apply?
Sure they can! And a linear and non-linear system can produce the same output for a particular input. For a specific input, you can design a literally infinite number of systems that will produce exactly the same output.My black box is a system described by a differential equation relating the input to the output.
So, if I change the input waveform, then the differential equation, i.e. the model, is the same. Only the output changes.
If the differential equation is a linear ODE, then the system is linear, no matter what input I apply.
And suppose that I do not know that the system is linear or not. I apply the unit step function and see the response. Having the input and the output in hand, surely I can deduce how the differential equation looks and if it is a linear ODE, and thus deduce if the system is linear. Right?
I do not think that for the same input, two different differential equations can produce the same output.
A system is linear if and only if it obeys superposition. NamelyThe usual RC series circuit where the input is a voltage source and the initial voltage on the capacitor is 0... this circuit is definitely linear. I am not saying this. Is on the internet. It is linear in the sense that it satisfies the additivity and scaling properties.
If the initial voltage on the capacitor is not 0, then it is not a linear circuit anymore.
I am not trying to develop a new language. I am talking about the linear from linear, time-invariant systems.
Are there LTIs where the gain is not linear?
If yes, I can see that the "linear" word is used differently depending on the context.
Notice what they are saying and what they are not saying.Ok. I understand.
The thing that bugs me is that in this edx course, they say that the response of a system to the unit step function is a(n)*u(n).
And then they say that if the system is LTI then i know the response of the system to any linear combination of unit step functions.
Not based on a single input.So if a system is LTI or not depends on how the output waveform looks like.
Nope. Because a linear system's behavior depends on how it responds to different inputs. A single input tells you pretty much nothing.So there must be a general form for the output waveform of an LTI.
You need to make up your mind -- one instant you are talking about black boxes in which you don't know what the system looks like and now you are talking specifically about the case where you are relying on having full knowledge of what the system looks like. Which is it?(I say 'observe' but i'm not referring to viewing the output on the oscilloscope, but how its formula looks like. So Au(t) is different than Bu(t)^2 )
That is correct. But why do you say this?Hi,
An equation is linear if the ratio of the difference between any two x values and the difference between the corresponding two y values is the same for any two points, or:
for any two points (x1,y1) and (x2,y2), and K being a constant.
The function A*sin(w*t) is non linear in time, but it could be considered linear in A.
A linear system would satisfy:
The sum of the function of the individual inputs equals the function of the sum of the individual inputs, and
Multiplication of the input by a factor X results in a multiplication of the output by the same factor X.
This makes it clear that A*sin(w*t) cant be linear in time but it can be linear in A. It might also be called linear in sin(w*t) even though sin(w*t) is not considered linear itself.
So besides other things, we have to know what the independent variable is.
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by Jake Hertz
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