Consider a discrete-time system. We apply the unit step function to it. And we observe a response. And we conclude based on the response that the system is linear. For example, we see that the response is s(n) = constant * u(n). So it is linear. Based on this observation, can we conclude that the system is linear for all possible inputs? For example, if I apply the sine wave, would I obtain a response from which I can conclude too that the system is linear? Put it in other words, are there systems that are linear for only particular inputs? 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? To me it seems like yes. It is independent. But I would want some proof, or some little logic that leads to this conclusion. (and maybe besides the fact that a system is described by a differential equation relating the input and the output...so no matter what input you choose, the differential equation is the same)
What is your definition of "linear?" Do you mean that the gain is linear (multiplies the input by some constant value) to generate the voltage of the output. Then No. First, you need two points to even start assuming there is a linear response. If you have reason to assume a linear circuit (I.e. Linear gain), then there is a possibility that non- linearity occurs at the extremes of voltage near the rails (ground and Vcc). Essentially clipping and distortion.
The textbook should have a procedure on how to show that the system is linear. My textbook, Signal Processing & Linear Systems by B. P. Lathi covers this topic on page 79, and there is example on page 81.
Hi, Well, if you define linear as K*input then that's linear, period. That would be implying a strict definition of linear as in a purely theoretical system study. When you think about it, why would it be anything else when you have already defined it to be linear. The strict definition of linear is multiplication of the input by a factor A leads to an output that is also multiplied by the same factor A, and also we have zero output for zero input.. So if we have 5 as input and 10 as output, if we increase 5 to 10 we get 20 on the output. And if we have 0 input then we have 0 output. But in systems sometimes we are dealing with a changing input, and the output has to be able to follow that input to within some error limit. These systems are often called linear too, even though they dont respond in quite the same way. When we get to the practical circuit all kinds of stuff creeps in. The power supply limits are one of the most important because if we drive the input of the system by a voltage V and then increase it by a factor A and that would result in a theoretical output of 20 volts and we only have a 10 volt power supply, then the system saturates and does not obey the definition of linear anymore.
Not necessarily K*input. I just gave a particular example. Another example could be the RC series circuit which is linear and the response to the unit step has the form: a(t)*u(t). But is linear because it is not of the form a(t)*u(t) + constant. (If the capacitor would have had an initial charge on it, then that "constant" would have had appeared there, and the circuit would have not been linear anymore). I do not want to go into the practical realm. I only talk about theory here. In other words, I have this RC series circuit. I apply the unit step function to it, and I calculate the response and see that the circuit is linear. Can I conclude from this that the RC series circuit is linear for any input?
To quot my textbook: "If cause1-->effect1 and cause2-->effect2, then for all constants k1 and k2 k1c1+k2c2-->k1e1+k2e2" The point is that linearity has two properties: 1) scaling 2) additivity Right now you are using one input, the unit step. By doing this you can show the scaling property. But one input does not allow to show additivity. For that you need at least two inputs. If you can show the scaling and additivity of a system, then you prove that the system is linear. If you don't show both, then system is not linear. In other words. YOU MUST SATISFY BOTH PROPERTIES. Have you satisfied both properties?
How does this tell you that it's linear? Consider two systems: V1 = A·Vin V2 = B·(Vin)² A = 1 V/V B = 1 V/V² You now have an input Vin = Vo·u(t) If Vo = 1V, what is the output from both systems?
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?
To show scaling property, you would need two inputs: x(t) and c*x(t). You could say is the same input but scaled. But they are two inputs. I only apply one input to the black box, only x(t) and observe y(t). Just that.
No. All input values that ate important to you must be tested. If it is truly a Black Box, then it may contain a combination of linear and step function transformations. Besides just looking at the transfer function of the black box by applying various DC voltages to the input or a single sine wave input to the black box, you have to look at the dependence on frequency there may be high pass, low pass or band pass or notch filters (intentional or accidental) and these are all going to change the value of the output to make it nonlinear. Depending on the control system in the black box, literally anything can happen. Look at the programming that VW added to their diesel vehicles. Their black box guesses when the input was actually a regulatory test vs a driver going to the store- outputs were adjusted to change the performance (and emissions) of the vehicle. When you say something is a "Black Box" then you have to put some limits on the capabilities of the box. Is it a linear amplifier, is it a student project the sometimes generates a random number and sometimes generates a linear response over some frequency range? Almost anything can be created with electronics. Your guess of, yes, depends on how you describe what your imaginary black box can do. So, feel free to make your prediction true by describing the capabilities of the black box as you want.
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.
Either you can demonstrate the two properties of linearity or you can not. This is all textbook material, there is no need to even discuss it.
What do you mean by there is no need to even discuss it? In my personal opinion, textbooks are not teaching facts in an euclidean, socratic way. They are merely presenting facts. There is a big difference between teaching facts and presenting facts. Or, as Feynman put it, there is a big difference between knowing something and knowing just the name of something. How would you demonstrate those two properties of linearity given that you have at your disposal just only one input signal? Or how would you demonstrate that you cannot demonstrate those two properties given only one input signal?
I am not talking about linear gain. I think our definitions of linearity are different. Take for example the RC series circuit where the sine wave is applied at input. The response will be: V*f(frequency). This is linear circuit, independent of the frequency. If the capacitor had an initial voltage on it, then the response would have been: V*f(frequency)+something. This is not linear.
Maybe you have to explain the goal of your search for enlightenment. The problem to this point is that you are using the term "linear" and linear already has a definition in electronic amplifiers, essentially a fixed gain. If you want to discuss something else, then use an appropriate term. Also, clarify what feature if the black box's output you are looking at - voltage, frequency, phase angle... And what input are you looking at for the black box (voltage, frequency, ... Finally, you mention an "RC series" circuit. You will have to draw that for me, I would call an RC a filter and I would take the signal from the RC junction (therefore, not series). Also, I don't know of any RC circuits that are linear. They are normally plotted on Graphs with Log-axis to make them appear linear but, the need for a log axis should prove to you that they are not linear. If you are trying to develop a new language and make a new definition for the word in that language that is a false cognate for the English word "Linear", then feel free to do so. On the other hand, if you already have a preconceived notion of what "linear" means and we cannot teach you the meaning that the rest of the world agrees is the meaning of linear, let is know so we can stop trying.
The 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.
Apply a second input and demonstrate that you have additive property in addition to scaling property. Let me put it this way. Linearity has a definition. Your work so far does not meet that definition. End file.
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. Consider the function f(x)=ax. This function is linear because it looks like the way it looks. Now because it looks like so, we can see that it satisfies the two properties. But the definition of the linear function is f(x)=ax. With this definition I am satisfied. And from this definition, the two properties arises. We could have gone the other way around, of course, and say the definition is about the two properties. And from these two properties, we see that the function must have the form f(x)=ax. I want a similar definition for the system. And from this definition, I want the two properties to derive from it. Because, if I have this definition, and I apply an input, then I can see that the output is either of the form input*something1+something2 or input*something2. For ex, this something2 could be noise or offset. If I have offset then the system is not linear.