How do you define a linear system? If I remember correctly it is linear if the output of the system changes proportionately to the input of the system? Also it must have finite limits?
Is this correct? For the life of me I cant remember the simple rules of a linear system and how to test for linearity.
A linear system is one which can be mathematically modeled by an equation of the form:
y = ax + b
where a and b are constants. Because the highest power that x is raised to is the first, this is considered to be a linear equation (directly or inversely proportionate, just like you said)
If the system was modeled by:
y = ax[sup]2[/sup] + bx + c
then it would be a second order or quadratic equation, it continues on from here…
Stinkpalm, I too have the feeling that I’m forgetting some important basic properties, but your list should include:
BIBO - if the input never reaches infinity, the output shouldn’t reach infinity (you’ve already alluded to this)
‘transitivity’ (I’m drawing a blank for the correct terminology) basically a system is linear if f(x+y) = f(x) + f(y), basically you can evaluate each input separately. This can easily be used to disprove linearity.
Proving linearity requires proof of the above two for all inputs and a few other I believe (yeah, that’s a whole lot of help I know).
I’ll post more if I recall them.
IIRC, a system must meet two criteria for all inputs before it is considered linear: additivity and homogeneity. Additivity means that if an input x[sub]1/sub to the system yields an output y[sub]1/sub, and an input x[sub]2/sub yields an output y[sub]2/sub, then (x[sub]1/sub+x[sub]2/sub) will yield y[sub]1/sub+y[sub]2/sub. Homogeneity means that multiplying the input by a scalar value will produce an output that is multiplied by the same scalar, or 5x(t) --> 5y(t).
It is important to note that in this context the term “linearity” refers to the behavior of the system and not to the equation that represents or models it. For example, the system represented by y(t) = 5x(t) + 2 is not linear, because the system fails the additivity test. And the BIBO criterion is used to determine the stability of a system, which is different from linearity. It is quite possible to design a system that is linear but unstable.
If we’re talking about the “linear systems” found in chaos theory, then headshok’s definition is the correct one. I’m gonna double-check it with a book I have at home, though–just to be sure.
Just chiming in again to say that headshok’s got it. Probably not the text ultrafilter was checking up in, but Alan Oppenheim’s Discrete-Time Signal Processing confirms the requirements of additivity and homogeneity, as well as my mistake with BIBO.
FWIW, Mirage posted the definition of a linear equation.
Yeah, headshok is right, more or less. The definition I have is phrased in terms of the solutions to a system of differential equations, but it works out to be the same, I think.
A linear system is any system that conforms to the superposition principle. That is, when the input is increased by a multiplying factor K, the output is also multiplied by K, and if two inputs are applied simultaneously, the total response is the sum of the individual responses.
For example,
if x is the input, and y is the desired output,
x -> LINEAR SYSTEM (LS) -> y
K1*x1 -> LS -> K1*y1
K2*x2 -> LS -> K2*y2
K1x1 + K2x2 -> LS -> K1y1 + K2y2
where K1, K2 are constants
You can ALWAYS use this test to determine if an expression is linear.
or… yeah, what headshok said…
headshok could you explain why
y(t) = 5x(t) + 2 fails the additivity test?
This post is by RonT not Ring.
If x[sub]1/sub = 2 and x[sub]2/sub = 2, then y[sub]1/sub = 12 and y[sub]2/sub = 12. However, if x(t) = 4, then y(t) = 22, not 24.
Ah. So even though the equation would be defined as linear the system modeled by this equation is not. Is this correct?
If x = 1 then y = 7 or (1,7) = u
If x = 2 then y = 12 or (2,12)= v
And since k*(u + v) = ku + kv the equation must be linear?
This post is by RonT not Ring
See Mirage’s post for the definition of a linear equation.