Hello forum, a "philosophical" question about the validity of linear theory. A linear model is usually adopted when the amplitude (strength) of the oscillation, signal is generally "small". How small? Small relative to what? By linearity, the total final response to the sum of separate inputs is equal to the sum of the responses of separate individual inputs. This implies that the responses of the individual inputs don't affect each other....no sort of coupling.... Does any one have some sort of physical justification for this independence of the responses and the fact that the amplitude of the responses need to be small? thanks fisico32
justification of linearity
Started by ●November 6, 2009
Reply by ●November 6, 20092009-11-06
On Fri, 06 Nov 2009 11:16:54 -0600 "fisico32" <marcoscipioni1@gmail.com> wrote:> Hello forum, > > a "philosophical" question about the validity of linear theory. > A linear model is usually adopted when the amplitude (strength) of the > oscillation, signal is generally "small". > > How small? Small relative to what? > > By linearity, the total final response to the sum of separate inputs > is equal to the sum of the responses of separate individual inputs. > This implies that the responses of the individual inputs don't affect > each other....no sort of coupling.... > > Does any one have some sort of physical justification for this > independence of the responses and the fact that the amplitude of the > responses need to be small? > > thanks > fisico32Yes, but it doesn't apply to y'all number crunchers on this group. The concept of small signal linearity comes from those of us who have to design on analog hardware. For small perturbations around a set-point, the transfer function of a transistor can be considered to be "linear", i.e., all of the terms past the linear in the Taylor series expansion of what's _actually_ happening are considered to contribute "negligably" to the situation. Once you begin applying negative feedback with substantially more linear elements, such as resistors, the feedback elements dominate the linearity calculations. Keeping a analog system's non-linearities 60dB down is trivial, 80dB is reasonable, and 100+dB is doable if you're paying close attention. Digitally, fixed point calculations are inherantly linear (unless they saturate). Floating point calculations aren't; in any reasonable system you wind up picking up truncation errors such that H(A+B) is no longer exactly the same as H(A) + H(B). -- Rob Gaddi, Highland Technology Email address is currently out of order
Reply by ●November 6, 20092009-11-06
On 6 Nov, 18:16, "fisico32" <marcoscipio...@gmail.com> wrote:> Hello forum, > > a "philosophical" question about the validity of linear theory. > A linear model is usually adopted when the amplitude (strength) of the > oscillation, signal is generally "small". > > How small? Small relative to what?The key word in "linear model" is *model*. Linear models are representations of some phenomenon, not True Systems in their own right. Linear models are used because they are simple to analyze mathematically. Use the linear model - and enjoy its simplicity - if it represents the phenomenon under study 'well enough'. Use a nonlinear model to describe the same system if the linear model is too crude. Rune
Reply by ●November 6, 20092009-11-06
True, a model is just a model, a useful abstraction to describe a real phenomenon. I guess we could then state that a nonlinear model is the most general type of model. A linear model is then an approximation of a nonlinear model when the amplitudes in the game are small enough....>On 6 Nov, 18:16, "fisico32" <marcoscipio...@gmail.com> wrote: >> Hello forum, >> >> a "philosophical" question about the validity of linear theory. >> A linear model is usually adopted when the amplitude (strength) of the >> oscillation, signal is generally "small". >> >> How small? Small relative to what? > >The key word in "linear model" is *model*. > >Linear models are representations of some phenomenon, not True >Systems in their own right. Linear models are used because they are >simple to analyze mathematically. Use the linear model - and enjoy >its simplicity - if it represents the phenomenon under study 'well >enough'. Use a nonlinear model to describe the same system if the >linear model is too crude. > >Rune >
Reply by ●November 6, 20092009-11-06
fisico32 wrote:> Hello forum, > > a "philosophical" question about the validity of linear theory. > A linear model is usually adopted when the amplitude (strength) of the > oscillation, signal is generally "small". > > How small? Small relative to what? > > By linearity, the total final response to the sum of separate inputs is > equal to the sum of the responses of separate individual inputs. > This implies that the responses of the individual inputs don't affect each > other....no sort of coupling.... > > Does any one have some sort of physical justification for this > independence of the responses and the fact that the amplitude of the > responses need to be small?Think graphically. Those who are not conversant with analog design are deprived of key insights. "Linearity" means the same when applied to circuits, to curves, and to the algebraic equations that represent them. A transfer function can be graphical. Let the ordinate be an input value and the abscissa be the corresponding output value. The system is linear if and only if the plot is straight. No plot is straight everywhere. Eventually, one runs out of paper or the system it represents reaches a limit. The system behaves linearly as long as the input signal remains on the straight portion. Even if the plot is curved everywhere, a "sufficiently small" portion approximated its tangent "well enough" to be treated as linear. Too large = no good. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 6, 20092009-11-06
Jerry Avins <jya@ieee.org> wrote: (snip)> Think graphically. Those who are not conversant with analog design are > deprived of key insights. "Linearity" means the same when applied to > circuits, to curves, and to the algebraic equations that represent them.Except when it doesn't. (Which took me a while to figure out.) A linear operator is one where, for one, a L(x) = L(ax) for operator L. That is not true for the algebraic linear equation y=mx+b when b is non-zero. (That is, for L(x)=mx+b, L is not a linear operator.) -- glen
Reply by ●November 6, 20092009-11-06
fisico32 wrote:> Hello forum, > > a "philosophical" question about the validity of linear theory. > A linear model is usually adopted when the amplitude (strength) of the > oscillation, signal is generally "small". > > How small? Small relative to what?Small relative to your age. The basics of calculus is taught in high school. Look wikipedia for definition of derivative. VLV
Reply by ●November 6, 20092009-11-06
glen herrmannsfeldt wrote:> Jerry Avins <jya@ieee.org> wrote: > (snip) > >> Think graphically. Those who are not conversant with analog design are >> deprived of key insights. "Linearity" means the same when applied to >> circuits, to curves, and to the algebraic equations that represent them. > > Except when it doesn't. (Which took me a while to figure out.) > > A linear operator is one where, for one, a L(x) = L(ax) for operator L. > > That is not true for the algebraic linear equation y=mx+b > when b is non-zero. > > (That is, for L(x)=mx+b, L is not a linear operator.)It's not a linear operator, but it describes a linear transfer curve. The b term introduces an offset, which is often a pain, but doesn't introduce harmonics or intermod. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 6, 20092009-11-06
On Fri, 06 Nov 2009 11:16:54 -0600, fisico32 wrote:> Hello forum, > > a "philosophical" question about the validity of linear theory. A linear > model is usually adopted when the amplitude (strength) of the > oscillation, signal is generally "small". > > How small? Small relative to what?Small relative to what you care about. One of the most explicit places you see this used is in the treatment of quantization as noise. You model quantization as the superposition of an ideal linear transfer curve and a random (if malicious) noise source. Then you check to see if your system will still behave correctly with that ideal linear element and the worst possible noise that you could inject at that point, with it's amplitude limited to the amplitude of your quantization. You can treat other nonlinearities the same way, although it gets tedious, it isn't always very accurate, and there are often reasons why it's easier to do the analysis in a different way.> By linearity, the total final response to the sum of separate inputs is > equal to the sum of the responses of separate individual inputs. This > implies that the responses of the individual inputs don't affect each > other....no sort of coupling.... > > Does any one have some sort of physical justification for this > independence of the responses and the fact that the amplitude of the > responses need to be small?Because many physical systems are continuous in their many states and parameters, which means that around any given operating condition the system behavior won't change 'much' if the perturbation is small. Note that some physical systems have nonlinearities which are discontinuous but limited in strength (think friction). In these cases a linear model works as long as certain parameters are _large_ enough that the nonlinearity's effect is swamped out by the linear behavior of the system, but breaks down when the system approaches the 'bad' zone. -- www.wescottdesign.com
Reply by ●November 6, 20092009-11-06
On Fri, 06 Nov 2009 20:35:05 +0000, glen herrmannsfeldt wrote:> Jerry Avins <jya@ieee.org> wrote: > (snip) > >> Think graphically. Those who are not conversant with analog design are >> deprived of key insights. "Linearity" means the same when applied to >> circuits, to curves, and to the algebraic equations that represent >> them. > > Except when it doesn't. (Which took me a while to figure out.) > > A linear operator is one where, for one, a L(x) = L(ax) for operator L. > > That is not true for the algebraic linear equation y=mx+b when b is > non-zero. > > (That is, for L(x)=mx+b, L is not a linear operator.) > > -- glenIn a system definition that makes the system nonlinear, but leaves it (if my terminology is right) affine. Since it's so easy to treat an affine system as linear they are almost always called such until you get to more advanced material. So, let the system h be such that d/dt x = f(x, u, t), y = g(x, u, t) defines it's behavior completely (and note that x, u, and y can all be vectors or even infinite-dimensional fields if you're a masochist. 't' is just time). If the system satisfies f(x1 + x2, u, t) = f(x1, u, t) + f(x2, u, t) for all x1, x2, u and t then it is linear in the states. If it satisfies f(x, u1 + u2, t) = f(x, u1, t) + f(x, u2, t) for all x, u1, u2, and t then it is linear in the states for the inputs. You can make a similar argument for the linearity in the outputs. If all four conditions are true, then the system is linear. If you can find an x0 for which the system satisfies f(x1 + x2 - x0, u, t) = f(x1, u, t) + f(x2, u, t) then the system is affine in the states; if you can find a u0 that satisfies f(x, u1 + u2 - u0, t) = f(x, u1, t) + f(x, u2, t) then it is affine in the states for the inputs, yadda yadda yadda. (I'm actually not sure that it's necessary for the two conditions to be proven separately -- it may well be that one proves the other, or that you just have to find the right x0, but who am I to say. At any rate, hopefully you get the idea, and clearly I need to refine my arguments.) -- www.wescottdesign.com
