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Q & A: Linear systems

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Most recent answer: 10/22/2007
Q:
what is a linear system?
- John Smith (age 11)
Tyalla Primary school, Australia
A:
Hi John,

There are two commonly used, and closely related, meanings to the phrase "linear system" and as I'm not sure which you mean, I'll describe both briefly.

Systems in the real world respond in some way to stimuli, and are characterized by the relationships between "input" parameters and their "outputs." An electrical system, like the amplifier in a stereo set, is an example of such a system -- it has signal inputs from CD players, tape decks, etc., and outputs to speakers. It is even linear, because its output is proportional to the input. If you draw a graph of the output voltage of an amplifier on one axis and its input on another, the graph will be a straight line, hence the name "linear system."

Linear systems can be more complicated than an amplifier, however, and some are quite useful. The general definition of a linear system is any system whose output function F() satisifies the following equation (now don't panic!):

F(a*g(t) + b*h(t)) = a*F(g(t)) + b*F(h(t))

where a and b are constant real numbers, and g and h are any arbitrary functions of an independent variable t (commonly time in many
applications, but it can also be something like position). The reason that equation looks so messy is just because it is trying to say two simple things at once:

1) You can "factor out" constant multiplicative scale factors (the "a" and "b" above), and

2) The response of the system to a sum of inputs is the sum of the responses to each individual input separately.

These two nice properties allow a whole range of tools to be applicable in designing linear systems and predicting their behavior. Some more examples of linear systems in real life:

a) Frequency filters -- circuits which only pass low frequencies and reject high, or vice-versa.

b) Delays are linear. Echos from faraway canyons are linear. Shout twice as loud, get an echo twice as loud. Two people shouting at the same time comes back as two people echoing at the same time.

c) Many different kinds of economic systems -- looking at the apple juice production (output) vs. the apple crop yield, for example.

d) Limiting cases of non-linear systems for small inputs: Even if the system's response may not satisfy the equation above exactly, it often will well enough for small enough inputs. In this case, even if the number of apples bought by consumers, say, is inversely proportional to the price of apples, you can still model small changes around a reference price with linear systems (but beware when the inputs get large!).

Another example of a nonlinear system is turning up a cheap audio amplifier way too loud. The amplifier will "saturate" -- producing sound waves that have their tops chopped off because there is a limit to how much power the amp can drive. You may hear different sounds than were on the original recording if the volume is too high (they tend to be screechy). All real amplifiers have their limitations for how loud they can make any sound, and are nonlinear, but if the signal is small enough, they will be very good approximations to linear (which is good, otherwise they would distort the music).

e) Most optical systems are linear -- lenses and optical fibers all have linear responses to how much light is put in. They may distort the input light in other ways (delays, changing the angle depending on color), but they follow the equation above. Some aren't linear: they can change the frequency of the light that hits them (fluorescent materials for example).

Often these systems are studied using sets of coupled linear equations which must be solved. These are also called "linear systems" in the jargon of those who use them.

In this context a linear system is a set of equations which can be solved together for the values of variables, with some restrictions made on the kinds of equations there are.

Each equation consists of a sum of terms -- fixed coefficients times the unkown variables and these sums add up to a fixed number. Here is a concrete example of a linear system where the variables x, y, and z are initially unknown but can be computed by solving the linear system.

4x + 5y + z = 0
3x + 2y - z = 7
-x + 112y + 3z = 127

(no, I'm not going to solve it). If the last equation were

xy + z = 127

then it would not be a linear system any more, it would be quadratic, because two variables are multiplied together.

In general, you need at least as many equations as you have unknown variables in order for the system to have just one solution. If you have more equations than variables you may have no solutions. Here's an example:

x + y = 3
x - y = 0
x + 2y = -1

(you can work it out, the first two equations have the solution x = y = 1.5, which doesn't fit the third -- the system is *inconsistent*). You can also have no solutions if the coefficients are badly chosen:

x + y = 3
3x + 3y = 10

is not consistent. If 3x + 3y = 9 were the second equation, it would add no new information and not help solve for x and y and then you would not have just one solution but many.

There is a whole field of study of linear equations because they are so useful in solving all sorts of problems (even physics problems)!

Tom

(published on 10/22/2007)

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