After getting a few projects done today I had a bit a free time and for some reason remembered a story my Linear Systems professor told my class about a year ago.  Here’s the gist: he was listening to a Master’s thesis defense that involved some sort of system.  The master’s candidate had, for the entirety of the thesis research, assumed that his/her system was linear.  They based this assumption on the fact that it could be modeled close to something like $y=ax+b$.  My professor quickly pointed out that it is not, in fact, a linear system but the candidate vehmently argued that the equation was linear therefore the system was linear.  It ended in the person not actually getting a Master’s degree but the point of the story is:

just because a system can be represented by a linear equation does NOT make it linear!

Why is this important?  Many equations and theories rely on the assumption that a system is linear.  If the system you’re working on isn’t linear, your life just became a bit more complicated and the techniques you used to use in school may not actually apply.  Being able to recognize if a system is linear or not is a fairly useful tool to have so I’ll go over the above example where you have a system whose output is defined as $ ax + b$ where $x$ is the input to the system. As a note, I take all derivation here from the textbook Signals and Systems by Alan V. Oppenheim and Alan S. Willsky.  It is the second edition book and the section that deals with this begins on page 53.

In order for a system to be linear, it must obey the property of superposition.  That is, if I have the input to a system as the sum of two signal, $ X_{1} + X_{2}$, the output will be $Y = Y_{1} + Y_{2}$.  Easy, right? Personally, I learn best by examples so I will offer the first one where I have a system whose input and output is related by $ Y = aX $.  First, let’s say we have two inputs $ X_{1}$ and $X_{2}$ so that we have $Y_1 = aX_1$ and $Y_2 = aX_2$. Now, let’s define a third signal that is a linear combination of our two inputs $X_1$ and $X_2$: $X_3 = bX_1 + cX_2$. (b and c are arbitrary scaling constants). Finally we simply need to check if the system is linear.

If we have an input $X_3$ we know that the system’s output and input will be related like so: $Y_3 = aX_3$.  Let’s plug in for $X_3$ to see if the system is linear:

++Y_3 = a(bX_1 + cX_2)
Y_3 = abX_1 + acX_2
Y_3 = b(aX_1) + c(aX_2)
Y_3 = bY_1 + cY_2++

Excellent!  Superposition holds true so we know that the system must be linear!   Now, let’s try that for a linear equation $Y = aX + b$. Let’s define the output for two different signals: $Y_1 = aX_1 + b$ and $Y_2 = aX_2 + b$.  And let’s also define a third signal as the sum of the first two inputs: $X_3 = mX_1 + nX_2$.  (Like before, m and n are arbitrary constants). So for an input of $X_3$ we should expect $Y_3 = aX_3 + b$.  Plugging in for X_3 yields:

++Y_3 = a(mX_1 + nX_2) + b++

Now here I’m going to do something a bit different than the first example.  I’m going to solve for $X_1$ and $X_2$ and plug them into the previous equation.

++Y_3 = a(m\frac{Y_1-b}{a} + n\frac{Y_2-b}{a}) + b
Y_3 = m(Y_1-b) + n(Y_2-b) + b
Y_3 = mY_1-mb + nY_2-nb + b
Y_3 = mY_1 + nY_2 + b(1-m-n)++

As can be seen: $mY_1 + nY_2 + b(1-m-n) eq mY_1 + nY_2$ So this system is not linear despite the fact that the system is a linear equation! A pretty important concept (and sure to be a test question on an “Intro to Signals”-type test)!


Tags:  theory systems