# Determining if a System is Linear

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)!