Suppose we have a linear system which takes an input function, and transforms it into an output function.
For the exact same input, a different linear system will produce a different output.
Let’s now keep the system the same, but try a different input function.
For any given linear system, our goal is to be able to predict the output function, for any possible input function.
The input function can be any size and any shape.
The fact that the system is linear means that if we think of the input as the sum of two different input functions…
Then the output will be the sum of the two associated output functions.
The fact that the system is linear means that if we think of the input as the sum of two different input functions…
Then the output will be the sum of the two associated output functions.
This means that to know the system’s response to any input, we only need to know its response to an input consisting of a unit impulse.
Unit impulse is a function with an area exactly equal to one and a width which approaches zero.
Since the area of the pulse must equal one, as the width becomes very small, the height becomes very large.
If this "unit impulse" is the input to our system, then we call the output of the system the system’s "Unit Impulse Response."
Any input to the system can be thought of as an infinite sum of impulse functions.
The output of the system can then be thought of as an infinite sum of output functions.
Each output function is multiplied by the height of the associated input pulse.
"Time" is a constant fixed at the moment we are interested in.
The total output at this moment in time is the sum of the areas of all the red rectangles.
Let us invent a new variable, called "tau" to help us calculate the area of each red rectangle.
Each red rectangle is associated with a particular value of tau.
"Time" is a constant fixed at the moment we are interested in,
and each red rectangle is associated with what the value of the input function was at the moment “Time minus Tau.”
"Time" is a constant fixed at the moment we are interested in,
and each red rectangle is associated with what the value of the input function was at the moment "Time minus Tau."
Each red rectangle is also associated with the value of the unit impulse response function at the moment where time equals Tau.
Since the height of each red rectangle depends on both of these factors, the height of each red rectangle is described by the following equation.
The width of each rectangle is represented by “d tau” and the value of “d tau” approaches zero.
The area of each rectangle is its height multiplied by its width, which is given by the following formula.
The sum of the areas of all the red rectangles is then represented as follows.
This equation for the total area represents the value of the total output at this moment in time.
Note that as "d tau" approaches zero, the area of each of the individual red rectangles approaches zero.
Hence the contribution of each of the individual output functions approaches zero.
This is because each of the input impulses shown here only have a finite height, whereas a "true" unit impulse input function would have a height that approaches infinity.
The equation shown here is what we refer to as "Convolution."
The output function can also instead be computed by making use of the Laplace Transform as shown.
The Laplace Transform of the Unit Impulse Response is what we refer to as the "Transfer Function" of the system.
The Transfer Function is very useful in studying the frequency response and stability of a system.
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