Waves have a lot of nice properties that make them easy to analyze.
So when a scientist or engineer needs to analyze a signal that's not exactly a wave, they often approximate it as a wave.
It turns out that vector spaces show us the "best" possible way to do this!
In this quiz, we'll sketch the basic ideas and fill the details in later.
Note on Notation:
means "evaluate at ," not "multiply by "
Any signal of interest to a scientist is a real-valued function of time Let's call the set of all such functions
Waves live in too, but they're special since they repeat. Put all the waves, i.e. functions obeying in a subset called
We can add functions and scalar-multiply with the following rules:
Which of the following statements is true?
Note: For reference, you can find the definition of a vector space here.
Both and are vector spaces, so we can rephrase our original problem of approximating a signal by a wave this way:
What vector in the subspace is as close to as possible?
We call a subspace because it is a space that's also a subset.
There's a way of solving this problem based solely on vector ideas. That means we can develop a strategy for solving it in, say, the plane (which is easier to visualize) and carry what we learn there over to the space
In the picture below, the point plays the role of the signal, the line plays the role of and the Cartesian plane corresponds to
Before we can take the analogy too seriously, we have to prove that the Cartesian plane is, in fact, a vector space.
The plane is made up of all real number pairs so we can define addition and scalar multiplication naturally as With these definitions, is the Cartesian plane a vector space?
With and the Cartesian plane is a vector space.
As a set, the line in the picture below consists of all pairs where is the slope.
It inherits addition and scalar multiplication from the plane, but to be a vector subspace, it has to be closed under these operations. This means
- the sum of any two points on the line is another point on the line;
- scaling a point on the line produces another point on the line.
Select the correct statement from the list of options.
So far, our analogy works quite well: both the Cartesian plane playing the role of and the line playing the role of are vector spaces.
Now, we need to decide how to tell if the point (representing the signal) is "close" to a point on the line.
The natural way to do this is to use the distance between and which is the length of the line segment connecting them.
The interactive below lets you pick a point on the line with slope It then computes the distance between it and
(We choose these numbers just to be concrete!)
Is there a smallest possible distance?
With a bit more work, we can actually show that the line segment minimizing the distance is perpendicular to the original line.
So here's the strategy we should import to the signal problem:
- Define distances between vectors in and
- Minimize the distance from to the signal
It'll take some work, but in later chapters we'll show there's a purely vector-based way to do these things!
We can't fill in the details here, but we can certainly peek ahead and see the result of applying this strategy to a slightly simpler problem.
Suppose instead that 's functions are defined on the smaller interval and is built up from the waves where is a positive integer.
The interactive below shows the result of approximating a step function by a combination of these sine waves:
The more waves we use, the better, but even with just a few waves the signal approximation is pretty good.
The scheme we outlined in this quiz applies to many real-world problems from physics and statistics, as we'll see.
This is just one example of how useful the vector space concept can be.