![]() Notice that the red vector is aligned with the “standard” basis while the other three are not aligned with either axis. Consider the following vectors:įigure 1: Four two-dimensional vectors in the “standard” basis. Note, however, the x, y, and z axes are not always the most convenient axes for expressing the vector. The numbers in the vectors then represent how much of each direction is needed to “build” the vector. For example, in the three vector, (1,1,1), the first position might represent a one-unit step in the x-axis, the second position might represent the one-unit step in the y-axis, and the third position might represent the one-unit step in the z-axis. Basis vectorsĪn implicit assumption of linear algebra is that each number in a vector has some physical meaning. Now that we know the punchline, let’s take some time to really understand what we mean by orthonormal bases and transformations. So here it is: the Fourier transform is a (linear) transformation between the standard orthonormal basis and a second orthonormal basis (the Fourier basis). I also quite like the explanation given by The PunchlineĪs with my introduction of interferometry, I’m going to start with the punchline and then work backwards to understand what it means and why it’s important. I also recommend the following learning resources: Steve Smith’s Book and Brian Douglas’ YouTube series. Fear not, however, you can find a good explanation of the (discrete) Fourier transform in my last blog post although I will probably repeat some of the points that I made there. Once you gain an appreciation for this connection, many of the subtleties of the Fourier transform become almost obvious when observed through this lens 1.Ī word of warning before we start: If you’ve never seen the Fourier transform before, this may not be the best place to start your journey. Rather, my objective is to show you a way of understanding the Fourier transform from a linear algebraic perspective. My objective with this blog post is not to belabor the mechanics of the Fourier transform or attempt to explain every intricate property of the Fourier transform. It took me a long time to gain a physical intuition for the Fourier transform, but now that I have it, I’ve decided that its too exciting not to share. Now, the Fourier transform is one of tools that I use most often. No explanation of why the mathematics should work the way they did. To me, the synthesis and decomposition equations appeared almost magically. And I couldn’t fathom how anybody could have just sat down and wrote down the equations. I didn’t understand why the mathematics worked the way they did. ![]() An Intuitive Interpretation Of The Fourier Transform (or The Link Between Fourier Analysis And Linear Algebra)
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