Hacker News Reader

Comment by eridius | Hacker News Reader

The isolated phrase "spinning a signal in a circle" is confusing, but the accompanying visualization of tracking a point on a spinning speaker cone is something that can be visualized.

More generally, your explanation may make sense to you, but it makes no sense to a layperson (i.e. me). But the linked short description, along with the speaker cone visualization, does make a reasonable amount of sense.

I've also seen a similar description in the past with accompanying animated GIFs illustrating the point, instead of the prose about a spinning speaker cone. I thought the animated GIFs made it even easier to understand. Sadly, I don't remember where I saw that.

View on HN

Is it this https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_...

I've seen that before, and it's neat, but I don't think that's what I'm remembering.

I think it's this one, which was also posted on HN:

http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...

Agreed a layperson could benefit, but karpathy insight is also very nice, it allows to leapfrog from elementary linear methods to a good intuition of the real thing.

Oh sure, I bet karpathy's explanation is great for the people it makes sense to. It just assumes a lot more specialized knowledge on the part of the reader, knowledge that I personally don't have.

Let me try a (hopefully unspecialized) basis view explanation. I think this is more of an explanation of Fourier Series, but hopefully it gets the idea across:

Consider the space around you. Pick some frame of reference with a fixed origin point in space. To specify an object, you must specify the distance along each independent direction (x, y, and z in 3d space) to arrive at the object from the fixed origin. These independent directions are referred to as a basis for the space. Often these independent bases are perpendicular (or orthogonal).

For higher dimensional spaces its the same idea. It turns out we can even have infinite dimensional spaces whose bases are functions that can be perpendicular in a certain sense. It also turns out that complex exponentials (waves) are an orthogonal basis for a certain space of functions. This is really cool! It means we can refer to any function in this space using the complex exponential basis functions just like we referred to objects in 3d space using three independent directions.

Why is this useful? The magnitude in a particular independent direction in this space tells us about frequency content of that function. So instead of the first coordinate telling you how much x distance there is, it says how much of a certain frequency there is.

That helps, thanks. But while that's an explanation that may make a certain amount of intellectual sense, it still doesn't make any intuitive sense. Perhaps if I'd worked with basis views a lot more, then it would become intuitive, but I haven't. The nice thing about the spinning signal idea is I can visualize the speaker, and it makes some intuitive sense that if I spin it at a given frequency, and the speaker cone's average movement is zero, then the signal has no content at that frequency.

OK, I'm curious about how well this explanation works for you. I came up with it, and I want to know if someone like you thinks it's any good:

The Fourier transform is usually used to flip from the time domain representation to the frequency domain representation.

"Time domain representation" means when you plot that information, the x-axis is in units of time; seconds or fractions of seconds, usually, for something humans can perceive as sound. The line tells us how loud the signal is at a given instant of time.

"Frequency domain representation" means when you plot that information, the x-axis is in units of frequency; hertz or kilohertz, usually, for something humans can perceive as sound. The line tells us how loud the signal is at a given frequency.

Here's the bit about basis vectors: The frequency domain representation is possible because we can make any graph by adding enough sine functions, assuming we're allowed to do things like increase or decrease their magnitude, increase or decrease their frequency, and shift them left or right relative to each other (adjust their phase).

Therefore, we can describe the graph were interested in using only sine waves, which are pure frequencies, and when we've done that, we'll know which specific frequencies contributed most to the information we represented in the original graph.

There is, of course, an inverse Fourier transform, which takes a frequency domain representation to the time domain, so we can do things like record a sound (time domain), take it into the frequency domain, remove certain frequencies (typically all frequencies above and/or below some cutoff) by zeroing parts of the graph, and take the result back into the time domain, so it can be played back as a recording.

Well this helped me! I'm trying to wrap my head around FFT (or DFT) for my course in speech recognition, and while I still wouldn't be able to derive FFT or understand maths, I now have an idea what it does ;) Thanks!

Sure, I agree. Its a mathematical sort of intuition - one on abstract objects (in particular Hilbert spaces). This abstraction provides something sort of fundamental and poetic. Of course, this is better motivated by something concrete or physical I think before stepping in this direction.

One small note is that independent vectors are ALWAYS orthogonal in their space. That's by definition of linear independence. If they were not mutually orthogonal, then at minimum you could remove 1 vector from the set and still have the same space representation.

[1, 0] and [1, 1] are linearly independent but not mutually orthogonal. Unless I am understanding what you mean by "in their space".

I don't see how it is so specialized. It is an inner product, which can be seen as analogous to a dot product that pretty much everybody sees at some point in high school.

Reply on news.ycombinator.com