So we have a rough idea what the frequency domain looks like; now we need some way of actually transforming data in the time domain (raw from the microphone) into frequency domain data. We'll do that through the creative application of interference.

For this trick to work, we must remember that the average value of a sinusoid is zero --- half the wave is positive, half the wave is negative, and it all cancels out nicely.

The other thing we need to know is how sinusoids multiply. So imagine a sinusoid, plodding along; a nice, slow wave with a long wavelength. Now imagine a fast, short wave sinusoid right on top of it. For each point along the x-axis of our imaginary graph, multiply the value of the slow wave with the value of the fast wave; and take care --- I said multiply, not add.

If your imagination is in smooth working order, you might piece together that the result is a periodic, oscillating wave. If your imagination is exceptionally good, you might even be able to notice that this curve is made up of two summed sinusoidal components. Truth be told, my imagination didn't hold up to the strain --- now I've got to find a new one. I recommend this for when your function imagination breaks down.

(In the meantime, we can examine a little math and discover that the two components are, in fact, sine waves whose frequencies are the sum of our original two sine waves and their difference, respectively.)

But regardless of what the components are, they are sinusoidal --- and we already said that the average value of a sine is zero.

What happens, though, if we take two identical sine waves, and multiply them? Every place that one is positive, the other is positive as well; a positive times a positive is another positive... and every place one is negative, the other is as well; negative times negative is also positive.

All of a sudden the average value is no longer zero!

More exciting still, the trick still works even when we use a sum of multiple sinusoids for our first wave --- if the sinusoid we're multiplying by is exactly the same frequency as one of the components of our first wave, then the average value of the product is non-zero --- and that value is proportional to the amplitude of the component wave. We've found a way to pick a single sine wave out of a sum of them! Our transform! Right?

There's an insidious evil lurking behind this joy.

Notice what happens when our two waves fall ninety degrees out of phase, so that the peaks of one wave occur simultaneous to the zeros of the other. When we multiply, BAM. another simple sine wave. That's no help --- its average is still zero. As it turns out, the average value of the product of same-frequency sines is not simply proportional to the amplitude of the original wave; it's proportional to the amplitude and phase shift. Shoot.

There is one hope, however: cosine. Cosine's already ninety degrees out of phase with sine, right? So whenever multiplying by a sine wave results in a false negative, multiplying by a cosine will get a full-strength output.

And sure enough, in combination, multiplying by sine, separately multiplying by cosine, and then adding the two results, negates the phase-based part of the new wave. Loverly. Here, we've invented the full form of the Fourier Transform (for that is indeed what it is): The amplitude of a given frequency component of a wave is equal to the average of the sum of a unit cosine at the given frequency times the original wave times a unit sine at the given frequency times the original wave.

That's it --- easy as pie, right?


Well, no, actually. Nowadays, no one uses that kind of notation --- we transform everything into the complex domain before we even begin. Even worse, there are huge pitfalls as soon as you make the whole thing digital --- when you no longer have smooth curves but individual samples the whole mess gets much more involved, plus there's the issue of efficiency; the method we used is about as fast as a mole on a paddleboat. It can't even reach the pedals, poor thing.


The practical application of all this comes right back to the Livescribe pulse. You remember that, right? All this has been about that. What we have now is a way to determine which frequency dominates a given audio sample. By making the pen play individual, pre-chosen frequencies, we can send data from the pen's speaker, through the PC microphone, and then decode the sounds back into labeled frequencies. Peachy. Now all we need is a way to encode data as a list of frequencies, and we're all set!

That will be the topic of my next post, where, with luck, I will be able to declare success in sending data from the pen to my PC without the use of Livescribe's desktop software.

So stay tuned.

Labels: livescribe, modem, pulse, sam