To go back to the original signal, we need to use another concept known as the inverse Fourier transform, and after applying this operation, we have effectively removed the high-pitched ringing noise from the signal. If we remove the spike by smooshing it out on the frequency plot, we will have a sound that is almost identical to the recording, but without the ringing. To keep track of this effect, let's draw the x-coordinate for the center of mass as the winding frequency changes. In most cases, the center of mass stays relatively close to the origin, but when the winding frequency is the same as the frequency for our signal, the center of mass is unusually far to the right. We'll revisit what we mean by "center of mass" here a bit further down, but for the moment we're looking for an intuitive way to measure how much this wound up graph is off-balance. Now put a dot at the center of mass location, and as we change the winding frequency, the center of mass will kind of wobble about. The rotating vector is always longer when it's pointing to the right, and shorter when it's pointed to the left, because the frequency with which its length changes is identical to the frequency with which it rotates around the circle.Ĭan we leverage this to build a frequency unwinding machine? To systematically identify a frequency in the original signal, especially in the case when it's jumbled together with many other frequencies? We actually can!įirst, imagine that the graph was made of something with some weight, like a metal wire. Notice, the resulting curve it draws out is off-balance to the right. To start, let's draw a sine wave at 3 beats per second from 0 to 4.5 seconds: Writing down what this machine does as a formula will give us the Fourier transform. Moreover, it should be able to detect the presence of that frequency even if it's been mixed together with others, for example recognizing that the messy waveform above has a pure A440 hiding inside it. The general strategy will be to build for ourselves a mathematical machine that treats signals with a given frequency differently from how it treats other signals. Or as Kalid Azad nicely phrased it, "Given a smoothie, it finds the recipe." So pulling them back apart feels akin to unmixing multiple paint colors that have all been stirred up together. So our central question is how can you take a signal like this, and decompose it into the pure frequencies that make it up? Pretty interesting, right? Adding up those signals really mixes them all together. A microphone recording any sound just picks up on the air pressure at many different points in time.
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