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The Nyquist Criterion

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The aim of this text is to understand (not to prove) the Nyquist sampling criterion. We shall explain why, when recording digital audio, the sampling rate should always be at least twice as high as the highest frequency present in the recorded analog signal.

First of all, let's refresh our memory about digitization
(if necessary, you can go and read again Feedback's article on the subject. It's here)

The idea is to store a sound in the shape of numbers. This process is called digitization (because numbers are a series of digits, just in case you stopped your studies before primary school !! ;Op). If you've gone a bit further, you must have noticed that, when your teacher asked you to draw a curve, you had to choose a certain amount of points and link them together. The problem was that you had to imagine the curve shape between each point. The curve was discontinuous. The principle of digitization is exactly the same.

Now, let's get to the heart of the matter

Cinema is another discontinuous recording/reconstruction process. It's just a series of photographs following each other at a rate of 24 pictures per second. Our eyes, far too slow, are completely taken in. The discontinuous flow of pictures is perceived as a continuous animation.

Yet, there are sometimes problems : it's almost sure that all of you have already watched a Western !! You know, with Indians galloping behind a coach !! Well, actually, horses are usually galloping, but, anyway...! You often get the impression that the coach wheels are turning very slowly or even backwards. This optical illusion is due to the camera's shooting speed.

Let's assume that the wheels have only one spoke (I reckon it's not very realistic, but....). If a wheel does one complete lap in 1/24th of a second, both pictures will be identical and we'll get the impression that the wheel didn't turn. If this wheel does 3/4 of a lap in the same interval, the spectator will see it going backwards by 1/4 of a lap !! Indeed, for our brains, it's easier to think about this backwards 1/4 lap than a forward 3/4 lap. You can check this out by yourself in the example below :

Bring your mouse over the wheel to turn it by 3/4 of a forward lap

In the case of a half lap, we're on the edge, and we won't be able to know if the coach is moving forward or backwards as the rotation is the same in both ways. Now, if the wheel does just a bit less than half a lap, our brains will think that the wheel is turning forward (the shortest way between each photograph). Well, at last, we're moving into the right direction !!

Same thing, but for just a bit less than half a lap

From all this we can infer that the camera must take AT LEAST 2 photographs per lap if we want to see in which direction the wheel is really going. Where do I want to get ?????? You should start to figure it out, by now !

Replace the wheel laps by oscillations, the camera by an analog to digital converter and your eyes by your ears. What we've said above becomes :

So that our ears are not mistaken, the converter must take AT LEAST 2 photographs per oscillation. This is exactly what the Nyquist Sampling Criterion says.

OK, all that stuff is very nice on paper but what really happens with such an "acoustic illusion" ? I'll get to it right away :

You all understood that an analog to digital converter (ADC) is going to regularly take photographs of a sonic amplitude at a precise given time. This ADC is not smarter than a camera. With the camera, we had the impression that the wheels were moving slowly or backwards. With the ADC, the oscillations, which had a certain original frequency (vibration), will vibrate with a slower frequency and therefore have a lower-pitched sound after being recorded/digitized !

Let's take an example. If you record an analog 12-Khz frequency sound with a sample rate of 10 Khz, when you'll play the sound back, it will be lower and the original spectral content will be spoiled. More exactly, you'll hear the symmetrical frequency to the original one, in relation to the sampling frequency. In this case, 8 Khz. This phenomenon is known as "aliasing". Therefore, converters will logically filter the signal before digitizing it.

As a conclusion, notice that the sample rate used for CD's (44,1 Khz) does follow the Nyquist criterion as we, poor human beings, cannot hear sounds higher than 20 Khz.

Note : the Nyquist criterion can attributed to Shannon, too. And also to a Russian guy whose name I forgot.

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Pipine, on the 20-08-1998

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