Nyquist-Shannon theorem – a milestone in digitisation

Party knowledge or a step towards a deeper understanding of analogue to digital signal conversion? Both!

Nyquist? Shannon? Who's that? Theorem? What's that? Never heard of it... But a beyond a shadow of a doubt each of us deals with it every day. We have Harry Nyquist and Claude Shannon to thank for the digitisation of music. But before we bombard you with formulas and explanations (and we could do that), let's keep it simple. Nyquist-Shannon theorem, or sampling theorem for short, states: If an analogue signal is sampled at a certain minimum frequency, it can be digitised and later made audible again. Just imagine you want to unlock your screen with your finger. For this purpose, your analogue fingerprint is scanned once beforehand and stored digitally. In theory, the signal can be reconstructed almost exactly and without errors, but this is not so easy in practice. Depending on the selected sampling rate, the analogue signal will be digitised in a better or worse way. Important if you want to record a demo in home recordingThe better the signal has been digitised, the better it can be made audible again. Let's now look at how this works without anything going wrong along the way.

Samples convert the signal from analogue to digital

Let's assume that an analogue signal oscillates as a sine curve around a horizontal axis. The signal is now sampled at certain intervals, i.e. at certain points in time – hence the name sampling theorem. That happens extremely often. A digital signal results from these sampling points (samples). The digital signal can then be reconstructed and made audible. Similar to the "connect the dots" game, where the connected dots create a picture.

Thanks to the Nyquist-Shannon theorem, signals can be stored digitally

Admittedly, the sampling method doesn't exactly sound complicated. We said we'd spare you formulas. But in order to take a quick look into the black box on the way from analogue to digital signal, we have to delve a little deeper into the matter and can't avoid a bit of complexity.

Because the Nyquist-Shannon theorem states specifically:

A band-limited signal can only be described and reconstructed without errors if it is sampled at more than twice the maximum frequency.

Yes, there it is, the incomprehensible stuff. So let's look at the individual terms related to the sampling theorem and put the puzzle together step by step. We want to show you which processes are involved when you, for example record vocals and want to convert your voice through a microphone.

 

Band limitation

The human ear only registers frequencies up to 20 kHz. If your dog suddenly raises its head attentively, it has heard something outside this spectrum that we are no longer able to perceive. That means our hearing is limited to a certain frequency range, or is band limited.

 

Low Pass Filter

A low-pass filter comes into play for the upper band limitation. It eliminates all frequencies above a certain frequency. For our hearing, such a low pass would remove all frequencies above 20kHz. We wouldn't hear them anyway, so get rid of them!

Let's assume you want produce music. The highest frequency in your recording is also equal to 20 kHz. How realistic that would be remains to be seen. Now it is sampled at more than twice the maximum frequency. So the sampling frequency must be greater than 40 kHz. That would mean your recording is sampled more than every forty-thousandth of a second. The samples are now only a forty-thousandth of a second apart. Insanity! Thanks to this sampling rate, no mistakes are made when digitising your recording. Let's look at it this way: If our analogue signal was a solid line, our sampled signal is a dotted line. There is now an empty space between our samples that we need to fill again if we want to hear our recording.

 

Sampling

But how does the sample become a signal again? By being connected again one after the other and without deviations. That way there is no mess. When everything is connected, we have a new signal. This signal is very similar to the original, but no longer the same signal. However, it is so similar that we do not notice the differences.

There is only one mathematical solution to the band-limited signal that runs through all the samples.

Sampling rate

But why is sampling rate so important? That's simple: If it is too low, undersampling may occur. Errors (aliasing) then occur in the sampled signal and it cannot be reproduced correctly. If the sampling rate is higher, more points of our recording will be stored digitally. This helps if we want to make the digital data audible again.

The Nyquist-Shannon theorem in practice

Please take a deep breath. We would like to show you another field where the sampling theorem occurs in everyday life and give you a little "aha! knowledge" to take with you:

The sampling rate of the CD standard is 44.1 kHz. This means that a range up to 22.05 kHz is possible. If the human ear can only perceive a range up to 20kHz, why are there such high sampling rates? Well, sampling rate isn't the only piece in digitisation, there are other complex factors at play.

And these factors need these slightly higher bandwidths. But what about the higher sampling rates of 88.2 kHz, 96 kHz or even 192 kHz? In general one can say: Higher sampling rates do not mean better quality. Rates up to 96 kHz can improve quality. Any rates above that will have next to no impact on quality.

Pretty clever how Nyquist and Shannon have influenced digitisation in so many areas thanks to their theorem. Because, last but not least:

Sampling or screening according with the Nyquist-Shannon theorem is also applied to images and videos.

Of course, the sampling theorem is not solely responsible for ensuring that our digital recording can be reproduced clearly and audibly. The so-called bit depth, which enables the amplitude to be roughly or finely divided, is also important. However, the theorem is an important component and indispensable for good recordings.

Headergraphik: ©  AdobeStock/ Davizro Photography

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