# Nyquist–Shannon sampling theorem

bwmac
14 May 2007

Nyquist–Shannon sampling theorem

I find this very confusing

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

I find this very confusing

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

## Comments

### If you are interested in the math, I'd look at the section of "S

If you are interested in the math, I'd look at the section of "Shannon's Original Proof" in the Wiki article to start rather than the version they start with. The original proof uses only Fourier transforms - you don't really need the distribution theory used in the proof the article starts with just to understand the theorem. (Distributions are crucial to the way electrical engineers think about sampling, but they are not needed for this.)

Here is a nice way to think of the theorem -

> If you know a function is a straight line you only need two samples to determine all the points exactly.

> If you know a function is a parabola you only need three samples.

> If you know a function is a cubic you need four samples, etc.

The Nyquist-Shannon theorem says that if you know a function is band limited, you only need to sample at twice the bandwidth or higher to determine all points of the function exactly. It is really the same type of idea and proof as the theorems mentioned above except with Fourier transforms instead of polynomials.

Here is a nice way to think of the theorem -

> If you know a function is a straight line you only need two samples to determine all the points exactly.

> If you know a function is a parabola you only need three samples.

> If you know a function is a cubic you need four samples, etc.

The Nyquist-Shannon theorem says that if you know a function is band limited, you only need to sample at twice the bandwidth or higher to determine all points of the function exactly. It is really the same type of idea and proof as the theorems mentioned above except with Fourier transforms instead of polynomials.

### An interesting article on the Nyquist criterion: http://www.tec

An interesting article on the Nyquist criterion:

http://www.techonline.com/article/pdf/showPDF.jhtml?id=1992042431

Edit: Sorry, it looks as though that article is only available to subscribers. I'll see if I can find another link to it.

2nd edit: Try this: http://www.wescottdesign.com/articles/Sampling/sampling.html

http://www.techonline.com/article/pdf/showPDF.jhtml?id=1992042431

Edit: Sorry, it looks as though that article is only available to subscribers. I'll see if I can find another link to it.

2nd edit: Try this: http://www.wescottdesign.com/articles/Sampling/sampling.html

### It doesn't need to be confusing if all you want is to know enoug

It doesn't need to be confusing if all you want is to know enough for practical purposes. If you want to study it as part of information theory or signal processing, then you need to get into the maths. However, few audio engineers need to do this, at least on a day-to-day basis, unless they are studying for exams.

Nyquist-Shannon is all about the maximum amount of information that can be represented uniquely as a set of digital numbers. For practical purposes, this reduces to the well-known Nyquist criterion: to represent a band-limited continuous signal as digital numbers in a way that can be reconstructed exactly, you need to take a digital sample (to infinite resolution) at a rate of at least twice the highest frequency present in the signal.

PS I don't think the Mastering Sound forum is really the right place for this thread.

Nyquist-Shannon is all about the maximum amount of information that can be represented uniquely as a set of digital numbers. For practical purposes, this reduces to the well-known Nyquist criterion: to represent a band-limited continuous signal as digital numbers in a way that can be reconstructed exactly, you need to take a digital sample (to infinite resolution) at a rate of at least twice the highest frequency present in the signal.

PS I don't think the Mastering Sound forum is really the right place for this thread.

## Thanks Thanks Boswell, That helps a bit by hearing it in layman

ThanksThanks Boswell, That helps a bit by hearing it in laymans terms.