# Shannon sampling theorem and Nyquist

1. May 6, 2007

### leright

I understand that it was Nyquist who proved that a channel of bandwidth B can transmit at a baud rate of 2B. Also, Shannon, about 10 years later showed that if a signal is sampled at a rate of at least 2B, where B is the bandwidth of the bandlimited signal, then the signal can be reconstructed from the samples. I read that Nyquist's theorem implies shannon's theorem, but I do not see how this is the case.

It is for this reason that the sampling theorem is sometimes called the 'shannon-nyquist theorem'. However, I cannot make the connection between the two theorems. The only connection I see is the '2B' part. For this reason, I don't really see why Nyquist deserves any credit for the sampling theorem...

2. May 7, 2007

### Maxwell

Taken from Wikipedia:

"The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result[1], and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).

The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication")."

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

3. May 7, 2007

### leright

I saw that article. I understand they are duals, but WHY are they duals? I cannot make the connection.

4. May 8, 2007

### Maxwell

It doesn't seem that there's much more to it than that. Nyquist introduced the concept of a sampling rate... or a sampled signal, while Shannon deduced this result to explain that you need to sample 2B to reconstruct a continuous signal out of samples. I don't think Nyquist considered the problems of aliasing, etc. Shannon must have.

5. May 8, 2007

### chroot

Staff Emeritus
Sending a signal with baud rate 2B is the inverse operation of sampling a signal with sample frequency 2B.

- Warren