I'm reading Computer Networks: A Systems Approach by Peterson and Davies. One of the examples demonstrates the relationship between link capacity and the Shannon-Hartley Theorem.

We can find the channel capacity by the formula:

$$C = B \log_2 \left( 1+\frac{S}{N} \right)$$

In the example of the book, they define bandwidth of the channel to be 3000Hz and the signal to noise ratio to be 30 dB, which they say would imply that S/N = 1000.

$$C = 3000 \times \log_2 (1001)$$

However, I don't understand how a signal to noise ratio of 30 dB is equivalent to 1000? How is this worked out? It's not explained in the example.

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    \$\begingroup\$ See Decibel 30db is 1000, and it's explained there. \$\endgroup\$ – gbulmer Oct 3 '14 at 20:45
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    \$\begingroup\$ Oh I see, so its the power ratio - since decibels is logarithmic? \$\endgroup\$ – user54505 Oct 3 '14 at 20:55
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    \$\begingroup\$ @gbulmer, with a couple more sentences of explanation, your comment could be an answer. \$\endgroup\$ – The Photon Oct 3 '14 at 20:57
  • \$\begingroup\$ @ThePhoton - I am still new here. However, I thought ee.se was not about collecting answers that are already satisfied by trivial web searches, i.e. type one word into wikipedia and get a definitive answer. I was only being helpful in my comment. I had expected this question to be closed because it fell into that category of "very well-answered on the web with minimal effort already" question. Am I wrong? \$\endgroup\$ – gbulmer Oct 4 '14 at 11:32
  • \$\begingroup\$ @gbulmer, you're right it's not a great question. But it's borderline and hasn't gotten any close votes yet. So it's better we get an answer posted and accepted to keep the question from re-appearing on the front page. \$\endgroup\$ – The Photon Oct 4 '14 at 15:46

In the formula, S/N is the power ratio of signal to noise. If this ratio is expressed as 30 dB, then we have 10log(S/N) = 30 which results in a value for S/N of 1000.


Shannon: considers noise. Define the signal-to-noise ratio, SNR or S/N. Often measured in dB. Then: C = B log_2(SNR + 1) B = bandwidth C = max channel capacity

    Example: 3000Hz voice bandwidth, s/n = 30 dB. or a ratio of 1000.
    **C = 3000*log_2(1000) = 3000*10 = 30kbps**

Note that increasing signal strength does tend to increase noise as well.
Also, increasing bandwidth increases noise more or less in proportion.
    So: increasing B does lead to more thermal noise, and thus
    by Nyquist's formula SNR will decrease.

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