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Link Capacity and the Shannon-Hartley Theorem

There has been an enormous body of work done in the related areas of signal processing and information theory, studying everything from how signals degrade over distance to how much data a given signal can effectively carry. The most notable piece of work in this area is a formula known as the Shannon-Hartley theorem Simply stated, this theorem gives an upper bound to the capacity of a link, in terms of bits per second (bps); as a function of the signal-to-noise ratio of the link, measured in decibels (dB); and the bandwidth of the channel, measured in Hertz (Hz). (As noted previously, bandwidth is a bit of an overloaded term in communications; here we use it to refer to the range of frequencies available for communication.)

As an example, we can apply the Shannon-Hartley theorem to determine the rate at which a dial-up modem can be expected to transmit binary data over a voice-grade phone line without suffering from too high an error rate. A standard voice-grade phone line typically supports a frequency range of 300 Hz to 3300 Hz, a channel bandwidth of 3 kHz. The theorem is typically given by the following formula:

C = B log2 (1 + S/N )

where C is the achievable channel capacity measured in bits per second, B is the bandwidth of the channel in Hz (3300 Hz − 300 Hz = 3000 Hz), S is the average signal power, and N is the average noise power. The signal-to-noise ratio (S/N , or SNR) is usually expressed in decibels, related as follows:

SNR = 10 × log10 (S/N )

Thus, a typical signal-to-noise ratio of 30 dB would imply that S/N = 1000. Thus, we have C = 3000 × log2 (1001) which equals approximately 30 kbps.

When dial-up modems were the main way to connect to the Internet in the 1990s, 56 kbps was a common advertised capacity for a modem (and continues to be about the upper limit for dial-up). However, the modems often achieved lower speeds in practice, because they didn’t always encounter a signal-to-noise ratio high enough to achieve 56 kbps. The Shannon-Hartley theorem is equally applicable to all sorts of links ranging from wireless to coaxial cable to optical fiber. It should be apparent that there are really only two ways to build a high-capacity link: start with a high-bandwidth channel or achieve a high signal-to-noise ratio, or, prefer- ably, both. Also, even those conditions won’t guarantee a high-capacity link—it often takes quite a bit of ingenuity on the part of people who design channel coding schemes to achieve the theoretical limits of a channel. This ingenuity is particularly apparent today in wireless links, where there is a great incentive to get the most bits per second from a given amount of wireless spectrum (the channel bandwidth) and signal power level (and hence SNR).


Published on Thu 08 September 2011 by Alistair Pinter in Networking with tag(s): link capacity