Shannon-Hartley theorem Guide, Meaning , Facts, Information and Description
In information theory, the Shannon-Hartley theorem states the maximum amount of error-free digital data (that is, information) that can be transmitted over a communication link with a specified bandwidth in the presence of noise interference. The law is named after Claude Shannon and Ralph Hartley. The Shannon limit or Shannon capacity of a communications channel is the theoretical maximum information transfer rate of the channel.
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Proved by Claude Shannon in 1948, the theorem describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption. The theory doesn't describe how to construct the error-correcting method, it only tells us how good the best possible method can be. Shannon's theorem has wide-ranging applications in both communications and data storage applications. This theorem is of foundational importance to the modern field of information theory.
If we had such a thing as an infinite-bandwidth, noise-free analog channel we could transmit unlimited amounts of error-free data over it per unit of time. However real life signals have both bandwidth and noise-interference limitations.
Shannon and Hartley asked: How do bandwidth and noise affect the rate at which information can be transmitted over an analog channel? Surprisingly, bandwidth limitations alone do not impose a cap on maximum information transfer. This is because it is still possible (at least in a thought-experiment model) for the signal to take on an infinite number of different voltage levels on each cycle, with each slightly different level being assigned a different meaning or bit sequence. If we combine both noise and bandwidth limitations, however, we do find there is a limit to the amount of information that can be transferred, even when clever multi-level encoding techniques are used. This is because the noise signal obliterates the fine differences that distinguish the various signal levels, limiting in practice the number of detection levels we can use in our scheme.
Considering all possible multi-level and multi-phase encoding techniques, Shannon's theorem gives the theoretical maximum rate of clean (or arbitrarily low bit error rate) data C with a given average signal power that can be sent through an analog communication channel subject to additive, white, Gaussian-distribution noise interference:
Simple schemes such as "send the message 3 times and use a best 2 out of 3 voting scheme if the copies differ" are inefficient users of bandwidth and thus are far from the Shannon limit. Advanced techniques such as Reed-Solomon codes and, more recently, Turbo codes come much closer to reaching the theoretical Shannon limit, but at a cost of high computational complexity. With Turbo codes and the computing power in today's digital signal processors, it is now possible to reach within 1/10 of one decibel of the Shannon limit.
The V.34 modem standard advertises a rate of 33.6 kbit/s, and V.90 claims a rate of 56 kbit/s, apparently in excess of the Shannon limit (telephone bandwidth is 3.3 kHz). In fact, neither standard actually reaches the Shannon limit, but closely approaches it. The speed improvement of V.90 was made possible by the elimination of an additional step of analog to digital conversion by the use of fully digital equipment at the other end of a modem connection. This improves the S/N ratio, which in turn produces the required headroom to exceed 33.6 kbit/s which was otherwise near the Shannon limit.
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If S/N > 1, C = 0.332 · BW · SNR (in dB).Examples
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