p {\displaystyle {\mathcal {Y}}_{1}} {\displaystyle p_{1}} y Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA, USA. 0 1 He called that rate the channel capacity, but today, it's just as often called the Shannon limit. H In 1949 Claude Shannon determined the capacity limits of communication channels with additive white Gaussian noise. X 1 By definition of mutual information, we have, I Note Increasing the levels of a signal may reduce the reliability of the system. p , 1 p ( C { , y P {\displaystyle (X_{1},Y_{1})} This similarity in form between Shannon's capacity and Hartley's law should not be interpreted to mean that p ), applying the approximation to the logarithm: then the capacity is linear in power. Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. Similarly, when the SNR is small (if , Shannon's discovery of {\displaystyle (X_{2},Y_{2})} X where N , h X Y Shannon Capacity Formula . 2 2 x The amount of thermal noise present is measured by the ratio of the signal power to the noise power, called the SNR (Signal-to-Noise Ratio). E Its signicance comes from Shannon's coding theorem and converse, which show that capacityis the maximumerror-free data rate a channel can support. = Y + Y {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)}. 2. , y Following the terms of the noisy-channel coding theorem, the channel capacity of a given channel is the highest information rate (in units of information per unit time) that can be achieved with arbitrarily small error probability. ( But such an errorless channel is an idealization, and if M is chosen small enough to make the noisy channel nearly errorless, the result is necessarily less than the Shannon capacity of the noisy channel of bandwidth 2 R 2 10 2 Idem for X n sup Keywords: information, entropy, channel capacity, mutual information, AWGN 1 Preface Claud Shannon's paper "A mathematical theory of communication" [2] published in July and October of 1948 is the Magna Carta of the information age. | This website is managed by the MIT News Office, part of the Institute Office of Communications. If the signal consists of L discrete levels, Nyquists theorem states: In the above equation, bandwidth is the bandwidth of the channel, L is the number of signal levels used to represent data, and BitRate is the bit rate in bits per second. X The prize is the top honor within the field of communications technology. | x 1. , 1 p ( Noisy channel coding theorem and capacity, Comparison of Shannon's capacity to Hartley's law, "Certain topics in telegraph transmission theory", Proceedings of the Institute of Radio Engineers, On-line textbook: Information Theory, Inference, and Learning Algorithms, https://en.wikipedia.org/w/index.php?title=ShannonHartley_theorem&oldid=1120109293. , and the corresponding output , X B 2 p By definition 1 Within this formula: C equals the capacity of the channel (bits/s) S equals the average received signal power. x 2 , X Shannon showed that this relationship is as follows: 1 y ) Y 2 , which is unknown to the transmitter. X Y : Hartley's name is often associated with it, owing to Hartley's rule: counting the highest possible number of distinguishable values for a given amplitude A and precision yields a similar expression C = log (1+A/). 2 X The key result states that the capacity of the channel, as defined above, is given by the maximum of the mutual information between the input and output of the channel, where the maximization is with respect to the input distribution. symbols per second. , and analogously chosen to meet the power constraint. ) 1 2 Hartley argued that the maximum number of distinguishable pulse levels that can be transmitted and received reliably over a communications channel is limited by the dynamic range of the signal amplitude and the precision with which the receiver can distinguish amplitude levels. 1 2 ( , Analysis: R = 32 kbps B = 3000 Hz SNR = 30 dB = 1000 30 = 10 log SNR Using shannon - Hartley formula C = B log 2 (1 + SNR) C in Eq. 1 ) {\displaystyle Y} pulse levels can be literally sent without any confusion. = . ) Bandwidth and noise affect the rate at which information can be transmitted over an analog channel. ) , ) ( 2 h ) Y as = ( 1 ( The computational complexity of finding the Shannon capacity of such a channel remains open, but it can be upper bounded by another important graph invariant, the Lovsz number.[5]. with these characteristics, the channel can never transmit much more than 13Mbps, no matter how many or how few signals level are used and no matter how often or how infrequently samples are taken. 2 ) {\displaystyle p_{X}(x)} , X {\displaystyle C(p_{1}\times p_{2})=\sup _{p_{X_{1},X_{2}}}(I(X_{1},X_{2}:Y_{1},Y_{2}))} ( 1 {\displaystyle R} When the SNR is small (SNR 0 dB), the capacity 2 ( x If the requirement is to transmit at 5 mbit/s, and a bandwidth of 1 MHz is used, then the minimum S/N required is given by 5000 = 1000 log 2 (1+S/N) so C/B = 5 then S/N = 2 5 1 = 31, corresponding to an SNR of 14.91 dB (10 x log 10 (31)). {\displaystyle N=B\cdot N_{0}} {\displaystyle R} [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small. p 1 the SNR depends strongly on the distance of the home from the telephone exchange, and an SNR of around 40 dB for short lines of 1 to 2km is very good. ( [W], the total bandwidth is p , given 2 Y p ) X {\displaystyle B} ) B By taking information per pulse in bit/pulse to be the base-2-logarithm of the number of distinct messages M that could be sent, Hartley[3] constructed a measure of the line rate R as: where 1 {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} {\displaystyle N} ) 2 1 The ShannonHartley theorem establishes what that channel capacity is for a finite-bandwidth continuous-time channel subject to Gaussian noise. 2 S | In 1948, Claude Shannon carried Nyquists work further and extended to it the case of a channel subject to random(that is, thermodynamic) noise (Shannon, 1948). = ) ) 2 X x Y P ( We define the product channel 2 Y 1 Channel capacity is proportional to . 1 ( Such noise can arise both from random sources of energy and also from coding and measurement error at the sender and receiver respectively. The . | = 1 0 We first show that 1 Y and an output alphabet 1 ) Y and A generalization of the above equation for the case where the additive noise is not white (or that the ( {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{2})} | Though such a noise may have a high power, it is fairly easy to transmit a continuous signal with much less power than one would need if the underlying noise was a sum of independent noises in each frequency band. = The capacity of an M-ary QAM system approaches the Shannon channel capacity Cc if the average transmitted signal power in the QAM system is increased by a factor of 1/K'. y 2 This section[6] focuses on the single-antenna, point-to-point scenario. 2 , X 2 | X Y 1 In a fast-fading channel, where the latency requirement is greater than the coherence time and the codeword length spans many coherence periods, one can average over many independent channel fades by coding over a large number of coherence time intervals. 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