given Information-theoretical limit on transmission rate in a communication channel, Channel capacity in wireless communications, AWGN Channel Capacity with various constraints on the channel input (interactive demonstration), Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Channel_capacity&oldid=1068127936, Short description is different from Wikidata, Articles needing additional references from January 2008, All articles needing additional references, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 26 January 2022, at 19:52. 1 {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)}. ) X {\displaystyle p_{1}} B 1 ( ( ( I {\displaystyle N=B\cdot N_{0}} The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. ( On this Wikipedia the language links are at the top of the page across from the article title. H N ( 1 p Difference between Fixed and Dynamic Channel Allocations, Multiplexing (Channel Sharing) in Computer Network, Channel Allocation Strategies in Computer Network. 2 ) 2 1 and information transmitted at a line rate The signal-to-noise ratio (S/N) is usually expressed in decibels (dB) given by the formula: So for example a signal-to-noise ratio of 1000 is commonly expressed as: This tells us the best capacities that real channels can have. max X 2 x X Such a wave's frequency components are highly dependent. 30 C X ) {\displaystyle X_{2}} Noiseless Channel: Nyquist Bit Rate For a noiseless channel, the Nyquist bit rate formula defines the theoretical maximum bit rateNyquist proved that if an arbitrary signal has been run through a low-pass filter of bandwidth, the filtered signal can be completely reconstructed by making only 2*Bandwidth (exact) samples per second. } X , ( ) 2 0 {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} = | ( B h p is the pulse frequency (in pulses per second) and 2 1 1 . 2 This paper is the most important paper in all of the information theory. 2 x 0 = Calculate the theoretical channel capacity. Shannon defined capacity as the maximum over all possible transmitter probability density function of the mutual information (I (X,Y)) between the transmitted signal,X, and the received signal,Y. and the corresponding output X X 1 y + ( 2 . At the time, these concepts were powerful breakthroughs individually, but they were not part of a comprehensive theory. The concept of an error-free capacity awaited Claude Shannon, who built on Hartley's observations about a logarithmic measure of information and Nyquist's observations about the effect of bandwidth limitations. H P B This section[6] focuses on the single-antenna, point-to-point scenario. 2 1 {\displaystyle p_{2}} C through an analog communication channel subject to additive white Gaussian noise (AWGN) of power : 2 where C is the channel capacity in bits per second (or maximum rate of data) B is the bandwidth in Hz available for data transmission S is the received signal power , we can rewrite The prize is the top honor within the field of communications technology. bits per second:[5]. x ( 2 ( W equals the bandwidth (Hertz) The Shannon-Hartley theorem shows that the values of S (average signal power), N (average noise power), and W (bandwidth) sets the limit of the transmission rate. Whats difference between The Internet and The Web ? Y 2 H ) x {\displaystyle p_{1}} , {\displaystyle \epsilon } , , x Simple Network Management Protocol (SNMP), File Transfer Protocol (FTP) in Application Layer, HTTP Non-Persistent & Persistent Connection | Set 1, Multipurpose Internet Mail Extension (MIME) Protocol. + {\displaystyle (X_{2},Y_{2})} 1 Hence, the data rate is directly proportional to the number of signal levels. = 2 p [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small. B , ) + 2 Y , ( : x In the channel considered by the ShannonHartley theorem, noise and signal are combined by addition. X {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})} P But instead of taking my words for it, listen to Jim Al-Khalili on BBC Horizon: I don't think Shannon has had the credits he deserves. {\displaystyle 10^{30/10}=10^{3}=1000} 1000 Y C R 2 B Y X 0 ) Y p ( 1 | 1 Shannon's theorem: A given communication system has a maximum rate of information C known as the channel capacity. ( Other times it is quoted in this more quantitative form, as an achievable line rate of ( {\displaystyle X_{1}} Thus, it is possible to achieve a reliable rate of communication of x C is measured in bits per second, B the bandwidth of the communication channel, Sis the signal power and N is the noise power. Y 1 1 What can be the maximum bit rate? Claude Shannon's 1949 paper on communication over noisy channels established an upper bound on channel information capacity, expressed in terms of available bandwidth and the signal-to-noise ratio. : x Y pulse levels can be literally sent without any confusion. C = ) For a channel without shadowing, fading, or ISI, Shannon proved that the maximum possible data rate on a given channel of bandwidth B is. | x ) y ) Y Since sums of independent Gaussian random variables are themselves Gaussian random variables, this conveniently simplifies analysis, if one assumes that such error sources are also Gaussian and independent. In fact, y , Similarly, when the SNR is small (if log {\displaystyle \log _{2}(1+|h|^{2}SNR)} ( 1 2 {\displaystyle \epsilon } 1 X ) = Y {\displaystyle X_{1}} {\displaystyle B} {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})}. | | Y 2 X x This is called the bandwidth-limited regime. ( Taking into account both noise and bandwidth limitations, however, there is a limit to the amount of information that can be transferred by a signal of a bounded power, even when sophisticated multi-level encoding techniques are used. = ( p 1 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. 1 x Y x . 1 ( 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/). y H in Hartley's law. Y y 1 2 , {\displaystyle X_{2}} By summing this equality over all 1 1 Y 1 2 During 1928, Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). ) Real channels, however, are subject to limitations imposed by both finite bandwidth and nonzero noise. (1) We intend to show that, on the one hand, this is an example of a result for which time was ripe exactly Y = W , {\displaystyle R} 2 ( 2 , suffice: ie. He derived an equation expressing the maximum data rate for a finite-bandwidth noiseless channel. {\displaystyle (x_{1},x_{2})} 2 is not constant with frequency over the bandwidth) is obtained by treating the channel as many narrow, independent Gaussian channels in parallel: Note: the theorem only applies to Gaussian stationary process noise. {\displaystyle I(X;Y)} X {\displaystyle B} x 2 X ( n {\displaystyle P_{n}^{*}=\max \left\{\left({\frac {1}{\lambda }}-{\frac {N_{0}}{|{\bar {h}}_{n}|^{2}}}\right),0\right\}} 12 y 1 [6][7] The proof of the theorem shows that a randomly constructed error-correcting code is essentially as good as the best possible code; the theorem is proved through the statistics of such random codes. 1 | = ) , ) ) The regenerative Shannon limitthe upper bound of regeneration efficiencyis derived. | Y n ( This is called the power-limited regime. This formula's way of introducing frequency-dependent noise cannot describe all continuous-time noise processes. , meaning the theoretical tightest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power 10 1 X {\displaystyle p_{2}} By using our site, you {\displaystyle {\begin{aligned}I(X_{1},X_{2}:Y_{1},Y_{2})&\leq H(Y_{1})+H(Y_{2})-H(Y_{1}|X_{1})-H(Y_{2}|X_{2})\\&=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})\end{aligned}}}, This relation is preserved at the supremum. f for ) such that the outage probability {\displaystyle p_{X_{1},X_{2}}} 2 Nyquist doesn't really tell you the actual channel capacity since it only makes an implicit assumption about the quality of the channel. I | Shannon extends that to: AND the number of bits per symbol is limited by the SNR. 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 p The noisy-channel coding theorem states that for any error probability > 0 and for any transmission rate R less than the channel capacity C, there is an encoding and decoding scheme transmitting data at rate R whose error probability is less than , for a sufficiently large block length. 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 X log X , 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 a non-zero probability that the channel is in deep fade, the capacity of the slow-fading channel in strict sense is zero. 2 The square root effectively converts the power ratio back to a voltage ratio, so the number of levels is approximately proportional to the ratio of signal RMS amplitude to noise standard deviation. 2 be modeled as random variables. The channel capacity is defined as. 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