, with [2] This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity. ) p The Shannon-Hartley theorem states that the channel capacity is given by- C = B log 2 (1 + S/N) where C is the capacity in bits per second, B is the bandwidth of the channel in Hertz, and S/N is the signal-to-noise ratio. P + Y R = , , and {\displaystyle X} {\displaystyle 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. N {\displaystyle p_{1}} 2 H 1 C in Eq. 0 ) {\displaystyle \mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})=\mathbb {P} (Y_{1}=y_{1}|X_{1}=x_{1})\mathbb {P} (Y_{2}=y_{2}|X_{2}=x_{2})} ( 2 ) + Y {\displaystyle (x_{1},x_{2})} 2 {\displaystyle 2B} Y 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]. Note Increasing the levels of a signal may reduce the reliability of the system. 1 ( 2 = Some authors refer to it as a capacity. and 7.2.7 Capacity Limits of Wireless Channels. and an output alphabet 1 1 p } 1 H ) ( Channel capacity is proportional to . 1 2 1 Such a channel is called the Additive White Gaussian Noise channel, because Gaussian noise is added to the signal; "white" means equal amounts of noise at all frequencies within the channel bandwidth. {\displaystyle C(p_{1})} 2 Y 2 ( He derived an equation expressing the maximum data rate for a finite-bandwidth noiseless channel. Y 1 {\displaystyle B} 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'. Real channels, however, are subject to limitations imposed by both finite bandwidth and nonzero noise. ) {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})\geq I(X_{1}:Y_{1})+I(X_{2}:Y_{2})} P C I I : 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)). ( p Let ( ( 2 1 ) , , 2 x X | ) C = 1 X , 1 2. = ( 1 x there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. Y ) 2 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. 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. P 1 ) {\displaystyle \pi _{1}} The prize is the top honor within the field of communications technology. {\displaystyle f_{p}} , The basic mathematical model for a communication system is the following: Let 2 {\displaystyle M} X ) p X 2 as: H . : 2 ( is the pulse frequency (in pulses per second) and {\displaystyle S/N} ( Y , bits per second:[5]. When the SNR is small (SNR 0 dB), the capacity Y ( {\displaystyle {\begin{aligned}H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})&=\sum _{(y_{1},y_{2})\in {\mathcal {Y}}_{1}\times {\mathcal {Y}}_{2}}\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})\log(\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2}))\\&=\sum _{(y_{1},y_{2})\in {\mathcal {Y}}_{1}\times {\mathcal {Y}}_{2}}\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})[\log(\mathbb {P} (Y_{1}=y_{1}|X_{1}=x_{1}))+\log(\mathbb {P} (Y_{2}=y_{2}|X_{2}=x_{2}))]\\&=H(Y_{1}|X_{1}=x_{1})+H(Y_{2}|X_{2}=x_{2})\end{aligned}}}. pulse levels can be literally sent without any confusion. Let N N p 2 2 2 X 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. = S This may be true, but it cannot be done with a binary system. X Y , 2 ( / ) 2 2 x ) How many signal levels do we need? 2 x 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. S 2 ) {\displaystyle X_{1}} Building on Hartley's foundation, Shannon's noisy channel coding theorem (1948) describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption. Bandwidth is a fixed quantity, so it cannot be changed. X {\displaystyle {\bar {P}}} {\displaystyle {\mathcal {X}}_{1}} {\displaystyle R} X X p 2 + 2 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. , the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. C , 2 Y ) {\displaystyle (Y_{1},Y_{2})} Y . X ) ( pulses per second, to arrive at his quantitative measure for achievable line rate. X x bits per second. ( Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA, USA. , | M Y Y Y {\displaystyle p_{X}(x)} ) Since S/N figures are often cited in dB, a conversion may be needed. This addition creates uncertainty as to the original signal's value. For years, modems that send data over the telephone lines have been stuck at a maximum rate of 9.6 kilobits per second: if you try to increase the rate, an intolerable number of errors creeps into the data. {\displaystyle p_{1}} 1 2 ( ) X Surprisingly, however, this is not the case. X ) 2 For better performance we choose something lower, 4 Mbps, for example. During 1928, Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). ( {\displaystyle p_{Y|X}(y|x)} X , The results of the preceding example indicate that 26.9 kbps can be propagated through a 2.7-kHz communications channel. {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} 2 : X Calculate the theoretical channel capacity. 2 X C 2 Also, for any rate greater than the channel capacity, the probability of error at the receiver goes to 0.5 as the block length goes to infinity. x ) 1 1 ( The MLK Visiting Professor studies the ways innovators are influenced by their communities. 1 Y ) For example, ADSL (Asymmetric Digital Subscriber Line), which provides Internet access over normal telephonic lines, uses a bandwidth of around 1 MHz. ( ( 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. y ) Output2 : 265000 = 2 * 20000 * log2(L)log2(L) = 6.625L = 26.625 = 98.7 levels. and {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{2})} | ) = For example, consider a noise process consisting of adding a random wave whose amplitude is 1 or 1 at any point in time, and a channel that adds such a wave to the source signal. p 1 P {\displaystyle (X_{1},X_{2})} Y In the simple version above, the signal and noise are fully uncorrelated, in which case 2 Y W y C as 1 Shannon's formula C = 1 2 log (1 + P/N) is the emblematic expression for the information capacity of a communication channel. 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