p . [ 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. ) ( ) 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. ( X The SNR is usually 3162. defining ( Y Its signicance comes from Shannon's coding theorem and converse, which show that capacityis the maximumerror-free data rate a channel can support. 1 {\displaystyle Y} 2 , {\displaystyle n} 1 p + The Shannon-Hartley theorem states the channel capacityC{\displaystyle C}, meaning the theoretical tightestupper bound on the information rateof data that can be communicated at an arbitrarily low error rateusing an average received signal power S{\displaystyle S}through an analog communication channel subject to additive white Gaussian ) Such a wave's frequency components are highly dependent. The channel capacity is defined as. 1 : 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. | 1 The channel capacity formula in Shannon's information theory defined the upper limit of the information transmission rate under the additive noise channel. {\displaystyle Y} 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. W ) Solution First, we use the Shannon formula to find the upper limit. , P {\displaystyle (Y_{1},Y_{2})} , C in Eq. X ) 1 2 and the corresponding output h y In 1927, Nyquist determined that the number of independent pulses that could be put through a telegraph channel per unit time is limited to twice the bandwidth of the channel. What is EDGE(Enhanced Data Rate for GSM Evolution)? Y N ( 2 1 | Shannon capacity is used, to determine the theoretical highest data rate for a noisy channel: Capacity = bandwidth * log 2 (1 + SNR) bits/sec In the above equation, bandwidth is the bandwidth of the channel, SNR is the signal-to-noise ratio, and capacity is the capacity of the channel in bits per second. . x {\displaystyle {\mathcal {Y}}_{1}} ( 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). | {\displaystyle 2B} 2 The bandwidth-limited regime and power-limited regime are illustrated in the figure. log / For large or small and constant signal-to-noise ratios, the capacity formula can be approximated: When the SNR is large (S/N 1), the logarithm is approximated by. p Hartley did not work out exactly how the number M should depend on the noise statistics of the channel, or how the communication could be made reliable even when individual symbol pulses could not be reliably distinguished to M levels; with Gaussian noise statistics, system designers had to choose a very conservative value of p In information theory, the ShannonHartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. 2 X {\displaystyle X_{1}} {\displaystyle {\begin{aligned}I(X_{1},X_{2}:Y_{1},Y_{2})&=H(Y_{1},Y_{2})-H(Y_{1},Y_{2}|X_{1},X_{2})\\&\leq H(Y_{1})+H(Y_{2})-H(Y_{1},Y_{2}|X_{1},X_{2})\end{aligned}}}, H ( and information transmitted at a line rate In the 1940s, Claude Shannon developed the concept of channel capacity, based in part on the ideas of Nyquist and Hartley, and then formulated a complete theory of information and its transmission. 2 ( 1 1 ( p 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)). 2. X ) | Y ( {\displaystyle X_{1}} , which is an inherent fixed property of the communication channel. ( and 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. Y pulses per second as signalling at the Nyquist rate. , two probability distributions for So no useful information can be transmitted beyond the channel capacity. Idem for p , 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. X X What can be the maximum bit rate? Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. 2 X 2 Equation: C = Blog (1+SNR) Represents theoretical maximum that can be achieved In practice, only much lower rates achieved Formula assumes white noise (thermal noise) Impulse noise is not accounted for - Attenuation distortion or delay distortion not accounted for Example of Nyquist and Shannon Formulations (1 . ) x Channel capacity is proportional to . N X 1 {\displaystyle \mathbb {E} (\log _{2}(1+|h|^{2}SNR))} ( Some authors refer to it as a capacity. 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. N pulses per second, to arrive at his quantitative measure for achievable line rate. p 1 1 S ) Perhaps the most eminent of Shannon's results was the concept that every communication channel had a speed limit, measured in binary digits per second: this is the famous Shannon Limit, exemplified by the famous and familiar formula for the capacity of a White Gaussian Noise Channel: 1 Gallager, R. Quoted in Technology Review, 1 X {\displaystyle C} , 1 {\displaystyle {\mathcal {Y}}_{1}} log : ( 1 When the SNR is large (SNR 0 dB), the capacity B Bandwidth and noise affect the rate at which information can be transmitted over an analog channel. That means a signal deeply buried in noise. is the bandwidth (in hertz). Y ) y Y ( Y and 1 = Y ( , depends on the random channel gain 1 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. Shannon's theorem shows how to compute a channel capacity from a statistical description of a channel, and establishes that given a noisy channel with capacity 1 P 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. Y Y At the time, these concepts were powerful breakthroughs individually, but they were not part of a comprehensive theory. ( f 1 {\displaystyle Y_{1}} p {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})} and = X , , we can rewrite 2 = X ( 2 How Address Resolution Protocol (ARP) works? 0 , 1 = X X It is also known as channel capacity theorem and Shannon capacity. {\displaystyle \forall (x_{1},x_{2})\in ({\mathcal {X}}_{1},{\mathcal {X}}_{2}),\;(y_{1},y_{2})\in ({\mathcal {Y}}_{1},{\mathcal {Y}}_{2}),\;(p_{1}\times p_{2})((y_{1},y_{2})|(x_{1},x_{2}))=p_{1}(y_{1}|x_{1})p_{2}(y_{2}|x_{2})}. 1 Y + W If there were such a thing as a noise-free analog channel, one could transmit unlimited amounts of error-free data over it per unit of time (Note that an infinite-bandwidth analog channel couldnt transmit unlimited amounts of error-free data absent infinite signal power). ( ) Note Increasing the levels of a signal may reduce the reliability of the system. ) + 1 through the channel C {\displaystyle p_{X_{1},X_{2}}} It connects Hartley's result with Shannon's channel capacity theorem in a form that is equivalent to specifying the M in Hartley's line rate formula in terms of a signal-to-noise ratio, but achieving reliability through error-correction coding rather than through reliably distinguishable pulse levels. ) Y , x . 2 P ( 2 : 2 {\displaystyle p_{2}} = 2 When the SNR is small (SNR 0 dB), the capacity For SNR > 0, the limit increases slowly. {\displaystyle W} {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=\sum _{(x_{1},x_{2})\in {\mathcal {X}}_{1}\times {\mathcal {X}}_{2}}\mathbb {P} (X_{1},X_{2}=x_{1},x_{2})H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})}. ) C 2 ) 10 X 1 {\displaystyle X_{2}} X 2 B Shannon capacity bps 10 p. linear here L o g r i t h m i c i n t h i s 0 10 20 30 Figure 3: Shannon capacity in bits/s as a function of SNR. h 1 there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. = is independent of Y x {\displaystyle (x_{1},x_{2})} S and X 2 2 N Shannon-Hartley theorem v t e Channel capacity, in electrical engineering, computer science, and information theory, is the tight upper boundon the rate at which informationcan be reliably transmitted over a communication channel. ( They become the same if M = 1 + S N R. Nyquist simply says: you can send 2B symbols per second. , } ( (4), is given in bits per second and is called the channel capacity, or the Shan-non capacity. 1 , I 0 and , , in Hertz and what today is called the digital bandwidth, {\displaystyle p_{X,Y}(x,y)} 2 = , with P {\displaystyle (x_{1},x_{2})} Y This may be true, but it cannot be done with a binary system. p S | . x and More about MIT News at Massachusetts Institute of Technology, Abdul Latif Jameel Poverty Action Lab (J-PAL), Picower Institute for Learning and Memory, School of Humanities, Arts, and Social Sciences, View all news coverage of MIT in the media, David Forneys acceptance speech on receiving the IEEEs Shannon Award, ARCHIVE: "MIT Professor Claude Shannon dies; was founder of digital communications", 3 Questions: Daniel Auguste on why successful entrepreneurs dont fall from the sky, Report: CHIPS Act just the first step in addressing threats to US leadership in advanced computing, New purification method could make protein drugs cheaper, Phiala Shanahan is seeking fundamental answers about our physical world. X log The capacity of the frequency-selective channel is given by so-called water filling power allocation. 1 ( as: H 1 ( = | = This addition creates uncertainty as to the original signal's value. {\displaystyle C} 1 . = 2 H 1 and , We can apply the following property of mutual information: H 2 p X p W [2] This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity. ) X p P , suffice: ie. + x 1 ) the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of Bandwidth limitations alone do not impose a cap on the maximum information rate because it is still possible for the signal to take on an indefinitely large number of different voltage levels on each symbol pulse, with each slightly different level being assigned a different meaning or bit sequence. 2 ) 1 1. 2 , in bit/s. 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 , For a given pair 1 and ( 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. Therefore. 2 ( {\displaystyle p_{X}(x)} , and {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} , ) = N {\displaystyle N=B\cdot N_{0}} So far, the communication technique has been rapidly developed to approach this theoretical limit. t Y ( 2 Y X {\displaystyle p_{1}} 2 + ) X . Nyquist published his results in 1928 as part of his paper "Certain topics in Telegraph Transmission Theory".[1]. = , P | 1 ( X {\displaystyle C(p_{2})} {\displaystyle S/N} Y {\displaystyle 2B} 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. Y The theorem does not address the rare situation in which rate and capacity are equal. The Shannon capacity theorem defines the maximum amount of information, or data capacity, which can be sent over any channel or medium (wireless, coax, twister pair, fiber etc.). 1 = 1 p p + 2 through Now let us show that y This capacity is given by an expression often known as "Shannon's formula1": C = W log2(1 + P/N) bits/second. 2 A generalization of the above equation for the case where the additive noise is not white (or that the {\displaystyle p_{out}} {\displaystyle {\bar {P}}} 1 The basic mathematical model for a communication system is the following: Let 1 12 2 {\displaystyle M} Let ) {\displaystyle C(p_{1}\times p_{2})=\sup _{p_{X_{1},X_{2}}}(I(X_{1},X_{2}:Y_{1},Y_{2}))} , ( {\displaystyle p_{1}} x x {\displaystyle p_{1}\times p_{2}} X Bandwidth is a fixed quantity, so it cannot be changed. y ( R {\displaystyle S} is the total power of the received signal and noise together. x x Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. X Given a channel with particular bandwidth and noise characteristics, Shannon showed how to calculate the maximum rate at which data can be sent over it with zero error. 1 H , ) p 1 H = ) 15K views 3 years ago Analog and Digital Communication This video lecture discusses the information capacity theorem. ) 2 , | H Y , N ) : C x , 2 Y B the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. ( 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 H 2 Output1 : BitRate = 2 * 3000 * log2(2) = 6000bps, Input2 : We need to send 265 kbps over a noiseless channel with a bandwidth of 20 kHz. ARP, Reverse ARP(RARP), Inverse ARP (InARP), Proxy ARP and Gratuitous ARP, Difference between layer-2 and layer-3 switches, Computer Network | Leaky bucket algorithm, Multiplexing and Demultiplexing in Transport Layer, Domain Name System (DNS) in Application Layer, Address Resolution in DNS (Domain Name Server), Dynamic Host Configuration Protocol (DHCP). Difference between Fixed and Dynamic Channel Allocations, Multiplexing (Channel Sharing) in Computer Network, Channel Allocation Strategies in Computer Network. 1 ( y {\displaystyle \epsilon } During 1928, Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). p , 2 Similarly, when the SNR is small (if 2 x 2 , which is the HartleyShannon result that followed later. 1 X How many signal levels do we need? ) 2 {\displaystyle X_{1}} p ( x B {\displaystyle {\mathcal {X}}_{1}} ) where 1 This value is known as the be two independent channels modelled as above; 2 Y X 1 2 1 {\displaystyle B} 1 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). 1 ( , X ( 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. 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