2 Data rate governs the speed of data transmission. The ShannonHartley theorem establishes what that channel capacity is for a finite-bandwidth continuous-time channel subject to Gaussian noise. 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. 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. C B The prize is the top honor within the field of communications technology. {\displaystyle C} 1 Shannon limit for information capacity is I = (3.32)(2700) log 10 (1 + 1000) = 26.9 kbps Shannon's formula is often misunderstood. More formally, let {\displaystyle B} {\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})} 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. X This website is managed by the MIT News Office, part of the Institute Office of Communications. x {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)}. = X R = for ) {\displaystyle N_{0}} X R = | 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, X The theorem does not address the rare situation in which rate and capacity are equal. 2 ) X ) We can now give an upper bound over mutual information: I 0 ) 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. 1 the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. X 1 C 1 1 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.). B How many signal levels do we need? . 1 I Y 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. C y {\displaystyle R} ( R and due to the identity, which, in turn, induces a mutual information This means that theoretically, it is possible to transmit information nearly without error up to nearly a limit of 0 information rate increases the number of errors per second will also increase. 1 and 1 2 2 = ( p 1 2 Y ( 2 {\displaystyle p_{2}} 1 X p ( P B {\displaystyle p_{2}} , X + Y {\displaystyle X} ) Calculate the theoretical channel capacity. 1 1 0 1 = , {\displaystyle X_{2}} R 2 ) The Shannon's equation relies on two important concepts: That, in principle, a trade-off between SNR and bandwidth is possible That, the information capacity depends on both SNR and bandwidth It is worth to mention two important works by eminent scientists prior to Shannon's paper [1]. x 2 u 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 ( What is EDGE(Enhanced Data Rate for GSM Evolution)? X = R The bandwidth-limited regime and power-limited regime are illustrated in the figure. ( {\displaystyle (x_{1},x_{2})} {\displaystyle M} Solution First, we use the Shannon formula to find the upper limit. In the simple version above, the signal and noise are fully uncorrelated, in which case ) is the bandwidth (in hertz). p are independent, as well as 2 1 p X 2 log 30 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 X X n | Y x I ) Output1 : BitRate = 2 * 3000 * log2(2) = 6000bps, Input2 : We need to send 265 kbps over a noiseless channel with a bandwidth of 20 kHz. p Other times it is quoted in this more quantitative form, as an achievable line rate of This may be true, but it cannot be done with a binary system. y y 1 C The notion of channel capacity has been central to the development of modern wireline and wireless communication systems, with the advent of novel error correction coding mechanisms that have resulted in achieving performance very close to the limits promised by channel capacity. 2 [1][2], Information theory, developed by Claude E. Shannon in 1948, defines the notion of channel capacity and provides a mathematical model by which it may be computed. By definition of mutual information, we have, I ) X ( 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 . {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})}. ) : The Advanced Computing Users Survey, sampling sentiments from 120 top-tier universities, national labs, federal agencies, and private firms, finds the decline in Americas advanced computing lead spans many areas. X ) {\displaystyle I(X;Y)} 2 1 2 where the supremum is taken over all possible choices of If the average received power is 1 2 / ( : Y 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). Shannon's theory has since transformed the world like no other ever had, from information technologies to telecommunications, from theoretical physics to economical globalization, from everyday life to philosophy. 2 1 , C ( ( 2 X | Y | X {\displaystyle f_{p}} 1 1 = , log p Y {\displaystyle X_{1}} ( 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. Shannon capacity 1 defines the maximum amount of error-free information that can be transmitted through a . = , 1 . What will be the capacity for this channel? p : C be modeled as random variables. 2 ) (1) We intend to show that, on the one hand, this is an example of a result for which time was ripe exactly The mathematical equation defining Shannon's Capacity Limit is shown below, and although mathematically simple, it has very complex implications in the real world where theory and engineering rubber meets the road. {\displaystyle p_{1}} Shannon's discovery of , 1 2 such that the outage probability Y 2 2 ) {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})\geq I(X_{1}:Y_{1})+I(X_{2}:Y_{2})} X = ) 1 The input and output of MIMO channels are vectors, not scalars as. X X They become the same if M = 1 + S N R. Nyquist simply says: you can send 2B symbols per second. The SNR is usually 3162. ( ( 2 X 2 2 and an output alphabet 1 3 1 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. ) y ) {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} ( ) 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. completely determines the joint distribution Y 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 2 Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. X , + H , : p ) 2 I | It has two ranges, the one below 0 dB SNR and one above. With a non-zero probability that the channel is in deep fade, the capacity of the slow-fading channel in strict sense is zero. 1 ( Let 1 x through chosen to meet the power constraint. X 2 ) X , with What can be the maximum bit rate? X Boston teen designers create fashion inspired by award-winning images from MIT laboratories. Y Y , This is called the power-limited regime. Hartley then combined the above quantification with Nyquist's observation that the number of independent pulses that could be put through a channel of bandwidth ) Y Y 2 ( Shanon stated that C= B log2 (1+S/N). be some distribution for the channel 2 x 1 , {\displaystyle (X_{2},Y_{2})} ( 1 | : Y Let 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. y P 1 ( (1) We intend to show that, on the one hand, this is an example of a result for which time was ripe exactly Shannon Capacity The maximum mutual information of a channel. p If the SNR is 20dB, and the bandwidth available is 4kHz, which is appropriate for telephone communications, then C = 4000 log, If the requirement is to transmit at 50 kbit/s, and a bandwidth of 10kHz is used, then the minimum S/N required is given by 50000 = 10000 log, What is the channel capacity for a signal having a 1MHz bandwidth, received with a SNR of 30dB? 2 Output2 : SNR(dB) = 10 * log10(SNR)SNR = 10(SNR(dB)/10)SNR = 103.6 = 3981, Reference:Book Computer Networks: A Top Down Approach by FOROUZAN, Capacity of a channel in Computer Network, Co-Channel and Adjacent Channel Interference in Mobile Computing, Difference between Bit Rate and Baud Rate, Data Communication - Definition, Components, Types, Channels, Difference between Bandwidth and Data Rate. to achieve a low error rate. ) ln , and ) In the channel considered by the ShannonHartley theorem, noise and signal are combined by addition. 1 ) 2 ( As early as 1924, an AT&T engineer, Henry Nyquist, realized that even a perfect channel has a finite transmission capacity. ( ) 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. 1 1 1 {\displaystyle X_{1}} ) given {\displaystyle 2B} 1 y | , | {\displaystyle p_{X_{1},X_{2}}} Sampling the line faster than 2*Bandwidth times per second is pointless because the higher-frequency components that such sampling could recover have already been filtered out. X 1 S 0 , p ( , which is the HartleyShannon result that followed later. I Bandwidth and noise affect the rate at which information can be transmitted over an analog channel. X ( and = 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. ) ( ( {\displaystyle Y} ( ( 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. 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 ) p 2 1 y 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]. This is called the bandwidth-limited regime. is the pulse frequency (in pulses per second) and When the SNR is large (SNR 0 dB), the capacity ) ) H The Shannon bound/capacity is defined as the maximum of the mutual information between the input and the output of a channel. P 1 X 2 y x Such a wave's frequency components are highly dependent. x Y , A very important consideration in data communication is how fast we can send data, in bits per second, over a channel. X 1 log = ) {\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})}. . , n | 1 | 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. ) x x 2 N equals the average noise power. 2 ) X , 1 {\displaystyle C(p_{1})} x | Program to calculate the Round Trip Time (RTT), Introduction of MAC Address in Computer Network, Maximum Data Rate (channel capacity) for Noiseless and Noisy channels, Difference between Unicast, Broadcast and Multicast in Computer Network, Collision Domain and Broadcast Domain in Computer Network, Internet Protocol version 6 (IPv6) Header, Program to determine class, Network and Host ID of an IPv4 address, C Program to find IP Address, Subnet Mask & Default Gateway, Introduction of Variable Length Subnet Mask (VLSM), Types of Network Address Translation (NAT), Difference between Distance vector routing and Link State routing, Routing v/s Routed Protocols in Computer Network, Route Poisoning and Count to infinity problem in Routing, Open Shortest Path First (OSPF) Protocol fundamentals, Open Shortest Path First (OSPF) protocol States, Open shortest path first (OSPF) router roles and configuration, Root Bridge Election in Spanning Tree Protocol, Features of Enhanced Interior Gateway Routing Protocol (EIGRP), Routing Information Protocol (RIP) V1 & V2, Administrative Distance (AD) and Autonomous System (AS), Packet Switching and Delays in Computer Network, Differences between Virtual Circuits and Datagram Networks, Difference between Circuit Switching and Packet Switching. , P , Y 1.Introduction. {\displaystyle \epsilon } is the gain of subchannel = = 2 P and | 2 For SNR > 0, the limit increases slowly. 30dB means a S/N = 10, As stated above, channel capacity is proportional to the bandwidth of the channel and to the logarithm of SNR. . 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