p where . For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of {\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 X ) I p x H [3]. = By definition , ) 2 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. ( ( : , , which is the HartleyShannon result that followed later. . x 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 x there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. We can now give an upper bound over mutual information: I A very important consideration in data communication is how fast we can send data, in bits per second, over a channel. {\displaystyle C(p_{1}\times p_{2})=\sup _{p_{X_{1},X_{2}}}(I(X_{1},X_{2}:Y_{1},Y_{2}))} x and log {\displaystyle 2B} Y 2 ( 2 symbols per second. {\displaystyle p_{X_{1},X_{2}}} be the conditional probability distribution function of = ) log ( x p X {\displaystyle (X_{1},X_{2})} ) Y 2 p later came to be called the Nyquist rate, and transmitting at the limiting pulse rate of 2 -outage capacity. = Y 0 2 , 1 ( | ( pulse levels can be literally sent without any confusion. Shannon's formula C = 1 2 log (1 + P/N) is the emblematic expression for the information capacity of a communication channel. X 1 2 Let | In the channel considered by the ShannonHartley theorem, noise and signal are combined by addition. 2 be two independent random variables. To achieve an , we can rewrite {\displaystyle N_{0}} 2 ( 1 ( Assume that SNR(dB) is 36 and the channel bandwidth is 2 MHz. N , {\displaystyle C} {\displaystyle p_{2}} {\displaystyle (X_{1},Y_{1})} For better performance we choose something lower, 4 Mbps, for example. {\displaystyle |h|^{2}} = {\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. X {\displaystyle Y_{2}} ) 1 ) | Since and X {\displaystyle |{\bar {h}}_{n}|^{2}} 1 0 . This result is known as the ShannonHartley theorem.[7]. , [4] 2 to achieve a low error rate. ) p Then the choice of the marginal distribution C is the pulse rate, also known as the symbol rate, in symbols/second or baud. W The SNR is usually 3162. . | Data rate governs the speed of data transmission. ( y be the alphabet of Y | | Y ) | Shannon capacity 1 defines the maximum amount of error-free information that can be transmitted through a . , 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. X At a SNR of 0dB (Signal power = Noise power) the Capacity in bits/s is equal to the bandwidth in hertz. p , Y = 1 He called that rate the channel capacity, but today, it's just as often called the Shannon limit. , 2 ( 1 achieving X During 1928, Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). Comparing the channel capacity to the information rate from Hartley's law, we can find the effective number of distinguishable levels M:[8]. If the information rate R is less than C, then one can approach Calculate the theoretical channel capacity. x H 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). 1 {\displaystyle M} 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. C 1 x x An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the ShannonHartley theorem: C is measured in bits per second if the logarithm is taken in base 2, or nats per second if the natural logarithm is used, assuming B is in hertz; the signal and noise powers S and N are expressed in a linear power unit (like watts or volts2). Y The basic mathematical model for a communication system is the following: Let 1 ( Thus, it is possible to achieve a reliable rate of communication of 1 Y X Y pulses per second, to arrive at his quantitative measure for achievable line rate. p Y {\displaystyle Y_{1}} P x {\displaystyle \pi _{2}} | ( 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. Let | ) ( Output2 : 265000 = 2 * 20000 * log2(L)log2(L) = 6.625L = 26.625 = 98.7 levels. x , } 1 : , hertz was ) y p {\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})}. ( | 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 y {\displaystyle (X_{2},Y_{2})} in which case the system is said to be in outage. p | | . + + , N ( M 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. However, it is possible to determine the largest value of ) For a channel without shadowing, fading, or ISI, Shannon proved that the maximum possible data rate on a given channel of bandwidth B is. , 2 X 2 2 X 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'. 1. 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 . ) 1 {\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})} ) 1 . 2 On this Wikipedia the language links are at the top of the page across from the article title. If the receiver has some information about the random process that generates the noise, one can in principle recover the information in the original signal by considering all possible states of the noise process. 1 ) , = 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? 1 , ) Y = ) X as ) 2 Y More formally, let ) 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. Idem for I n P 2 The capacity of the frequency-selective channel is given by so-called water filling power allocation. 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). where the supremum is taken over all possible choices of Real channels, however, are subject to limitations imposed by both finite bandwidth and nonzero noise. 1 p be some distribution for the channel , in bit/s. p ( 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 In the case of the ShannonHartley theorem, the noise is assumed to be generated by a Gaussian process with a known variance. B B ( , {\displaystyle M} the probability of error at the receiver increases without bound as the rate is increased. 2 Y 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.). The quantity 10 . X Data rate depends upon 3 factors: Two theoretical formulas were developed to calculate the data rate: one by Nyquist for a noiseless channel, another by Shannon for a noisy channel. Y Y Y {\displaystyle Y_{1}} f , 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 {\mathcal {Y}}_{1}} ) X , 2 y 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. 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