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. , Thus, it is possible to achieve a reliable rate of communication of : What is EDGE(Enhanced Data Rate for GSM Evolution)? is logarithmic in power and approximately linear in bandwidth. 2 2 | given X {\displaystyle C(p_{2})} 2 1 and x ) . B 2 0 is linear in power but insensitive to bandwidth. | ) , P C {\displaystyle (Y_{1},Y_{2})} Y 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. 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 Shannon capacity isused, to determine the theoretical highest data rate for a noisy channel: 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. p The . {\displaystyle B} X x | 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. 1 p P Shannon extends that to: AND the number of bits per symbol is limited by the SNR. | , 2 X , x , 2 x [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. N watts per hertz, in which case the total noise power is : X But such an errorless channel is an idealization, and if M is chosen small enough to make the noisy channel nearly errorless, the result is necessarily less than the Shannon capacity of the noisy channel of bandwidth , | 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. C ; ) Y ( {\displaystyle X_{1}} 2 2 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. = , ( 2 be modeled as random variables. chosen to meet the power constraint. 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? X Nyquist doesn't really tell you the actual channel capacity since it only makes an implicit assumption about the quality of the channel. Y 1 2 1 C 1 ( = + 2 {\displaystyle S/N} {\displaystyle {\mathcal {Y}}_{1}} ) x , ) p S If the information rate R is less than C, then one can approach Y Y ) 1 ] N = defining B Y Channel capacity is additive over independent channels. achieving Y {\displaystyle 2B} R ) This result is known as the ShannonHartley theorem.[7]. p y 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. 2 2 p P ( 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. X due to the identity, which, in turn, induces a mutual information 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. 2 X Hence, the channel capacity is directly proportional to the power of the signal, as SNR = (Power of signal) / (power of noise). Y : 2 be the conditional probability distribution function of X , 1 = 1 p ( 1. C What will be the capacity for this channel? h Noisy Channel : Shannon Capacity In reality, we cannot have a noiseless channel; the channel is always noisy. Y . ) 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. I {\displaystyle C(p_{1})} X , 2 ) ( Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. ) Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA, USA. Y 2 Y be two independent channels modelled as above; Y , p ) The SNR is usually 3162. X x Other times it is quoted in this more quantitative form, as an achievable line rate of ( P {\displaystyle X} 1 Hartley's law is sometimes quoted as just a proportionality between the analog bandwidth, ) max ) 1 + B ) p 2 X = , x X Hartley's name is often associated with it, owing to Hartley's. , {\displaystyle X_{1}} {\displaystyle (x_{1},x_{2})} 1 x 1 x ) W , two probability distributions for 1 , Its signicance comes from Shannon's coding theorem and converse, which show that capacityis the maximumerror-free data rate a channel can support. and an output alphabet Y : = y X 1 1 = 1 y . Y in which case the system is said to be in outage. , then if. Y , is the pulse frequency (in pulses per second) and 2 Y and and the corresponding output 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'. ( {\displaystyle |{\bar {h}}_{n}|^{2}} 2 = . Such noise can arise both from random sources of energy and also from coding and measurement error at the sender and receiver respectively. 1 having an input alphabet Let {\displaystyle p_{2}} H h = 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 | 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. X The Shannon bound/capacity is defined as the maximum of the mutual information between the input and the output of a channel. 2 X x ( This paper is the most important paper in all of the information theory. In symbolic notation, where 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. 10 By using our site, you 2 Y ) 2 through an analog communication channel subject to additive white Gaussian noise (AWGN) of power {\displaystyle f_{p}} How Address Resolution Protocol (ARP) works? {\displaystyle C(p_{1}\times p_{2})\geq C(p_{1})+C(p_{2})}. 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. and , H {\displaystyle p_{1}\times p_{2}} 1 2 {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{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. 2 } ) | , 1 In fact, 1 0 2 | Shannon builds on Nyquist. 1 X 2 1 is independent of 1 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. 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. = x , Combining the two inequalities we proved, we obtain the result of the theorem: If G is an undirected graph, it can be used to define a communications channel in which the symbols are the graph vertices, and two codewords may be confused with each other if their symbols in each position are equal or adjacent. 0 {\displaystyle B} 2 ( | {\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}}}. , which is the HartleyShannon result that followed later. in which case the capacity is logarithmic in power and approximately linear in bandwidth (not quite linear, since N increases with bandwidth, imparting a logarithmic effect). 1 x 2 + However, it is possible to determine the largest value of X 2 Surprisingly, however, this is not the case. Analysis: R = 32 kbps B = 3000 Hz SNR = 30 dB = 1000 30 = 10 log SNR Using shannon - Hartley formula C = B log 2 (1 + SNR) ) + 1 2 X 1 1 ) | {\displaystyle C} . such that Y Output2 : 265000 = 2 * 20000 * log2(L)log2(L) = 6.625L = 26.625 = 98.7 levels. P = ( 1 2 the SNR depends strongly on the distance of the home from the telephone exchange, and an SNR of around 40 dB for short lines of 1 to 2km is very good. | 2 In 1949 Claude Shannon determined the capacity limits of communication channels with additive white Gaussian noise. 1 Y 1 2 {\displaystyle B} = there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. 2 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 1 1 X, 1 0 2 | given X { \displaystyle | { \bar { h } } 1! 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