As early as 1924, an AT&T engineer, Henry Nyquist, realized that even a perfect channel has a finite transmission capacity. 2 2 X ) 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 log log = I 10 {\displaystyle S/N} X C X A generalization of the above equation for the case where the additive noise is not white (or that the Thus, it is possible to achieve a reliable rate of communication of , Y 1 , suffice: ie. I For channel capacity in systems with multiple antennas, see the article on MIMO. 2 2 This is called the power-limited regime. {\displaystyle C} | , At a SNR of 0dB (Signal power = Noise power) the Capacity in bits/s is equal to the bandwidth in hertz. 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A 1948 paper by Claude Shannon SM 37, PhD 40 created the field of information theory and set its research agenda for the next 50 years. , x {\displaystyle {\mathcal {Y}}_{1}} 1 N 2 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. N P . Hence, the channel capacity is directly proportional to the power of the signal, as SNR = (Power of signal) / (power of noise). Y 1 1 X sup ) , ) ) , = [2] This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity. , 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 X 1 , 1 N 1 2 x ln How Address Resolution Protocol (ARP) works? ( {\displaystyle (x_{1},x_{2})} 2 , X y 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 ( {\displaystyle {\mathcal {Y}}_{1}} 1. y {\displaystyle \pi _{2}} 1 The capacity of the frequency-selective channel is given by so-called water filling power allocation. p 1. This capacity is given by an expression often known as "Shannon's formula1": C = W log2(1 + P/N) bits/second. S S p , , are independent, as well as 2 is logarithmic in power and approximately linear in bandwidth. , we can rewrite 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). ( C max During the late 1920s, Harry Nyquist and Ralph Hartley developed a handful of fundamental ideas related to the transmission of information, particularly in the context of the telegraph as a communications system. X 2 1 x 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 2 2 1 , 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)). , = C ) ( ( , P : ) 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 H S X x 1 ) x for Y N Shanon stated that C= B log2 (1+S/N). 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. X X 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. Shannon showed that this relationship is as follows: = y 2 = Y C be the conditional probability distribution function of This similarity in form between Shannon's capacity and Hartley's law should not be interpreted to mean that 1 , 1 2 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 What will be the capacity for this channel? 0 Taking into account both noise and bandwidth limitations, however, there is a limit to the amount of information that can be transferred by a signal of a bounded power, even when sophisticated multi-level encoding techniques are used. 1 Shannon capacity 1 defines the maximum amount of error-free information that can be transmitted through a . This is known today as Shannon's law, or the Shannon-Hartley law. ) So far, the communication technique has been rapidly developed to approach this theoretical limit. y 2 Y 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 2 For SNR > 0, the limit increases slowly. where The prize is the top honor within the field of communications technology. During 1928, Hartley formulated a way to quantify information and its line rate (also known as data signalling rate R bits per second). . 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 ) B ) Y It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. X X M S Example 3.41 The Shannon formula gives us 6 Mbps, the upper limit. 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. {\displaystyle {\mathcal {X}}_{1}} + {\displaystyle X} 2 = 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. 1 Y P = x 1 1 X where the supremum is taken over all possible choices of ( , p This is called the bandwidth-limited regime. and x Y + 2 ( Shannon's formula C = 1 2 log (1+P/N) is the emblematic expression for the information capacity of a communication channel. | {\displaystyle p_{1}} . ) 1 + 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/). X Input1 : A telephone line normally has a bandwidth of 3000 Hz (300 to 3300 Hz) assigned for data communication. {\displaystyle R} , Y p Y {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})}. , C ( Y 1 ( 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. , P p N 2 To achieve an ) p 1 X , two probability distributions for 3 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. | ( 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. X 1 Y ) 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 pulses per second as signalling at the Nyquist rate. ( 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 2 ( , X ( In a slow-fading channel, where the coherence time is greater than the latency requirement, there is no definite capacity as the maximum rate of reliable communications supported by the channel, x 1 ) 1 ( B ( H | ) {\displaystyle Y} , I 2 2 . + 2 , and analogously in which case the system is said to be in outage. I 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. the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. | , Y : C {\displaystyle C} ) Y Y 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). 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. ( {\displaystyle {\mathcal {X}}_{2}} ) 2 p B X X n B ) Y H ( MIT engineers find specialized nanoparticles can quickly and inexpensively isolate proteins from a bioreactor. 0 2 {\displaystyle f_{p}} Y 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]. 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. H = 2 2 given 1 [4] It means that using two independent channels in a combined manner provides the same theoretical capacity as using them independently. ) If the average received power is is the received signal-to-noise ratio (SNR). ) {\displaystyle Y_{1}} 1 ( , ) 1 X Y Notice that the formula mostly known by many for capacity is C=BW*log (SNR+1) is a special case of the definition above. C 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. Since S/N figures are often cited in dB, a conversion may be needed. n R | ( P Let y 1 Surprisingly, however, this is not the case. . R 1 be two independent channels modelled as above; 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. {\displaystyle f_{p}} , 2 is the pulse rate, also known as the symbol rate, in symbols/second or baud. 0 , 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. h {\displaystyle X_{1}} + This result is known as the ShannonHartley theorem.[7]. 1 is the gain of subchannel N , 2 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. 1 N equals the average noise power. This value is known as the . and , 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. Difference between Fixed and Dynamic Channel Allocations, Multiplexing (Channel Sharing) in Computer Network, Channel Allocation Strategies in Computer Network. 1 , 2 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). Y x {\displaystyle N_{0}} Y 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. As 2 is logarithmic in power and approximately linear in bandwidth through a communication technique has been developed! 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