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 The SNR is usually 3162. 1 C B , then if. 2 [bits/s/Hz] and it is meaningful to speak of this value as the capacity of the fast-fading channel. ( be two independent random variables. X , 1 ) and , = X 1 X be the alphabet of x How DHCP server dynamically assigns IP address to a host? 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. {\displaystyle n} ( log 0 ( 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. , we can rewrite ) log Example 3.41 The Shannon formula gives us 6 Mbps, the upper limit. Y 1 X 1.Introduction. y At the time, these concepts were powerful breakthroughs individually, but they were not part of a comprehensive theory. This section[6] focuses on the single-antenna, point-to-point scenario. symbols per second. 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. 2 The quantity , Bandwidth is a fixed quantity, so it cannot be changed. Y , depends on the random channel gain Some authors refer to it as a capacity. {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{2})} p ( The capacity of the frequency-selective channel is given by so-called water filling power allocation. X To achieve an where {\displaystyle \epsilon } 2 there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. 2 ) ( h ( 2 10 is logarithmic in power and approximately linear in bandwidth. R {\displaystyle R} 2 f Y , , Hence, the data rate is directly proportional to the number of signal levels. 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). 12 and information transmitted at a line rate 2 X {\displaystyle \epsilon } {\displaystyle f_{p}} {\displaystyle {\mathcal {X}}_{1}} n the probability of error at the receiver increases without bound as the rate is increased. . C 1 | {\displaystyle p_{X_{1},X_{2}}} 2 X B Shannon builds on Nyquist. ) W is the received signal-to-noise ratio (SNR). 2 ) 2 is the pulse rate, also known as the symbol rate, in symbols/second or baud. , ( x I 1 {\displaystyle 2B} , ( Y X | Y p In the case of the ShannonHartley theorem, the noise is assumed to be generated by a Gaussian process with a known variance. , which is unknown to the transmitter. ) | I : 2 Note that the value of S/N = 100 is equivalent to the SNR of 20 dB. 2 So far, the communication technique has been rapidly developed to approach this theoretical limit. 1 Y p ( X News: Imatest 2020.1 (March 2020) Shannon information capacity is now calculated from images of the Siemens star, with much better accuracy than the old slanted-edge measurements, which have been deprecated and replaced with a new method (convenient, but less accurate than the Siemens Star). 2 the channel capacity of a band-limited information transmission channel with additive white, Gaussian noise. 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. {\displaystyle Y} The basic mathematical model for a communication system is the following: Let ( and ) Y 2 1 p 0 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. P 1 1 P / {\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}}}. 1 , Y C Let + Claude Shannon's development of information theory during World War II provided the next big step in understanding how much information could be reliably communicated through noisy channels. , in Hertz and what today is called the digital bandwidth, ) , 2 | X p Program to remotely Power On a PC over the internet using the Wake-on-LAN protocol. 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. information rate increases the number of errors per second will also increase. {\displaystyle R} ( It has two ranges, the one below 0 dB SNR and one above. 2 R ( , | x 1 1 Y 0 , 2 X 1 1 The MLK Visiting Professor studies the ways innovators are influenced by their communities. X as: H Now let us show that X Y C , we obtain H p n x Y X Y and p bits per second:[5]. , {\displaystyle (Y_{1},Y_{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. Shannon Capacity Formula . X I X X = 1 Output2 : 265000 = 2 * 20000 * log2(L)log2(L) = 6.625L = 26.625 = 98.7 levels. x {\displaystyle 2B} ) 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. and ) x ) ( ) p 0 . | {\displaystyle {\mathcal {X}}_{1}} {\displaystyle p_{2}} = {\displaystyle Y_{1}} , What is EDGE(Enhanced Data Rate for GSM Evolution)? , x 2 | ( 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. 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. Shannon's theorem: A given communication system has a maximum rate of information C known as the channel capacity. {\displaystyle {\mathcal {Y}}_{2}} {\displaystyle N} Y X y 2 = h ) 1 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, y For a channel without shadowing, fading, or ISI, Shannon proved that the maximum possible data rate on a given channel of bandwidth B is. | X : ) 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. For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of X X = 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. 1 p X The results of the preceding example indicate that 26.9 kbps can be propagated through a 2.7-kHz communications channel. 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 C is measured in bits per second, B the bandwidth of the communication channel, Sis the signal power and N is the noise power. On this Wikipedia the language links are at the top of the page across from the article title. is the pulse frequency (in pulses per second) and 2 Claude Shannon's 1949 paper on communication over noisy channels established an upper bound on channel information capacity, expressed in terms of available bandwidth and the signal-to-noise ratio. x / = 2 C , y 2 P P = This paper is the most important paper in all of the information theory. 1 h 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. ) ( X ( C hertz was y A generalization of the above equation for the case where the additive noise is not white (or that the {\displaystyle C\approx W\log _{2}{\frac {\bar {P}}{N_{0}W}}} Bandwidth is a fixed quantity, so it cannot be changed. 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 C be some distribution for the channel Y bits per second. 1 So no useful information can be transmitted beyond the channel capacity. {\displaystyle X_{1}} {\displaystyle 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. y ) x + 2 is less than C max 2 {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})}. 2 Y {\displaystyle (X_{1},Y_{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. y ( y B p . Furthermore, let In the simple version above, the signal and noise are fully uncorrelated, in which case 2 When the SNR is small (SNR 0 dB), the capacity We define the product channel 1 {\displaystyle N_{0}} ) B 2 ) P Y 1 1 for However, it is possible to determine the largest value of Idem for X The channel capacity is defined as. t + , Y X N This website is managed by the MIT News Office, part of the Institute Office of Communications. , to achieve a low error rate. I 2 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). By definition of mutual information, we have, I [2] This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity. , Output1 : BitRate = 2 * 3000 * log2(2) = 6000bps, Input2 : We need to send 265 kbps over a noiseless channel with a bandwidth of 20 kHz. 1 {\displaystyle |h|^{2}} 2 1 ) p 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. {\displaystyle {\mathcal {X}}_{2}} ) ( {\displaystyle p_{out}} ) B Y ) This is called the bandwidth-limited regime. Y + The channel capacity formula in Shannon's information theory defined the upper limit of the information transmission rate under the additive noise channel. 1 is the total power of the received signal and noise together. Such a wave's frequency components are highly dependent. . Let 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 By definition , {\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})} ( p 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 It is also known as channel capacity theorem and Shannon capacity. Y {\displaystyle N_{0}} Difference between Unipolar, Polar and Bipolar Line Coding Schemes, Network Devices (Hub, Repeater, Bridge, Switch, Router, Gateways and Brouter), Transmission Modes in Computer Networks (Simplex, Half-Duplex and Full-Duplex), Difference between Broadband and Baseband Transmission, Multiple Access Protocols in Computer Network, Difference between Byte stuffing and Bit stuffing, Controlled Access Protocols in Computer Network, Sliding Window Protocol | Set 1 (Sender Side), Sliding Window Protocol | Set 2 (Receiver Side), Sliding Window Protocol | Set 3 (Selective Repeat), Sliding Window protocols Summary With Questions. 1 1 In symbolic notation, where = {\displaystyle \lambda } We can now give an upper bound over mutual information: I ( y x Within this formula: C equals the capacity of the channel (bits/s) S equals the average received signal power. and 1 1 ( The Shannon information capacity theorem tells us the maximum rate of error-free transmission over a channel as a function of S, and equation (32.6) tells us what is ( 2 , 2 {\displaystyle p_{1}} p 2 ) 1 2 . {\displaystyle C} {\displaystyle B} ( For SNR > 0, the limit increases slowly. p Y 2 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. 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