# Shannon-Hartley Law

In communications engineering, the Shannon-Hartley law describes the theoretical upper limit of the bit rate of a transmission channel as a function of bandwidth and signal-to-noise ratio , over which error-free data transmission is possible with a certain probability. It is named after Claude Elwood Shannon and Ralph Hartley .

In practice, the achievable bit rate is influenced by properties such as channel capacity and methods such as channel coding . The Shannon-Hartley law supplies the theoretical maximum that can be achieved with a hypothetical optimal channel coding, without giving information about the method with which this optimal can be achieved.

## introduction

Theoretically, unlimited amounts of data could be transmitted over a perfect (i.e. interference-free) transmission channel. However, since real existing channels are both limited in their bandwidth and subject to interference such as interference , thermal noise , finite resistance of the conductor, etc., the maximum possible transmission rate is limited. The transfer rate is limited by both factors:

1. The bandwidth of the transmission path determines the maximum possible symbol rate , i.e. how many individual symbols can be safely transmitted per unit of time.
2. The strength of the interference generated or captured on the transmission path, described by the signal-to-noise ratio, limits the maximum information content of a symbol, i.e. That is, how many different symbols can still be reliably distinguished on the receiver.

To put it simply, the bandwidth determines how often the voltage can be changed per unit of time when transmitting through a cable, and the signal-to-noise ratio determines how many different voltage levels can be reliably differentiated at the receiver.

The Shannon-Hartley law expresses more precisely the fact that, in the case of a transmission channel disturbed by interference such as noise, error-free data transmission that can be achieved by means of channel coding is possible with a probability of δ> 0 if the bit rate C R realized is less than that achieved by the Shannon Hartley law boundary formed C S is. As a special feature, no statement is made as to which specific channel coding or which technical method this case can be achieved with. If the realized bit rate C R is above the limit of C S , the probability of freedom from errors is δ = 0, which means that, regardless of the method used, no error-free data transmission is possible.

## Mathematical description

### Noise-free transmission channel

The maximum data transmission rate C N for an interference-free transmission channel with bandwidth B is given by:

${\ displaystyle C_ {N} = 2 \ cdot B}$ The bandwidth B is specified in Hertz , the data transmission rate in symbols per second ( Baud ).

In the case of a binary symbol alphabet with only two characters, the bit rate , measured in bit / s (bps), is equal to the symbol rate , measured in baud. If L symbols are available, ld ( L ) bits per symbol can be represented:

${\ displaystyle C_ {N} = 2 \ cdot B \ cdot \ mathrm {ld} (L),}$ where the expression ld (·) denotes the logarithm to base 2.

Example: With a bandwidth of 1000 Hz, a maximum of 2000 baud can be transmitted. If the symbols consist of one bit, e.g. B. "0" or "1", a data rate of 2000 bit / s is achieved. If there are 26 characters of the German alphabet (without special characters), the data rate of 9400 bit / s is higher by a factor of ld (26). By choosing a sufficient number of different symbols, any high bit rate can be achieved on a noise-free, band-limited transmission channel.

### Transmission channel with noise

Claude Shannon generalized this theorem. For a channel disturbed with additive white Gaussian noise , abbreviated AWGN channel , the Shannon-Hartley law takes the following form:

${\ displaystyle C_ {S} = B \ cdot \ mathrm {ld} \ left (1 + {\ frac {S} {N}} \ right) = B \ cdot \ mathrm {ld} \ left (1 + {\ frac {S} {N_ {0} \ cdot B}} \ right)}$ C S represents the maximum possible bit rate (bits per second) (in this channel model), S the signal power. The parameter N represents the spectrally constant noise power , the ratio S / N is also referred to as the signal-to-noise ratio (SNR). The noise power N can also by the energy of the noise power density N 0 over the bandwidth B will be expressed.

The maximum possible bit rate represents an upper limit, assuming white noise. With channel models that do not correspond to the AWGN channel, as well as with different spectral noise power densities, different relationships result.

Example: A line with an SNR of 20  dB can transmit a maximum of 6.7 kbit / s with an available bandwidth of 1000 Hz  .

Invoice:

1. Conversion of SNR to S / N: SNR = 10 · log 10 (S / N) → 20  dB = 10 · log 10 (x) ↔ 2  dB = log 10 (x) ↔ 10² = x ↔ x = 100 → S / N = 100
2. Calculation of the transmission capacity: C S = f max · log 2 (1 + S / N) = 1000 Hz · ln (1 + 100) ÷ ln (2) bit = 1000 · ln (101) ÷ ln (2)  bit / s ≈ 6658  bit / s ≈ 6.7  kbit / s

This bit rate can be approximately achieved in practice , for example, by means of appropriate channel coding such as turbo codes .

Further values ​​for estimation with a bandwidth of B = 1 Hz (values ​​rounded):

SNR C s SNR C s SNR C s
-30 dB 0.001442 bit / s 0 dB 1,000 bit / s +20 dB 6.658 bit / s
-20 dB 0.014355 bit / s +3 dB 1.582 bit / s +40 dB 13.288 bit / s
-10 dB 0.137503 bit / s +6 dB 2.316 bit / s +60 dB 19.932 bit / s
-6 dB 0.323299 bit / s +10 dB 3.459 bit / s +80 dB 26.575 bit / s
-3 dB 0.586104 bit / s +15 dB 5.028 bit / s +100 dB 33.219 bit / s

#### Limits

If only the bandwidth B is increased with the signal power S kept constant and the spectral noise power density N 0 constant , the maximum possible bit rate C S can be reduced to

${\ displaystyle \ lim _ {B \ to \ infty} C_ {S} = {\ frac {S} {N_ {0}}} \ mathrm {ld} (e) \ approx 1 {,} 442 \ dots {\ frac {S} {N_ {0}}}}$ increase. This means that the maximum possible bit rate can only be increased to a limited extent even if the bandwidth B of the transmission channel is extended towards infinity .

In real transmission systems, the energy E b that has to be expended for the transmission of one bit can also be varied. This requires a signal power S for the duration T for transmission . The actual bit rate C R is always below the maximum possible bit rate C S :

${\ displaystyle {\ frac {C_ {R}} {B}} = x <{\ frac {C_ {S}} {B}} = \ mathrm {ld} \ left (1 + {\ frac {S} { N_ {0} \ cdot B}} \ right) = \ mathrm {ld} \ left (1 + {\ frac {C_ {R}} {B}} \ cdot {\ frac {E_ {b}} {N_ { 0}}} \ right)}$ This equation describes the Shannon limit (engl. Shannon limit ) as a function of E b / N 0 : x ( E b / N 0 ). The relation C R / B = x describes how many bit / s per Hertz bandwidth can be transmitted with a certain transmission technology, depending on the SNR, and is referred to as the spectral efficiency . In the illustration on the right, the courses of different transmission methods are shown in blue, red and green.

As a borderline case when equating the above inequality and when the spectral efficiency approaches 0 bit / s per Hz bandwidth, the lower limit of the SNR results after conversion to:

${\ displaystyle {\ frac {E_ {b}} {N_ {0}}} = \ lim _ {x \ to 0} {\ frac {2 ^ {x} -1} {x}} = \ ln (2 ) \ approx 0 {,} 693 \ approx -1 {,} 6 \, \ mathrm {dB}}$ This expresses the fact that under a ratio of E b / N 0 = −1.6 dB, error-free data transmission is not possible in an AWGN channel. This ratio is not about the S / N, but about the energy E b that has to be expended minimally for the transmission of the amount of information of one bit with the spectral noise power density N 0 . It is shown in the diagram on the right for the ratios of E b / N 0 for different channel codings as a vertical black border line.

That threshold applies to small spectral efficiency with x «1. Such signals are also known as performance-limited designated signals in which the bandwidth, but the power available is large limited. For example, communication with space probes is limited in power, while radiation takes place on a large frequency band. This section of the Shannon border is shown in black in the illustration on the right.

For the spectral efficiency x ≫ 1, however, the bandwidth B is the limiting factor; these signals are referred to as band-limited signals. For example, terrestrial digital radio links with spectrally efficient modulation methods such as 1024-QAM are typical bandwidth-limited signals.

## Shannon's geometrical-stochastic approach

In the work “Communication in the presence of noise” , Claude Elwood Shannon modeled the transmission channel as a real vector space. Every transferable symbol is a coordinate in this vector space. Since any number of symbols can be transmitted in time, the vector space is infinitely dimensional. Each coordinate corresponds to a basic signal, i.e. H. a real-valued function dependent on time. For the sake of simplicity of the model, the base signals should repeat themselves periodically, with the copies only differing by a time shift. For example, the (k + nD) -th base signal could be identical to the k -th base signal except for a time shift of nT . Here D is the number of “elementary” base signals, the sequence of which is repeated with period T. It can then be said that a number of nD symbols can be transmitted in the period nT .

It is assumed that the base signals assigned to the coordinates are orthogonal to one another and, overall, span an orthonormal base of the signal vector space. Any signal is then an (infinite) linear combination of these basic signals. The coefficients of this linear combination, which correspond to the transmitted symbols, can now be recovered by forming the scalar products of the signal with the basic signals.

In the important, theory-guiding example of band-limited transmission channels, the symbol rate is limited to 2 W by the maximum frequency W. In a time interval of finite length T , only a finite number D of symbols can therefore be transmitted. These span a sub-vector space of dimension D in the signal vector space . The maximum dimension D = 2WT according to the sampling theorem is assumed.

### Examples of basic signals

In the following some mathematical, i.e. H. idealized transmission channels with their systems of basis functions that meet the above assumptions for a signal vector space. These are all band-limited, whereby, in addition to the “elementary” baseband channel, systems of base signals can also be specified for channels with a minimum frequency other than zero.

#### Cardinal series

Shannon used the baseband signals with a highest frequency W as the simplest signal model . According to the WKS sampling theorem (for Whittaker-Kotelnikow-Shannon, see Nyquist-Shannon sampling theorem ), 2WT symbols can be transmitted in this channel in the period T , the basic signals are sinc functions

${\ displaystyle g_ {n} (t) = \ operatorname {sinc} (2Wt-n) = {\ frac {\ sin \ pi (2Wt-n)} {\ pi (2Wt-n)}}}$ ,

n = ..., -1, 0, 1, ... These each have at its center or maximum that is, the symbol rate is 2 W . This orthonormal system is the ideal theoretical modeling of the frequency-limited PCM method ( pulse code modulation ). ${\ displaystyle t_ {n} = {\ frac {n} {2W}}}$ #### QAM

The ideal QAM system (quadrature amplitude modulation) transmits data at a symbol rate W on the frequency band [FW / 2, F + W / 2] . It must the average carrier frequency F is an integer multiple of bandwidth W to be. The symbols here are complex numbers , ie points in the Gaussian plane of numbers . So again 2WT real numbers are transmitted in the period T. There must also be two basic functions per complex symbol; these can be combined to form a complex-valued function: ${\ displaystyle A_ {n} + iB_ {n}}$ ${\ displaystyle g_ {n} (t) = g_ {Q, n} (t) + ig_ {I, n} (t) = \ operatorname {sinc} (Wt-n) \ cdot e ^ {i2 \ pi \ , Ft} ​​= {\ frac {\ sin \ pi (Wt-n)} {\ pi (Wt-n)}} (\ cos (2 \ pi \, Ft) + i \ sin (2 \ pi \, Ft ))}$ ,

n =…, −1, 0, 1,… Each signal is then the sum of . ${\ displaystyle A_ {n} g_ {Q, n} (t) + B_ {n} g_ {I, n} (t)}$ #### OFDM

The ideal OFDM system (Orthogonal Frequency Division Multiplexing) transmits with symbol rate W / M a complex vector of dimension M on the frequency band [FW / (2M), F + W + W / (2M)] . F must be an integer multiple of the data rate W / M to be. There must therefore be 2M real-valued base signals per vector-valued symbol, which can be combined into M complex-valued functions

${\ displaystyle g_ {j, n} (t) + ig_ {M + j, n} (t) = \ operatorname {sinc} (Wt / Mn) \ cdot e ^ {- i2 \ pi \, (F + jW / M) t}}$ ${\ displaystyle = {\ frac {\ sin \ pi (Wt / Mn)} {\ pi (Wt / Mn)}} (\ cos (2 \ pi \, (F + jW / M) t) -i \ sin (2 \ pi \, (F + jW / M) t))}$ ,

j = 0, ..., M -1, n = ..., −1, 0, 1, ...

Since the sinc function cannot be implemented technically, other solutions have to be found. The orthogonality of the basic signals is destroyed by frequency filters, mutual interference occurs within the symbol (ICI) and between the symbols (ISI). If the clock rate of the signal generation is increased without increasing the data rate, the freedom gained can be used to form a signal that is frequency-restricted even without filtering. A variant of this uses wavelet packet trees .

### Transmission in the noisy channel

Let the real base signals be numbered consecutively with a single index and a period T fixed in such a way that D = 2WT base signals lie within this period . Uniform noise restricted to the transmission channel can be simulated by linear combinations of these same basic signals with normally distributed, mutually independent random coefficients of the variance . ${\ displaystyle \ sum _ {n} \ varepsilon _ {n} g_ {n} (t)}$ ${\ displaystyle \ varepsilon _ {n}}$ ${\ displaystyle \ sigma ^ {2} = N}$ A code of length D , i.e. H. a tuple of real numbers is sent as a continuous signal . During the transmission, an interference is linearly superimposed on this, which is the received, interfered signal ${\ displaystyle x_ {1}, \ dots, x_ {D}}$ ${\ displaystyle f (t) = x_ {1} g_ {1} (t) + \ dots + x_ {D} g_ {D} (t)}$ ${\ displaystyle {\ tilde {f}} (t) = (x_ {1} + \ varepsilon _ {1}) g_ {1} (t) + \ dots + (x_ {D} + \ varepsilon _ {D} ) g_ {D} (t)}$ .

#### Geometry of the signal points

Let the signal be limited to an average power , where power corresponds directly to the amplitude square. This is permissible because in the end only the ratios of different services are compared, so other constant factors are reduced. Since the base signals are orthonormal, the continuous signal has the square sum of its coefficients as power, i.e. H. . ${\ displaystyle P}$ ${\ displaystyle \ | f \ | _ {2} ^ {2} = | x_ {1} | ^ {2} + \ dots + | x_ {D} | ^ {2} = DP}$ In other words, the code is a point on a D -dimensional sphere with radius . ${\ displaystyle (x_ {1}, \ dots, x_ {D})}$ ${\ displaystyle R_ {0}: = {\ sqrt {DP}}}$ According to the law of large numbers, the sum of squares of the D independent errors is close to its expected value DN . This means that there is a very high probability that the received code lies within a sphere of the radius with the transmitted code as the center. Since the interference is assumed to be independent of the signal, the sum of squares of the received code is very likely to be close to the expected value , i.e. H. near the sphere with the radius around the origin. ${\ displaystyle (\ varepsilon _ {1}) ^ {2} + \ dots + (\ varepsilon _ {D}) ^ {2}}$ ${\ displaystyle r: = {\ sqrt {DN}}}$ ${\ displaystyle R_ {1} ^ {2}: = DP + DN}$ ${\ displaystyle R_ {1}}$ #### Random configuration

Let a configuration of M = 2 DB randomly selected codes with average power P be fixed, which should correspond to M different digital messages, i.e. H. bits are coded using D basic signals or B bits per basic signal. ${\ displaystyle DB}$ Of the small spheres with a radius around the codes of the configuration, maximally fit ${\ displaystyle r = {\ sqrt {DN}}}$ ${\ displaystyle M = {\ frac {R_ {1} ^ {D} \ operatorname {vol} (K_ {D})} {r ^ {D} \ operatorname {vol} (K_ {D})}} = \ left ({\ frac {P + N} {N}} \ right) ^ {\ frac {D} {2}}}$ Piece into the big ball of receivable signals, d. H. for the maximum bit rate applies (with D / 2 = WT )

${\ displaystyle 2WB = {\ frac {\ log _ {2} (M)} {T}} \ leq W \ log _ {2} \ left (1 + {\ frac {P} {N}} \ right) }$ .

#### Estimation of the transmission error

For a very large D , the codes sent are on a sphere with a radius and the received codes are most likely in spheres with a radius around them and on the sphere with a radius . So you can compare the received code with all codes from the configuration in order to determine the one that has a distance less than r . ${\ displaystyle R_ {0} = {\ sqrt {DP}}}$ ${\ displaystyle r = {\ sqrt {DN}}}$ ${\ displaystyle R_ {0} = {\ sqrt {D (P + N)}}}$ The error sphere with radius r and centered on the sphere of the received codes sweeps over an area in the sphere of the transmitted codes, which in turn lies within a sphere with a radius . The probability that a random code falls outside this range is therefore greater than ${\ displaystyle h = {\ sqrt {\ frac {DNP} {N + P}}}}$ ${\ displaystyle 1 - {\ frac {h ^ {D} \ operatorname {vol} (K_ {D})} {R_ {0} ^ {D} \ operatorname {vol} (K_ {D})}} = 1 - \ left ({\ frac {N} {P + N}} \ right) ^ {\ frac {D} {2}}}$ .

The probability that all M-1 codes of the configuration that differ from the code sent are outside this range therefore has a probability that is greater than

${\ displaystyle \ left (1- \ left ({\ frac {N} {P + N}} \ right) ^ {\ frac {D} {2}} \ right) ^ {M-1} \ geq 1- M \ left ({\ frac {N} {P + N}} \ right) ^ {\ frac {D} {2}}}$ .

If the error probability e is to be fallen below, d. H. If the above expression is greater than 1-e , you get the bit rate after changing

${\ displaystyle {\ frac {\ log _ {2} (M)} {T}} \ leq W \ log _ {2} \ left (1 + {\ frac {P} {N}} \ right) + { \ frac {\ log _ {2} (e)} {T}}}$ .

${\ displaystyle \ log _ {2} (e)}$ in the second summand is negative and its magnitude is very large, but the contribution of the second summand can be made as small as desired if the time period T and thus also the thickness M of the configuration are large enough.

This means that as the length of the signals in the configuration increases, the bit rate can be brought as close as desired to the ideal bit rate. However, managing the configuration and searching for the signal most closely resembling the receiving one place rapidly growing demands against direct practical application.

## literature

• John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Prentice Hall, Upper Saddle River NJ 2002, ISBN 0-13-095007-6 .

## Individual evidence

1. ^ Ralph VL Hartley: Transmission of Information . Bell System Technical Journal, 1928 ( dotrose.com [PDF]).
2. ^ Claude E. Shannon: The Mathematical Theory of Communication . University of Illinois Press, 1949.
3. John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Pearson Education International, ISBN 0-13-095007-6 .