Limiting efficiency of transmission systems and Shannon bound

Indicators b and g make sense a specific rates, and inverse values b¢ =1/b and g¢ =1/g define specific expenses of corresponding resources on an information transferring with unity rate (1 bit per second). For the Gaussian channel with frequency band F _ch, the ratio of signal to noise r = P _s/ P_n and channel capacity it is possible to establish, that these efficiency indicators are connected by the relation:

and (12.4)

For ideal system (h =1) limiting equationcan be defined. According to Shannon theorem by the corresponding transmission methods (coding and modulation) and receiving (demodulation and decoding), the value h can be as much as close to unit. Thus the error can be made as much as small. In this case by a condition h = 1 it is received limiting equation between b and g:

. (12.5)

This formula defines of energy efficiency from the frequency efficiency for the ideal system ensuring equality of a information rate to a channel capacity. It is convenient to represent this equation in the form of a curve on a plane b = f (g) (figure 12.1, a curve «Shannon bound»). This curve is limiting and reflects the best interchanging between b and g in the continuous channel (СC).

It is necessary to notice, that frequency efficiency g varies in limits from 0 to ¥, energy efficiency is bounded above by magnitude:

. (12.6)

Differently, energy efficiency of any information transmitting system in a Gaussian channel can not exceed the magnitude

. (12.7)

Similar limiting curves can be constructed and for any other channels if in formulas (12.2) and (12.3) instead of a rate R _chan to substitute expressions for a channel capacity of the corresponding channel. So, in particular, on fig. 12.1 the curve for limiting equation b = f (g) the is discrete-continuous channel (D-CC) is shown.

It "is enclosed" in a curve of the continuous channel (CC) that confirms know result of an information theory according to which DN channel capacity of D-CC always is less a than channel capacity of the continuous channel (CC) which is a basis for construction of corresponding D-CC. In real digital systems error probability p always has a final value and informational efficiency is less then a limiting value h_max. In these cases for the fixed error probability p = const it is possible to define efficiency ratio b to g and to construct curves b  = f (g).

In coordinates (b, g) to each variant of a transmission system there will corresponds a point on a plane. All these points (curves) should place below a limiting curve of «Shannon bound». The place of these curves depends on an aspect of signals (modulations), a codes (coding methods) and a method of the elaborating of a signals (demodulation/decoding). About perfection of the digital telecommunication methods judge on a degree of placing of real efficiency of to the limiting values.

Concrete data about the efficiency of various modulation/coding methods and also their combinations are given in following section.