Energy coding gain. As well as by an estimation of a decoding noise immunity of the block codes (see Section 8) in a case of convolution codes use concept of an energy coding

As well as by an estimation of a decoding noise immunity of the block codes (see Section 8) in a case of convolution codes use concept of an energy coding gain.

The energy coding gain g is equal to a difference between of values necessary to get the given error probability p by the absence and by the coding use.

Values of error probability level at which the gain is defined depends on the requirements to fidelity of the transmitted digital information. For digital telephony systems a acceptable level of a bit error probability usually makes p _acc = (10^–5... 10^–6). In systems of digital TV transmission try to ensure p _acc = (10^–10... 10^–11). acceptable

The value of coding gain at the given bit error probability p _acccan be defined by comparing the arguments of function Q (z) in a formulas for error probability (11.4) and (1.5) for identical probabilities p _d = p _BPSK = p _acc. Calculations show, that gain depends from level of error probability p _acc on which it is defined. It is well visible on the curves figure 11.1 representing calculation results from Exercise 11.1. Value of a gain with decreasing of a probability p _acc aspires to the limit which in the coding theory name as asymptotic coding gain:

A-gain = lim g (p _acc → 0). (11.6)

Comparing arguments in the expressions (11.5) and (11.4) we come to wide used in energy calculations of transmission systems to expression for A-gain in logarithmic units:

A-gain = 10lg(R _code d _f) (dB). (11.7)

table 11.1 – Calculation of decoding noise immunity

Eb / N 0, dB	Bit error probability on demodulator BPSK output	Bit error probability on decoder output
Code (5, 7)	code (133, 171)
z	p BPSK	z	p d	z	p d
	1,59	5,8∙10–2	2,51	6,1∙10–3	3,55	6,9∙10–3
	1,78	3,9∙10–2	2,82	2,4∙10–3	3,98	1,2∙10–3
	2,00	2,4∙10–2	3,16	7,8∙10–4	4,47	1,5∙10–4
	2,24	1,3∙10–2	3,54	1,9∙10–4	5,01	1,1∙10–5
	2,52	6,1∙10–3	3,98	3,5∙10–5	5,62	4,1∙10–7
	2,82	2,4∙10–3	4,46	4,2∙10–6
	3,55	7,9∙10–4	5,62	1,1∙10–8
	3,99	2,0∙10–4
	4,47	4,2∙10–6

As A-gain is upper bound of a gain g for fast comparison and a choice of codes use A-gain. Values of this A-gain often include in the code tables (see tables of Attachment А.3). In table 11.1 for an example data about convolution codes with various lengths of a code length ν and rate R _code are cited. Values of a A-gain are shown. More detailed data are given in tables А.3…А.6 from the Attachment А.3.

table 11.1 – Characteristics of a convolutional codes

Code rate R code	Code constraint length ν = 4	Code constraint length ν = 6
Code	A-gain, dB	Code	A-gain, dB
1/3	25, 33, 37	6,02	133, 145, 175	6,99
1/2	31, 33	5,44	133, 171	6,99
2/3	31, 33, 31	5,23	133, 171, 133	6,02
3/4	25, 37, 37, 37	4,78	135, 163, 163, 163	6,73

Comparison of a gain values is ensured by the cyclic coding (see table 8.3 and figure 8.1) with similar parameters for convolutional codes (see table 11.1 and figure 11.1) shows, that convolution codes in a combination to Viterbi decoding algorithm ensure considerably more gain in comparison with block codes. It explains wide using of convolution codes in transmission systems for a noise immunity increasing. Typical a using of the code (133, 171) ensuring A-gain = 6,99 dB by the rate R _code = 0,5 here is, i.e. at two-multiple expansion of a frequency band of the coded signal. The codecs of such code are developed in the form of the big chips are serially emitted.