2.3 Coding gain

Coding gain (CG) shows how many decibels lower required SNR at the input of the demodulator in a transmission system with a error-control code, rather than in a transmission system without error-control code for a given value р_d.

Let SNR (dB) at the input of the demodulator in a transmission system without error-control code , and in a transmission system with error-control code with the same symbol error probability. Then, if the signal to noise ratio expressed in decibels,

CG =  –  . (1)

We remind that a ratio is determined as

 =   =  , (2)

where P_s/N₀ is ratio of average power of signal to noise power density at the demodulator input; T_b is duration of information binary symbol (bit) at the transmission system input.

CG can be calculated in theory or measured experimentally. For example (figure 2), in a transmission system is required to ensure the symbol error probability p = 10^–5. In a transmission system without error-control code SNR  = 9,6 dB is required, and in a transmission system with error-control code SNR = 6,3 dB is required, then CG = 9,6 – 6,3 = 3,3 dB.

If CG > 0 dB (on figure 2 at р_d < 10^–2), such error-control code allows to decrease the signal/noise ratio at the demodulator input, if CG  0 dB, such error-control code does not allow to decrease the signal/noise ratio at the demodulator input, its application is worsened by quality of reception.

Dependencies р = f ( ) and р_d = f ( ) can be obtained experimentally by the method described in the laboratory work 3.3. In this laboratory we study noise immunity of transmission system with error-control code using a model of digital communication channel with errors (figure 3).

The order of experimental determination of the CG when using the model of digital communication channel with errors.

1. A series of values of error probability at the channel output р in a certain range of values, such as, р = 10^–1–10^–3 is setting (model of the digital communication channel with errors should allow such settings). For each value of p the error probability at the output of the decoder is determined р_d.

2. Must be specified modulation type which is used in digital communication channel. For transmission system without error-control code and with this type of modulation р = f ( ) is plotted. Figure 2 this dependence is built for BPSK and QPSK signals.

3. Graphing р_d = f ( ) is produced in the following order. For one of the values of p on plot р = f ( ) determine the value, which for the transmission system with a error-control code will be equal to the ratio signal/noise

 =   =  , (3)

where T_s is duration of the binary symbols at the modulator input in a transmission system with error-control codes (the output of the error-control code encoder). As , so in a transmission system with an error-control code . In the presentation in decibels . This conversion takes into account the fact that at the error-control code encoding in nonredundant code word n – k checking symbols are introduced which lead to a decrease in a duration of all symbols in the code word and a corresponding decrease in signal energy.

In the resulting value and the corresponding value р_d point depending is set р_d = f ( ). The same procedure is repeated for all other values of p. Points are connected by a smooth curve and get dependence р_d = f ( ).

4. At a given level of error probability determine the values and . CG is calculated by the formula (1).

Example calculation of points on the graph р_d = f ( ). (31, 26) code is used; BPSK modulation; error probability р = 510^–3 and р_d = 10^–5. On the value of p and the graph in figure 2 h² = 5,5 dB is defined. Calculate

= 5,5 + 10lg(31/26) = 6,3 dB.

Lay off the point schedule р_d = f ( ), whose coordinates is (6,3; 10^–5).