Correcting ability of error-control codes

Correcting ability of error-control code is expressed by the guaranteed correctable errors value q _cor £ (d _min – 1)/2 and by the guaranteed detectable errors value q _det £ d _min – 1, where d _min – code distance.

Noise immunity of transmission system with a error-control code is described by the р = f (E_b / N ₀), where р is bit error probability, E_b / N ₀ is the ratio of signal energy expended on the transfer of one bit to the power density of noise (SNR) at the input of the demodulator.

Quantitative measure of correcting ability (noise immunity) of error-control code decoder is the dependence of the symbols error probability on the output of the decoder p _out of error probability on its input р _in, so р _out = f (р _in), and for illustration represent graphically using for р _in and р _out logarithmic scale.

Examples of dependences р = f (h _b²) and р _out = f (р _in) for transmission systems without error-control coding and with BCH (1023,733) code are shown in Fig. 2 and Fig. 3, respectively.

Dependence р _out = f (р _in) is determined by ECC parameters such as:

– code distance d _min, that the multiplicity of errors q _cor, which are corrected by decoder;

– length of code word n, because the probability of falling single, double, etc. errors on different code length n is different.

Error probability p in the transmission without code is calculated using Q -function. For BPSK modulation

p _BPSK = p _in= , (3)

where

Q (z) = . (4)

For the calculations convenient to use the approximation of this function

Q (z)=0,65exp(–0,44(z +0,75)²). (5)

Using error-control code dependence of symbols error probability on the output of the decoder of error probability on its input is the next:

p _{out ,} (6)

where p _outisbit error probability at the output of decoder, d _min is code distance, d _min = 2 q _cor+1.

In sum (6) for calculations one can only use the first term and drop the multiplier , that is close to unity:

p _{out ,} (7)

For BDPSK modulation error probability at the input decoder increases twice.

2.4 Coding gain

Coding gain (CG) is a generalized characteristic of noise immunity of ECC in a given transmission system [4], that shows the difference between signal/noise ratio (dB) at the demodulator input without error-control code and with it at the same symbols error probability.

CG = – . (8)

We remind that a ratio is determined as [3]

= = , (9)

where P _s/ N ₀ 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.

Figure 3 – Correcting ability of (1023,733) BCH code

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 _all = 10^–5. In a transmission system without error-control code SNR = 9,6 dB is required, and in a transmission system with BCH (1023,733) error-control code SNR = 6 dB is required, then CG = 9,6 – 6 = 3,6 dB.

If CG > 0 dB, 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 _in = f () and р _out = f () can be obtained experimentally by the method described in the laboratory work 2.5.

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