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# Bit error ratio

In telecommunication, an error ratio is the ratio of the number of bits, elements, characters, or blocks incorrectly received to the total number of bits, elements, characters, or blocks sent during a specified time interval.

The most commonly encountered ratio is the bit error ratio (BER) - also sometimes referred to as bit error rate.

Examples of bit error ratio are (a) transmission BER, i.e., the number of erroneous bits received divided by the total number of bits transmitted; and (b) information BER, i.e., the number of erroneous decoded (corrected) bits divided by the total number of decoded (corrected) bits.

The test time for a 95% confidence interval at several speed links is shown here:

Please note that the above sample time is based on BER=10-10. Source: from Federal Standard 1037C and from MIL-STD-188

The test time t can be calculated using Gaussian error distribution to:

$t = -frac\left\{ln\left(1-c\right)\right\}\left\{b*r\right\}$

where c is the degree of confidence level, b = upper bound of BER and r = bit rate.

See the following technical article for measuring BER for High-speed serial communication. http://www.analogzone.com/nett1003.pdf

People usually plot the BER curves to describe the functionality of a digital communication system. In optical communication, BER(dB) vs. Received Power(dBm) is usually used; while in wireless communication, BER(dB) vs. SNR(dB) is used.

Curve fitting for such BER curve is a topic, attracting many research efforts.

## Mathematical draft

The BER is the likelihood of a bit misinterpretation due to electrical noise $w\left(t\right)$. Considering a bipolar NRZ transmission, we have $x_1\left(t\right) = A + w\left(t\right)$ for a "1" and $x_0\left(t\right) = -A + w\left(t\right)$ for a "0". Each of $x_1\left(t\right)$ and $x_0\left(t\right)$ has a period of $T$.

Knowing that the noise has a bilateral spectral density $frac\left\{N_0\right\}\left\{2\right\}$,

$x_1\left(t\right)$ is $mathcal\left\{N\right\}left\left(A,frac\left\{N_0\right\}\left\{2T\right\}right\right)$

and $x_0\left(t\right)$ is $mathcal\left\{N\right\}left\left(-A,frac\left\{N_0\right\}\left\{2T\right\}right\right)$.

Returning to BER, we have the likelihood of a bit misinterpretation $p_e = p\left(0|1\right) p_1 + p\left(1|0\right) p_0$.

$p\left(1|0\right) = 0.5, operatorname\left\{erfc\right\}left\left(frac\left\{A+lambda\right\}\left\{sqrt\left\{N_o/T\right\}\right\}right\right)$ and $p\left(0|1\right) = 0.5, operatorname\left\{erfc\right\}left\left(frac\left\{A-lambda\right\}\left\{sqrt\left\{N_o/T\right\}\right\}right\right)$

where $lambda$ is the threshold of decision, set to 0 when $p_1 = p_0 = 0.5$.

We can use the average energy of the signal $E = A^2 T$ to find the final expression :

$p_e = 0.5, operatorname\left\{erfc\right\}left\left(sqrt\left\{frac\left\{E\right\}\left\{N_o\right\}\right\}right\right).$

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