Bit error ratio: Information from Answers.com

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The BER is a measure of signal quality, and is a function of a quantity called Eb/N0, the energy per bit to noise power spectral density ratio of the signal for QPSK signal. In an additive white-gaussian-noise (AWGN) channel, the BER is given by: $B E R = 1 / 2 e r f c (E b / N 0)$ . This formula relates the BER of any signal to its Eb/N0. 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:

40 Gbit/s (STM-256 or OC-768): 1 s
10 Gbit/s (STM-64 or OC-192): 3 s
2.5 Gbit/s (STM-16 or OC-48): 12 s
622 Mbit/s (STM-4c or OC-12): 48 s
155 Mbit/s (STM-1 or OC-3): 3.2 min
64 Mbit/s (STM-1 or stnd) : 7.8 min

Please note that the above sample time is based on BER=10⁻¹⁰.

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

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

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.

Measuring the bit error ratio helps people choose the appropriate forward error correction codes. Since most such codes correct only bit-flips, but not bit-insertions or bit-deletions, the Hamming distance metric is the appropriate way to measure the number of bit errors. Many FEC coders also continuously measure the current BER.

A more general way of measuring the number of bit errors is the Levenshtein distance. The Levenshtein distance measurement is more appropriate for measuring raw channel performance before frame synchronization, and when using error correction codes designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes.^[1]

Mathematical draft

The BER is the likelihood of a bit misinterpretation due to electrical noise $w (t)$ . Considering a bipolar NRZ transmission, we have

$x 1 (t) = A + w (t)$ for a "1" and $x 0 (t) = - A + w (t)$ for a "0". Each of $x 1 (t)$ and $x 0 (t)$ has a period of $T$ .

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

$x 1 (t)$ is $\mathcal{N}\left(A,\frac{N_0}{2T}\right)$

and $x 0 (t)$ is $\mathcal{N}\left(-A,\frac{N_0}{2T}\right)$ .

Returning to BER, we have the likelihood of a bit misinterpretation $p e = p (0 | 1) p 1 + p (1 | 0) p 0$ .

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

where $λ$ 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 :

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

References

^ "Keyboards and Covert Channels" by Gaurav Shah, Andres Molina, and Matt Blaze (2006?)

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188).

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Bit error ratio

Mathematical draft

See also

References