The Bit Error Rate (BER) of a digital optical receiver indicates the probability of an incorrect bit identification. In other words, the BER is the ratio of bits received in error to the total number of bits received. Below lists different values for BER and their corresponding errors per bits and over time.
As we know that, the photocurrent is converted to a voltage then measured. The measurement procedure involves a decision as to whether the bit received is a 1 or a 0. The BER is a not only a function of the noise in the receiver and distortion in the system, but also on the decision level voltage,VD that is the threshold level above which the signal is classified as a 1 and below which the signal is classified as a 0. Even an ideal signal with no noise nor distortions has a non-zero BER if the decision level is set too high or too low. For example, if VD is set above the voltage of the 1 bit, the BER is 0.5, assuming equal probability of receiving a one and a zero.

BER	Error per 10E^-15 bits	@ 10Gbps, One error in
1×10^-6	10,00,00,000	0.1 msec
1×10^-9	1,00,000	0.1 sec
1×10^-12	100	1.7 min
1×10^-15	1	1.2 days

Mathematically, the Bit Error Rate is expressed as

BER = p(1)P(0 ⁄ 1) + p(0)P(1 ⁄ 0)

where p(1) and p(0) are the probabilities of receiving a 1 and a 0, respectively. P(0/1) is the probability of deciding a 0 when the bit is actually a 1, and P(1/0) is the probability of deciding a 1 when the bit is a 0.

The mathematical relations to BER for non-FEC operation when the threshold is set to the optimum value are:

where:

A commonly used approximation for this function is:

An alternative expression that gives accurate answers over the whole range of Q is expressed as:

Minimum BER as a function of Q where both formulas are compared.

BER to Q relation

e.g: BER of 10^–12, is Q » 7.03.

What is Q-factor ?

A Q-factor measurement occupies an intermediate position between the classical optical parameters (power, OSNR, and wavelength) and the digital end-to-end performance parameters based on BER.A Q-factor is measured in the time domain by analyzing the statistics of the pulse shape of the optical signal. A Q-factor is a comprehensive measure for the signal quality of an optical channel taking into account the effects of noise, filtering, and linear/non-linear distortions on the pulse shape, which is not possible with simple optical parameters alone.

Definition 1:

The Q-factor, a function of the OSNR, provides a qualitative description of the receiver performance. The Q-factor suggests the minimum signal-to-noise ratio (SNR) required to obtain a specific BER for a given signal. OSNR is measured in decibels. The higher the bit rate, the higher the OSNR ratio required. For OC-192 transmissions, the OSNR should be at least 27 to 31 dB compared to 18 to 21 dB for OC-48.

Definition 2:

The Quality factor is a measure of how noisy a pulse is for diagnostic purposes. The eye pattern oscilloscope will typically generate a report that shows what the Q factor number is. The Q factor is defined as shown in the figure: the difference of the mean values of the two signal levels (level for a “1” bit and level for a “0” bit) divided by the sum of the noise standard deviations at the two signal levels. A larger number in the result means that the pulse is relatively free from noise.

Definition 3:

Q is defined as follows: The ratio between the sums of the distance from the decision point within the eye (D) to each edge of the eye, and the sum of the RMS noise on each edge of the eye.

This definition can be derived from the following definition, which in turn comes from ITU-T G.976 (ref. 3).

where m_1,0 are the mean positions of each rail of the eye, and s_1,0 are the S.D., or RMS noise, present on each of these rails.

For an illustration of where these values lie within the eye see the following figure:

As Q is a ratio it is reported as a unit-less positive value greater than 1 (Q>1). A Q of 1 represents complete closure of the received optical eye. To give some idea of the associated raw BER a Q of 6 corresponds to a raw BER of 10^-9.

Q factor as defined in ITU-T G.976

The Q factor is the signal-to-noise ratio at the decision circuit in voltage or current units, and is typically expressed by:

(A-1)

where µ_1,0, is the mean value of the marks/spaces voltages or currents, and s_1,0 is the standard deviation.

The mathematic relations to BER when the threshold is set to the optimum value are:

(A-2)

with:

(A-3)

The Q factor can be written in terms of decibels rather than in linear values:

(A-4)

Calculation of Q-Factor from OSNR

The OSNR is the most important parameter that is associated with a given optical signal. It is a measurable (practical) quantity for a given network, and it can be calculated from the given system parameters. The following sections show you how to calculate OSNR. This section discusses the relationship of OSNR to the Q-factor.

The logarithmic value of Q (in dB) is related to the OSNR by following Equation

In the equation, B₀ is the optical bandwidth of the end device (photodetector) and B_c is the electrical bandwidth of the receiver filter.

Therefore, Q(dB) is shown in

In other words, Q is somewhat proportional to the OSNR. Generally, noise calculations are performed by optical spectrum analyzers (OSAs) or sampling oscilloscopes, and these measurements are carried over a particular measuring range of B_m. Typically, B_mis approximately 0.1 nm or 12.5 GHz for a given OSA. From Equation showing Q in dB in terms of OSNR, it can be understood that if B₀ < B_c, then OSNR (dB )> Q (dB). For practical designs OSNR(dB) > Q(dB), by at least 1–2 dB. Typically, while designing a high-bit rate system, the margin at the receiver is approximately 2 dB, such that Q is about 2 dB smaller than OSNR (dB).

The Q-Factor, is in fact a metric to identify the attenuation in the receiving signal and determine a potential LOS and it is an estimate of the Optical-Signal-to-Noise-Ratio (OSNR) at the optical receiver. As attenuation in the receiving signal increases, the dBQ value drops and vice-versa. Hence a drop in the dBQ value can mean that there is an increase in the Pre FEC BER, and a possible LOS could occur if the problem is not corrected in time.

Reference:

ITU-T G.976