## 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 Equation 4-11.

#### Equation 4-11

In the equation, B_{0} 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 Equation 4-12.

#### Equation 4-12

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_{m }is approximately 0.1 nm or 12.5
GHz for a given OSA. From Equation 4-12, showing Q in dB in terms of OSNR, it
can be understood that if B_{0} < 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).

### Calculation of OSNR for a Point-to-Point Link

Consider a physical link AB, as shown in Figure 4-5. Assume this to be a long-haul fiber WDM link (a link that is several hundred kilometers). Amplifiers are placed periodically at repeated intervals to boost signal power. Therefore, a signal can reach much farther than the maximum allowable accumulated loss due to the fiber (υL). However, in doing so, each amplifier stage adds its own component of amplified spontaneous emission (ASE) noise and degrades the OSNR further. Moreover, every amplifier amplifies the already present noise. Note that noise is omnipresent throughout the spectra and almost impossible to be removed. Therefore, it is imperative to devise a method to calculate the OSNR (output) at the end of an N stage-amplified system and see if the value N is still valid.

**Figure 4-5 A Multiple
Stage Amplified DWDM System Deployed in a Point-to-Point Topology**

In an OSNR-based design, we must ensure that OSNR of the final stage is in compliance with system OSNR requirements and hence the BER requirements. To make the system support a particular BER, it is necessary to make the OSNR system design compliant.

The OSNR of each stage is shown in Equation 4-13.

#### Equation 4-13

In Equation 4-13, NF_{stage} is the noise figure of the stage, h is
Plank's constant (6.6260 × 10^{-34}), ν is the optical
frequency 193 THz, and Δf is the bandwidth that measures the NF (it is
usually 0.1 nm).

The total OSNR for the system can be considered by a reciprocal method and is shown in Equation 4-14.

#### Equation 4-14

for the 'N' stage system. That summarizes to Equation 4-15.

#### Equation 4-15

A slight detailed analysis provides a more appropriate equation for OSNR. For a single amplifier of gain G, the OSNR is shown in Equation 4-16.

#### Equation 4-16

In Equation 4-16, n_{sp} is the population inversion parameter that
is shown in Equation 4-17 and is the ratio of electrons in higher and lower
states.

#### Equation 4-17

In Equation 4-17, N_{2} is the number of electrons in a higher state
and N_{1} is the number of electrons in the lower state. (Refer to
Chapter 2 for more details.)

The population inversion parameter is also shown in Equation 4-18.

#### Equation 4-18

For an N amplifier stage system, with each amplifier compensating for the loss of the previous span where the span loss in dB is , you have the relationship for final stage OSNR in Equation 4-19.

#### Equation 4-19

Taking logarithm to the common base (10), we get Equation 4-20.

#### Equation 4-20

From the previous section, we get Δf = 0.1 nm, or 12.5 GHz. Substituting this, we get Equation 4-21.

#### Equation 4-21

The following is assumed:

The NF of every amplifier is the same. (we assume uniformity of products; therefore, NFs are the same for all amplifiers.)

is the span loss and is same. (This is a generic assumption and can be changed, as shown later in this section.)

Noise is totaled over both states of polarization. In short, it is unpolarized noise.

Equation 4-21 provides the actual mathematical calculation of OSNR. This calculation method has quite a few approximations in which we can still find the system OSNR to a great degree of accuracy. In a multichannel WDM system, the design should consider OSNR for the worst channel (the one that has the worst impairment). The worst channel is generally the first or last channel in the spectrum.

### OSNR Improvements by Raman Amplification

If we look at Equation 4-21, we can see that the EDFA gain factor G is not considered. That is because OSNR is a ratio, and the gain acts equally on signal and noise, canceling the gain factor in the numerator and denominator. In other words, although EDFAs alleviate the upper bound on transmission length due to attenuation, by cascading EDFAs in a series, the OSNR is continuously degraded with transmission length and ASE (from EDFAs). This degradation can be lessened somewhat by distributed Raman amplifiers (DRAs). As can be seen from Chapter 3, "Networking with DWDM -2," Raman amplification is inherently a result of stimulated Raman scattering of a high intensity pump signal at a different frequency (compared to the signal frequency). This produces a gain because of creation of a Stokes wave, which in turn produces a gain feeding wave of a wide bandwidth.

Consider a hybrid system as shown in Figure 4-6.

**Figure 4-6 Multistage
Raman Amplifier-Based System**

From the preceding system, the OSNR of the final stage is shown as in Equation 4-22.

#### Equation 4-22

Equation 4-23 gives the OSNR value of each stage.

#### Equation 4-23

As we can see from Equation 4-23, the factor G_{RA }in the numerator
actually enhances the OSNR of the system.
Figure 4-7 shows the variation of Raman gain with pump power.

**Figure 4-7 Variation
of Raman Gain with Pump Power**