Quote of the Day

The stock market is a device for transferring money from the impatient to the patient.

— Warren Buffet. I have worked to pass this wisdom on to my sons. The patient investor will sleep much better than the impatient investor.

## Introduction

I was reviewing an industry standard when I saw Figure 1, which clearly looked wrong – the asymptotes seem like they are in the wrong place. I decided to take a closer look at this figure and see if I could determine what the correct version of this chart would be.

The chart shows how adding multiple wavelengths (i.e. colors) onto a fiber will impact the performance of a system. The metric used to measure this impact is called the power penalty, which is a function of the number of wavelengths and their level of isolation from each other. In this post, I will using a Mathcad model to show how to generate a clearer version of this chart. It was an interesting diversion (~15 minutes) from my usual workload of budgeting and planning.

There were several reasons why I feel this work is worth documenting here:

- It shows the kind of challenges being faced by those who are putting multiple wavelengths on a Passive Optical Networks (PONs).
- It shows how one can easily compute the location of asymptotes using a computer algebra system.
- I used Mathcad scriptable components to help annotate my version of Figure 1.

For those who are interested, my Mathcad source and its PDF are included here.

## Background

### PON Background

Figure 2 shows how a current generation Gigabit Passive Optical Network (GPON) works. Here are the key points to understand:

- Every home has an Optical Network Terminal (ONT). Every home in the network has an ONT for converting the optical signal into voice, video, and data services.
- The ONT connect to a central office through an Optical Line Terminal (OLT).
- The ONT and OLT are connected to each other over a set of components called the Optical Distribution Network (ODN) – the ODN requires no power as it is composed entirely of glass components.
- The central office connects every home to the internet backbone.

While today's PONs only use two wavelengths (1490 nanometer[nm] and 1310 nm) for digital services, future PONs (e.g. NGPON2) will use many wavelengths. The use of multiple wavelengths will allow us to enormously increase the amount of data carried over existing fiber optic cables. Unfortunately, each wavelength added will impair the performance of the other wavelengths on the fiber. This post calculates the magnitude of this impairment.

### Definitions

- Wavelength Division Multiplexing (WDM)
- In fiber-optic communications, WDM is a technology which multiplexes a number of optical carrier signals onto a single optical fiber by using different wavelengths (i.e., colors) of laser light. This technique enables bidirectional communications over one strand of fiber, as well as multiplication of capacity. (Source)
- Channel
- A range of wavelengths assigned to a particular beam of light, which is generated by a laser. The wavelength of laser moves as a function of the laser's temperature and laser. Feedback control systems are put in place to modulate the laser's temperature to ensure that its wavelength stays in its define channel range.
- Extinction Ratio (
*ER*) - The ratio of the logic "1" optical power to the logic "0" optical power.
*ER*is a measure of how hard you are driving the laser. Large*ER*values will cause the laser to have wider spectral width, which will increase dispersion and co-channel interference.

- Wanted Channel (WC)
- The channel for which we want to detect the performance degradation attributable to the presence of other channels.
- Disturbing Channel (
*DC*) - The channels that are degrading the performance of the WC.
- Adjacent Channel (
*AC*) - The DCs occupying the channels on either side of the WC. ACs usually cause more degradation in the WC because they are closer and the effective isolation levels are lower.
- Non-Adjacent Channel (
*NAC*) - These are DCs that are not adjacent to the
*WC*. Because their greater wavelength separation from the WC, their effective isolation levels tend to be higher. - Channel Crosstalk (
*C*)_{C} - Sometimes called inter-channel crosstalk, it is the ratio of total power ingress from the DCs into the WCs. In general, the receivers cannot distinguish between photons of different wavelengths and the DCs appear as Gaussian noise to the WC's receiver.
- Power Penalty (
*PP*) - For this discussion, power penalty is the reduction in signal power (and SNR) due to a specific impairment. Some people define power penalty as the amount of signal power increase needed to compensate for a specific impairment.
- Isolation (
*I*) - The amount of power reduction between the signal power in the DCs and the WC. This reduction is normally provided by optical filtering using thin films.

### C_{C} Model

In a WDM system, there are multiple wavelengths (i.e. colors) traveling on the fiber. Each wavelength contains a separate data transmission – this means that each wavelength needs a separate receiver (Figure 3). We use a device, called a demultiplexer, to separate the colors and direct each wavelength to its receiver. The demultiplexer contains a filter that reduces power of the DCs by an amount I_{A} or I_{NA}, which we call the isolation. If the filter was perfect, the isolation would be infinitely large.

Figure 4 shows how we will model the wavelength separation process using four parameters (all expressed in dB):

- Each ONT will transmit at a slightly different power because of imperfections in their laser power control systems (
*ΔP*)._{ONT} - Each ONT can be at a different range, which means the light received from each ONT can incur a different amount of distance-dependent attenuation (
*d*)._{Max} - Each AC will be attenuated by
*I*._{A } - Each NAC will be attenuated by
*I*._{NA}

Given these assumptions, we can express the channel crosstalk as shown in Equation 1.

Eq. 1 |

where

*C*is total cross-channel interference power (dB)._{C}*N*is number of wavelengths in our WDM system.

### Power Penalty

Equation 2 is the model used for the power penalty. The derivation of Equation 2 is not simple, and I will not derive it here.

Eq. 2 |

where

*C*is total cross-channel interference power (dB)._{C}*ER*is extinction ratio of the WDM signals (linear). The extinction ratio is defined as

, where*P*is the power of logic one and_{1}*P*is the power of logic zero._{0}*BER*is the desired bit error level.*Q*is the Q-function evaluated at the BER level ,i.e. .

One interesting aspect of Equation 2 is that it has an asymptote where the argument of its logarithm goes to zero. The asymptote tells us that there is a threshold level of crosstalk above which we will not be able to communicate. This makes sense – have you ever been in a noisy, crowded room and not be able to understand the people next to you?

## Analysis

### Determine the Asymptotes

Determining the locations of the power penalty asymptotes is simple enough.

- The asymptote will occur when Equation 2's logarithm has an argument of zero.
- Take the argument of the logarithm of Equation 2 and determine the value of C
_{c}that makes it zero.

Figure 5 shows how I determined the asymptote locations on the crosstalk axis. My results are different than are shown in the specifications – the standard shows asymptotes at C_{c} = 0.4 dB (N=4) and C_{c} = 1.8 dB (N=8).

In Figure 6, I show values of isolation (adjacent and non-adjacent) that will produce a crosstalk value with infinite power penalty.

I did try to determine what math error the ITU might have made. While the C_{c} values they compute are theoretically possible, it could only be an asymptote for a negative ER, which is not physically possible (Figure 7).

### My Version of Figure 1

Figure 8 shows what I believe to be the correct plot. It shows Equation 2 graphed over the same C_{c} range as used in the standard, and I get the same curves shown in the standard. However, my asymptotes look much more reasonable. This graph was generated in Mathcad, with the text boxes done using scriptable components. I added the lines using a graphic editor (PicPick).

## Conclusion

I frequently am asked to review specifications. In this case, I did all my analysis in Mathcad. When I was done, I had a complete report ready to go. All that I needed to add were a few graphic lines to show key relationships.