Mathcad Example Using Decibels

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I learned filmmaking by studying the Old Masters - and by that I mean John Ford, John Ford and John Ford.

— Orson Welles

Introduction

I have been working on a software requirements document that involves optical power levels. I thought it might be worthwhile showing you how I use Mathcad as part of my requirements analysis process. This particular example shows how I computed the minimum optical power levels that a circuit will need detect. I used decibels for the calculations, which are not like standard "multiplicative units". Decibels are not really a unit -- they are more of a scaling. However, Mathcad handles decibels very cleanly.

Background

Decibel Basics

The Mathcad folks recommend defining special functions for doing dB and dBm conversions and that is exactly what I do. Figure 1 illustrates the definitions I used in this analysis.

Figure 1: Defining a Set of Decibel Units.

Optical Power Basics

P₀ and P₁ Power Levels

Binary digital communication systems work by sending streams of bits (0s and 1s) across a communications channel. These bits have to be transferred in a form that is compatible with the channel. In the case of copper wire-based systems, we use current or voltage. In the case of fiber optic systems, we use optical power to represent the 0s and 1s. I will assume that we are using "positive logic", which means that the power used to represent a logic 1 (called P₁) is higher than the power used to represent a logic (called P₀). Unfortunately, measuring P₀ and P₁ directly requires expensive test equipment (e.g. oscilloscope with optical probe).

Because this measurement is expensive, we define two alternative parameters, average power and extinction ratio. Average power is easy to measure, which makes it the most commonly reported optical parameter. Extinction ratio is difficult to measure (i.e. also requires an oscilloscope and optical probe), but it provides us additional information as to degree of wavelength purity we will see in the laser output (i.e. smaller extinction ratio means less optical power off of our desired wavelength).

Average Power

Many inexpensive handheld instruments do a fine job of taking an average power measurement. The definition of average power is , assuming that P₀ and P₁ are equally probable. The measurement instrumentation involved is pretty straightforward:

• a photodiode to convert optical power to current
• a low-pass filter circuit to determine the average of the current from the photodiode.

In most cases, the average power measurement will give us enough information to understand the condition of the fiber network (e.g. ensuring that the loss levels on the plant are within specification).

Extinction Ratio

The extinction ratio is defined as . I do not know of an easy way to measure extinction ratio. It is usually measured with oscilloscope with optical probe. However, the extinction ratio is important because lasers work best with lower extinction ratios. High extinction ratios (>10) cause problems like:

• chirp

  High extinction ratio means that charge density inside the laser's optical channel is varying greatly. This can change the effective index of refraction within the channel and will cause the laser's wavelength to change during the transmission. This is bad because a varying wavelength causes dispersion.

• overshoot/undershoot

  High extinction ratio means that the laser is driven hard on and off. This can mean that the power level will overshoot its P₁ or undershoot its P₀ level. Overshooting the P₁ level is bad because it be difficult for the receiver circuit to deal with the widely varying signal level. Undershooting the P₀ level usually means driving the laser below threshold (i.e. the laser stop lasing). This can result in some bizarre problems, including very slow laser switching speeds.

Formulas for Translating Between (P₀,P₁) and (, P_Ave)

Figure 2 shows how you can use the definitions for P_Ave and to compute P₀ and P₁.

Figure 2: Deriving Equations for P1 and P2 in Terms of Extinction Ratio and Average Power.

PON Basics

Most of my work involves Passive Optical Networks (PONs). The details of the PON are not important to understand the gist of the post. It is important to know that there are two common types of PON (I am only concerned in this example with ONT-to-OLT receive power levels):

• B+
  • PON loss is 28 dB
  • ONT P_Ave= 2.5 dBm
  • ONT
• C+
  • PON loss is 32 dB
  • ONT P_Ave= 2.5 dBm
  • ONT

Figure 3 shows the loss model I used for this post.

Figure 3: Upstream PON Power Model.

Analysis

B+ PON

Figure 4 shows how I calculated the 0 and 1 levels at the OLT receiver for a B+ PON.

Figure 4: Determining the Power of the "0" and "1" Levels for a B+ PON.

C+ PON

Figure 5 shows how I calculated the 0 and 1 levels at the OLT receiver for a C+ PON.

Figure 5: Determining the Power of the “0″ and “1″ Levels for a C+ PON.

Conclusion

I just wanted to show folks how I document my requirements work using Mathcad. I do this sort of analysis every day.

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