Statistical Range Analysis in RF Systems: the "Range Fallacy"

A few days later they arrive at your office, with a year's worth of gray skies in tow, to tell you "it doesn't work " you haven't got the range we need; you said the range would be 5 meters but we tried it at that distance and we could make it fail." What has gone wrong? You had read every paper you could get your hands on to do with propagation and path loss at the frequency you're using, and you had carefully included all the known (and measured) losses, antenna gains, power outputs and sensitivities in a detailed link budget which said, yep, this thing will do 5 meters.

Sounds familiar? You're not alone. You've probably committed what I'll call the "range fallacy" which, simply put, is to believe (wrongly) that range is a number with a value that you can calculate accurately as long as you have really accurate information about each hop in the complicated path from transmitter to receiver.

Range Calculations
Thinking of range as just a number, while sitting in a lab with $100k worth of test equipment — signed off by your management — leads you to believe that it can be measured. When you find out that it's hard to make that measurement, instead you measure the performance of all the blocks that make up your signal path, and drop these numbers into a 'link budget'. This tells you the distance up to which your system will work.

This isn't the place for a tutorial on the fundamentals of propagation (and I'm not the person to write that tutorial). The core of it is this:

• you have a transmitter which can deliver a certain RF power
• some of that power gets lost on the way to the antenna
• some of the remaining power reflects back off the antenna termination
• all that's left escapes into free space around the antenna and propagates away in the form of an electromagnetic wave, whose strength varies with direction and distance away from the antenna...
• then, at the other end of the link, the receive antenna is exposed to a small fraction of the power that the transmit antenna — smaller, the further apart the two ends are. The drop in power level on this leg is called the "path loss" and is expressed by an equation which is some function of the distance (and environment) between the transmitter and receiver.
• the receive antenna is able to extract some power from the incoming EM wave
• some of that power reflects back from the port where your electronics attaches to the antenna, and the rest enters your electronics, where some more of it may be lost before it does useful work on the actual first stage amplification
• the quality of data recovery from your link has a predictable relationship with this final power level (the sensitivity of the system for a given rate of link errors is just the input power needed such that you get this average error rate).

The link budget is just a way of accounting for the losses at each of the stages in the link where power gets diverted from the useful purpose to which the receiver will put it. The normal definition of range is just the distance that you need to enter into the particular path loss equation you've chosen in order that the power at the receiver works out to be exactly what you need to get the link performance you wanted. We can also solve the link budget for the range by working out how much path loss needs to be added (we're talking dB here) to all the other fixed gains and losses so that the starting power at the transmitter is reduced to exactly the sensitivity power at the receiver. We equate that path loss with the known equation and the range value can immediately be calculated from it. Hey Presto! A number for the range — what could possibly be wrong with that?

The "Range Fallacy"
So what is the "Range Fallacy"? It is just that each component of this link budget is not a single number, it's a statistical variable—ultimately, every parameter has an individual distribution quantifying the probability that the parameter will have a certain value.

And if the link budget components are distributions and we solve the link budget to find the range, the range will be a distribution too. In other words:

For any given ensemble of systems, the inherent statistical variability of the mechanisms that affect power transmission and reception result in a statistical variability in the range that individual systems from that ensemble will deliver. In the limiting case, where the variability becomes Gaussian, the normally calculated scalar range value will be the mean of the distribution and 50% of systems in the ensemble will not achieve it.

Now no one should be surprised that variations in things like the loss of a SAW filter will cause variations in the link budget. Normally, a worst-case design approach is used to ensure that the performance will be better than calculated. But small variations like this, 0.5dB here and there, are only the "thin end of the wedge." We need to take into account every source of variability. This is the reason for the word 'ensemble' in the formulation above. It's not just the variation we get when we build a thousand units that needs to be accounted for. It's also the variation when a thousand customers (your marketing department, perhaps) take out any given unit and try it. Or when one customer takes out a unit and tries it in a thousand different environments.

All the variations compound together to give a resultant distribution of the probability that the unit will 'work' (i.e. deliver the targeted error rate) when a trial is carried out.

When developing this methodology to use on a recent system, I used the term "% chance of failure" to express the potential for disappointment when the system doesn't perform. This is the probability that, given a large number of trials that exercise all of the forms of variability — just what would happen if you made lots of units, gave them to lots of people and they used them in lots of places — you wouldn't get the performance you were expecting.

It's a function of the distance between the transmitter and receiver; the curve intersects the distance equal to the conventionally calculated range at a 50% probability of disappointment. Any marketing person will tell you that this is rather a poor showing — you don't want half your customer engagements to end in dissatisfaction with the product. The real eye-opener was calculating, for realistic values of all parameter variations, how much the achievable distance was reduced before the probability of disappointment was reduced to a more tractable value of say 1% or 0.1%. This is the 'range' over which it's unlikely that any trial will catch any system out — but it's a much, much smaller distance than the range the engineers originally calculated! Here's a sample set of curves from that recent product development. Individual product details aren't important; suffice it to say that the various coloured traces indicated various differences in some factor than influenced a component of the link budget.

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