From spectrum efficiency metrics to parameter spaces

In my post FCC white paper shows that “spectrum efficiency” is meaningless I argued that spectrum efficiency metrics are not very helpful.

They won’t go away, though, because engineers and economists instinctively characterize systems numerically. Both tribes strive to separate a problem into smaller independent parts, each described quantitatively; metrics are just a symptom.  The goal is to convert a complex mess into a problem amenable to objective analysis, yielding an incontrovertible answer. No more messy politics! 

Since politicians always look for cover behind engineers and economists, simplistic metrics will always be with us – not least in radio regulation.  Given that reality, I’m going to dig into spectrum metrics a little more. I conclude that it could be more productive to define a series of axes in a parameter space than a single metric.


Any metric is a model, i.e. an abstraction. It leaves a lot of stuff out in order to provide a useful thumbnail. As George Box said, “All models are wrong, but some are useful.” For example, to quote from the TAC “spectrum efficiency metrics” white paper (DOC), “communications systems must often meet basic user needs in a number of quality of service (QoS) measures, including latency/access time, coverage/reliability, information error rates, and peak-loading requirements.  Maintaining this service level or even improving it in some of these areas may have a negative impact on spectral efficiency metrics, but may be required for particular system applications. ”

There are lots of ways to measure radio operation. It’s worth distinguishing between rulers and ratios:

Rulers: a measure of a quantity of interest. Broadly speaking of two kinds:

1. Inputs: e.g. MHz, sq. miles, POPs, $ infrastructure investment, $ backhaul cost, $ device cost, maximum transmit power, maximum PFD, antenna size, consumed field of view or orbital arc or geographic region (for satellites); combinations of these like MHz*POP
2. Outputs: e.g. bps, $ surplus, system response time, bit error rate, device size, coverage, peak loading requirement. Note that some important goods that flow from radio operation cannot be quantified, e.g. public safety, distributional equity, basic knowledge.

Ratios: built using rulers, e.g. $/MHz*POP, bps/Hz, $ surplus / $ investment.

Rather than picking any single one of these, it could be more productive to define a series of axes in a parameter space. One could then plot different services in this space; rather than requiring that all of them are in the "top right" of the space, one could look at the various clusters and try to move services withing clusters out to the leading edge in parameter space.

A parameter space is a bunch of axes or dimensions that can be rulers or ratios. The paradigmatic one in wireless is electrospace (Hinchman 1969; Matheson 2003 et seq.) that defines radio field strength in a seven-dimensional space {frequency, time, spatial location, direction-of-travel}; it’s defined using input rulers as axes.

A given ratio metric is a surface in a parameters space. For example, a given body mass index (weight divided by height squared) parameterizes a parabola if you plot weight against height. A bunch of BMI’s define boundaries between regions that represent being underweight, normal, overweight and obese. If you then plot BMI against age, you get growth curves for BMI-for-age percentiles (Wikipedia).

In order to get a flavor of how this might work in radio regulation, here are the metrics in the TAC white paper (omitting the “Additional Efficiency Considerations”). I’ve translated the dimensions into SI units; they vary quite a bit in the document.

• Satellite broadcast systems: bits / (second * Hz) within each common program area
• Point-to-point satellite systems: bits / (second * Hz * meter^2)
• Terrestrial Broadcast Systems: bits / (second * Hz) within each common geographic area * the average number of users simultaneously served
• Personal Communications Systems (aka PCS): bits / (second * Hz * meter^2)
• Point-to-Point Terrestrial Systems: (transmitted distance) * bits / (second * Hz* meter^2)
• Hybrid Terrestrial Systems: (info bits / second / Hz) * meter^2 * the average number of users simultaneously served. 

(Note that the area in Hybrid Terrestrial Systems seems to be in the numerator, not the denominator; this probably relates to the fact that for public safety systems, similar to terrestrial broadcasting systems, as the number of users increases, the spectrum efficiency improves when compared to point-to-point systems where each additional user consumes additional capacity.)

In could represent all these metrics as surfaces in a parameter space with these dimensions:

• bits / (second * Hz); this could be split into two dimensions, bits/second and Hz
• area (meter^2)
• transmitted distance (meters)
• average number of users simultaneously served

Note that the surfaces depend on the kind of service. Thus, for PCS systems, the figure of merit goes as bits/(second*Hz) / meter^2, whereas for terrestrial broadcast and public safety it’s bits/(second*Hz) * meter^2. Increasing the figure of merit for PCS increases the slope, whereas increasing it for broadcast pushes it out towards the top-right. It’s therefore obvious that it’s hard to compare different categories of service with each other.

Constant-value surfaces for system performance metrics

For systems that aren’t measured on a particular axis (e.g. the satellite broadcast systems metric omits area), the representation of the metric becomes a section of the space, e.g.