Measurement Abstraction

It is a sad irony that focusing on the measurement for the purpose of making life simple for the test engineer leads to abstraction, and a new abstraction is never simple for anyone to accept at first. I would go so far as to say that the most formidable human problems facing the introduction of synthetic instrument design concepts in real-world applications is the fact that synthetic instrumentation embodies an abstract model of measurements. When people are accustomed to dealing with concrete things (for example, a specific set of instruments that they use to make a specific measurement), it becomes very difficult for them to let go of this conception of things and try to imagine any other way to accomplish the measurement.

This fact about human nature is why virtual instruments are so popular. With virtual instruments, you can imagine that your favorite old instruments are still being used to make your measurement. You can have comforting little virtual knobs on cute virtual front panels constituting your virtual rack of virtual instruments. This collection of instruments are then programmed in a quite familiar manner that mimics the way a corresponding set of physical instruments would be applied to do the measurement.

But actually, if you want to measure something, Yogi Berra might say that all you really need is a thing that measures what you want to measure. If you want to measure A and B vs. C and D, you need an (A,B) = f(C,D) measuring instrument, an AB vs. CD meter, so to speak. Nothing else is really the “right” thing. Yes, maybe you have always measured A and B separately as functions of C and D and stitched it together, but fundamentally what you really want to do (if you don’t mind me putting words in your mouth) is measure the AB vector over the CD manifold¹.

This idea becomes crucial when C and D are not separable. A function of several variables:

f(x₀, x₁, x₂, …)

is separable if it can be expressed as a product of functions of the individual variables.

g₀(X₀) · g(x₂) · g₂(x₂) · …

Here’s an example: Suppose you have an ultrasonic transducer. You would like to measure its response versus frequency and its response versus signal input level. These two measurements may not be separable. You could vary input level and plot output at some fixed frequency. Then you could vary frequency and plot output at some fixed input level. If the response was separable, you could then multiply those two functions and have the whole thing.

But it may be the case that the shape of the frequency response curve changes at different power levels. It may also be the case the power transfer curve is shaped differently at different frequencies. You really need to measure the response over the joint domain manifold of frequency response and power response to fully characterize the sensor.

Figure 6-1 is an example of a manifold f(x,y) that isn’t separable.

Synthetic instrumentation approaches don’t yield their biggest payoff unless you are willing to think about measurements themselves, as pure measurements, especially multidimensional measurements. You need to divorce your thought from particular instrumentation used to make the measurements and think only about what you want to measure, in and of itself. Failing to do this will inevitably shift the focus from the measurement to the instrument being used and result in the introduction of myriad irrelevant and extraneous considerations that would not otherwise appear.

To the end of attempting to get everyone thinking about measurements abstractly, the following section attempts to introduce some vocabulary that does not shackle us to instrument-specific ideas. The vocabulary is based on mathematical concepts that, for the most part, are studied by teenagers in high school.

General Measurements

In a synthetic instrument, measurements are performed by software running on generic hardware. Ideally, this software is completely flexible. Any sort of measurement is possible to define, so long as it falls within the capabilities of the hardware. Unfortunately, if measurements are flexible without limit, one is wandering without guideposts in this total freedom. The system can do anything, so there’s no structure to provide a handle on what can be accomplished.

For example, there is no finite set of distinct measurement parameters that fully describe the required inputs to all these possible measurements. Moreover, there is no possible universal parameter format, type, or structure that can cover all cases. Similarly, in general there is no standard data type to cover all possible measurement results. Nor is there a finite and standard calibration set that can be applied to relate these arbitrary measurements to physical units.

It should be clear, therefore, that some sort of structure must be imposed on this vast abstract possibility in order for our finite human resources to be applied. One type of structure is the virtual instrument structure. This introduces the accustomed structure of everyday instrumentation in order to make our options reasonably finite.

But virtual instruments are not unlike the way stops on a church organ mimic classic musical instruments. The organ can synthesize various classic instruments, but if you wanted something else not in the set of organ stops—something new—you couldn’t have it. This isn’t because of any inherent limitation in the organ, but rather because of a limitation in the model used to parameterize and limit what the church organ can be asked to synthesize.

What is needed is a structure on possible measurements that is limiting enough to result in a tractable system design, but generic enough to allow the full range of possibilities. Returning to the organ metaphor, introduce the idea of a Fourier series, and allow organ stops to be specified as Fourier coefficients. Now the goal is reached. The Fourier series provides a handy and compact structure without introducing practical limitations. The full freedom of synthesizing any past instrument lives along with the possibility of synthesizing future instruments.

Abscissas and Ordinates

I will now propose a system for describing measurements that is free of specific instrumentation focus, and thereby does not require reference to virtual instruments, or any other legacy crutch. It is compactly and usefully structured, but the full freedom of synthesizing any past instrument lives along with the possibility of synthesizing future instruments.

In this system, I consider only the subset of all possible measurements that I call stimulus response measurement map (SRMM) measurements. I will show that with this conception, a uniform format for parameters, data, and calibration is possible. This has far-reaching significance because the broad class of SRMM measurements comprises all the typical measurements made with conventional instrumentation, as well as much of what is possible to do with any instrumentation.

Ports and Modes

I’ve talked endlessly about measurements. How do these relate to a DUT? What is the link between the two? A bridging concept that I use to relate a DUT to its measurement is the concept of a port. A port is a plane of interaction between the measurement system and the DUT—a precisely specified data interface between the signal conditioner and the DUT. Often it is a physical cable interface to a sensor or a connector on the DUT, but it might be an abstract plane in space. It depends on what you are measuring and how.

Many schemes have been developed for managing ports and linking logical ports to the real physical wiring interfaces, registers, or commands they represent. These schemes tend to be proprietary and associated with particular measurement systems. Others are more generic. I will present a sketch of a minimal scheme in the section titled “Describing the Measurement System with XML” that uses XML syntax.

Ports can have many attributes, but foremost is if the port is an input to the DUT, or if it is an output from the DUT. This will determine if the measurement system applies a stimulus to the DUT at a given port, or measures a response.

In my model of a synthetic measurement system, ports must be associated with physical point or planes, and must be categorized as either stimulus or response. They can’t be both, or neither, although it is certainly possible to define two ports, one stimulus and one response that both connect to the same place, or no place.

Ports are a logical concept, but ultimately they connect to the real world. Stimulus ports are, ultimately, controlled by a register that is written or a command that is sent; response ports are likewise wired to a register that is read or a command that is received.

The port is not the only bridging concept that links from the measurement to the hardware. A mode denotes states of the system itself independent of its physical interfaces.

The distinction between ports and modes is a fuzzy one. Perhaps the signal conditioner has two separate physical interfaces: One through an amplifier and one direct, with a switch internal to the conditioner selecting between them. I might call that switch a mode switch. On the other hand, suppose these two conditioner ports are connected to different DUT ports. Maybe then the switch is really a port switch. But what if the switch matrix outside the conditioner is able to route either of these signal-conditioning interfaces to the same or different interfaces on the DUT. Now it may be unclear if the gain selection switch should be considered a port or a mode.

It might be argued that implementing a mode by means of a physical interface, mixing the two concepts of mode and port, might be considered a hardware design mistake in the same way as a GOTO statement is considered, by some, to be a mistake in software design. This would be true if there were no other considerations, but the reason hardware is designed a certain way (or software, for that matter) is frequently based on the optimization of certain aspects of performance, along with considerations of safety, cost, reliability, and so forth. There may be good reasons to use separate physical interfaces for different modes in a signal conditioner that override any paradigm purity considerations.

A common voltmeter is a good example, where the high-voltage measurement input is often a different interface than the normal, low-voltage input. This is done in order to reduce the chance of damaging the low-voltage circuitry with an accidental high-voltage input, as well as eliminating the need for an expensive high-voltage switch in the meter.

In general, it is wise to distinguish and separate modes from ports as much as possible. In a well-designed SMS, there will be a site configuration document and associated software layer that can disentangle many of these overlaps between ports and modes.

Map Manipulations

If you have followed me so far, you should see that the description of a measurement and the results of measurements are mappings. Using the stimulus response measurement map model of measurements allows us to see exactly what aspects of each particular measurement are unique to that measurement, and what parts are generic aspects. Looking at the map as a whole allows us to see beyond the particular list of abscissas and ordinates associated with a test and focus on the measurement itself. This measurement focus leads inevitably to a compact implementation as a synthetic instrument.

In regards to abscissas, in the common case of a separable domain, the process of applying stimuli to a device can be reduced to defining the individual abscissa scales. With the addition of banded and locked domains, all usual domain cases are comprised by a small set with compact descriptions.

There is a distinction between a map description and the map data itself. The map description is present before the measurement is made. The map data is acquired after the measurement is made. Together they represent a fully documented measurement map. Often I will discuss the two collectively using just the one word map. In cases where the distinction is relevant, I will explicitly specify map description or map data.

Maps are more than just a way to define measurements and record the results of measurements. Maps can change. They can be processed and manipulated both before the measurement and after the measurement in ways that are isometrological—manipulations that do not affect what is ultimately measured. In fact, they must be processed and manipulated isometrologically in order to apply to many common measurements, particularly relative measurements, or measurements that include a calibration ordinate. This isometrologic manipulation process is called canonicalization. It is described in detail in the section titled “Canonical Maps” and it is the paramount benefit of the Stimulus response measurement map viewpoint.

Maps can be interpreted (with some restrictions) as a multidimensional entity, as a manifold themselves. As such, all the concepts associated with manipulating objects in space work as tools for manipulating maps. For example, you might take a slice through a particular plane. A slicing action represents holding the value of some variable constant, either an abscissa or ordinate. Imagine a measurement of an amplifier gain and power supply drain as a function of input power and input frequency. Maybe you might like the gain versus frequency at constant power supply current. That would be a slicing operation. Slicing operations usually require interpolation because the plane that slices through the data may fall between measured or controlled points.

Processing may also rotate the map. This may be done to reorder the axes, or perhaps to remove dependencies between axes—to orthogonalize axes. An example of orthogonalizing would be a DUT that has two inputs and two outputs. Both inputs affect both outputs, but the inputs affect the outputs in different ways. Imagine, for example, the hot and cold knobs on your sink. They control the temperature and flow rate of the water out of the spigot. Two abscissas, two ordinates. There is an interaction between the two. Turning one knob changes both the flow and the temperature. Maybe you would like to know how to control just the temperature at a constant flow, or the flow at constant temperature. That is the process of orthogonalization applied to measurements. In this case, it involved finding a rotation transformation to apply that orthogonalizes the abscissas.

When rotating more than just the abscissas, the rotation can be interpreted as an inversion of the map. For example, if you have a map y = f(x) and you find a function g that exchanges y for x, allowing you to get x = g(y), then you have rotated and flipped the x-y plane. That is to say, you have rotated abscissas and ordinates together, as one unit. Inversions, like slicing, require interpolation in order to make the new abscissa fall on nice, uniformly gridded points.

Sometimes it’s necessary to flatten a map. This process removes an abscissa or ordinate. Calibration is often accompanied by flattening. For example, suppose you are measuring the gain of an amplifier. You may do that by measuring its input power, then its output power, then dividing the two, yielding gain. After the division, if you no longer want input and output power, you can flatten the map data manifold by combining two of its dimensions with a calculation. In that case, I call gain a calculated ordinate as compared to the directly measured ordinates of input and output power.

Maybe you want to expand (or maybe better would be thicken) the map data with the gain calculation result, keeping input and output power in place. This is fine too. Expanding is the dual of flattening. It’s common to add new dimensions to map data during post-processing with additional calculated results; it’s also common to add them in preprocessing to include calibration ordinates or abscissas. Sometimes these expansions are paired with their dual, a flattening, on the opposite side of the data acquisition.

An alternative to expanding and flattening is to make a child map. When creating a child map, it is generated from one or more parent maps by a calculation. It’s a good idea to maintain a link between child and parent so that later you can figure out what the calculated data was based on. This is often done in a nested or hierarchical tree-like manner that can easily be expressed in XML or HDF.

Other dual-pair manipulations that are useful are rastering and raveling. Both of these ideas are implicit in the way domains manifolds are sampled. Normally, where there is more than one abscissa, the abscissa is sampled in raster order. That is to say, one of the axis is sampled so as to vary fastest in an innermost loop with the other axes held constant. Then, once the innermost abscissa is completely sampled across its domain, the next innermost abscissa is incremented to its next domain sample point, and then the innermost repeats its scan. Thus, all the axes are sampled through their ranges in this way.

An alternative to rastering is to ravel the abscissa points in some other order. All points are still sampled, but the samples are ordered differently. Normally, the order in which the abscissa set is sampled does not affect the measurement, but when hysteresis is present, or when certain axes are slow and others are fast to measure, the ravel or raster order can be essential to the success of the measurement.

All these manipulation techniques, and more, can be used on measurement maps. They can be used individually, or in combination. What is less obvious that should be emphasized is that they can also be used either on map data, or on map descriptions. This may be surprising to you if you had been thinking of all these cutting, flattening, and interpolation operations as after-the-fact post-processing of map data.

Operations on map descriptions are performed differently than operations on map data, that is true. But the same set of techniques are generally available. One can expand the map description to include extra abscissas or ordinates before the measurement; similarly, one can interpolate, invert, or otherwise reshape.

Consider the gain measurement example again, beginning with a map description that gives gain as an ordinate. One of the pre-processing steps might be to expand the map into canonical form with an atomic input power and output power ordinate.² This would be done by canonicalization so that during post processing the operation could be reversed, yielding gain.

In fact, it is a general rule that calibration strategy considerations often will required a transform of the map description prior to the measurement. Any time you make a relative measurement, for example, the relative ordinate you want to measure must be split into multiple measurements, which are divided or subtracted or otherwise combined through some calculation to compute the desired computed ordinate.

Calibration Strategy and Map Manipulations

Why did I bother to create the stimulus response measurement map model of measurements? What good is it, really? Maybe it seems like an interesting way to describe measurements, but is there any big payoff? These are good questions. It may not be obvious why any of the math stuff is worth the trouble. Abscissas are just fancy loops. Ordinates are measurement subroutines. Yes, I see how this ties data together with the measurement in a neat package. That’s nice. But is there a bigger payoff?

While I have already pointed out the many small payoffs that derive from the use of the SRMM approach in general, and the benefits of XML schema for describing measurements in particular, the jackpot payoff is with the concept of calibration strategy. Without the concept of calibration strategy, and related concepts, like compound and atomic ordinates and abscissas, the formalizing of measurement descriptions under the SRMM stance has no more benefit than other more generic object-oriented approaches—other approaches that, although they have nothing to do with test and measurement, may be more familiar to software engineers.

Calibration strategy is a method for specifying how maps should be isometrologically transformed both before and after physical interactions with the DUT. Calibration strategy will rewrite the measurement to a new form, changing them from what the user originally specified for the test into what the synthetic measurement system actually can do. After the raw measurement is made, calibration strategy guides the post processing manipulations that occur, transforming the map back to what the user wanted in the first place.

I have already discussed map manipulations in some detail in the section titled “Map Manipulations.” I gave an example of a map manipulation that would be applied in the case of a gain measurement. The gain ordinate versus some abscissa, is rewritten into the power-in and power-out ordinates versus the same abscissa. That map data is acquired. Then the result is transformed back, calculating the ratio to collapse the power-in and power-out axes, yielding the desired gain map.

Surely, the same thing be accomplished by simply writing a new ordinate called gain. Such an ordinate would operate the measurement hardware explicitly so as to make power-in and power-out measurements, it would divide the result, and it would return gain. What’s wrong with that?

Nothing is really wrong, in the sense that this could certainly be made to work. In fact, I’ve seen many systems where test engineers do exactly this: write a new test script every time they want to measure something new.

Fine, but there are a bunch of methodological problems here. Now you have a growing list of idiosyncratic ordinates to maintain. Improvements made to power-in and power-out may or may not be reflected in an improved gain ordinate. Worse yet, it’s not exactly clear what depends on what. Maybe the gain ordinate is written first and later on power-in and power-out versions are abstracted. This creates a labyrinth of dependencies. Axes, unit conversions, calibration, calculation, post processing are all embedded in “ordinates” which no longer are worthy of the name. Ultimately, a collection of hand coded, ad hoc measurement scripts accumulate that don’t form any sort of coherent, reusable system.

Don’t get me wrong. I’m as much in favor of hand coding, ad hoc, hacked up, extreme programming as the next guy, but I don’t want to create a big system entirely this haphazard way unless I am planning to quit just after CDR. The concept of calibration strategy that I have been describing in this book leads to a better place where I can keep (and maybe enjoy) my job.

Canonical Maps

Instead of “just coding a new ordinate,” there is a fundamentally better way to deal with what I will call a compound ordinate like gain.

Processing a Measurement

How does the SRMM description of a measurement get translated into an actual measurement in a synthetic instrument? This question is a variation on the oft-heard refrain: “But how do you really do a measurement?” What is the measurement algorithm?

Given the map model of measurements, all possible measurement algorithms can be described generically by one specific, unchanging, high-level algorithm. In my effort to object-orient everything, I recast the measurement algorithm, something that might be seen as inherently procedural, as a collection of algorithm components that can be dealt with independently, and reused.

Reuse is urgent because nothing is more expensive to produce, per pound, than software. Anything ensuring a software job is done just once is immensely valuable. What I am saying, therefore, is if you want to do a stimulus response measurement map measurement (or you can recast your time-honored measurement in a SRMM framework), I can give you a way to automatically generate software to do that measurement using a generic template.

Establishing an algorithm template therefore leads us toward object-oriented (OO) techniques. You no longer have to specify the parts of the custom measurement problem that are the same as a standard measurement; all you need to specify are the differences. In the parlance of OO design, establish a base or parent class that describes what all measurement algorithms have basically in common. From that base class create child classes for specific variations. Thus, the variations are created without risk of losing the tested, reliable functionality of the parent.