Understanding logic level conventions

Over lunch John clarified a few things for me about the nomenclature used for transfer functions.

A "transfer function" is the model of how a gate/amplifier behaves. Given an input level (voltage for an electrical device or molarity for a chemical one) the model describes the equilibrium (or steady-state) output level. The above graph illustrates a hypothetical transfer function.

The main point of confusion for me was "What exactly is the definition of 'gain'?" and "By what convention are logic levels defined?"

John pointed out that the word "gain" is an over-used / abused word. Many people over-simplify the transfer function graph above and use 'gain' to mean different things. The gain is the slope -- but as you can see the slope of the function is different at different input levels so these is no such thing as "the" gain for a gate.

In the middle, linear range, the slope is roughly constant over an input domain. When building analog devices it is this roughly-linear region that is of interest and so an analog engineer would probably refer to the approximately-constant slope in this linear region as "the gain".

However a digital engineer uses the wider non-linear range to encode a binary variable. In this case, we must now have a convention that defines the logic levels. The electrical convention for this is that the two places where the slope, aka the "incremental gain", are equal to 1 are the places that define the inside bounds of the logic levels. Anything outside of these bounds are considered valid logic levels. Anything inside of them are considered "undetermined". The nominal values (the desired levels to be obtained by any gates) are defined by a "noise margin" outside of these inc. gain=1 points.

Glazed fountain spouts

My friends John and Chris Gray at Clayworks glazed and fired the spouts I built last week. They look very nice. Next step is to mortar them in to place -- which is going to be a pain because it is now 103 degrees outside!

Edge Detector Paper

The genetic edge detector paper just came out in Cell! Congratulations to Jeff and everyone else who worked on this paper. This is a project that we hatched about 3 or 4 years ago and it is finally out after a lot of hard work by everyone involved except me :-).

The project engineered bacteria to act as a communal signal processor implementing an "edge detector". You can think of the engineered bacteria as being like very simple computers, each running the following program: "Am I in the light? If so, shout to my neighbors. Otherwise, if I'm in the dark and I hear my neighbors shouting then raise my hand." Those that are in the dark and hear a neighbor must be near an edge. The "shout" is implemented by having the cells produce a small molecule which diffuses to their neighbors to be "heard" (actually, "smelled" is a better description). The "raise my hand" is implemented by having the cells produce a dark sugar which is visible to the naked eye when you look at the plate.

Interestingly, natural cells implement this edge-detection algorithm: the retina -- the first stages of image processing in the eye is to extract edges by a similar algorithm. The Nobel prize was awarded in 1967 to Hartline for this discovery in the retinas of the horseshoe crab.

Fountain spouts

Today I sculpted the spouts for the fountain on my back porch. The white boards under each are scaffolds supporting them until the dry at which time I will remove, glaze, and fire them. The firing should cause them to shrink so then I will be able to mortar them back into position.

Bed stain

Stained the bed today (not what you're thinking!). Almost done with this project! It's taken way too long.

Bed

I haven't been posting house progress for a while. My night-time project for the last week has been construction of a bed with a little bit of a floating cloud theme. I laminated four 4x8 sheets of maple plywood together and then cut out circles of various radii.


After Alex's sanding for many hours...


Here's the rough-cut end table before I cut out the circles...

Molecular model transfer function

Today I got around to trying out a simplified molecular version of the gate model that will replace my hyperbolic function.



The kinetics are all arbitrary for the model, but the shape of the transfer function looks even better than the made-up model from before. There's an almost perfectly linear section in the middle -- it looks more made-up than my made-up model! This is assuming that all three reactions have the same strength. Next, I need reasonable terms for the three reaction rates.