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Optical Bench REDUX: Digital Switching can have Analog Functions!

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...and while this current explanation is in limits, for an OCTAL count output, the same applies, generally, in a full decimal format, (0 - 9).  That is, a separate process must, upon 'underflow' of digit counting, (coordinated elsewhere in the process of counting down), must supply the full-loaded '9', or even a '10' for the immediate count-down to a 'nine' state. 
   Only difference, with OCTAL use is that the upper limit is skipping the last two integers, (9 and 8), that are not used in OCTAL count-downs.

   Chaining a few decrements can be done, in absence of any rounding action in between each.  About like predicted, any 'un-rounded' attempts should only last a few trials, at best.  Earlier methods had been using such a sloppy multiplication factor that some would 'skip' such as going '9' down to a '7'.  That particular instance had the multiplier bringing the '9.0' down, to '7.2',...(which could be rounded down to a '7').
   Keep in mind, I hadn't even really expected a single decrement-style action to work...but this should work and in complete absence of any active components or substantial complexity...just a (fast) light 'beam' coded by amplitude and subject to a couple of attenuating steps, (usually done by simple dropping off of portions  of the data BUS).

   Of course, in this set of schemes there's going to be construction errors / tolerances, initially guessed or gauged at somewhat near to +/- 5 %.  In that sense, a physical encoded count, of '5.0' could be offset by equivalent actual error of +/- 0.25 which is half the step size, when using this 1/2 counts per step, (in the
OCTAL 8 counts from 0 to 7 logical).
Or, in other words, an error of a half-step, plus or minus.

   Other considerations include noise immunity, which relies on keeping out of lower (physical) levels, in the representative 'coded' values.

   Readers might have noticed, earlier; that while the current 'universal' multiplier is around near '0.92' X, when using 1/2 count sized steps...restricting the range to be from 4.0 (code for zero logical), up to '7.5' as the code for logical '7'.
   The older methods used multipliers anywhere from about '0.633' to '0.833', and operating on 8 steps F.S. at one full count each.

   One structural aspect here that I've not explicitly mentioned is that some of the analog functions have been tailored specifically to do the (limited) job performed for integer math, digit by digit.  The main limitation is the segmentation of ranges (I.E. digital numerical 'macro' columns in base ten), where each separate logical logical range keeps a same or similar physical range:
   What this means is, a multi-digits number like '241', for instance, has each 'macro-column' separate and, likely, same range values.  Then, in that case, if you could consider that 'ones' column as analog 0 thru 1, (such as in a lumen scaled calc), then the 'TENS' column is represented at one-tenth the real or expected amplitude, (and the 'hundreds' column is even worse, at another 10X too small).

   That's not so bad, when processing that example '241', in three separate, distinct actions, while ignoring any needed CARRY or BORROW process.  For a decent BORROW action, though, digit down to lower macro column, you've got to have the expected 10 X relation.
That way, a borrow of a 'one', from upper column, will result in 'ten' units being deposited.

   To help illustrate my point, the diagram shows how the scope or domain of an analog value is actually only for the immediate purpose, of representing DIGITAL values, or integers, as closely as permits.  Aside from the more simple decrements, (or pseudo), a fully analog computer would have shown that example number in full; that is as value '1234'.
Instead, showing a single analog value as one instance of a full, digital multiplication.  So, a more fully expressed multiply would look like '1234 X 7', as a single operation.  Still the methods used in doing a decrement, ratiometrically, are interesting and can be applied elsewhere, as things develop.

   One nice aspect, I've discovered, in doing that kind of 4 by 1 multiply is that any multiply, with '9999 X 9' being the maximum, there isn't any need for carry propagation in that limited case.  Nice as the carry or borrow situation needs to act across columns and thus, as mentioned, you've really got to have the conventional 10X relation, column to column, for carry to be proper amplitude.
   (You can see, at bottom of enclosed diagram, the case for the maximum four digit multiply).


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