While desktop computers accept bags of accretion ability and storage, some baby CPUs don’t accept a lot of amplitude to abundance things. What’s added is some CPUs don’t do multiplication and analysis actual well. Same can be said for FPGAs. So accept we are activity to grab a agglomeration of three-digit decimal numbers from, say, a consecutive port. We appetite to abundance as abounding as we can, and we don’t appetite to do a lot of algebraic because we can’t, it is slow, or conceivably it keeps our processor alive best and we appetite to beddy-bye to conserve power. We appetite a way to backpack the numbers as abutting to the abstract best as we can but with little or no math.
The simple access is to abundance the numbers as ASCII. Abundant for processing aback they are apparently in ASCII already. If they aren’t, you aloof add 30 hex to anniversary chiffre and you are done. That’s abominable for accumulator space, though, aback we can abundance 999 in 10 $.25 if it were bifold and now we are application 24 bits! Autumn in bifold isn’t a acceptable option, if you comedy by our rules, by the way. You charge to accumulate by 10 and 100 (or 10 twice) to get the encoding. Granted, you can change that to two accouterment and an add (8x 2x=10x) but there’s no accessible way to do the analysis you’ll accept to do for the decode.
Of course, there’s no acumen we can’t aloof abundance decimal digits. That’s alarm bifold coded decimal or BCD and that has some advantages, too. It is appealing accessible to do algebraic on BCD numbers and you don’t get rounding problems. Some CPUs alike accept specific instructions for BCD manipulation. However, three digits will crave 12 bits. That’s bigger than 24, we agree. But it isn’t as acceptable as that abstract maximum. After all, if you anticipate about it, you could abundance 16 audible codes in 4 bits, and we are autumn alone 10, so that 6 positions lost. Accumulate that by 3 and you are crumbling 18 codes.
But there is a way to hit that ten-bit ambition after accomplishing any math. Its alleged DPD or densely arranged decimal. You can catechumen three decimal digits into ten $.25 and afresh aback afresh with no absolute algebraic at all. You could apparatus it with a baby lookup table or aloof do some actual simple multiplexer-style argumentation which agency it is bargain and accessible to apparatus in software or onboard an FPGA.
This packing of $.25 was the botheration that Thedore Hertz and Tien Chi Chen both noticed about 1969-1971. Hertz formed for Rockwell and Chen formed for IBM and consulted with addition IBM employee, Irving Tze Ho. Hertz and Chen apart developed what would become accepted as Chen-Ho encoding. A bit later, Michael Cowlinshaw published a clarification of the encoding alleged DPD or densely arranged decimal that became allotment of the IEEE amphibian point standard.
Both Hertz and Chen acclimated hardly altered encodings, but the Cowlinshaw arrangement had several advantages. You can calmly abound above 3 digits easily. Decimal numbers from 0 to 79 map to themselves. Bit aught of anniversary chiffre is preserved so you can do things like analysis for alike and odd numbers after unpacking.
Think of three BCD digits accounting in binary. Because anniversary chiffre is in the ambit [0-9], it can be represented with four bits, so our three digits attending like Xabc Ydef Zghi. For instance, if the aboriginal chiffre is 9 afresh Xabc = 1001. Aback we’re alone encoding numbers up to nine, if X is 1, afresh a and b charge be 0, abrogation us some amplitude to backpack added $.25 into. Now accede XYZ as its own three-digit bifold number. This leads to eight audible cases. The table beneath shows the encoding for all eight cases, with a lower case x indicates a don’t care:
Notice that c, f, and i consistently canyon through. The added bit positions alter depending on the case, but you don’t charge any algebraic to artlessly adapt the $.25 and add the anchored indicator $.25 in $.25 6-5 and 3-1 of the 10-bit encoding.
In the archetype figure, 105 matches case 000 — all arch $.25 are aught — so the encoding is 0010000101 or 085 hex. If the cardinal had been, say, 905 that would bout case 100 and would encode as 1010001101 or 28D hex. Decoding is aloof a amount of active the table backward. If bit 3 is zero, that’s case 000. Otherwise, attending at $.25 2 and 1. Unless they are 11, you can anon acquisition the agnate row in the table. If they are 11, you’ll accept to added attending at $.25 6 and 5 to acquisition the appropriate entry. Afresh you aloof disentangle the $.25 according to the table.
I was mostly absorbed in this for FPGA designs, so I wrote some simple Verilog to do the assignment and you can try it online. The testbench included runs through all 1,000 accessible numbers and outputs the DPD cipher in hex, the 3 ascribe digits, a slash, and the 3 achievement digits like this:
The encoding is actual straightforward. Here’s a atom for two rows of the table:
You’ll apprehension the cipher acclimated no clocks — it is authentic argumentation and makes all-encompassing use of the case and casez statements. These are like about-face statements in C although the casez account can use ? as a don’t affliction back matching.
I larboard both Verilog and C implementations on GitHub for you. Both are appealing naive. For example, the Verilog cipher doesn’t booty advantage of the actuality that some of the $.25 “pass through.” However, a acceptable compiler or synthesizer may about-face out some appealing acceptable cipher anyway. But if you were absolutely afraid about aspersing floorplan or cipher amplitude or aspersing ability consumption, you’ll charge to tune these to fit your architectonics and your needs.
It acclimated to be back you abstruse about computers one of the aboriginal things you were accomplished was algorithms and abstracts structures. I’m not abiding that’s the case anymore. But the apple is abounding of abstruse algorithms we ability use if we alone knew them. I’m afraid there aren’t added catalogs of algorithms like [Sean Anderson’s] acclaimed bit twiddling hacks page. Apparently the best one for aggregate — which agency it is cutting — is the NIST DADS dictionary. Be warned! You can absorb a lot of your day browsing that site. Alike as big as that armpit is, they accurately exclude business, communications, operating system, AI, and abounding added types of algorithms.
I mean, sure, best of us apperceive a balloon array and a carapace sort, but do you apperceive a cocktail array or how to do a Richardson-Lucy deconvolution? The DADS doesn’t either, although Wikipedia is helpful. We couldn’t acquisition constant aerial byte capacity or Chen-Ho in either place. Seems like this would be a abundant AI activity to archive and advice locate algorithms and abstracts structures for accurate uses. But for now, you can at atomic add close arranged decimal to your bag of accepted tricks.
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