Ring counter

{{Short description|Type of counter}} {{Use dmy dates|date=May 2020|cs1-dates=y}} A '''ring counter''' is a type of [[counter (digital)|counter]] composed of [[flip-flop (electronics)|flip-flop]]s connected into a [[shift register]], with the output of the last flip-flop fed to the input of the first, making a "circular" or "ring" structure.

There are two types of ring counters: * {{anchor|Straight ring counter|Overbeck counter}} A '''straight ring counter''', also known as a '''[[one-hot]] counter''', connects the output of the last shift register to the first shift register input and circulates a single one (or zero) bit around the ring. * {{anchor|Twisted ring counter|Switch-tail ring counter|Walking ring counter|Johnson counter|Möbius counter|Johnson code}} A '''Johnson counter''', also called '''twisted ring counter''', '''switch-tail ring counter'''{{cn|date=May 2026}}, '''walking ring counter'''{{cn|date=May 2026}}, or '''[[Möbius band|Möbius]] counter'''{{cn|date=May 2026}}, connects the last shift register's complemented output to the input of the first register and circulates a stream of ones followed by zeros around the ring.

== Four-bit ring-counter sequences ==

{|class="wikitable" style="text-align:center" |- ! colspan="5"|Straight ring counter || || colspan="5"|Johnson counter |- ! | State || Q0 || Q1 || Q2 || Q3 || || State || Q0 || Q1 || Q2 || Q3 |- ! 0 | style="background:#0FF"|1 ||0 || 0 || 0 |   ! 0 | 0 || 0 || 0 || 0 |- ! 1 | 0 || style="background:#0FF"|1 || 0 || 0 |   ! 1 | style="background:#0FF"|1 || 0 || 0 || 0 |- ! 2 | 0 || 0 || style="background:#0FF"|1 || 0 |   ! 2 | style="background:#0FF"|1 || style="background:#0FF"|1 || 0 || 0 |- ! 3 | 0 || 0 || 0 || style="background:#0FF"|1 |   ! 3 | style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 || 0 |- ! ''0'' | style="background:#0FF"|1 || 0 || 0 || 0 |   ! 4 | style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 |- ! ''1'' | 0 || style="background:#0FF"|1 || 0 || 0 |   ! 5 | 0 || style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 |- ! ''2'' | 0 || 0 || style="background:#0FF"|1 || 0 |   ! 6 | 0 || 0 || style="background:#0FF"|1 || style="background:#0FF"|1 |- ! ''3'' | 0 || 0 || 0 || style="background:#0FF"|1 |   ! 7 | 0 || 0 || 0 || style="background:#0FF"|1 |- ! 0 | style="background:#0FF"|1 || 0 || 0 || 0 |   ! ''0'' | 0 || 0 || 0 || 0 |}

==Properties==

Ring counters are often used in larger [[finite-state machines]]. A binary counter requires [[adder (electronics)|adder]] circuits which have propagation delays that increase as the number of bits increases, whereas the propagation delays within a ring counter are constant regardless of the number of bits.

The straight and twisted forms have different properties, and relative advantages and disadvantages.

A disadvantage of ring counters is that they are lower density codes than [[binary encoding]]s of state numbers. A binary counter has 2<sup>''N''</sup> states, where ''N'' is the number of bits in the code, whereas a straight ring counter has only ''N'' states and a Johnson counter has only 2''N'' states. This may be an important consideration in hardware implementations where registers are more expensive than combinational logic.

Johnson counters are sometimes favored because they offer twice as many states from the same number of bits, and because they are able to self-initialize from the all-zeros state. The Johnson counter generates a [[Gray code]] in which adjacent states differ by only one bit (i.e., have a [[Hamming distance]] of 1), which can be useful if the bit pattern will be asynchronously sampled.<ref name="Pedroni_2013"/>

When a fully decoded or [[one-hot]] representation of the counter state is needed, as in some sequence controllers, the straight ring counter is usually preferred. The one-hot property means that the set of codes are separated by a [[minimum Hamming distance]] of 2,<ref name="Mengibar_2003"/> so any single-bit error is detectable.

Sometimes bidirectional shift registers are used (using multiplexors to take the input for each flip-flop from its left or right neighbor), so that bidirectional or up–down ring counters can be made.<ref name="Stan_1997"/>

==Logic diagrams==

The straight ring counter has the logical structure shown here:

[[File:Overbeck Counter 4bit.svg|400px|border|4-bit ring counter using four [[Flip-flop (electronics)#D flip-flop|D-type flip flops]]. Synchronous clock and reset line shown.]]

Instead of the reset line setting up the initial [[one-hot]] pattern, the straight ring is sometimes made self-initializing by the use of a distributed feedback gate across all of the outputs except that last, so that a 1 is presented at the input when there is no 1 in any stage but the last.<ref name="Holdsworth_2002"/>

A Johnson counter, named for [[Robert Royce Johnson]], is a ring with an inversion; here is a 4-bit Johnson counter:

[[File:Johnson Counter 4bit.svg|400px|border|4-bit Johnson counter using four [[Flip-flop (electronics)#D flip-flop|D-type flip flops]]. Synchronous clock and reset line shown.]]

Note the small bubble indicating inversion of the Q signal from the last shift register before feeding back to the first D input, making this a Johnson counter.

==History==

Before the days of digital computing, digital counters were used to measure rates of random events such as radioactive decays to alpha and beta particle. Fast "pre-scaling" counters reduced the rate of random events to more manageable and more regular rates. Five-state ring counters were used along with divide-by-two scalers to make decade (power-of-ten) scalers before 1940, such as those developed by [[C. E. Wynn-Williams]].<ref name="Lewis_1942"/>

Early ring counters used only one active element (vacuum tube, valve, or transistor) per stage, relying on global feedback rather than local bistable flip-flops, to suppress states other than the one-hot states, for example in the 1941 patent filing of [[Robert E. Mumma]] of the [[National Cash Registor Company|National Cash Register Company]].<ref name="Mumma_1941"/> [[Wilcox P. Overbeck]] invented a version using multiple anodes in a single vacuum tube,<ref name="Overbeck_1943"/><ref name="Dayton"/> In recognition of his work, ring counters are sometimes referred to as "Overbeck rings".<ref name="RAMAC_1959"/><ref name="US_1960"/>

The [[ENIAC]] used decimal arithmetic based on 10-state one-hot ring counters. The works of Mumma at [[NCR Voyix|NCR]] and Overbeck at [[MIT]] were among the prior art works examined by the patent office that invalidated the patents of [[J. Presper Eckert]] and [[John Mauchly]] for the ENIAC technology.<ref name="Randall_2014"/>

By the 1950s, ring counters with a two-tube or twin-triode flip-flop per stage were appearing.<ref name="Higinbotham_1949"/>

Robert Royce Johnson developed a number of different shift-register-based counters with the aim of making different numbers of states with the simplest possible feedback logic, and filed for a patent in 1953.<ref name="Johnson_1953"/> The Johnson counter is the simplest of these.

=={{anchor|1-2-1 code|1-of-10 code}}Applications==

Early applications of ring counters were as frequency prescalers (e.g. for [[Geiger counter]] and such instruments),<ref name="Lewis_1942"/> as counters to count pattern occurrences in cryptanalysis (e.g. in the [[Heath Robinson (codebreaking machine)|Heath Robinson codebreaking machine]] and the [[Colossus computer]]),<ref name="Copeland_2010"/> and as accumulator counter elements for decimal arithmetic in computers and calculators, using either [[bi-quinary coded decimal|bi-quinary]] (as in the Colossus) or ten-state one-hot (as in the [[ENIAC]]) representations.

Straight ring counters generate fully decoded one-hot codes to that are often used to enable a specific action in each state of a cyclic control cycle. One-hot codes can also be decoded from a Johnson counter, using one gate for each state.<ref name="Langholz_1998"/><ref group="nb" name="NB1"/>

{{anchor|Libaw–Craig code}} Besides being an efficient alternative way to generate one-hot codes and frequency pre-scalers, a Johnson counter is also a simple way to encode a cycle of an even number of states that can be asynchronously sampled without glitching, since only one bit changes at a time, as in a [[Gray code]].<ref name="Holten_1982"/> Early [[computer mice]] used up–down (bidirectional) 2-bit Johnson or Gray encodings to indicate motion in each of the two dimensions, though in mice those codes were not usually generated by rings of flip-flops (but instead by electro-mechanical or optical [[quadrature encoder]]s).<ref name="Lyon_1981"/> A 2-bit Johnson code and a 2-bit Gray code are identical, while for 3 or more bits Gray and Johnson codes are different. In the 5-bit case, the Johnson counter's code is the same as the {{ill|Libaw–Craig code|de|Libaw-Craig-Code}} for decimal digits, from "a non-counting decimal- coded shaft digitizer".<ref name="Libaw-Craig_1953"/><ref name="Powell_1968"/><ref name="Dokter_1973"/><ref name="Dokter_1975_1"/><ref name="Dokter_1975_2"/><ref name="Steinbuch_1962"/><ref name="Steinbuch_1967"/><ref name="Steinbuch-Weber_1974"/>

A Johnson counter and a few resistors can produce a glitch-free approximation of a sine wave having an output frequency determined solely by the counter's clock frequency.<ref> Don Lancaster. [https://www.tinaja.com/ebooks/tvtcb.pdf "TV Typewriter Cookbook"]. ([[TV Typewriter]]). 1976. p. 180-181. </ref>

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style="background:lightgray"|3 !! style="background:lightgray"|2 !! style="background:lightgray"|1 |- | 0 || 0 || 0 |- | 0 || 0 || style="background:#0FF"|1 |- | 0 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || 0 |- | style="background:#0FF"|1 || 0 || 0 |- | style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|0 |- | style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|1 |- | style="background:lightgray"|0 || style="background:lightgray"|1 || style="background:lightgray"|1 |- | style="background:lightgray"|1 || style="background:lightgray"|1 || style="background:lightgray"|1 |- |} | {| class="wikitable" style="text-align:center;" |+ {{nowrap|4-bit Johnson}} ! style="background:lightgray"|4 !! style="background:lightgray"|3 !! style="background:lightgray"|2 !! style="background:lightgray"|1 |- | 0 || 0 || 0 || 0 |- | 0 || 0 || 0 || style="background:#0FF"|1 |- | 0 || 0 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | 0 || style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || style="background:#0FF"|1 || 0 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || 0 || 0 |- | style="background:#0FF"|1 || 0 || 0 || 0 |- | style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|0 |- | style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|0 || style="background:lightgray"|1 |- |} | {| class="wikitable" style="text-align:center;" |+ Libaw–Craig ! style="background:lightgray"|5 !! style="background:lightgray"|4 !! 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style="background:lightgray"|5 !! style="background:lightgray"|4 !! style="background:lightgray"|3 !! style="background:lightgray"|2 !! style="background:lightgray"|1 |- | style="background:#0FF"|1 || 0 || 0 || 0 || style="background:#0FF"|1 |- | 0 || 0 || 0 || 0 || style="background:#0FF"|1 |- | 0 || 0 || 0 || style="background:#0FF"|1 || style="background:#0FF"|1 |- | 0 || 0 || 0 || style="background:#0FF"|1 || 0 |- | 0 || 0 || style="background:#0FF"|1 || style="background:#0FF"|1 || 0 |- | 0 || 0 || style="background:#0FF"|1 || 0 || 0 |- | 0 || style="background:#0FF"|1 || style="background:#0FF"|1 || 0 || 0 |- | 0 || style="background:#0FF"|1 || 0 || 0 || 0 |- | style="background:#0FF"|1 || style="background:#0FF"|1 || 0 || 0 || 0 |- | style="background:#0FF"|1 || 0 || 0 || 0 || 0 |- |} | {| class="wikitable" style="text-align:center;" |+ [[1-of-10 code|1-of-10]] ! style="background:lightgray"|10 !! style="background:lightgray"|9 !! style="background:lightgray"|8 !! style="background:lightgray"|7 !! 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== See also == * [[Counter (digital)]] * [[Ring oscillator]] * [[Linear-feedback shift register]]

== Notes == {{reflist|group="nb"|refs= <ref group="nb" name="NB1">Johnson counter circuits with single states decoded in this way can be found in the original [[IBM]] [[Monochrome Display Adapter|MDA]] and [[Color Graphics Adapter|CGA]] video display adapter designs, in the timing sequencer logic: one or two [[74174|74x174]] hex D-type flip-flop ICs are wired as a shift register, fed back with inversion to form a Johnson counter, and 2-input [[NAND gate]]s (in the MDA) or [[XOR gate]]s (in the CGA) are used to decode states used as signals such as +RAS (Row Address Strobe [to [[DRAM]]]) and S/-L (Shift / NOT Load). Source: IBM Personal Computer Options & Adapters Technical Reference, Monochrome Display and Printer Adapter, logic diagrams; IBM Personal Computer Options & Adapters Technical Reference, Color Graphics Monitor Adapter, logic diagrams.</ref> }}

== References == <references> <ref name="Dokter_1973">{{cite book |title=Digital Electronics |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |date=1973-06-18 |series=Philips Technical Library (PTL) / Macmillan Education |publisher=[[The Macmillan Press Ltd.]] / [[N. V. Philips' Gloeilampenfabrieken]] |edition=Reprint of 1st English |location=Eindhoven, Netherlands |sbn=333-13360-9 |isbn=978-1-349-01419-4 |doi=10.1007/978-1-349-01417-0 |page=43 |url=https://books.google.com/books?id=hlRdDwAAQBAJ |access-date=2020-05-11 }} (270 pages)</ref> <ref name="Dokter_1975_1">{{cite book |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |title=Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik |language=de |series=Philips Fachbücher |publisher=[[Deutsche Philips GmbH]] |location=Hamburg, Germany |volume=I |date=1975 |orig-year=1969 |edition=improved and extended 5th |isbn=3-87145-272-6 |pages=52, 58, 98}} (xii+327+3 pages)</ref> <ref name="Dokter_1975_2">{{cite book |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |title=Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Anwendung der digitalen Grundschaltungen und Gerätetechnik |language=de |series=Philips Fachbücher |publisher=[[Deutsche Philips GmbH]] |location=Hamburg, Germany |volume=II |date=1975 |orig-year=1970 |edition=4th |isbn=3-87145-273-4 |page=169}} (xi+393+3 pages)</ref> <ref name="Steinbuch_1962">{{cite book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first=Karl W. |editor-last=Steinbuch |editor-link=Karl W. Steinbuch |date=1962 |edition=1 |publisher=[[Springer-Verlag OHG]] |location=Karlsruhe, Germany |publication-place=Berlin / Göttingen / New York |lccn=62-14511 |pages=71–72, 74}}</ref> <ref name="Steinbuch_1967">{{cite book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Siegfried W. |editor-last2=Wagner |date=1967 |orig-year=1962 |edition=2 |publisher=[[Springer-Verlag OHG]] |location=Berlin, Germany |id=Title No. 1036 |lccn=67-21079 }}</ref> <ref name="Steinbuch-Weber_1974">{{cite book |title=Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Wolfgang |editor-last2=Weber  |editor-first3=Traute |editor-last3=Heinemann |date=1974 |orig-year=1967 |edition=3 |volume=2 |work=Taschenbuch der Nachrichtenverarbeitung |publisher=[[Springer Verlag]] |location=Berlin, Germany |isbn=3-540-06241-6 |lccn=73-80607 }}</ref> <ref name="Pedroni_2013">{{cite book |author-last1=Pedroni |author-first1=Volnei A. |title=Finite State Machines in Hardware: Theory and Design |date=2013 |publisher=[[MIT Press]] |isbn=978-0-26201966-8 |page=50 |url=https://books.google.com/books?id=SSkTDgAAQBAJ&pg=PA50}}</ref> <ref name="Mengibar_2003">{{cite book |title=Integrated Circuit and System Design. Power and Timing Modeling, Optimization and Simulation: Proceedings of the 13th International Workshop, PATMOS 2003, Torino, Italy, 10–12 September 2003 |chapter=State Encoding for Low-Power FSMs in FPGA |author-first1=Luis |author-last1=Mengibar |author-first2=Luis |author-last2=Entrena |author-first3=Michael G. |author-last3=Lorenz |author-first4=Raúl |author-last4=Sánchez-Reillo |volume=13 |date=2003 |publisher=[[Springer Science & Business Media]] |page=35 |isbn=9783540200741 |chapter-url=https://books.google.com/books?id=JEEmfObxnrAC&pg=PA35}}</ref> <ref name="Stan_1997">{{cite journal |author-last=Stan |author-first=Mircea R. |title=Synchronous up/down counter with clock period independent of counter size |journal=Proceedings 13th IEEE Symposium on Computer Arithmetic |date=1997 |pages=274–281 |url=http://www.acsel-lab.com/arithmetic/arith13/papers/ARITH13_Stan.pdf}}</ref> <ref name="Holdsworth_2002">{{cite book |author-last1=Holdsworth |author-first1=Brian |url=https://books.google.com/books?id=o7enSwSVvgYC&pg=PA192 |title=Digital Logic Design |author-last2=Woods |author-first2=Clive |date=2002 |publisher=[[Newnes Books]] / [[Elsevier Science]] |isbn=((0-7506-4588-2)) |edition=4 |pages=191–192 |access-date=2020-04-19}} (519 pages) [https://web.archive.org/web/20200419213939/http://s2.bitdownload.ir/Ebook/Electronics/Holdsworth%20-%20Digital%20Logic%20Design%204e%20HQ%20(Newnes,%202002).pdf]</ref> <ref name="Lewis_1942">{{cite book |author-last=Lewis |author-first=Wilfrid Bennett |author-link=Wilfrid Bennett Lewis |title=Electrical Counting: With Special Reference to Counting Alpha and Beta Particles |date=1942 |publisher=[[Cambridge University Press]] |page=90 |isbn=9781316611760 |url=https://books.google.com/books?id=5B5CDAAAQBAJ&pg=PA90}}</ref> <ref name="Mumma_1941">[https://patents.google.com/patent/US2405096 "Electronic accumulation", Robert E. Mumma's US Patent No. 2405096, filed in 1941]</ref> <ref name="Overbeck_1943">[https://patents.google.com/patent/US2427533 "Electronic switching device", Wilcox P. Overbeck's US Patent No. 2427533, filed in 1943]</ref> <ref name="Dayton">[http://daytoncodebreakers.org/depth/42_res_rpt/ Dayton Codebreakers: 1942 Research Report, mentioning "A new high speed counter by Mr. Overbeck, January 8, 1942"]</ref> <ref name="RAMAC_1959">{{cite book |title=RAMAC 305 - IBM Customer Engineering Manual of Instruction |publisher=[[IBM]] |date=1959 |url=http://www.ed-thelen.org/RAMAC/IBM-227-3534-0-305-RAMAC-r.pdf |quote=[…] The Overbeck ring is used to supply timed pulses within computer circuits much as cam operated circuit breakers supply timed pulses on mechanical machines. It consists of a set of triggers with a common input from the ''ring drive line'' which carries pulses supplied by the process drum. […] Initially the triggers are reset OFF with the exception of the ''home'' trigger, which is ON. Each negative input pulse will turn OFF the trigger that is ON. The fall of the voltage at pin 10 of the trigger being turned OFF will grid flip the next trigger ON. This continues through a closed ring […]}}</ref> <ref name="US_1960">{{cite book |title=Electrical Technology - A Suggested 2-Year Post High School Curriculum |series=Technical Education Program Series |publisher=United States, Division of Vocational and Technical Education |date=1960 |issue=1–5 |page=52 |url=https://books.google.com/books?id=0zoUAAAAIAAJ&q=%22overbeck+ring%22}}</ref> <ref name="Randall_2014">{{cite book |editor-first=Nicholas |editor-last=Metropolis |author-first=Brian |author-last=Randall |chapter=The Origins of Digital Computers: Supplementary Bibliography |title=History of Computing in the Twentieth Century |date=2014 |publisher=Elsevier |pages=651–652 |isbn=9781483296685 |chapter-url=https://books.google.com/books?id=AsvSBQAAQBAJ&pg=PA652}}</ref> <ref name="Higinbotham_1949">[[William Alfred Higinbotham]], [https://patents.google.com/patent/US2536808A/en "Fast impulse circuits"], US Patent No. 2536808, filed in 1949</ref> <ref name="Johnson_1953">[[Robert Royce Johnson]], [https://patents.google.com/patent/US3030581A/en "Electronic counter"], US Patent No. 3030581, filed in 1953</ref> <ref name="Copeland_2010">{{cite book |author-last1=Copeland |author-first1=B. Jack |title=Colossus: The Secrets of Bletchley Park's Code-breaking Computers |date=2010 |publisher=[[Oxford University Press]] |isbn=978-0-19957814-6 |pages=123–128}}</ref> <ref name="Langholz_1998">{{cite book |author-first1=Gideon |author-last1=Langholz |author-first2=Abraham |author-last2=Kandel |author-first3=Joe L. |author-last3=Mott |title=Foundations of Digital Logic Design |date=1998 |publisher=World Scientific |isbn=978-9-81023110-1 |pages=525–526 |url=https://books.google.com/books?id=4sX9fTGRo7QC&pg=PA525}}</ref> <ref name="Lyon_1981">{{citation |title=The Optical Mouse, and an Architectural Methodology for Smart Digital Sensors |author-first=Richard F. |author-last=Lyon |author-link=Richard F. Lyon |date=August 1981 |id=VLSI 81-1 |type=Report |publisher=[[Xerox Corporation]] |location=Palo Alto Research Center, Palo Alto, California, USA |url=http://www.bitsavers.org/pdf/xerox/parc/techReports/VLSI-81-1_The_Optical_Mouse.pdf |access-date=2020-05-23 |url-status=live |archive-url=https://web.archive.org/web/20200523093939/http://www.bitsavers.org/pdf/xerox/parc/techReports/VLSI-81-1_The_Optical_Mouse.pdf |archive-date=2020-05-23 |quote=The counters needed for X and Y simply count through four states, in either direction (up or down), changing only one bit at a time (i.e., 00, 01, 11, 10). This is a simple case of either a Gray-code counter or a Johnson counter (Moebius counter).}} (41 pages)</ref> <ref name="Libaw-Craig_1953">{{cite journal |title=A Photoelectric Decimal-Coded Shaft Digitizer |author-first1=William H. |author-last1=Libaw |author-first2=Leonard J. |author-last2=Craig |date=October 1953 |orig-year=September 1953 |journal=[[Transactions of the I.R.E. Professional Group on Electronic Computers]] |issn=2168-1740 |eissn=2168-1759 |volume=EC-2 |issue=3 |pages=1–4 |doi=10.1109/IREPGELC.1953.5407731 |url=https://www.researchgate.net/publication/224112055 |access-date=2020-05-26 }} (4 pages)</ref> <ref name="Powell_1968">{{cite book |title=A short note on useful codes for Fluidic Control Circuits |chapter=Codes particularly useful for analogue to digital conversions |author-first=E. Alexander |author-last=Powell |date=June 1968 |id=CoA Memo 156 |publisher=[[College of Aeronautics, Cranfield|The College of Aeronautics]], Department of Production Engineering |publication-place=Cranfield, UK |s2cid=215864694 |page=10 |url=https://dspace.lib.cranfield.ac.uk/bitstream/handle/1826/9559/COA_Memo_156_June_1968.pdf |access-date=2020-12-15 |url-status=live |archive-url=https://web.archive.org/web/20201215124905/https://dspace.lib.cranfield.ac.uk/bitstream/handle/1826/9559/COA_Memo_156_June_1968.pdf |archive-date=2020-12-15}} (18 pages) (NB. The paper names the [[#Glixon|Glixon code]] ''modified Gray code'' and misspells [[Richard W. Hamming]]'s name.)</ref> <ref name="Holten_1982">{{cite magazine |date=August 1982 |title=Digital dividers with symmetrical outputs - The author uses Johnson counters with controlled feedback to give symmetrical even and odd-numbered divisions of a clock pulse. |url=https://worldradiohistory.com/hd2/IDX-UK/Technology/Technology-All-Eras/Archive-Wireless-World-IDX/80s/Wireless-World-1982-08-OCR-Page-0024.pdf |url-status=live |archive-url=https://web.archive.org/web/20210221205532/https://worldradiohistory.com/hd2/IDX-UK/Technology/Technology-All-Eras/Archive-Wireless-World-IDX/80s/Wireless-World-1982-08-OCR-Page-0024.pdf |archive-date=2021-02-21 |access-date=2021-02-20 |magazine=[[Wireless World]] |publisher=[[IPC Business Press Ltd.]] |location=Delft Technical University, Delft, Netherlands |pages=43–46 |publication-place=Sutton, Surrey, UK |volume=88 |issn=0043-6062 |author-first=Cornelius |author-last=van Holten |number=1559}} [https://web.archive.org/web/20210221205421/https://worldradiohistory.com/hd2/IDX-UK/Technology/Technology-All-Eras/Archive-Wireless-World-IDX/80s/Wireless-World-1982-08-OCR-Page-0025.pdf] [https://web.archive.org/web/20210221205307/https://worldradiohistory.com/hd2/IDX-UK/Technology/Technology-All-Eras/Archive-Wireless-World-IDX/80s/Wireless-World-1982-08-OCR-Page-0026.pdf] (4 pages)</ref> </references>

[[Category:Digital circuits]] [[Category:Digital registers]]