Offset binary

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Offset binary,[1] also referred to as excess-K,[1] excess-N, excess-e,[2][3] excess code or biased representation, is a digital coding scheme where all-zero corresponds to the minimal negative value and all-one to the maximal positive value. There is no standard for offset binary, but most often the offset K for an n-bit binary word is K = 2n−1. This has the consequence that the "zero" value is represented by a 1 in the most significant bit and zero in all other bits, and in general the effect is conveniently the same as using two's complement except that the most significant bit is inverted. It also has the consequence that in a logical comparison operation, one gets the same result as with a true form numerical comparison operation, whereas, in two's complement notation a logical comparison will agree with true form numerical comparison operation if and only if the numbers being compared have the same sign. Otherwise the sense of the comparison will be inverted, with all negative values being taken as being larger than all positive values.

One historically prominent example of offset-64 (excess-64) notation was in the floating point (exponential) notation in the IBM System/360 and System/370 generations of computers. The "characteristic" (exponent) took the form of a seven-bit excess-64 number (The high-order bit of the same byte contained the sign of the significand).[4]

The 8-bit exponent in Microsoft Binary Format, a floating point format used in various programming languages (in particular BASIC) in the 1970s and 1980s, was encoded using an offset-129 notation (excess-129).

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) uses various sizes of exponent, but also uses offset notation for the format of each precision. Unusually however, instead of using "excess 2n−1" it uses "excess 2n−1 − 1" (i.e. excess-15, excess-127, excess-1023, excess-16383) which means that inverting the leading (high-order) bit of the exponent will not convert the exponent to correct two's complement notation.

Offset binary is often used in digital signal processing (DSP). Most analog to digital (A/D) and digital to analog (D/A) chips are unipolar, which means that they cannot handle bipolar signals (signals with both positive and negative values). A simple solution to this is to bias the analog signals with a DC offset equal to half of the A/D and D/A converter's range. The resulting digital data then ends up being in offset binary format.[5]

Most standard computer CPU chips cannot handle the offset binary format directly. CPU chips typically can only handle signed and unsigned integers, and floating point value formats. Offset binary values can be handled in several ways by these CPU chips. The data may just be treated as unsigned integers, requiring the programmer to deal with the zero offset in software. The data may also be converted to signed integer format (which the CPU can handle natively) by simply subtracting the zero offset. As a consequence of the most common offset for an n-bit word being 2n−1, which implies that the first bit is inverted relative to two's complement, there is no need for a separate subtraction step, but one simply can invert the first bit. This sometimes is a useful simplification in hardware, and can be convenient in software as well.

Table of offset binary for four bits, with two's complement for comparison[6]

Offset binary code, K = 8 Decimal code Two's complement binary
1111 7 0111
1110 6 0110
1101 5 0101
1100 4 0100
1011 3 0011
1010 2 0010
1001 1 0001
1000 0 0000
0111 −1 1111
0110 −2 1110
0101 −3 1101
0100 −4 1100
0011 −5 1011
0010 −6 1010
0001 −7 1001
0000 −8 1000

Offset binary may be converted into two's complement by inverting the most significant bit. For example, with 8-bit values, the offset binary value may be XORed with 0x80 in order to convert to two's complement. In specialised hardware it may be simpler to accept the bit as it stands, but to apply its value in inverted significance.

Related codes

[math]z = \left[\left(\sum_{i=1}^n p_i \times b_i\right) - k\right] / q[/math][2][3][7]
Code comparison[2][3][7]
Code Type Offset k Width n Factor q Weights Distance Checking Complement Groups of 5 Simple addition
8421 code n[8] 0 4 1 8 4 2 1 1..4 No No No No
Nuding code[8][9] 3n + 2[8] 2 5 3 N/A 2..5 Yes Yes (9) Yes Yes
Stibitz code[10] n + 3[8] 3 4 1 8  4 −2 −1 1..4 No Yes (9) Yes Yes
Diamond code[8][11] 27n + 6[8][12][13] 6 8 27 N/A 3..8 Yes Yes (9) Yes Yes
25n + 15[12][13] 15 8 25 N/A 3+ Yes Yes ? Yes
23n + 24[12][13] 24 8 23 N/A 3+ Yes Yes ? Yes
19n + 42[12][13] 42 8 19 N/A 3..8 Yes Yes (9) Yes Yes
Decimal
 
0
1
2
3
4
5
6
7
8
9
8421
4 3 2 1
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
Stibitz[10]
4 3 2 1
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
Nuding[8][9]
5 4 3 2 1
0 0 0 1 0
0 0 1 0 1
0 1 0 0 0
0 1 0 1 1
0 1 1 1 0
1 0 0 0 1
1 0 1 0 0
1 0 1 1 1
1 1 0 1 0
1 1 1 0 1
Diamond[8]
8 7 6 5 4 3 2 1
0 0 0 0 0 1 1 0
0 0 1 0 0 0 0 1
0 0 1 1 1 1 0 0
0 1 0 1 0 1 1 1
0 1 1 1 0 0 1 0
1 0 0 0 1 1 0 1
1 0 1 0 1 0 0 0
1 1 0 0 0 0 1 1
1 1 0 1 1 1 1 0
1 1 1 1 1 0 0 1
19n + 42[12][13]
8 7 6 5 4 3 2 1
0 0 1 0 1 0 1 0
0 0 1 1 1 1 0 1
0 1 0 1 0 0 0 0
0 1 1 0 0 0 1 1
0 1 1 1 0 1 1 0
1 0 0 0 1 0 0 1
1 0 0 1 1 1 0 0
1 0 1 0 1 1 1 1
1 1 0 0 0 0 1 0
1 1 0 1 0 1 0 1

See also

References

  1. 1.0 1.1 "2.5.2: Data Representation: Offset binary representation (Excess-K)". COMPSCI 210S1T 2006. Department of Computer Science, The University of Auckland, NZ. 2006-03-07. p. 18. http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf. Retrieved 2016-02-04. 
  2. 2.0 2.1 2.2 Digital Electronics. Philips Technical Library (PTL) / Macmillan Education (Reprint of 1st English ed.). Eindhoven, Netherlands: The Macmillan Press Ltd. / N. V. Philips' Gloeilampenfabrieken. 1973-06-18. p. 44. doi:10.1007/978-1-349-01417-0. ISBN 978-1-349-01419-4. https://books.google.com/books?id=hlRdDwAAQBAJ. Retrieved 2018-07-01.  (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)
  3. 3.0 3.1 3.2 "2.4.4.4. Exzeß-e-Kodes" (in de). Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik. Philips Fachbücher. I (improved and extended 5th ed.). Hamburg, Germany: Deutsche Philips GmbH. 1975. pp. 51, 53–54. ISBN 3-87145-272-6.  (xii+327+3 pages) (NB. The German edition of volume I was published in 1969, 1971, two editions in 1972, and 1975. Volume II was published in 1970, 1972, 1973, and 1975.)
  4. IBM System/360 Principles of Operation Form A22-6821. Various editions available on the WWW.[page needed]
  5. ((Electrical and Computer Science Department, Southeastern Massachusetts University, North Dartmouth, MA, USA)) (1988). Chen, Chi-hau. ed. Signal Processing Handbook. New York, USA: Marcel Dekker, Inc./CRC Press. ISBN 0-8247-7956-8. https://books.google.com/books?id=10Pi0MRbaOYC. Retrieved 2016-02-04. 
  6. "Data Conversion Binary Code Formats". Intersil Corporation. May 1997. http://www.intersil.com/data/an/an9657.pdf. 
  7. 7.0 7.1 "10.5.3.5 Excess-e-Code" (in de). Elektronik: Digitale Schaltungen und Systeme. Studium Technik. 3 (revised 2nd ed.). Friedrich Vieweg & Sohn Verlagsgesellschaft mbH. January 1997. pp. 120–121. doi:10.1007/978-3-322-85053-9. ISBN 978-3-528-13366-5. https://books.google.com/books?id=VcmeBgAAQBAJ&pg=PA120&lpg=PA118. Retrieved 2020-05-26.  (xviii+393 pages)
  8. 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 "Checking Codes for Digital Computers". Proceedings of the IRE. Correspondence (New York, USA) 43 (4): 483–490 [487–488]. April 1955. ISSN 0096-8390. https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4055437. Retrieved 2020-05-26.  (2 pages) (NB. The results discussed in this report are based on an earlier study carried out by Joseph M. Diamond and Morris Plotkin at Moore School of Engineering, University of Pennsylvania, in 1950–1951, on contract with the Burroughs Adding Machine Co.)
  9. 9.0 9.1 "Ein Sicherheitscode für Fernschreibgeräte, die zur Ein- und Ausgabe an elektronischen Rechenmaschine verwendet werden" (in de). Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM). Kleine Mitteilungen 39 (5–6): 429. 1959-01-01. doi:10.1002/zamm.19590390511. https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19590390511.  (1 page)
  10. 10.0 10.1 "Complex Computer". 1954-02-09. https://patents.google.com/patent/US2668661?oq=US2668661.  [1] (102 pages)
  11. "Binary Codes with Specified Minimum Distance". IRE Transactions on Information Theory IT-6 (4): 445–450. September 1960. doi:10.1109/TIT.1960.1057584. ISSN 0096-1000.  (NB. Also published as Research Division Report 51-20 of University of Pennsylvania in January 1951.)
  12. 12.0 12.1 12.2 12.3 12.4 "Error Detecting and Correcting Binary Codes for Arithmetic Operations". IRE Transactions on Electronic Computers EC-9 (3): 333–337. September 1960. doi:10.1109/TEC.1960.5219855. ISSN 0367-9950. 
  13. 13.0 13.1 13.2 13.3 13.4 "15.3 Arithmetic Codes / 15.6 Self-Complementing AN + B Codes". written at Honolulu, Hawaii. Error-Correcting Codes (2 ed.). Cambridge, Massachusetts, USA: The Massachusetts Institute of Technology (The MIT Press). 1972. pp. 454–456, 460–461 [456, 461]. ISBN:978-0-262-16-039-1. ISBN 0-262-16-039-0.  (xii+560+4 pages)

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