1.
DISCUSSION ON
'ORTHOGONAL CODES'*
Dr. D. A. Bell (communicated): The idea of two independent elements per unit of time-bandwidth is fundamental to communi- cation theory.^ and the possibility of choosing these in the form of sine and cosine functions having an integral number of periods in a standard interval follows naturally, but Dr. Harmuth's paper is probably the first detailed description of a proposed complete system As the author recognizes, synchronization is the major practical problem. Is it intended that periods of ynchronizing signal should be interpolated in the message from Or is it intended that the synchronizing signal ime to time? should be transmitted continuously on a separate channel? The bandwidth needed for the transmission of the synchronizing signal would be governed by the steepness of transition between + and -1, which in the author's diagrams is portrayed as being infinite If the tolerable error of just under ±3 millisec is equated to a quarter-cycle of the maximum frequency trans- mitted, the necessary bandwidth is about 85c/s, which is 1-77 times the highest of the character fundamental frequencies.
The author refers to the difficulty of assessing the redundancy of a trigonometric code such as that of his Fig. 2, but it is easy to evaluate the related concept of 'packing factor'. A closely packed binary code capable of detecting 3 errors and having characters of length ♬ digits would contain. V distinct characters, where C
2"
M =
+(7) + (2) (1)
+
-
16, It is true that there is no such close-packed code for n = the first known case being Golay's code for 23. But as an indication of the quality of packing, one can say that for 16 the formula gives M ~ 94. This means that in fact one 64) in a 3-error- could only include 6 bits of information (M correcting code of total length 16 digits, and the value of 5 bits obtained in the orthogonal code does not seem too unreasonable. Another packing consideration is that an e-error-correcting code requires only a minimum distance of 2e + 1 digits between characters. A 7-element orthogonal code satisfies this require. ment for 3 errors but has only 14 characters. It is now apparent that the packing factor of orthogonal codes decreases rapidly as the length is increased. As an example, Golay's 23-digit code contains 12 information digits and therefore 4096 characters; and it can correct 3 errors. A 23-digit orthogonal code would contain only 46 characters and be capable of correcting 5 errors, since the minimum distance between characters would be at least 11 digits. According to the formula given above, a 23-digit code capable of correcting 5 errors should include 190 characters compared with the 46 of a 23-digit orthogonal code.
REFERENCES
(A) Gabor, D.: 'Communication Theory and Physics, Philoso-
phical Magazine, 1950, 41, p. 1161.
(B) BELL, D. A.: 'Information Theory and its Engineering
Applications', 2nd edition (Pitman, 1956).
(C) Hamming, R. W.: 'Error Detection and Error Correction
Codes', Bell System Technical Journal, 1950, 29, p. 147. (D) GOLAY, M. J. E.: 'Notes on Digital Coding', Proceedings of
the Institute of Radio Engineers, 1949, 17, p. 697.
• HARMUTH, M. F.: Monograph No. 349 E, March, 1960 (pon 187 C, p. 242).
at IES 1960
Mr. D. S. Blacklock (conimunicated); Fig 4 of this monograph shows that for acceptable levels of error probability there would be a 3 dB saving in signal power if a 5-bit ‘ortho-trig' code were used instead of normal 5-digit binary. Thus, only 6-7 dB signals would be needed for 99% freedom from error, 8-0 dB for 99·9% and 9.0 dB for 99-99% reliability.
Dr. Harmuth, commenting on this comparison, says that the gain is obtained at the cost of increased bandwidth. In order to meet the U.S. teleprinter standard transmission rate (six characters per second) the bandwidth would need to be widened from 24 c/s to 60 c/s.
Since Gabor and others have shown that signal strength can always be exchanged for extra bandwidth, irrespective of coding. it would be interesting to compare this extra 36 c/s of band- width with what would be needed for a 3dB saving when 5-digit binary is retained. I should then be able to assess the value of an ortho-trig code when all other factors were kept constant.
Dr. Harmuth distinguishes the more practical ortho-trig coding from the simpler ortho-step coding where the steps or bits can be classified as informational and redundant, 5 and 11 respectively for the 32 combinations of normal ABC coding. He asserts that this classification does not hold for the ortho-trig coding, but in commenting on Fig. 4 he does in fact refer to the 5 bits of information in the ortho-trig code. Does this reference not imply that the remaining 11 crests and troughs of ↓ 2 sin 1670 or v/2 cos 16′′ are in fact redundant?
With ortho-trig coding there are two square-wave characters that may be transmitted as an alternating series for the correct phasing of the receiver. Am I right in thinking that with normal 5-digit binary coding an extra digital load of 50% has to be transmitted for the correct phasing of the receiver?
I am puzzled by the absence of any claim of superiority for the orthogonal coding, other than the clairn of practicality for the Are these trig variant in comparison with the siep variant. codes with 5 or more bits of information better than the 5-digit binary code of everyday business?
I can, however, make a comparison within Dr. Harmuth's ortho-trig field, as illustrated in the communication-system block diagram of Fig. 5, between our outmoded ABC coding and a "Tunish' coding of an English language so modified as to make more intensive use of the 'th', 'sh' and other phonetic charac- teristics which already distinguish English from other languages. Tunish would provide the 3dB saving along with a bandwidth saving (using fundamental frequencies of 6c/s and 12c/s only) and a terminal-equipment saving of 75% in the count of oscil- lators, multipliers, integrators and sampling switches; these gains would be accompanied by a 10% gain in transmission rate (counting words, not characters) and by many educational and other benefits. The Simpler Spelling Association of Lake Placid, N.Y., claim a 16·6% saving in writing and printing costs for their 40-symbol alphabet. Tunish makes a 50% claim and enlarges its view by including telegraphy. Thus, the 7000 bits needed to transmit Lincoln's 266 words at Gettysburg would be reduced to 3630 for Tunish language and writing. Every new age requires a revolution in human thinking and the advent of electronics in communication merits a new set of symbols along with greater semantic consistency in the phonetics of our English language.
[ 528 ]
F
05
-05
+15
TRANSMITER CLOCK
RECEIVER
CLOCK
05
15
2
Fig. 6.-Synchronization by orthogonal functions.
may alternatingly transmit characters 0 and 33 of Fig. 2. The resulting square wave of period 2r is shown in Fig. 5. A similar squa— wave, delayed by i̟r, is produced in the receiver. Multi- plk. a of the two square waves and integration of the product over the period r yields zero if transmitter and receiver are in phase. A phase difference of 7 between the two square waves makes the output of the integrator vary like the sawtooth function in Fig $. Transmitter and receiver are in phase at
Table 1
'CHARACTERS OF A TELETYPE ALPHABET
Number
Character
Fundamental frequency
แก 10 d
228-
Limit for tolerable synchronization
c/
2 millisec
sin 2x0
6
20 8
COS 20
6
20.8
sin 4w0
12
10.4
COS 418
12
10.4
sin 60
18
6.94
COS бTO
6.94
Bin Bro
24
5-21
cos 8x8
5.21
4.17
cos 10w6
4-17
sin 128
3-47
12
cos 12 #6
3.47
13
sin 14-0
2.97
14
cos 14x8
2.97
sin 1670
2.60
COS 16x8
2.60
-cos 160
2.60
-sin 16=0
2.60
-cos 140
42
2 97
20
-in 14:0
42
2.97
-cos 12-0
36
3.47
22
-sin 120
3-47
23
-cos 10:0
4-17
24
-sin 10:0
4.17
25
-cos 8=0
3.21
-sin To
5.21
-cos 60
6.94
-sin 6:0
6.94
-COS 4.70
1
10-4
-sin 4TA
4
20 8
2018
==287=****KRAB=B.
-cus 2.0 - sin 20
HARMUTH: ORTHOGONAL CODES
the points T/T = 0, −1, ±2, ... The deviation of the integrator voltage from zero may be used to correct the phase of the receiver. The points T/T =0, +2, −4, ... of the sawtooth function are stable if the receiver is retarded by a positive integrator voltage and advanced by a negative one.
If the communication system transmits six characters per second (U.S. teletype standard), the oscillators must provide the frequencies shown in Table I. This Table also lists the tolerable synchronization error which is obtained by considering that a sine wave advanced by one-eighth of its period cannot be distinguished from a cosine wave of the same frequency retarded by one-eighth of its period.
(5) ACKNOWLEDGMENT
The author wishes to express his appreciation to Messrs. R. Schwartz, J. Navarro, and A. Konheim of the Advanced Electronics Center of the General Electric Company, and to Messrs. G. Franco and G. Lachs of the Stromberg-Carlson Company for several helpful discussions.
(6) REFERENCES
(1) Hamming, R. W.: 'Error-Detecting and Error-Correcting
Codes', Bell System Technical Journal, 1950, 29, p. 147.
(2) SLEPIAN, P.: 'A Class of Binary Signalling Alphabets", ibid.,
1956, 35, p. 203.
(3) RED, 1. S.: 'A Class of Multiple-Error-Correcting Codes and the Decoding Scheme', Transactions of the Institute of Radio Engineers, 1954, IT-4, p. 38.
(4) GREEN, J. H., and San Soucie, R. L.: 'An Error-Correcting Encoder and Decoder of High Efficiency', Proceedings of the Institute of Radio Engineers, 1958, 46, p. 1741.
(5) TITCHMARSCH, E. C.: 'Theory of Fourier Integrals' (Oxford
University Press, London).
(6) Rice, S. O.: 'Mathematical Analysis of Random Noise', Bell System Technical Journal, 1944, 23, p. 282 and 1945, 24, p. 46.
(7) HAYTON, T., HUGHES, C. J., and SANDERS, R. L.: 'Telegraph Codes and Code Convertors', Proceedings 1.E.E., Paper No. 1585 R, November, 1953 (101, Part III, p. 137). (8) FLOOD, J. E.: 'Noise-Reducing Codes for Pulse-Code
Modulation', ibid., Monograph No. 291 R, February," 1958 (105 C, p. 391).
(7) APPENDIX
The statistical independence of W(a,) and p(1 + «) follows from that of œ and œ;; i*k, m + 1 -k. To prove the statistical independence of ∞ and «, it is sufficient to prove that the relation [(1 + on)xx,) 0 holds, since ∞ and œ, have a normal distribution according to eqn. (21).
[(1 + œ4)x,) = W-2< [*F2(1){F2(1) + B(1)]dt
+
F,(1){Fa(1) + g{1)]di'>
--[[E][i
- w=3[<[F(1~g{r^>dr">
(didi'
~<[*[Emringuent>]
· w~2 [<q> + [** [F(1)F,(1){B(1)g(1')) dtdi