596 IEEETRANSACTIONSONINFORMATIONTHEORY,SEPTEMRER 1975 111 PI I31 [41 I51 161 [71 181 191 4 REFERENCE-S B. S. Bosik, “The spectral density of a coded digital signal,” Bell Syst. Tech. J., vol. 51, pp. 921-932, Apr. 1972. R. C. Houts and T. A. Green, “Comparing bandwidth requirementsfor binary baseband signals,” IEEE Trans. Commun. (Corresp.), vol. COM-21, pp. 776-781, June 1973. P. D. Shaft, “Bandwidth compaction codes for communications,” IEEE Trans. Commun., vol. COM-21, pp. 687-695, June 1973. C. V. Freiman and A. D.,,Wyner, “Optimum block codes for noiseless iingpt-restrtcted channels, Inform. Conrr., vol. 7, pp. 398415, Sept. W. H. Kautz, “Fibonacci codes for synchronization control,” IEEE Trans. Inform; Theory, vol. IT-l 1, pp. 284-292, Apr. 1965. D. T. Tane and L. R. Bahl. “Block codes for a class of constrained noiseless channels,” Inform. ‘Contr., vol. 17, pp. 435-461, Dec. 1970. P. A. Franaszek, “Sequence-state methods for run-length-limited coding,” IBM J. Res. Develop., vol. 14, pp. 376-383, July 1970. E. Gorog, “Redundant alphabets with desirable frequency spectrum properties,” IBM J. Res. Develop., vol. 12, pp. 234-241, May 1968. H. Kobayashi, “Schemes for reduction of intersymbol interference in data transmission systems,” IBM J. Res. Develop., vol. 14, pp. 343-353, July 1970. - An Upper Bound Associated with Errors in Gray Code STEPHAN R. CAVIOR Abstract--Suppose 0 I i, j I 2” - 1. We prove that, if i, j are encoded as binary Gray codewords whose Hamming distance is M 2 1, then Ii - jl < 2” - 2m/3. In [I], Yuen finds a lower bound on the signal error that produces an m-bit error in its Gray codeword. Denoting the Gray codeword for i by g(i), he proves that if the Hamming distance between g(i) and g(j) is m 2 1, then (i - j] > 2m/3. The object of this correspondence is to establish the related upper bound. Theorem: Suppose 0 5 i, j I 2” - 1. If the Hamming distance between g(i) and g(j) is m, m 2 1, then Ii - jl < 2” - 2”‘/3. Proof: Suppose that, in binary notation, i = (i,,-1 . . . i&, j = CL1 . . . jo)2, and that we define I = (I,,-, . . . I&, where 1, G ik + jk (mod 2), k = 0, l,..., n - 1. In [I], Yuen proves that 1 is an integer whose Gray codeword g(l) has weight m; moreover, he notes that if the m ones in g(l) occur in positions k, < k, < +. . < k,,,, then lk = 1 for k,,,-, c k s k,,,, lk = 0 for kmw2 < k I k,,,- i, and so forth. This is a consequence of the fact that for any integer j = (jnF1 . . . j,), encoded in Gray code as g(j) = (g,‘- i . . . g&, n-1 is = Lzs si' (mod 3, s = 0, 1,. * *, n - 1. See [2] for a proof of this formula. Without loss of generality, assume i > j. By the definition of 1, i - j is maximized, under the constraint of a distance m between g(i) and g(j), if i = 1 and j = 0. Furthermore, 1 is maximized if lk = 1, for m - 1 I k I n - 1, and if the bits l,,,-2rl,,,-3r~ . atlO alternate between 0 and 1 beginning with 0. For example, when m = 5, I = (1 . . . 10101); when m = 6,l = (1 . . . 101010). In general, if m = 2t + 1, we have 1 = 2” - 1 - A, where A = z1 + z3 + . . . + pvi, Manuscript received October 24, 1974. The author is with the Department of Mathematics, State University of New York at Buffalo, Amherst, N.Y. 14226. andifm=2t+2,wehave1=2”-1-B,where B = 2O + 2’ + . . . + 22f. Note that A + B = 22cf1 - 1 and b = 1 + 2A. Thus, p+1 - 2 22t+1 A= = 3 F-1 3 and it follows that p+2 - 1 22t+2 B= = 3 L-1 3 * ([xl denotes the greatest integer not exceeding x). Consequently, for all m 2 1, we have 1 = 2” - 1 - [2m/3] < 2” - 2”‘/3. REFERENCES 111 C. K. Yuen, “The separability of Gray code,” IEEE Trans. Inform. Theory (Corresp.), vol: IT-20, p. 668, Sept. 1974. 121 -. “Comments on ‘correction of errors in multilevel Grav coded --. data’,” I&EE Trans. Inform. Theory, vol. IT-20, pp. 283-284, Mar. 1974. _ A Sequential Approach to Heart-Beat Interval Classification ERNEST T. TSUI AND EUGENE WONG, FELLOW, IEEE Abstract-Application of sequential testing to a Markovian model of cardiac rhythm intervals is investigated. An approximate expression for the expected number of observations is obtained for Wald’s sequential test under dependent sampling. The interclass separability of three selected cardiac rhythms is then determined, and the results are used to evaluate the feasibility of an on-line implementation of a sequential classification procedure in a coronary care ward. I. IN-I~zoDuC~~N Because of developments in preventive therapy in cardiac intensive care wards, the problem of obtaining reliable detection of certain specific types of abnormalities in rhythm that fre- quently prelude serious arrhythmias has recently received much attention. Romhilt et al. in a recent survey [l] have indicated that the present methods of using high and low rate alarms, one minute electrocardiogram (ECG) printouts every hour, and continuous multibed supervision by skilled personnel, though very reliable in the detection of serious fatal arrhythmias, are unreliable in the detection of the premonitory rhythm changes. They cite delays of several hours in the detection of premonitory rhythms such as premature atria1 contractions, premature ventricular contractions, and various ventricular arrhythmias. Thus there appears to be a need for an economical on-line automated system or subsystem that can detect certain rhythm changes reliably. Such a system can be realized only if the num- ber of features extracted as well as the number of pattern classes considered can be minimized. Recently, a hardware monitoring system with artifact rejection and physician-controlled param- eters has been used by Dell’osso [2]. Several investigators [3]-[5] have proposed (see Fig. 1) that the extraction of only the R-wave interval feature not only can Manuscript receivedJune 14, 1974; revised October 11, 1974.This work was supported by the Army Research Office, Durham, under Contract DAHCO4-74-GO087. The authors are with the Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, Calif. 94720.