644 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1976

ACKNOWLEDGMENT

The author would like to thank Mrs. D. Vitello for the computer graphs and Mr. A. R. McCormick for his assistance with the experiments.

REFERENCES [ 11 S. C. Gupta, “Phase-locked loops,” Roc. ZEEE, vol. 63, pp. 291-

306. Feb. 1975. [2] A. Y. Viterbi, Principles of Coherent Communication. New

York: McGraw-Hill. 131 D. Richman, “Color-carrier reference phase synchronization

accuracy in NTSC color television,” Proc. IRE, vol. 42, pp. 106- 133, Jan. 1954.

[4] A. Acampora and A. Newton, “Use of phase subtraction to ex- tend the range of a phased locked demodulator,” RCA Rev., p. 577, Dec. 1966.

[SI A. W. Lahti and T. E. Beling, “Phase-locked-loop coherent FM detector with synchronized reference oscillator,” U.S. Patent 3 189 825, Tied Mar. 29,1962.

[6] K. Murakami, “A new phase lock demodulator with injection locking,” Electron. Commun., vol. 52, p. 119, Feb. 1969.

[7] B. N. Biswas, “Combination injection locking with indirect synchronization technique,” IEEE Trans. Commun. Technoi.,

[8] B. N. Biswas and P. Banerjee, “Range extension of a phase- locked loop,” IEEE Trans. Cornmun. Technoi., vol. COM-21,

VOI. COM-19, pp. 574-576, Aug. 1971.

pp. 293-296, Apr. 1973.

191 J. F. Oberst, “Generalized phase comparators for improved phase-locked loop acquisition,” ZEEE Trans. Commun. Technol., vol. COM-19, pp. 1142-1148, Dec. 1971.

[ lo] L. J. Greenstein, “Phase-locked loop pull-in frequency,” IEEE Trans. Cornmun., vol. COM-22, pp. 1005-1013, Aug. 1974.

[ l l ] R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE, vol. 34, pp. 351-357, June 1946; also, Proc. IEEE, vol.

[ 121 M. Armand, “On the output spectra of unlocked driven oscilla- tors,”Proc. IEEE, vol. 57, pp. 798-799, May 1969.

[ 131 E. Roza, “Analysis of phase-locked timing extraction circuits for pulse code transmission,” IEEE Trans. Commun., vol. COM- 22, pp. 1236-1249, Sept. 1974.

61, pp. 1380-1385, Oct. 1973.

*

of a group doing res€

Peter K. Runge was born ‘ i n Bremen, West Germany, on May 13, 1939. He received the Dip1.-Ing. degree in 1963 and. the Dr.-Ing. de- gree in 1967 in electrical engineering from the Technical University Braunschweig, Braun- schweig, West Germany.

Since 1967 he has been with Bell Labora- tories, Holmdel, NJ. He was engaged in research on gas and organic dye lasers and since 1972 has been concerned with research on fiber optic transmission systems. He is presently supervisor

:arch work on fiber optic systems.

Concise Papers

Symbol Error Probabilities for M-ary CPFSK: Coherent and Noncoherent Detection

THOMAS A. SCHONHOFF

Absfruct-Continuous-phase frequency shift keying (CPFSK) is discussed and theoretical predictions for symbol error probabilities are derived, where the memory inherent in the phase continuity is used to improve performance. Previously known results concluded that binary CPFSK can outperform coherently detected PSK at high SNR New results presented here show that Mary CPFSK outperforms more tranditionally used Mary modulation systems. Specifically, coherently detected quaternary CPFSK with a five-symbol interval decision can outperform coherent QPSK by 3.5 dB, and octal coherent CPFSK with a three- symbol decision can outperform octal orthogonal signaling by 2.6 dB at high SNR. Results for coherently detected and noncoherently detected CPFSK are derived. These performance improvements are

Paper approved by the Editor for Data Communication Systems of the IEEE Communications Society for publication after presenta- tion at the International Conference on Communications, San Fran- cisco, CA, June 16-18, 1975. Manuscript received May 12, 1975; revised November 18, 1975.

The author is with the Eastern Division of GTE Sylvania, Needham Heights, MA 02194.

estimates derived from symbol error probability upper bounds. Monte Carlo simulation was performed which then verified the results.

I . INTRODUCTION

Continuous-phase frequency shift keying (CPFSK) refers to an FSK modulation scheme wherein the phase is con- strained to be continuous during a symbol transition. This constraint of continuous phase affects the signal in two important ways.

1) Transient effects are lessened at the symbol transitions, thereby offering spectral bandwidth advantages [ 1 ] -[ 31.

2) Memory, imposed upon the waveform by continuous phase transitions, improves performance by providing for the use of several symbols to make a decision rather than the more common approach of making independent symbol-by-symbol decisions.

This modulation has been investigated in the case of binary signaling [41-[61, and if the parameters of the modulation and demoduation are chosen correctly [ 51 , [6] , b inary CPFSK offers advantages in EB/No over coherent antipodal PSK, where EB is the energy per bit and NO is the one-sided noise-power spectral density. From [6], in the case of co- herent detection, binary CPFSK has up to a 1.1-dB advantage

CONCISE PAPERS 645

PHASE

6nh

5nh

47 h

3w h

2nh

nh

/ The possible phase trajectories using the phase term

dinh( t - (i - l )T) /T + nh Zjj=idj + Cp are shown in Fig. 1 for quaternary modulation and q5 = 0. For quaternary modula- tion, d i = + 1 , +3. As can be seen, the phase trajectories have a tree-like quality. Furthermore, if h is a rational number the tree will eventually fold upon itself modulo 2n, producing a trellis-like depiction. This similarity with convolutional codes has been recognized previously [ 41 , [ 71 .

The procedure to improve performance is to observe the received signal for n symbol intervals, and then make a deci- sion on one of the n symbols. In the case of coherent detec- tion, the decision is made on the first symbol, and for non- coherent detection, an odd number of symbols is observed and a decision made on the middle one.

-nh

-2n h

-3mh

-4% h

-5nh

-6nh

Fig. 1. Phase trajectories for quaternary CPFSK.

over PSK, and in the case of noncoherent detection it has a 0.3-dB advantage over coherently detected binary PSK.

This paper extends the results to Wary modulation, with quaternary and octal modulations being emphasized. Both theoretical and simulation results are presented.

No attempt was made to investigate bandwidth expansion versus performance improvement. Such an effort has been investigated in [ 121.

11. MATHEMATICAL BACKGROUND

Much of the development in this and succeeding sections

We can model the received M-ary CPFSK signal during parallels the theoretical binary development of [ 61 .

the ith symbol as

r ( t ) = e cos ( W C t +- d inh( t - (i - 1)T)

T

111. OPTIMUM ML COHERENT RECEIVER

In determining the optimum receiver structure, we shall use a shorthand notation developed in [6] for the received signal

where d l is the first symbol, and D k is the ( n - I)-tuple D k = { d z , -, d n } . For coherent detection, the starting phase (J is assumed known, and hence will be assumed to be 0 with no loss in generality.

The optimum receiver will determine the M likelihood parameters [ 81

where

Es energy per symbol interval T ; W, carrier radian frequency; di digital information signal (*1,+3, -, *(M - 1)); M always even; h deviation ratio; Cp starting phase; n( t ) additive white Gaussian noise process.

In (1) it is understood that for i = 1, EjEi = 0. The signal is constructed such that two ad.jacent frequencies in the M- ary set are separated by h/T Hz.

where f ( D k ) is the discrete probability density function (PDF) of D k . The integral S A D k ) d D k is really the (n - 1 ) - fold integral

I/ .*- f ( d 2 If(da 1 f(d, 1 dd2 dd3 *.. d d ,

and each of the discrete PDF f ( d , ) is a series of Dirac-6 functions

Realizing that D k can have = m different possibilities, the integrals over D can be evaluated t o give the M parameters

646 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1976

m r( t )s( t , 1,Dj) d t

j = 1

Z2 = 2 exp (' I n T r ( t ) s ( t , -l ,Dj) d t j= 1 No 0

/

The optimum receiver then makes a decision on d l depending on the largest of these parameters. The optimum coherent receiver is depicted in Fig. 2.

Unfortunately, it is not possible to analyze the perfor- mance of this receiver exactly. It is possible, however, t o determine bounds on the performance which are tight at high or low SNR. In this report, only high SNR bounds will be generated.

. .

I I I

OPTIMUM RECEIVER PICKS LARGEST

1,

IV. HIGH-SNR COHERENT BOUND

SUBOPTIMUM HIGH SNR RECEIVER PICKS LARGEST

x . X J

If we let

Fig. 2. Optimum and high-SNR suboptimum CPFSK coherent receivers.

then the M likelihood parameters can be written as N f u

and averaging over v gives

For large SNR,

Furthermore the probability that one Gaussian is greater than another Gaussian is from [ 101

where X I \ is the largest of the xhj . Furthermore, since exp ( ) is a monotonic function, xA is an equivalent parameter to investigate.

This implies that a suboptimum receiver as depicted in Fig. 2 should have good performance for high SNR. This receiver simply calculates all x h j , X = 1, -., M ; j = 1, ... , m and makes a decision on dl depending on the largest of these.

T o evaluate the performance of this receiver, first we note that all the x h j are Gaussian variables [8, sec. 4.21. We will use a union bound technique which is tight for high SNR.

Given d l = v and ( d 2 , -e, d,} = D j , the probability of an error is by the union bound

where

Equation (10) can be used to arrive at an estimate of the probability of error for M-ary CPFSK, but two other con- siderations should be made.

First, even though it was not explicitly shown, the cor- relations P N J , ~ ~ are functions of h [refer to (l), (2), and (12)] . Hence some investigation as to which h , if any, is best is in order. The next paragraph addresses this. Secondly, for even modest values of M and n, the number of terms which must be evaluated in (10) becomes prohibitive. For

N # V

Averaging over D j results in

CONCISE PAPERS 647

example, for quaternary modulation ( M = 4) and a five- symboi observation interval (n = s), the number of terms in (10) is 786 432, so clearly the bounding technique must be more closely investigated. Thi:s is done in Section V.

T:he value of h (the deviation ratio) which should be chosen is otwiously the one that minimizes the probability of error (IO). Some authors in the palst have alternately examined the mini.mum distance criterion, wherein a value of h is chosen which maximizes the minimum distance between pairs of signals in the signal set [ 51, [ 11 1. In this report, an attempt was made to find the value of h for which PrM(E) is mini- mized. This was done numerically by programming the PrM(E) bound with h as an input parameter, and then at a suitably high SNR (e.g., EB/No = 6 dB), finding that value of h for whicb PrM(E) was smallest.

For quaternary modulation, the optimum h for h = 2 is h = 1.75, and for n > 3 it is h = 0.8. For octal modulation, the optimum h for n > 2 is h := 0.879. For binary modulation, h = 0.715 is optimum as was already known [ 4 ] - [ 6 ] . It is interesting to compare these values of h with those which are obtained by maximizing the minimum distance. For binary modulation; the values of h are the same, and this is true also for octal modulation [ 11.1. For quaternary modulation however, the h which maximizes minimum distance is h = 0.85, whereas for minimum probability of error it is h = 0.8 [ 111.

V. REEXAMINATION OF THE UPPER BOUND

The four summations of (1 01) contain

( M - 1 ) m m ~ = M ~ ~ - I ( M .- ) terms.

As mentioned previously, even for modest values of M and n, this represents a prohibitively large number of terms. A significant savings in the cornputation can be achieved by recognizing that the correlations of many of the terms are identicai, thereby allowing the combination of terms without separate addition.

If 'we do a symbol-by-symbol correiation of s(t,v,Di) and s(t,N,DJ), the first symbol correlation is

2 T 1PNJ,uj = --l cos a t cos c t d t

where

Nnh a = a c + -

T

and

vnh

T c = w, +-

Since 'we are constraining v f N, and assuming w, > N n h / T and 0,: 3 vnh / T

lPN.J,uj = sin nh(N - u)

nh(N - v)

From (14) it is seen that the parameter of interest is the difference 6 1 = N - v, and it is clear that any N s and v's which hlave the same 6 will have the same IPNJ, uj. If we denote DJ = { d s ~ , d s ~ , --, d n J } and Dj = {dzj,d3i, *-, d, j} , the second symbol correlation is

2PNJ,uj ?[ cos (u,t -I- - t + Nnh d 2 JTh

T T

which is

Thus 2 P N J,?j is a function of s1 and 6 2 where 62= d 2 J - d2j. Following this line of thought, the nth symbol correla- tion is -

sin nh{G, + ( b - d ) } - sin nh(b - d )

n-1 d = nhu + nh d,.

I = 2

Since

and also any other sequence of symbols which has the same 61, 6 2 , -, 6, will have the same correlation. There are 2M - 1 values that each of the terms 6 2 , .**, 6, can have, and 2M - 2 values that 6 1 can have (61 # 0 because N # v). Thus only (2M - 2)(2M - correlations need be calculated rather than M 2 , - I ( M - 1) which would be required by a direct computation of (10). An example of the savings this affords is the case of M = 4, n = 5, for which a direct computation of ( I O ) would require 786 432 terms, whereas only 14 406 correlations are required.

All of the correlations of (18) are not equally likely, but they have some probability

Pr (&,,A,, *e., 6,) = Pr (6,) Pr (a2 ) ... Pr (6,) (19)

because each successive symbol is assumed independent. Since the original information symbols d have a discrete uniform PDF, the random variables 6 i have a discrete tri-

648 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1976

angular PDF except for 61 which has a conditional PDF (conditioned upon h1 # 0). Hence calculation of (19) is straightforward.

In (IO) the number of terms that must be determined is ( M - 1)Mm2, .and the number of these with ;a particular correlation p(61,62, -, 6,) is ( M - l)Mm?Pr(61.,62, e*., 6,). Averaging over the 6's and using (10) and (1 1) and the argu- ments discussed in this section

This technique required considerably less computation than the direct evaluation of (1 0).

VI. THEORETICAL COHERENT RESULTS

The results for binary modulation are already published [SI , [ 61. They are included here for the sake of complete- ness. The h used is h = 0.715. Using this value of h , bit error probabilities Pr, ( E ) are shown as a function of EB/No in Fig. 3. Also shown are the error curves for coherent anti- podal PSK and differentially coherent PSK. The coherent antipodal. PSK is the best performance possible when deci- sions are made over 1 bit. As can be seen from Fig. 4, binary CPFSK offers up to 1.1-dB improvement for five-symbol decisions with respect to coherent PSK.

For quaternary modulation, the deviation ratio is h = 1.75 for n = 2 and h = 0.8 for IZ 2 3 . The symbol error prob- ability baunds Pr4 ( E ) are shown in Fig. 4. Results for co- herent QPSK are also shown. The n = 2 case offers a 2.5-dB low-error-rate improvement over coherent QPSK, and a five- symbol interval decision offers a further 1 .O-dB improvement.

For octal modulation, the optimum h is 0.879 and the symbol error probability bound is shown in Fig. 5. As well as comparing the results with octal coherent PSK, an error curve bound is also included for octal coherent orthogonal signaling. This bound is [8, sec. 4.2.51

As can be seen, octal coherent CPFSK offers a 1.9-dB advantage over orthogonal signaling for n = 2 and a 2.6-dB advantage for Iz = 3 .

VII. OPTIMAL ML NONCOHERENT RECEIVER

For noncoherent detection, the initial phase 4 in (1) is assumed unknown with uniform PDF from 0 t o 2n. Again, we shall use a shorthand notation for the received signal as

r ( t ) = dt ,dn+l ,Ak,$) + n ( t ) . ( 2 2 )

We are observing the signal for 211 i- 1 symbols, n is an integer, and then making a decision on the middle symbol d,+l. A h is a 2n-tuple consisting of

so k can contain up to p values where p = M Z n . The optimum

lOLOG E/No 10 8

Fig. 3. Probability of bit error for binary coherent CPFSK, differentially coherent PSK, and coherent PSK.

/- CIPSK

/

10 LOGlO (EB/No)

Fig. 4. Probability of symbol error for quaternary coherent, CPFSK and QPSK.

CONCISE PAPERS 649

case gives the M likelihood parameters

COHERENT CPFSK fh =0.8791

2 4 b 8

10 LOGlO iEB/No)

Fig. 5 , Probability of symbol error for octal coherent CPFSK, PSK, and orthogonal signaling.

ML receiver generates the M lik'elihood parameters '

where !(A) is the discrete PDF of A and f(@) is the PDF of @. As can be seen, these likelihood. parameters are similar to the coherent likelihood parameters of (3 ) except for the added integral over Cp. The integral J a f ( A ) d A is really 2N-fold integral

The average over the phase gives the zeroth-order Bessel func- tion, so we can write

N even (28)

N odd

N even.

(29)

Z N k is a Rician statistical variable. The optimum receiver structure is shown in Fig. 6. It is

worth mentioning again that this receiver structure is optimum independent of SNR. Unfortunately, as in the case of coherent detection, the performance of this receiver can not be esti- mated. Hence in the next section, a high-SNR approximation is made and a resulting high-SNR receiver structure which is analyzable is developed.

VIII. HIGH-SNR NONCOHERENT BOUND

A suboptimum receiver structure which examines all the Z N k , N = 1, . . e , M ; k = 1, *-, p and then makes a decision on

650 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1976

1

DETECTOR . z21 . Io{Z/No ( ) }

e . e e

SUBOPTIMUM

RECEIVER PICKS HIGH SNR

LARGEST 'xi Fig. 6 . Optimum and high-SNR suboptimum CPFSK noncoherent receivers.

d n + l depending on the largest will perform well for high SNR. This can be seen from the facts that ZO( ) is a montonic func- tion, and also that for large SNR

where Z N A is the largest of the Z N k . The suboptimum non- coherent large SNR i s shown in Fig. 6.

To evaluate the performance of this receiver, we again use the union bound. Given that d,+l = K and Ai = { d l , d 2 , --, d n ~ d n + 2 , *** , d z n f l }, the probability of an error is bounded by

NPK

Averaging over A, results in

N # K

and averaging over K gives

OPTIMUM RECEIVER PICKS LARGEST 1,

where

and

(2n + l)E,/No is the SNR of the whole sequence. of (2n + 1) symbols and p ~ ~ , ~ j is the complex correlation between s ( t , N , A J , @ ) and s( t ,K,Aj ,@). Q( ,) is the Marcum (2 function defined as

and accurate computer integration technqiues are available for its evaluation. Using (32):(34), an upper bound on the proba- bility of error can be determined.

CONCISE PAPERS 65 1

10 LOGlO (EB/No)

Fig. ?. Probability of bit error for binary noncoherent CPFSK and . coherent PSK.

As in the case of coherent detection, an investigation of the h for which PrM ( E ) is smallest: is in order. The search for the best Iz was accomplished by evaluating PrM ( E ) at a suitable high SNR (EBINo = 6 dB was chosen), and finding the h , by trial and error, which results in the smallest probability of error. The noncoherent values of h are similar to the coherent and arl:

M = = 2 h = 0.715

M == 4 11 = 0.8

M = : 8 k = 0.879

All results which are presented in the next section were gen- erated using these values of h .

Again the evaluation of (32) requires prohibitively many calculations if M and n are not small. Fortunately, as in the coherent detection case, the number of computations can be significantly reduced by recognizing and identifying the Rician pairs which have the same complex correlation. The discussion of this computational simplification parallels the development of Sect.ion V and will be omitted.

IX. THEORETICAL NONCOHERENT RESULTS

Binary results have already been published [61 and are include(l here for completeness. The optimum h is 0.7 15, and bit error probabilities are displayed in Fig. 7. Also shown is the bit error probability curve for ;antipodal coherently detected PSK. As pointed out in [6], the rather surprising result that noncoherent CPFSK outperfornls coherent PSK is seen. The improvetment over PSK is up t o 0.5 dB in E B I N ~ at high SNR.

In Fig. 8, quaternary results are shown for noncoherent

FOR NONCOHERENT CPFSK, h =0.8

Fig. 8. Probability of symbol error for quaternary noncoherent CPFSK and coherent QPSK.

CPFSK with three- and five-symbol observation times and also coherent QPSK. Again, the noncoherently detected CPFSK achieves better performance than the PSK signal, with im- provements in EB fNo of u p t o 2.8 dB possible.

Finally, Fig.'9, octal results for three-symbol interval deci- sions are presented. Error probability curves are also shown for octal PSK and octal orthogonal signaling, both coherently detected. Again, the memory inherent in CPFSK offers sig- nificant improvement in EB/No for low error probability.

X; SIMULATION RESULTS

To verify that the bounds are indeed upper bounds, and to get an estimate as to how tight the bounds are, Monte Carlo simulations of both the coherent and noncoherent results were obtained. The simulations were of the high-SNR receivers rather than the optimal receivers because the high-SNR re- ceivers would more likely be implemented., being of lower cost.

The results for quaternary CPFSK with three-symbol interval decisions are shown in Fig. 10. As can be seen, the bounds are very tight indeed at high SNR, giving further cre- dence to the theoretical results.

XI. CONCLUSIONS

The theory and simulations assumed perfect carrier and symbol synchronization. Hence, the results must be considered as a baseline comparison to other modulation systems which are also usually derived assuming perfect carrier and symbol synchronization.

The question of degradation caused by nonperfect imple- mentation is an important one, but naturally it depends critically on the implementation, and this is a subject for future consideration.

652 IEEE TRANSACTIONS ON COMMUNICATIONS. JUNE 1976

l € \

POR NONCOHERENT CPFSK. h =0.87’)

NONCOHERENT CPFSK

I 2 4 b 8 10 I? 14

10 LOGlO (E$No)

Fig. 9. Probability of symbol error for octal noncoherent CPFSK, coherent PSK, and coherent orthogonal signaling.

10 LOGlO (FB/N0)

Fig. 10. Monte Carlo simulation results and theoretical bounds for quaternary CPFSK with three-symbol decisions.

It has been shown that the performance improvements of CPFSK which were described in 161 for binary modulation also extend to quaternary and octal. For quaternary modula- tion, improvements over ,coherent QPSK are possible up to 3.5 dB for coherent detection and 2.8 dB for noncoherent detection. In the case of octal modulation, improvements over coherently detected FSK are possible up to 2.6 dB for coherent CPFSK and 1 .‘2 dB for noncoherent CPFSK.

ACKNOWLEDGMENT

The author wishes to thank J . Meyn and 2. Huntoon who were of great assistance in the development of Section V. Dr. M. B. Luntz gave.very helpful advice when the simulation pro- grams which generated Fig. 10 were developed. Their help is greatly appreciated.

REFERENCES W. R. Bennett and S. 0. Rice, “Spectral density and autocorrela- tion functions associated with binary frequency-shift keying,” BSTJ, pp. 2355-2385, Sept. 1963. M. G . Pelchat, “The autocorrelation function and power spec- trum of PCM/FM with random binary modulating waveforms,” IEEE Trans. Space Electron. Telem., pp. 39-44, Mar. 1964. R. R. Anderson and J . Salz, “Spectra of digital FM,”BSTJ, vol.

R. de Buda, “Coherent demodulation of frequency shift keying with low deviation ratio,” IEEE Trans. Commun. (Concise Paper), pp. 429-435, June 1972. M. G. Pelchat, R. C. Davis, and M. B. Luntz, “Coherent demod- ulation of continuous phase binary FSK signals,” in 1!271 Int. Telemetering Con$ Conv. Rec. (Washington, DC), pp. 181- 190. W. P. Osborne and M. B. Luntz, “Coherent and noncoherent detection of CPFSK,” IEEE Trans. Commun., vol. COM-22,

G . D. Forney, Jr., ‘The Viterbi algorithm,” hoc. ZEEE, vol. 61, pp. 268-278, Mar. 1973. J . J . Stiffler, Theory of Synchronous Communications. Engle- wood Cliffs, NJ: Prentice-Hall, 1971. Schwaxtz, Bennett, and Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966, sec. 8.2 and 8.3. A. D. Whalen, Detection of Signals in Noise, New York: Academic, 1971, sec. 6.2. T. A. Schonhoff, “Symbol error probabilities for M-ary (coherent continuous phase frequency shift keying (CPFSK),” presented at the Int. Conf. Communications (San Francisco, CA, June 16- 18,1975) .

presented at Nat. Telecommun. Conf., New Orleans, L.A, 1975.

44, pp. 1165-1189, July/Aug. 1965.

pp. 1023-1036, Aug. 1974.

- , “Bandwidth vs. performance considerations for CPFSK,”

Algorithms for Delayed Encoding in Delta Modulation with Speech-Like Signals

JAN UDDENFELDT AND LARS H. ZETTERBERG, SENIOR MEMBER, IEEE

Absfruct-This concise paper is concerned with the problem of im- proved delta-coding by using delayed decision instead of bit-by-bit decision. It is found that delayed encoding allows a fairly general pre- dictor to be used without causing instability problems. Simulations

Paper approved by the Editor for Data Communication Systems of the IEEE Communications Society for publication without oral presentation. Manuscript received July 1, 1975; revised January 21, 1976.

The authors are with the Department of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden.