Comments on and extensions of wolf's signal-to-channel noise formulas for delta-modulated systems

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 131 CHEEYSHEV PULSE, ~=0.2T (FAST FADING) CHEEYSHEV PULSE, (SLOW FADING) y=om 0 5 10 15 20 25 30 35 MEAN SIGNAL-TO-NOISE RATIO, P IN dB. Figure 3. Performance of Rayleigh Fast-Fading Channel with Synchronization Errors. CONCLUSION It is to be expected that the effects of intersymbol interference on fast-fading.channels would be more significant than on slowly fading channels. This hypothesis is clearly supported by the data shown in Figures 2 and 3, particularly in the latter case where the intersymbol interference is enhanced by a large synchronization error. When the signal-to-noise ratio is low, the intersymbol interference has little effect on the channel performance, but as the mean signal-to-noise exceeds 10 dB the ratio Pe,/Pe begins to increase rapidly. For mean signal-to-noise ratios in excess of 25 dB, P,,/P, approaches unity, i.e., the total biterror probability is almost entirely due to intersymbol interference. Thus the effect of the intersymbo1 interference on the fast-fading channel is to introduce an irreducible error probability that limits performance at high signal-to-noise ratios. This irreducible error probability is 7.38 X for the Gaussian pulse with no synchronization error. For the Chebyshev pulse it is 3.39 X with no synchronization error. While reliable communication over slowly fading channels can be obtained with large mean signal-to-noise ratios or diversity techniques, the data presented herein clearly indicate that intersymbol interference presents a very serious problem in Rayleigh fast-fading channels. Application of equalization techniques should also be considered to combat the performance degradation due to intersymbol interference in these cases. REFERENCES [I] E. Y. Ho and Y. S. Yeh. "A new approach for evaluating the error probability in the presence of intersymbol interferences and additive Gaussian noise," BeNSyst. Tech. J., Vol. 49, pp. 2249-2266, 1970. [2] 0. Shimbo and M. 1. Celebiler, "The probability of error due to intersymbol interference and Gaussian noise in digital communication systems," IEEE Truns. Commun. Technol.. Vol. COM-19, pp. 113119, April, 1971. 13) S. Benedetto. G. De Vincentis and A. Luvison, "Error probability in the presence of intersymbol interference and additive noise for multilevel digital signals," IEEE Trans. Commun., Vol. COM-21. pp. 181190. March 1973. [4] James C. Vanelli and Nazmi M. Shehadeh, "Computation of bit error probability using the trapezoidal integration rule," IEEE Trans. Commun.. Vol. COM-22, pp. 331-334, March 1974. [SI I. S. Gradshteyn and I. W. Ryzhik, Table oflntegruls. Series and Producrs, New York: Academic Press, 1965. [6] Howard H. Ma, "The performance of Rayleigh fast-fading channels with intersymbol interference and additive Gaussian noise," Master Thesis, Dept. of Elec. Eng., University of Houston, May 1978. [7] S. D. Poisson, "Memoire sur le Calcul numerigue des integrales defines." Mem. Acad. Sci. Ins?. Fr.. Vol. 6, pp. 57 1-602. 1823. Comments on and Extensions of Wolf's Signal-to-Channel Noise Formulas for Delta-Modulated Systems ANIBAL R. FIGUEIRAS-VIDAL, MEMBER, IEEE, JOSE B. MARIRO-ACEBAL, AND MIGUEL A. LAGUNAS-HERNANDEZ, MEMBER, IEEE AbsfmcGThe channel noise effects on linear delta modulation (LDM) systems have not yet been adequately analyzed. This paper presents a new and general formulation of these effects, based on the theoretical work by Wolf [I]. A comparative discussion of our formulas with previous results is also included. Finally, the application of our methods and the validity of our comments are illustrated by some numerical examples. I. INTRODUCTION This paper gives a method for calculating the signal-to-channe1 noise power ratio in a linear delta modulation (LDM) system. Although LDM systems do not offer a valid alternative to PCM for applications requiring a wide dynamic range [ 21, the technique described can be applied to other robust DM methods, especially digitally syllabic-companded delta modulation (DSCDM) systems. Paper approved by the Editor for Data Communication Systems of the IEEE Communications Society for publication without oral presentation. Manuscript received April 21, 1978; revised June 21, 1979. A. R. Figueiras-Vidal is with ETSI Telecomunicacion, Ciudad Universitaria, Madrid, Spain. J. B. Mariilo-Acebal and M. A. Lagunas-Hernandez are with ETSI Telecomunicacion, Jorge Girona Salgado, Barcelona, Spain. 0090-6778/80/0100-0131$00.75 0 1980 IEEE 132 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 In many cases, the channel noise effect is negligible with res(t) + n(t) spect to the quantization noise (granular and excess slope errors), but, in some applications, where the quantization noise is made relatively negligible in an otherwise noisy channel, this effect can be a critical quality parameter. Even in cases where the two noise components are of equal importance, one can calculate the channel noise effects by the techniques described herein. The quality criterion employed in this paper is the classical mean-squared error measure. Although in most applications of voice or image transmission subjective criteria may be more appropriate, the mean-squared expressions are a first-order indicator. Furthermore, they can be modified easily into a weighted mean-squared error criterion, which is in closer agreement with subjective evaluations. Finally, it should be pointed out that extensions of the given formulation are easily obtained for those cases in which the independent channel error model cannot be used (e.g., in commercial telephony) in the same way that Wolf indicated [ 11 for the Gilbert burst-noise model. The only modification necessary would be in the digital autocorrelation function E(ni ni+m) (see Section 11). " LINEAR DELTA DEMODULATOR Fig. 1: Linear delta-modulation receiver. and the power spectral densities: Here Rpp(t) and SDp(f) are the autocorrelation function and the energy spectral density of p(t), respectively. Then, the signal and noise powers at the output can be calculated with the help of the following formulas: 11. SIGNAL AND CHANNEL NOISE POWERS 1 &O) = 5 E(bibi+m)[Rpp(t) * Rhh(t)l t=mT Let bi indicate the ith transmitted symbol, where bj = +I. mZ-00 and ni the channel noise effect on this svmbol. where n.. = 0 The receiver detects and regenerates the incoming channel signal; if p(t) is the basic shape of the regenerated pulses, we exp (j2nmTf) df; will have, at the input to the LD demodulator, 00 and a noise term: n(t) = nip(t - iT) (2) j=-m where T1 is the symbol rate. The LD demodulator can be considered a linear system, having an equivalent impulse response h(t) and a transfer function H(f). Figure 1 shows the general situation just described. Since the output of the LD demodulator is a continuous process whose power level does not depend on the time origin, we can randomize the reference time for the pulses 131, obtaining the signal and noise terms: exp (j2nmTf) df; (10) where * indicates convolution, and Rhh(t) is the autocorrelation function of the impulse response of the LD demodulator. We have thus found the signal-to-channel noise formula: m m n(t)= 2 nip(t+ 8 -iT) where 0 is a random variable, uniformly distributed over [0, m TI . These signal and noise terms are stationary processes, with I(mT) & sp,(f) I H(f) l2 exp (i2nmTf) df. (12) respective autocorrelation functions [ 31 : 1This signal-to-channel noise formula is absolutely general R,s(t) = - x E(bibi+m)Rpp(t + mT) T m=-m 1T m=-- effects, but we will restrict our discussion to direct transmission. (5) for LDM systems. It can also be extended to adaptive (variable step) DM systems, using an appropriate redefinition of the signal and noise terms, although the digital source model will Rnn(t) = - 2 E(nini+m)Rpp(t + mT) (6) have to be changed. It is also possible to consider line coding IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 133 I I 1 1 I , I ~ -. Fig. 2: Markov system model. 111. WOLF'S MODEL FOR INDEPENDENT CHANNEL ERRORS The only problem remaining to be solved is the evaluation of E(bibi+,) and E(nini+,). Wolf's model [ 1 ] is appropriate for evaluating these mathematical expectations assuming independent channel errors. This model consist of two synchronized Markov chains for the digitized source and the channel errors; Figure 2 shows the model. The transition probabilities PT are equal, since we assume symmetry in this regard. Only PT is required for the evaluation of the digital autocorrelation functions, and we can obtain it from more structured (and realistic) Markov models for the quantized signal. We derive the signal-to-channel noise formula in Appendix I. The obtained result is: the analog message; we will denote the corresponding transition probability by PT'. Wolf's formula is exact when PT = 0.5, i.e., when P(so)/ P(no) = 1/4Pe, but it will be more and more inaccurate as PT approaches zero or unity. This appears to be the main reason for the progressive separation of Wolf's theoretical curve from Braun's experimental values [ 1 ] : the fitting for low Pe implied deviation for high Pe values. If PT approaches unity, P(so) will increase appreciably in practice with respect to the value obtained assuming sin2 nTf = 0, due to the term (1 - 2PT) sin2 nTf in the denominator of (El), considering that, in practice, the effective bandwidth corresponding to H(f) is less than 1/T. P(no) has a small variation because of the presence of the factor P, in the second term of (1.4). Consequently, P(so)/P(no) exceeds Wolf's value. Just the contrary occurs when PT approaches zero. The variations are larger when P, is small, i.e., in "good" channel cases. It is easy to see that our formulation gives P(so)/P(no) = 0 when PT = 0 or PT = 1 (signal absent), but it also implies that P(so)/P(no) # 0 when P, = 1/2. This incorrect result is due to the definition selected for signal-to-channel noise ratio, as Wolf indicated. Alternative definitions chosen in order to solve this problem present other difficulties [ 1 ] . Nevertheless, the cases in which P, % 1/2 correspond to transmissions over unusable channels (having a capacity near zero), and the previous P(so)/P(no) formulas are decreasing functions of P, which indicate the system performance in all practical situations. In practical systems, the final step in the LD demodulator is a bandpass filter, with lower and upper cutoff frequencies fcl, fc2, respectively, and 1/T > fc2. A first approximation would be to assume sin2 nTfe 0 in the effective band of integration. Then: and IV. EXAMPLES When the information consists of a voice signal and the system is approximately optimized with respect to granular and excess slope noises, Kikkert's simulation results [4] allow us to conclude that P, will not be very far from 0.5. In this case, the application of Wolf's formula will be acceptable. This remains true as long as 1 /Tis large enough to maintain sin2 nTfX 0 in the integration band. Nevertheless, we will present more general calculatjons to illustrate the departure from Wolf's results. Let us consider a final ideal bandpass filter having cutoff frequencies fcl = 300 Hz, fc2 = 3400 Hz, let us also assume T = 1/5600 symbols/s, and: P(0 = v wm (1 7) where: This is Wolf's formula. Note that Wolf did not consider the rqx)= 1, if Ix I< 1/2 final bandpass filter. We note that the approximation is ac0, if I x I > 1/2. ceptable when 1/T is large with respect tof,2. If we accept the approximation, the resulting formula will apply independent From the above expression of p(t): of the shape of the regenerated pulses. Wolf assumed a perfect integrator, but his result is also useful for leaky and double inspp(n = pp sinc2 ~f tegration systems and for delta-sigma systems. In the latter case, PT would correspond to a model of the digitized integrawhere: ted message signal. The same model previously indicated would apply to the process resulting from the integration of ~inc X 4 sin nx/nX. 134 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 Fig. 3: Perfect integration linear delta demodulator (T : delay; the fin? block is an ideal bandpass filter). Figure 3 shows a circuit that can be used as a perfect integration demodulator when p(t) is a full rectangular pulse. Its power transfer function may be written: where fc = (fcl -t fcz)/2 is the central frequency, and I, = fcz - fcl is the bandwidth of the final bandpass filter. Table I presents numerical results from our formulation, showing the small differences with respect to Wolf's values (in brackets) if P, is not very different from 0.5, and the increasing differences when PT goes to zero or 1. If a single RC (leaky) integrator is used, we will have: (22) where f3 is the cutoff frequency of the RC filter. Table 11 shows numerical values corresponding to a typical f3 equal to 150 Hz. If'PT = 0.5 and we put sinc2 Tf % 1 in [fcl,fcz], we will obtain Johnson's formula for P(n0) [6] [71. A double integrator is shown in Figure 4. Typical parameters allow us to write: TABLE I1 SIGNAL-TO-CHANNEL NOISE POWER RATIOS (IN dB), RC INTEGRATION '+, PT la' lo3 105 107 0.05 11.57 34.78 54.02 X02 0.20 6.73 29.80 ~BI 69.81 0.35 -1.69 17.90 37.97 57.97 0.60 1.51 2.30 41.30 61.30 0.65 3.98 23.96 43.90 63.98 0.50 620 26.63 4663 66.63 0.95 -8.36 n.21 3.21 51.21 1 I. Fig. 4: Double integrator. TABLE 111 SIGNAL-TO-CHANNEL NOISE POWER RATIOS (IN dB), DOUBLE INTEGRATION XI lo3 10-1 0.05 -0.37 11.20 3120 51.20 0.95 -1.70 17.97 3.97 57.97 0.00 1.50 21.30 41.30 61.30 0.65 3.98 23.98 43.98 63.98 0.50 6.30 26.65 46.65 66.65 0.35 8.80 29.90 49.4 69.91 0.20 11.83 3540 55.45 7545 where fl = 1/2nR,C1(1 + C2lCl) (24) fz = 1/2nR&z[ 1 -RZCz(l + R1/R2)'/4R1C1If (25) Table 111 presents numerical results corresponding to a typical fz equal to 1 kHz. If PT = 0.5 and we approximate sincz Tf by unity, our formulation will give an already known result for P(n0) [21: P(n 0) k 8pe vzfl T{ l/fc 1 - 1 /fc 2 - [tan-l(f,z/fz)- tan-lCrcl/fdl/f~). (26) In the case of delta-sigma modulation, the results obtained will not be directly comparable with the previous ones, since IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 135 TABLE IV SIGNAL-TOCHANNEL NOISE POWER RATIOS (IN dB), DELTA-SIGMA MODULATION PT Pe los 10" 104 to7 0.05 0.50 6.20 2653 4653 6653 0.35 8.36 29.28 4929 6929 0.20 9.88 31.54 51.56 7.56 1.53 2l.33 41.33 61.33 0.65 3.98 23.98 43.98 63.98 0.80 -1.65 18.02 38.01 58.a /* 0.95 -8.32 1125 3125 51.25 4) By properly redefining signal and noise terms, the method can be extended to coded LDM systems and to other practical DM systems (adaptive delta modulation, ADM, and, par&ularly, digitally syllabic companded delta modulation, DSCDM). We are obtaining results along these lines at the present time. APPENDIX I From the model, it is easy to obtain [ 1 ] : E(bibi+,) = (1 - 2PT)lm1 (1.1) PT' # PT in a general case. The receiver has: where P, is the error probability. Then, rearranging the power l.V11'=.(b)+fl(~). f -fc f +f, expressions, we can write: Some signal-to-channel noise ratio values are included in PT' = 0.5 and sinc2 TfX 1 in [fcl, fc2] will lead to Johnm=-m The case of linear delta-sigma modulation can be solved by pT(l -PT) Table IV. son's formula for P(no) [ 51 [ 61. calculating the corresponding Z(mT) in the time domain. Sineintegral functions appear in the resulting expression, which is difficult to manipulate and to interpret. Making sinc2 Tf 1, we obtain: m = x (1 - 2PT)lm I exp (j2nm Tf> = - - PT ' + (1 - 2PT) sin' nTf (1.3) and the resulting P(so)/P(no) can be expressed with the help of (1 1). The same approximate method can be applied in other cases, but the integralsl(mT) have to be calculated numerically. PT( 1 - 2PT) - (1 - PT) sin n PT2 + (1 - ;?PT) sin2 ' "1 (1.4) V. SUMMARY AND CONCLUSIONS where the prime sign in the summation indicates exclusion of The general formulation of the signal-tochannel noise power ratio in (direct) LDM systems has been presented and compared with earlier results. The main conclusions are: the zero index term. The ratio of the integrals of these functions multiplied by SppO I H(f) 1' will be the signal-tochanne1 noise ratio, formula (13) in the main text. 1) A slight modification of Wolf's work allows us to extend it to all LDM systems; 2) Wolf's formula is accurate enough in practical systems 111 having PT not very different from 0.5, but it fails when PT goes to zero or PT goes to 1, and especially when P, is small or 121 when Tis large. Some numerical results are comparatively presented. Additional comments are: 3) The method can be easily extended to correlated channel 151 [31 141 errors (e.g., to the case of burst errors accordingly to the Gilbert model [7], considered by Wolf), the only modification being the alteration of E(nini+m) values (not very problematic 171 161 in the case of the Gilbert model); REFERENCES WOLF, J. K.: Effects of Channel Errors on Delta Modulation. IEEE Trans. on Comm., vol. COM-14, pp. 2-7; Febr. 1966. BETTS, J. A,: Signul Processing, Modulation and Noise. London: Hodder and Stoughtoni 1975. FRANKS, L. E.: Teoria de IaSeiTal. Barcelona: Reverte; 1975. KIKKERT, C. J.: Digital Companding Techniques. IEEE Trans. on Comm.. vol. COM-22, pp. 75-78; Jan. 1974. JOHNSON, F. B.: Calculating %Ita Modulator Performance. IEEE Trans. on Audio and Electroacoust., vol. AU-16, pp. 121-129; Mar. 1%8. STEELE, R.: DeltaModulation Systems. London: Pentech Press; 1975. GILBERT, E. N.: Capacity of a Burst Noise Channel. Bell Sys. Tech. Jour., vol. 39, pp. 1253-1265; 1960.