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. Figu e 3. Pe o mance o Rayleigh Fas -Fading Channel wi h Syn- ch oniza ion E o s. CONCLUSION I is o be expec ed ha he e ec s o in e symbol in e - e ence on as - ading.channels would be mo e signi ican han on slowly ading channels. This hypo hesis is clea ly suppo ed by he da a shown in Figu es 2 and 3, pa icula ly in he la e case whe e he in e symbol in e e ence is enhanced by a la ge synch oniza ion e o . When he signal- o-noise a io is low, he in e symbol in e e ence has li le e ec on he channel pe o mance, bu as he mean signal- o-noise exceeds 10 dB he a io Pe,/Pe begins o inc ease apidly. Fo mean signal- o-noise a ios in excess o 25 dB, P,,/P, app oaches uni y, i.e., he o al bi e o p obabili y is almos en i ely due o in e symbol in e e ence. Thus he e ec o he in e sym- bo1 in e e ence on he as - ading channel is o in oduce an i educible e o p obabili y ha limi s pe o mance a high signal- o-noise a ios. This i educible e o p obabili y is 7.38 X o he Gaussian pulse wi h no synch oniza ion e o . Fo he Chebyshe pulse i is 3.39 X wi h no synch oniza ion e o . While eliable communica ion o e slowly ading channels can be ob ained wi h la ge mean signal- o-noise a ios o di e si y echniques, he da a p esen ed he ein clea ly indi- ca e ha in e symbol in e e ence p esen s a e y se ious p oblem in Rayleigh as - ading channels. Applica ion o equaliza ion echniques should also be conside ed o comba he pe o mance deg ada ion due o in e symbol in e e ence in hese cases. REFERENCES [I] E. Y. Ho and Y. S. Yeh. "A new app oach o e alua ing he e o p obabili y in he p esence o in e symbol in e e ences and addi i e Gaussian noise," BeNSys . Tech. J., Vol. 49, pp. 2249-2266, 1970. [2] 0. Shimbo and M. 1. Celebile , "The p obabili y o e o due o in e symbol in e e ence and Gaussian noise in digi al communica ion sys ems," IEEE T uns. Commun. Technol.. Vol. COM-19, pp. 113- 119, Ap il, 1971. 13) S. Benede o. G. De Vincen is and A. Lu ison, "E o p obabili y in he p esence o in e symbol in e e ence and addi i e noise o mul ile el digi al signals," IEEE T ans. Commun., Vol. COM-21. pp. 181- 190. Ma ch 1973. [4] James C. Vanelli and Nazmi M. Shehadeh, "Compu a ion o bi e o p obabili y using he apezoidal in eg a ion ule," IEEE T ans. Commun.. Vol. COM-22, pp. 331-334, Ma ch 1974. [SI I. S. G adsh eyn and I. W. Ryzhik, Table o ln eg uls. Se ies and P oduc s, New Yo k: Academic P ess, 1965. [6] Howa d H. Ma, "The pe o mance o Rayleigh as - ading channels wi h in e symbol in e e ence and addi i e Gaussian noise," Mas e Thesis, Dep . o Elec. Eng., Uni e si y o Hous on, May 1978. [7] S. D. Poisson, "Memoi e su le Calcul nume igue des in eg ales de ines." Mem. Acad. Sci. Ins?. F .. Vol. 6, pp. 57 1-602. 1823. Commen s on and Ex ensions o Wol 's Signal- o-Channel Noise Fo mulas o Del a-Modula ed Sys ems ANIBAL R. FIGUEIRAS-VIDAL, MEMBER, IEEE, JOSE B. MARIRO-ACEBAL, AND MIGUEL A. LAGUNAS-HERNANDEZ, MEMBER, IEEE Abs mcGThe channel noise e ec s on linea del a modula ion (LDM) sys ems ha e no ye been adequa ely analyzed. This pape p esen s a new and gene al o mula ion o hese e ec s, based on he heo e ical wo k by Wol [I]. A compa a i e discussion o ou o mulas wi h p e ious esul s is also included. Finally, he applica ion o ou me hods and he alidi y o ou commen s a e illus a ed by some nume ical examples. I. INTRODUCTION This pape gi es a me hod o calcula ing he signal- o-chan- ne1 noise powe a io in a linea del a modula ion (LDM) sys- em. Al hough LDM sys ems do no o e a alid al e na i e o PCM o applica ions equi ing a wide dynamic ange [ 21, he echnique desc ibed can be applied o o he obus DM me h- ods, especially digi ally syllabic-companded del a modula ion (DSCDM) sys ems. Pape app o ed by he Edi o o Da a Communica ion Sys ems o he IEEE Communica ions Socie y o publica ion wi hou o al p esen- a ion. Manusc ip ecei ed Ap il 21, 1978; e ised June 21, 1979. A. R. Figuei as-Vidal is wi h ETSI Telecomunicacion, Ciudad Uni e si a ia, Mad id, Spain. J. B. Ma iilo-Acebal and M. A. Lagunas-He nandez a e wi h ETSI Telecomunicacion, Jo ge Gi ona Salgado, Ba celona, 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, he channel noise e ec is negligible wi h e- s( ) + n( ) spec o he quan iza ion noise (g anula and excess slope e - o s), bu , in some applica ions, whe e he quan iza ion noise is made ela i ely negligible in an o he wise noisy channel, his e ec can be a c i ical quali y pa ame e . E en in cases whe e he wo noise componen s a e o equal impo ance, one can calcula e he channel noise e ec s by he echniques desc ibed he ein. The quali y c i e ion employed in his pape is he classical mean-squa ed e o measu e. Al hough in mos applica ions o oice o image ansmission subjec i e c i e ia may be mo e ap- p op ia e, he mean-squa ed exp essions a e a i s -o de indi- ca o . Fu he mo e, hey can be modi ied easily in o a weigh ed mean-squa ed e o c i e ion, which is in close ag eemen wi h subjec i e e alua ions. Finally, i should be poin ed ou ha ex ensions o he gi en o mula ion a e easily ob ained o hose cases in which he independen channel e o model canno be used (e.g., in comme cial elephony) in he same way ha Wol indica ed [ 11 o he Gilbe bu s -noise model. The only modi ica ion necessa y would be in he digi al au oco ela ion unc ion E(ni ni+m) (see Sec ion 11). " LINEAR DELTA DEMODULATOR Fig. 1: Linea del a-modula ion ecei e . and he powe spec al densi ies: He e Rpp( ) and SDp( ) a e he au oco ela ion unc ion and he ene gy spec al densi y o p( ), espec i ely. Then, he signal and noise powe s a he ou pu can be calcula ed wi h he help o he ollowing o mulas: 11. SIGNAL AND CHANNEL NOISE POWERS 1 &O) = 5 E(bibi+m)[Rpp( ) * Rhh( )l =mT Le bi indica e he i h ansmi ed symbol, whe e bj = +I. mZ-00 and ni he channel noise e ec on his s mbol. whe e n.. = 0 The ecei e de ec s and egene a es he incoming channel signal; i p( ) is he basic shape o he egene a ed pulses, we exp (j2nmT ) d ; will ha e, a he inpu o he LD demodula o , 00 and a noise e m: n( ) = nip( - iT) (2) j=-m whe e T1 is he symbol a e. The LD demodula o can be conside ed a linea sys em, ha ing an equi alen impulse e- sponse h( ) and a ans e unc ion H( ). Figu e 1 shows he gene al si ua ion jus desc ibed. Since he ou pu o he LD demodula o is a con inuous p ocess whose powe le el does no depend on he ime o igin, we can andomize he e e ence ime o he pulses 131, ob aining he signal and noise e ms: exp (j2nmT ) d ; (10) whe e * indica es con olu ion, and Rhh( ) is he au oco e- la ion unc ion o he impulse esponse o he LD demodu- la o . We ha e hus ound he signal- o-channel noise o mula: m m n( )= 2 nip( + 8 -iT) whe e 0 is a andom a iable, uni o mly dis ibu ed o e [0, m TI . These signal and noise e ms a e s a iona y p ocesses, wi h I(mT) & sp,( ) I H( ) l2 exp (i2nmT ) d . (12) espec i e au oco ela ion unc ions [ 31 : 1- This signal- o-channel noise o mula is absolu ely gene al R,s( ) = - x E(bibi+m)Rpp( + mT) T m=-m 1- T m=-- e ec s, bu we will es ic ou discussion o di ec ansmission. (5) o LDM sys ems. I can also be ex ended o adap i e ( a iable s ep) DM sys ems, using an app op ia e ede ini ion o he sig- nal and noise e ms, al hough he digi al sou ce model will Rnn( ) = - 2 E(nini+m)Rpp( + mT) (6) ha e o be changed. I is also possible o conside line coding IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 133 I I 1 1 I , I ~ -. Fig. 2: Ma ko sys em model. 111. WOLF'S MODEL FOR INDEPENDENT CHANNEL ERRORS The only p oblem emaining o be sol ed is he e alua ion o E(bibi+,) and E(nini+,). Wol 's model [ 1 ] is app op ia e o e alua ing hese ma hema ical expec a ions assuming in- dependen channel e o s. This model consis o wo syn- ch onized Ma ko chains o he digi ized sou ce and he chan- nel e o s; Figu e 2 shows he model. The ansi ion p oba- bili ies PT a e equal, since we assume symme y in his ega d. Only PT is equi ed o he e alua ion o he digi al au oco e- la ion unc ions, and we can ob ain i om mo e s uc u ed (and ealis ic) Ma ko models o he quan ized signal. We de i e he signal- o-channel noise o mula in Appendix I. The ob ained esul is: he analog message; we will deno e he co esponding ansi ion p obabili y by PT'. Wol 's o mula is exac when PT = 0.5, i.e., when P(so)/ P(no) = 1/4Pe, bu i will be mo e and mo e inaccu a e as PT app oaches ze o o uni y. This appea s o be he main eason o he p og essi e sepa a ion o Wol 's heo e ical cu e om B aun's expe imen al alues [ 1 ] : he i ing o low Pe implied de ia ion o high Pe alues. I PT app oaches uni y, P(so) will inc ease app eciably in p ac ice wi h espec o he alue ob ained assuming sin2 nT = 0, due o he e m (1 - 2PT) sin2 nT in he denomina o o (El), conside ing ha , in p ac ice, he e ec i e bandwid h co esponding o H( ) is less han 1/T. P(no) has a small a i- a ion because o he p esence o he ac o P, in he second e m o (1.4). Consequen ly, P(so)/P(no) exceeds Wol 's alue. Jus he con a y occu s when PT app oaches ze o. The a i- a ions a e la ge when P, is small, i.e., in "good" channel cases. I is easy o see ha ou o mula ion gi es P(so)/P(no) = 0 when PT = 0 o PT = 1 (signal absen ), bu i also implies ha P(so)/P(no) # 0 when P, = 1/2. This inco ec esul is due o he de ini ion selec ed o signal- o-channel noise a io, as Wol indica ed. Al e na i e de ini ions chosen in o de o sol e his p oblem p esen o he di icul ies [ 1 ] . Ne e heless, he cases in which P, % 1/2 co espond o ansmissions o e unusable channels (ha ing a capaci y nea ze o), and he p e ious P(so)/P(no) o mulas a e dec easing unc ions o P, which in- dica e he sys em pe o mance in all p ac ical si ua ions. In p ac ical sys ems, he inal s ep in he LD demodula o is a bandpass il e , wi h lowe and uppe cu o equencies cl, c2, espec i ely, and 1/T > c2. A i s app oxima ion would be o assume sin2 nT e 0 in he e ec i e band o in- eg a ion. Then: and IV. EXAMPLES When he in o ma ion consis s o a oice signal and he sys em is app oxima ely op imized wi h espec o g anula and excess slope noises, Kikke 's simula ion esul s [4] allow us o conclude ha P, will no be e y a om 0.5. In his case, he applica ion o Wol 's o mula will be accep able. This emains ue as long as 1 /Tis la ge enough o main ain sin2 nT X 0 in he in eg a ion band. Ne e heless, we will p esen mo e gene al calcula jons o illus a e he depa u e om Wol 's esul s. Le us conside a inal ideal bandpass il e ha ing cu o equencies cl = 300 Hz, c2 = 3400 Hz, le us also assume T = 1/5600 symbols/s, and: P(0 = wm (1 7) whe e: This is Wol 's o mula. No e ha Wol did no conside he qx)= 1, i Ix I< 1/2 inal bandpass il e . We no e ha he app oxima ion is ac- 0, i I x I > 1/2. cep able when 1/T is la ge wi h espec o ,2. I we accep he app oxima ion, he esul ing o mula will apply independen F om he abo e exp ession o p( ): o he shape o he egene a ed pulses. Wol assumed a pe ec in eg a o , bu his esul is also use ul o leaky and double in- spp(n = pp sinc2 ~ eg a ion sys ems and o del a-sigma sys ems. In he la e case, PT would co espond o a model o he digi ized in eg a- whe e: ed message signal. The same model p e iously indica ed would apply o he p ocess esul ing om he in eg a ion o ~inc X 4 sin nx/nX. 134 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 1, JANUARY 1980 Fig. 3: Pe ec in eg a ion linea del a demodula o (T : delay; he in? block is an ideal bandpass il e ). Figu e 3 shows a ci cui ha can be used as a pe ec in e- g a ion demodula o when p( ) is a ull ec angula pulse. I s powe ans e unc ion may be w i en: whe e c = ( cl - cz)/2 is he cen al equency, and I, = cz - cl is he bandwid h o he inal bandpass il e . Table I p esen s nume ical esul s om ou o mula ion, showing he small di e ences wi h espec o Wol 's alues (in b acke s) i P, is no e y di e en om 0.5, and he inc easing di e - ences when PT goes o ze o o 1. I a single RC (leaky) in eg a o is used, we will ha e: (22) whe e 3 is he cu o equency o he RC il e . Table 11 shows nume ical alues co esponding o a ypical 3 equal o 150 Hz. I 'PT = 0.5 and we pu sinc2 T % 1 in [ cl, cz], we will ob ain Johnson's o mula o P(n0) [6] [71. A double in eg a o is shown in Figu e 4. Typical pa am- e e s allow us o w i e: 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 in eg a o . 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 whe e l = 1/2nR,C1(1 + C2lCl) (24) z = 1/2nR&z[ 1 -RZCz(l + R1/R2)'/4R1C1I (25) Table 111 p esen s nume ical esul s co esponding o a ypical z equal o 1 kHz. I PT = 0.5 and we app oxima e sincz T by uni y, ou o mula ion will gi e an al eady known esul o P(n0) [21: P(n 0) k 8pe z l T{ l/ c 1 - 1 / c 2 - [ an-l( ,z/ z)- an-lC cl/ dl/ ~). (26) In he case o del a-sigma modula ion, he esul s ob ained will no be di ec ly compa able wi h he p e ious 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 o7 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 p ope ly ede ining signal and noise e ms, he me hod can be ex ended o coded LDM sys ems and o o he p ac ical DM sys ems (adap i e del a modula ion, ADM, and, pa &ula ly, digi ally syllabic companded del a modula ion, DSCDM). We a e ob aining esul s along hese lines a he p e- sen ime. APPENDIX I F om he model, i is easy o ob ain [ 1 ] : E(bibi+,) = (1 - 2PT)lm1 (1.1) PT' # PT in a gene al case. The ecei e has: whe e P, is he e o p obabili y. Then, ea anging he powe l.V11'=.(b)+ l(~). - c + , exp essions, we can w i e: Some signal- o-channel noise a io alues a e included in PT' = 0.5 and sinc2 T X 1 in [ cl, c2] will lead o John- m=-m The case o linea del a-sigma modula ion can be sol ed by pT(l -PT) Table IV. son's o mula o P(no) [ 51 [ 61. calcula ing he co esponding Z(mT) in he ime domain. Sine- in eg al unc ions appea in he esul ing exp ession, which is di icul o manipula e and o in e p e . Making sinc2 T 1, we ob ain: m = x (1 - 2PT)lm I exp (j2nm T > = - - PT ' + (1 - 2PT) sin' nT (1.3) and he esul ing P(so)/P(no) can be exp essed wi h he help o (1 1). The same app oxima e me hod can be applied in o he cases, bu he in eg alsl(mT) ha e o be calcula ed nume ically. PT( 1 - 2PT) - (1 - PT) sin n PT2 + (1 - ;?PT) sin2 ' "1 (1.4) V. SUMMARY AND CONCLUSIONS whe e he p ime sign in he summa ion indica es exclusion o The gene al o mula ion o he signal- ochannel noise powe a io in (di ec ) LDM sys ems has been p esen ed and compa ed wi h ea lie esul s. The main conclusions a e: he ze o index e m. The a io o he in eg als o hese unc- ions mul iplied by SppO I H( ) 1' will be he signal- ochan- ne1 noise a io, o mula (13) in he main ex . 1) A sligh modi ica ion o Wol 's wo k allows us o ex end i o all LDM sys ems; 2) Wol 's o mula is accu a e enough in p ac ical sys ems 111 ha ing PT no e y di e en om 0.5, bu i ails when PT goes o ze o o PT goes o 1, and especially when P, is small o 121 when Tis la ge. Some nume ical esul s a e compa a i ely p esen ed. Addi ional commen s a e: 3) The me hod can be easily ex ended o co ela ed channel 151 [31 141 e o s (e.g., o he case o bu s e o s acco dingly o he Gil- be model [7], conside ed by Wol ), he only modi ica ion being he al e a ion o E(nini+m) alues (no e y p oblema ic 171 161 in he case o he Gilbe model); REFERENCES WOLF, J. K.: E ec s o Channel E o s on Del a Modula ion. IEEE T ans. on Comm., ol. COM-14, pp. 2-7; Feb . 1966. BETTS, J. A,: Signul P ocessing, Modula ion and Noise. London: Hodde and S ough oni 1975. FRANKS, L. E.: Teo ia de IaSeiTal. Ba celona: Re e e; 1975. KIKKERT, C. J.: Digi al Companding Techniques. IEEE T ans. on Comm.. ol. COM-22, pp. 75-78; Jan. 1974. JOHNSON, F. B.: Calcula ing %I a Modula o Pe o mance. IEEE T ans. on Audio and Elec oacous ., ol. AU-16, pp. 121-129; Ma . 1%8. STEELE, R.: Del aModula ion Sys ems. London: Pen ech P ess; 1975. GILBERT, E. N.: Capaci y o a Bu s Noise Channel. Bell Sys. Tech. Jou ., ol. 39, pp. 1253-1265; 1960.