Adachi F.

Parsons J. D.

Payne F. P.

Williamson A. G.

IEEE Transactions on Communications

English

Parsons, J. D.

Williamson, A. G.

Adachi, F.

Payne, F. P.

Institutional Repositories DataBase (IRDB)

Comment on the calculation of the pdf of the output of a two-branch switch and stay diversity system

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Comment on the calculation of the pdf of theoutput of a two-branch switch and staydiversity system著者 安達  文幸journal orpublication titleIEEE Transactions on Communicationsvolume 34number 11page range 1150-1151year 1986URL http://hdl.handle.net/10097/464471150 10-1 10-2 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-34, NO. 11, NOVEMBER 1986 [31 [41 K.  S. Chung,  “Generalized tamed frequency modulation and its application for mobile radio  communications,” IEEE Trans. Veh. Technol., vol. VT-33, p. 103, Aug. 1984. K. Murota and K .  Hirade,  “GMSK modulation for digital mobile radio telephony,” IEEE Trans. Commun., vol. COM-29, pp. 1044-1050, July 1981. T. Aulin and C.  E.  Sundberg,  “Continuous phase modulation-Part 1: Full response signaling.” IEEE Trans. Commun., vol. COM-29, pp. 196-209, Mar. 1981. T.  Aulin, N.  Rydbeck, and C.  E.  Sundberg, “Continuous phase modulation-Part 2: Partial response signaling,” IEEE Trans. Com- mun., vol. COM-29, pp. 210-225. Mar. 1981. K. S. Chung. “A noncoherent receiver for  GTFM  signals,” in Proc. GLOBECOM, Miami. FL, Nov. 29-Dec. 2. 1982, paper B3.5. M.  Hirono, T .  Miki, and K.  Murota,  “Multilevel decision method for band-limited digital FM limiter-discriminator detection,” IEEE J. Select. Areas Commun., vol. SAC-2. pp. 484-506, July 1984. Y. Akaiwa, 1. Takase, S .  Kojima.  M.  Ikoma, and N.  Saegusa, “Performance of baseband bandlimited multilevel FM with discrimina- tor detection for digital mobile telephony,” Trans. IECE Japan, vol. T.  T. Tjhung,  C. S.  Ng. K.  K. Yeo, and P. H. Wittke,  “Error performance analysis  for narrow-band duobinary  FM with discrimina- tor detection.” IEEE Trans. Commun., vol.  COM-33, pp. 399-408, May 1985. T.  T. Tjhung and P. H. Wittke,  “Carrier transmission of binary data in a restricted band.” IEEE Trans. Commun., vol. COM-18, pp. 295- 304. Aug. 1970. E64, pp. 463-469. July 1981. Fig.  4. GMSK detectability results (h = 0.5 and h = 0.7 with Viterbi GMSK. algorithm) compared to binary FSK and GTFM, and multilevel decision We  chose  to  detect  GMSK  at  the  discriminator output  using the MLSE algorithm of [2]. By varying the premodulation Gaussian filter (3 dB) bandwidth BT,  the  index of modulation h ,  and the receiver IF  and  LPF  bandwidths, it is possible  to generate a whole class of modulation techniques. We  chose  to  look at BT/Rb = 0.25  and h = 0.5 (the  value used in [ 7 ] ) ,  and also h equal to 0.7. As seen in Fig. 4, the results for h = ,0.5 represent about  a 1 dB  improvement  over the multilevel decision method. The results for h = 0.7 are almost  similar  to those of classical binary  FSK of Tjhung  and Wittke [lo] but, of course, with a narrower  bandwidth. V. CONCLUSIONS It has been shown  that by properly  designing digital CPM signals, it is possible to  improve  BER  performance  for discriminator detected modulations in comparison to previ- ously reported results. In the “modified GTFM” case it is possible to  achieve  better  BER  performance  than that of similar  GTFM  schemes, with  only  a small  sacrifice in bandwidth. For GMSK ( h  = 0.5) the discriminator-MLSE detector  results in detectability performance  at  least 1 dB better than those previously described in the  literature  (for a discriminator-multilevel decision  detector).  The  GMSK  detec- tability results can be improved (using h = 0.7) with only a small increase in bandwidth,  due  to  the  larger  index of modulation.  These  results  are  almost  equal  to  those f classical binary FSK with optimum modulation index ( h  = 0.7). REFERENCES 111 F. de Jager and C. B. Dekker,  “Tamed frequency modulation-A novel method to achieve spectrum economy in digital transmission,” IEEE Trans. Cornmun., vol. COM-26, pp. 534-542, May 1978. Comment on the Calculation of the pdf of the Output of a Two-Branch Switch and Stay Diversity System J .  D. PARSONS, A. G. WILLIAMSON, F. ADACHI, AND F. P. PAYNE Expressions  for  the  pdf  of  the  output of a two-branch  switch and stay diversity system have been derived in both [I] and [Z]. The basic difference  is that the  analysis in [I]  is based on a continuous time approach, whereas that in [2]  is based on a discrete  time  approach. The  principal  purpose  of this letter  is  to  draw  attention  to  the fact that the pdf‘s obtained by the two methods are not the same; compare [ 1 ,  eq. (26)] to [2, eq. (2.17)]. Moreover, while  Blanco and  Zdunek  stated  [2,  sect. 11-B] that they  would compare  their  results  to  those  of [l], they did not do so. Having noted that the two pdf‘s are not the same, it is interesting  to  consider what different  assumptions  are  made in the  two  approaches,  and in what  ways  the  problems so analyzed  are  different. A major  assumption  made  in [l] is  that, having switched from r, to r2 (rl  and r2 being the two input signals), “ . . . when  we  switch  back  from r2 to r l ,  the  statistics of rl are  already  independent  of  that  portion of rl which we switched  out  initially.” A similar  assumption  is  made in [Z]. However, an additional assumption is also made, effectively between (2.8) and (2.9), concerning the statistical indepen- dence of successive  samples of the  same  signal.  While x;- I and x; (where x; and y ;  are  the  ith  samples of the  two input signals x and y )  may be statistically independent if the time between samples  is very long, it is  questionable  whether this is still a Paper approved by the Editor  for Signal Design,  Modulation,  and Detection of the IEEE  Communications  Society. Manuscript received October 10. 1985; revised May 21, 1986. The  authors  are with the Department of Electrical Engineering  and Electronics, University of Liverpool.  Liverpool,  L69  3BX,  England. IEEE Log Number  8610850. 0090-6778/86/1100-1150$01 .OO O 1986 IEEE Authorized licensed use limited to: TOHOKU UNIVERSITY. Downloaded on March 03,2010 at 02:17:36 EST from IEEE Xplore.  Restrictions apply. IEEE  TRANSACTIONS ON COMMUNICATIONS, VOL.  C M-34,  NO. 11, NOVEMBER 1986 1151 realistic  assumption  at a high  sampling  rate,  certainly  at  a  rate which is  sufficiently  high so that  the [x;] adequately  represent x. It  is  also  surprising  that  he  final  result  is  completely independent  of  the  sampling  rate. Another  difference  between  the  two  analyses  is  the  nature of the  output  signal. The output  of  the  system in [ 11 is  continuous in time.  It  is  n6t  clear  whether  the  output  of  the  system in [2] is a  series of discrete  samples,  or  whether  the  output  is continuous  but  the  testing  is  done  at  discrete  intervals. If the latter is the  case, it is  possible  for  the  switching  from  one  input signal  to  the  other  to be  delayed  by  up  to  one  sampling  period from the time when the input signal actually falls below the threshold d, in the  case when > d and x; < d .  REFERENCES [I] A. J. Rustako, Jr., Y. S. Yeh, and R. R. Murray, "Performance of feedback and switch space diversity 900  MHz  FM mobile radio systems with Rayleigh fading," IEEE Truns. Commun., vol. COM-21, pp. 1257-1268, 1973; see also  J.  D.  Parsons, "Experimental switched diversity system for VHF AM mobile radio," Proc. IEE, vol. 122, pp. [2] M. A. Blanco and K. J. Zdunek, "Performance and optimization of switched diversity systems for the detection of signals with Rayleigh fading," IEEE Truns. Commun., vol. COM-27, pp. 1887-1895, 1979. 780-784, 1975. A Numerical Method for Evaluating the Distortion of Angle-Modulated Signals in a Time Domain YASUNORI IWANAMI AND TETSUO IKEDA Abstract-As yet there exists no analytical method of obtaining the exact output  distortion  of BPF's for an angle-modulated  signal  input.  In this  paper, we have  introduced a numerical  method  in  which a filtering problem is described  by  the  state  space  method  and  then is converted  to  a time  domain  difference  equation. As a result,  evaluation  can  be  carried out  quickly  and  easily. I. INTRODUCTION The  problem of  evaluating  the  response of bandpass  filters (BPF's) by angle-modulated  signals  has  been  investigated  by many authors. J .  R. Carson  tackled  it  in  the 1930's. Due  to  the nonlinearity  involved in this  problem,  however,  there  exists no exact analytical method of expressing the response when an arbitrary  modulating  signal  is  used.  Consequently,  evaluation is made by either approximate analytical methods [1]-[3] or numerical  ones [4]-[SI. Numerical  methods  consist  of  Monte Carlo  .simulation  methods [4]-[6], which  are  mainly  used  for the  evaluation  of  the  interchannel  interference  distortion  in  the FM  trunk  radio  system,  and  numerical  techniques  to  evaluate the  distortion  of  periodically  modulated  signals [7], [ 8 ] .  Numerical  methods  can  be  roughly  divided  into  two categories-the  frequency domain [4], [7] and  the  time  domain [5 ] ,  [6],  [SI. The  former  uses  the  fast  Fourier  transform  (FFT) and  the latter  uses  convolutional  integrals  in  the  time  domain. The  method  which  uses  the  FFT  is  popular  because it is easy to understand and the error is small.  On  the  other  hand,  the Paper approved by the Editor for Computer-Aided Design of Communica- tions Systems of the IEEE Communications Society. Manuscript received July 31, 1985; revised March 28, 1986. The  authors  are with the Department of Electrical and Computer Engineer- ing, Nagoya Institute of Technology,  Nagoya,  466,  Japan. IEEE Log Number  8610856. 0090-6778/86/1100-115 method  which  uses  the  convolutional  integral is not so common  because  of  the  long  computation  time  required  for  the numericakhtegration. However, the method which uses the FFT- is  ordinarily  applied in cases  where  the  angle  modulation is periodic. It is difficult to apply when the modulation has transients  such  as  frequency  steps  and/or  frequency  ramps. In  this  correspondence  a  very  simple  iteration  method,  using the first-order difference equation derived by combining the state  variable  method [9] with  the  numerical  integration. formulas [IO], [ 1 I], is  introduced,  as  an  improved  version of the time domain convolutional method. In this method, the types  of BPF  and  angle-modulating  signal  shape  are  arbitrary and  the BPF type  is  specified  by  the  state  transition  matrix. In  comparing  this  method  to  the  FFT  method,  the  response of  a BPF  for  the  transiently  modulated  FM  can be  evaluated quickly  and  easily,  even  using  a  microcomputer  with  a  small amount of memory.  A  real-time  application of this algorithm may lead to the discovery of new digital FM demodulation techniques. 11. A  MODEL OF EVALUATION A .  A Description of the System by State Variables The model for the analysis is shown in Fig. 1 .  The input angle-modulated  signal uin ( t ) ,  the  transfer  function  of  the  BPF H ( j w ) ,  and  the  output  signal  of  the BPF uOut ( t )  are  expressed as vi&) = A  cos [oat+ 4i"(t)] H ( j w )  =' S m  h(t)d"' dt - m  u,ut(t) = B(t) cos [wet + 4 O U d t ) l  (1) where A ,  WO, and din ( t )  represent  the  amplitude,  the  angular frequency,  and  the  phase  of  the  input  angle-modulated  signal, and h(t) is the unit impulse response of the BPF. B(t) and 4out ( t )  denote  the  amplitude  and  the  phase of the  output  signal, respectively. Given uin ( t )  and H( j w )  as  shown in Fig. 1 ,  the  problem  is to  obtain uout(t). But wo is  arbitrary  and  the  idea  of  an equivalent  low-pass  model [12] is used as  shown in Fig. 2. In  Fig. 2, u(t) and u(t) denote  the  input  complex  low-pass signal and the output complex low-pass signal, respectively, and'C(t) = B(t ) /A,  and H L ( j w )  is the equivalent low-pass expression  of  the  transfer  function H ( j w )  which  is  written as where * denotes  the  complex  conjugate. form Ordinarily,  the  equivalent  low-pass  function H L ( j w )  has  the c ~ " - ~ + c , _ ~ s " - ~ + ' . ' + c ~ s + c ~  HL(S)=s~+a,s"-~+a,.Is"-Z+ e - .  +azs+al (3) which is  the  rational  function of s, where s = j w .  Thus,  from  Fig. 2 and (3), it follows that (I?" + anp"-' + * * + azp + al)u(t)  = (c,pn-'  + cn- l pn-2  + * + C ~ P  + cl)u(t) (4) where p denotes  the  operator d /d t .  Applying  the  state  variable representations  to (4), the  following  expressions are  obtained. I k(t) = A X ( t )  + Bu(t)  u ( t )  = CX(t )  ( 5 )  $01.00 @ 1986 IEEE Authorized licensed use limited to: TOHOKU UNIVERSITY. 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Comment on the calculation of the pdf of the output of a two-branch switch and stay diversity system

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