(bits/slot) Nhomakorabea (2)
recyived sigilal
lVe can select the optimum 2L-symbol sets mininiizing the probability of bit error from Q word patterns. .rZ h I P P h l system transmits data waveform, and h1I’Ph.I signal contains a frequency component of the timing clock of the transmitter. The receiver separates MPPM sigiial into clock waveform and data waveform, and tlieii slot clock recovery system regenerates the receiver clock to be synchronized with the clock of the transmitter [4]. M P P M decoder integrates the photodetector output d a t a signal over time slot and compares ex11 of the m integrated outputs to find p pulsed slots in order of the large output. When the received word does not correspond to any assigned symbols, the receiver decodes it as the assigned symbol whose distalice froln the received word is minimum. Tlius, h1PPM has the possibility that the symbol is decoded correctly by MPPM decoder. In practical systems, however, the clock recovery is imperfect, and imparftxt slot synchronization between the transmitter add the receiver affects the integrations over the time slots. Therefore, we need to analyze the performance of MPPM with slot synchronization.
1.
INTRODUCTION
In optical communication systems using a photon countiiig technique, pulse position modulation (PPM) has a possibility of a very low energy communication [I]. However, high energy information efficiency with PPM requires exponential increase of the transmission bandwidth, which is a serious problem of PPM. To solve this problem, multi-pulse PPM (MPPM) h a s , been proposed to reduce the transmission bandwidth to about half of that of PPM at the same transmission efficiency 121, and convolutional codes is employed in MPPM to obtain further improvement [3]. In a practical system, to achieve the synchronization between the transmitter and the receiver, the transmitted signal contains a spectral component at the time slot frequency and the receiver regenerates a clock waveform which is synchronized with the clock at the transmitter. Once the correct -slot timing is almost provided, word synchronization can be maintained unless the recovered clock jitter is so large that a cycleever, some .of the signal photons transmitted in signal time slot will be counted i n an adjacent slot, which
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and the transnussion rate R, is
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AT
PERFORMANCE ANALYSIS O F ( m ,2 ) MPPM WITH IMPERFECT SLOT SYNCHRONIZATION
Kazunii Sato, Tomoaki Ohtsuki, Iwan Sasase and Sliinsaku nlori Dept. of Electrical Engineering, Keio University 3-14-1 Hiyoshi, Kohoku, Yokohama, 223 Japan
3. WORD ERnOR PROBABILITY
I :
1 :
I :
I :
I
We assume that a (m, p ) MPP-M word has p rectangular pulses, and each pulse occupies the slot of duration T,. In ( n i , 2) MPPM, the performance of the word w i t h two continuous pulses is different from that of the word witli two separated pulses when the timing offset AT occurs at the receiver. Timing diagrams o the word with continuous pulses and the word with f separated pulses are shown in Fig. 2(a) and Fig. 2(b), respectively. In Fig. 2(a), IC, and IC* are the expected photocounts over a slot period and background noise, respectively, k l and kz are the numbers of detected photons i n signal slots, k3 is number of the detected photons in the adjacent slot in which the number of photons is increased i n the presence of the timing offset, and k., is the vector of the numbers of photons detected i n the other slots. We define that E = AT/T, is tlie normalized timing offset, anaassume that E < 1 [4]. In the case with the word with continuous pulses, at the input of the photodetector, the number of photons in one of two signal slots is reduced to (1 - €)I<,, but that i n the other signal slot is constant, even if the timing offset occurs. When the timing offset E occurs at the receiver, word error probability ( W E P ( e ) )of the word witli continuous pulses can be derived as
ever, the performance of MPPM in the presence of timing error has not been analyzed. In this paper, we focus on MPPM with two pulses i n one block consisting m slots denoted as ( m , 2 ) M P P h l , and analyze the performance of (m, 2 ) MPPM w i t h imperfect slot synchronization. First, we evaluates the probability of word error i n the preseiicc of t i i i i i i i K clffset. I V e also assign the optimum symbol sets id ( 7 , 2) hlPPM and (8, 2 ) MPPM mininuzing the probiibilities of bit error. I t is s h o w n that tlie woril ivitli separated pulses has larger probability o \sori1 c r r w f than tlie word with continuous pulses, and the L i i t error probability of (m, 2 ) hlPPAl. is smaller tlraii t l l i L t of I’PRI, at the same transmission bandwidth and the saint transmission efficiency.