Study of trellis coded overlapping ppm and overlapping multi-pulse

Study of trellis coded overlapping ppm and overlapping multi-pulse ppm schemes for deep space optical communication systems Manish Choudhary1, Ranjan Bose, V.K. Jain* Electrical Engineering Department, Indian Institute of Technology, Delhi, New Delhi 110016, India ABSTRACT The unguided optical communication system in free space are deployed to establish, long distance links when both the terminals has to be mobile i.e., inter satellite links or deep space mission. Such system generally employs intensity modulation/direct detection (IM/DD) scheme. The M-ary pulse position modulation is a preferable modulation scheme over unguided channels since it is energy efficient. However, it results in an exponential increase of transmission bandwidth. As inter satellite links are severely bandwidth limited, this issue becomes quite significant in such systems. In this paper, we discussed the performance of Trellis coded overlapping PPM and overlapping multi-pulse PPM schemes for deep space optical communication systems, and their potential impact to overcome the BER, bandwidth and throughput constraints to envisage high data rate requirements for intersattellite link and deep space missions. Keywords: Overlapping pulse position modulation, Overlapping multi -pulse position modulation, Trellis coded modulation, Manhattan distance, Convolution encoder, Set partitioning.

1. INTRODUCTION The deep space optical communication systems are expected to supplement the microwave based systems [1-2] for satellite to satellite (when both the terminals are mobile), space to satellite communication (when the distances are exceptionally large). In addition to this, optical systems can carry more traffic than microwave based systems. There is already significant research and commercial interest in this area. The advantages of unguided optical communication are: (i) smaller size and lighter weight of the terminal, (ii) less transmitter power, (iii) higher immunity to interference, (iv) large data rates and (v) smaller beam divergence angle. There are several key advantages of PPM scheme [3] over OOK that are critical to its deployment. • Requirement of the high peak laser power to survive the huge deep space loss. For this reason Q switched laser are typically used, but current technology does not support toggling of this laser between ON and OFF state at a high rate thus severely limiting the data rate that can be supported using OOK scheme. Hence M-ary PPM with large M is more suitable than its OOK counterpart for deep space application. • M-ary PPM is more energy efficient. • M-ary PPM, having a pulse in a symbol frame benefit the clock recovery subsystem, whereas OOK system may suffer a synchronization loss if a long sequence of 0’s is encountered. In future, when higher data rates are envisaged the M-ary PPM may not be well suited because of its throughput limitations. The only way to increase the throughput is to make the time slot narrow in which optical pulse is placed. However, there is a practical limit in narrowing the pulse width. So overlapping PPM and overlapping multi-pulse PPM are obvious choice for higher data rate requirements.

* 1

Corresponding author, Fax: +91-11-26581606; [email protected] Address : C-DoT, C-DoT Campus, Mandi Gaon Road, Chattarpur, New Delhi 110030; [email protected]

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2.

SYSTEM MODEL OF OVERLAPPING PPM

In OPPM[5], the symbol duration of Tsym is subdivided into W chips of equal duration of τ (=Tsym /W ). The information is conveyed by the position of the pulse of duration Tslot in one of the overlapping time slots. The pulse is constrained to be consecutive for N time slots (or chips). The motivation of constraining N 1’s to be consecutive is decreased bandwidth, but the price paid is the reduced alphabet size. If we define Q as the alphabet size of the OPPM signal set with no overlap, N the overlapping index and J as the alphabet size with overlap, which is related to Q and N by J=N(Q-1)+1

(1)

The number of chips, W in symbol duration can be given as W=QN

(2)

and the alphabet size as compared to orthogonal PPM is reduced from Symbols (J) 1

Tslot 1

2

W

c N to (W-N+1).

τ 3

……………………..w Tsym N

2

3

4 Fig. 1. OPPM signaling scheme with Q=2, N=3 and J=4 The OPPM signal sets are no more orthogonal, which implies poorer BER performance. The throughput of the system can be increased by increasing the number of overlaps i.e., the value of N, but at the expense of the BER performance and increased complexity of the synchronization subsystem. In the following analysis, we have assumed that synchronization is perfect to the extent that its degrading effect on the performance is negligible.

2.1 PERFORMANCE EVALUATION WITH UNCODED AND TRELLIS CODED OPPM Under Poisson channel assumption [6], let N be the number of photons observed in N slots that overlap with the pulsed slots. The receiver counts the number of photons in the different overlapping chip duration corresponding to the different symbols and makes a decision by choosing that symbol interval which has the largest number of photons count. In case of symbols with equal count, a random choice is made among these symbols. Assuming equally likely OPPM symbols, the error probability, given that symbol j was sent and some other symbol was received is given by

Pe =

1 J

J

∑ P (e / j )

(3)

j =1

In the above equation, P(e / j) is the pair-wise error probability of making a decision in favor of the other symbol, say, kth symbol when jth symbol is being transmitted and Pe is the symbol error rate of the uncoded OPPM. This error event [6] is possible only when the integrated count of the kth slot exceeds that of the jth slot. The pair wise error probability can be given by

2

P(e / j) = Pr( E k , j ) = Pr [{Nj ≤ Nk } / j]

(4)

The union bound may be obtained as Pr(Ek,j) ≤

Υ Pr[{N

≤ N k } / j ],

j

(5)

k≠ j

Here Nj and Nk are independent Poisson random variables with mean d jk (λ n + λ s )Tslot / N and d jk λ n T slot / N , respectively, λs the average number of signal photons per second due to signal impinging on the photodetector and λn the noise photons due to background light. The distance metric can be defined as (6)

d jk =| j − k |, if | j − k |≤ N ;

= N , if | j − k | >N Equation (6) is derived by considering that between symbols j and k there is an overlap of ( N - | j-k | ) ≥ 0 and that the symbols become orthogonal (no overlap) when | j-k |>N. It follows from Eqs. (6) [7].

 2λ n T slot d jk 2 d jk (λ n + λ s )T slot Pr[ N j ≤ N k / j ] = Q1  , N N  where

Q 1 (α , β ) = exp( −

α

2

+β 2

2

∞

)∑ ( k =0

  

(7)

β k ) I k (αβ ) α

Here Q1 is the Macrum’s Q function [8]. Further, Eqn.(7) can be simplified by using Chernoff bound to yield  d jk  Pr[ N j ≤ N k / j ] ≤ exp  − ( λ n T slot + λ S T slot − λ n T slot ) 2  (8)  N  From Eqs. (3), (4), and (8) we obtain ∞ ∞   d Pe ≤ 1 exp− jk ( (λn + λs )Tslot − λnTslot ) 2  ∑∑ J j =1 k ≠ j N  

(9)

The numerical results computed from Eqn.(9) are shown in the Figs. 2 and 3. In Fig.2 the variation of Pe with N (=1,3,5,7) for fixed Q(=2) is given and in Fig. 3 the variation of Pe with Q(=2,4,6) for fixed N(=3) is depicted. It is noted from Fig. 2 that for fixed alphabet size, increase in N degrades the performance, as the signal sets no more remain orthogonal and from Fig. 3 it is observed that for fixed overlap N, with increase in Q , leads to performance improvement, but the price paid is increase in bandwidth since increasing Q decreases the pulse width for the fixed symbol duration [9].

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Fig. 2.

Probability of symbol error versus signal photons for different values of overlap (N=1,3,5,7) and fixed alphabet size (Q=2)

Fig. 3. Probability of symbol error versus signal photons for different values of alphabet size (Q=2,4,6) and fixed number of overlap (N=3) As mentioned earlier, TCM is used to improve BER performance without compromising the data rate. The TCM can also be used to overcome the impact of pulse spreading arising due to large area detector. Such detectors when used in deep space system collects large number of signal photons and give rise to more received power [10]. For the Trellis coded OPPM, the upper bound on symbol error probability can be obtained using Chernoff bound as  d  (10) Pe ≤ exp − free ( λnTslot + R ln( J ) µ − λnTslot ) 2   N  Let us consider an example of OPPM signaling with J=8, N=7 and Q=2, and applying 2/3 Trellis code on the expanded signal set to increase the minimum distance among symbol set so as to achieve performance gain without bandwidth expansion. The signal sets are partitioned with increasing distance metric as shown in the Fig. 4.

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Set partitioning of (2,7,8) OPPM No. of symbols= 8

Distance (djk)

1,2,3,4,5,6,7,8

1,3,5,7

1

2,4,6,8

1,5

3,7

2

2,6

4,8

4

Fig. 4. The set partitioning of (2,7,8) symbol set The alphabet size of OPPM is 8 (3 bits), which allows one to encode 2 data bits per T-PPM symbol. We are using 1/2 rate convolution encoder followed by a mapper. The use of TCM over expanded sets of OPPM signal can result in both the performance and throughput gain without a reduction in pulse width as depicted in Fig.5.

Fig.5. Symbol error probability versus signal photons for uncoded and Trellis coded OPPM Table 1 Performance comparison for different background noise photons Signal photons ( λ s ) for N=7

Pe

λ.n =1.0 Uncoded 10-3 10-6 10-9 10-12

25 54 80 115

TCM coded 17 34 47 64

λ.n =10.0 Uncoded 45 90 115 145

TCM coded 35 60 75 90

λ.n =25.0 Uncoded 56 105 145 170

TCM coded 45 71 91 115

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2. SYSTEM MODEL OF OVERLAPPING MULTI-PULSE PPM (OMPPM) In OMPPM [11], laser is pulsed in P slots in one frame consisting of total M slots. A symbol of duration Tsym sec, is subdivided into Q=MN subintervals, of equal duration of τ , referred as a sequence of chips. Each slot with duration of Tslot is divided into N chips and duration of one slot or one optical pulse is represented as Tslot= n τ , and this scheme is represented by (M,P,N) OMPPM. By having multiple pulse positions per pulsewidth, we can increase the number of symbols of (M,P,N). J=(M-P)N+1

(11)

Where J is the number of symbols, M the number of slots in one frame, P the number of pulsed chips in one frame and N the number of chips in one slot.

Symbols 1…………………….

……..P

1 Tslot 2

τ

(1 chip)

3

4 0

1…………………………

…………M

Tsymbol

Fig. 6. Signaling model of overlapping multi-pulse pulse position modulation (OMPPM) 3.1 PERFORMANCE EVALUATION WITH UNCODED AND TRELLIS CODED OMPPM To find the uncoded error probability of OMPPM, the upper bound on symbol error probability can be obtained using Chernoff bound as Pe ≤

1 J

J

∑∑ j =1 k ≠ j

 d  exp − jk ( (λs + λn )Tslot − λnTslot )2   N 

(12)

The distance between the symbols j and k, is d j ,k and is defined as the number of non overlapping pulsed chips it is expressed as d j ,k = PN − V

(13)

where V = N-|j-k| |j-k|N (15) Here V is the number of overlapping pulsed chips between the symbols j and k. Large djk , implies small number of overlapping pulse chips between symbols.

Similarly the Trellis coded symbol error probability can be approximated using the Chernoff bound as

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 d free  1 J (16) exp  − ( (λ N + λ s )T slot − λ n T slot ) 2  ∑ ∑ J j =1 k ≠ j  N  The first step to apply the TCM to the given optical channel is to identify the distance metric that will be used for partitioning the OMPPM signal set. And then applying the heuristic approach of Ungerboeck rules for maximizing the free Euclidean distance. P (e) ≤

The distance metric is computed by finding the Manhattan distance between the two pulses i.e., between symbols j and k and is defined as (for two pulses case)

a = ws − y s + we − y e

(17)

b = x s − z s + xe − z e

(18)

The Manhattan distance is given by dj,k=

a2 + b2

(19)

where ws and ys is the beginning time and we and ye the ending time of the pulse of symbol ‘j’ and similarly xs and zs and xe and ze are the beginning and ending times of the pulse of symbol ‘k’. For mapping the symbol set in two-dimensional plane, each symbol set is assigned the pair (x,y), where x is the beginning of the signal pulse and y the ending time of the signal pulse. Now the Ungerboeck rule of set partitioning is applied to this mapping so as to maximize the symbol set in terms of the Euclidean distance. The free Euclidean distance [12] of the TCM encoder is given by dfree = min[ ∆ m +1, d free ]

(20)

Here we have considered an example of improving the performance of the (7,2,3) OMPPM by using TCM, where the PPM alphabet size (J) is 16, number of overlap (N) is 7, the number of pulses in one frame i.e., P=2, and applying 3/4 Trellis code on the expanded signal set to tradeoff some of the throughput gain for performance. The set partitioning of (7,2,3) OMPPM No. of symbols=(M-P)N + 1 = 16

Distance D=2 2

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)

1,3,5,7,9,11,13,15

1,5,9,13

1,9

5,13

3,7,11,15

3,11

D=2 2

2,4,6,8,10,12,14,16

7,15

2,6,10,14

2,10

6,14

4,8,12,16

4,12

8,16

D=4 2

D= 40

Fig. 7. Set partitioning of OMPPM with M=7, P=2,N=3 Here the concern is 3/4 convolution encoder with eight state Trellis. The alphabet size of the OMPPM is 16 (4 bits), which allows encoding of 3 data bits per Trellis coded OMPPM symbol. The decrease in data rate is ¾ which is a small price to pay to circumvent the substantial energy spreading caused by imperfect pulse shapes.

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A1 C1

A2

C2

A3

Natural C3 mapping C4 (16 Symbols) Fig. 8. The equivalent systematic (3/4) TCM encoder State S0:s4,s0,s6,s2

S0

S0

S0

State S1:s4,s0,s6,s2 State S2:s4,s0,s6,s2

S6

S6

State S3:s5,s1,s7,s3 State S4:s2,s6,s0,s4

S7

State S5:s3,s7,s1,s5 State S6:s6,s2,s4,s0 State S7:s7,s3,s5,s1 Fig. 9. Eight state Trellis diagram for the 2/3 convolution encoder The minimum Euclidean distance between the two points in the signal constellation of 16 symbols is 2 2 . The squared free Euclidean distance between the two non parallel path as shown in Trellis diagram is

d 2free = d E2 ( S 0 , S 6 ) + d E2 ( S 0 , S 7 ) + d E2 ( S 0 , S 6 ) = ∆21 + ∆20 + ∆21 = 5( 2 2 ) 2

(21)

The distance between any four parallel paths from the second stage of the set-partitioning tree is given as ∆ 2 = (4 2 ) 2

(22)

Hence, the free Euclidean distance for this TCM scheme at the second stage of the set partitioning is given by (23) d 2free = min[ 5( 2 2 ) 2 , ( 4 2 ) 2 ] = ( 4 2 ) 2 The distance between any two parallel paths from the last stage of the set-partitioning tree will be ∆ 23 = (8 ) 2

(24)

8

The free Euclidean distance for the last stage of this set-partitioning scheme is given by

d 2free = min[5( 2 2 ) 2 , (8) 2 ] = 5(2 2 ) 2 = 40

(25)

Table. 2 Asymptotic coding gain with respect to the uncoded OMPPM scheme Number of symbols in each set 16

Number of partitioned sets -

8

2

4

4

2

8

dfree (uncoded)

dfree (coded)

2 2 2 2

2 2 2 2 4 2 40

2 2 2 2

Asymptotic coding gain (dB) 0.0 0.0 3.0 3.5

Fig. 9. Probability of symbol error for (16,2) uncoded and Trellis coded OMPPM versus number of signal photons Fig.9 shows the performance improvement of the OMPPM using eight states Trellis coding for the symbol set of 16 in the presence of background noise. From the above figure, it is evident that with the use of TCM scheme we can achieve the symbol error rate with lesser number of signal photons without any bandwidth expansion. 4.

CONCLUSIONS

In this paper, we have analyzed the performance of the Trellis coded overlapping PPM and overlapping multi-pulse PPM schemes for deep space optical communication systems. These schemes are bandwidth as well as throughput efficient. By allowing overlap among the adjacent pulse positions and multiple pulses in one symbol duration, the throughput of the system increases, but because of non orthogonality of signal set the BER performance of the system becomes poorer. In this case, usage of simple Ungerboecks Trellis codes over expanded signal sets of the OPPM signals can improve the performance without sacrificing the throughput and bandwidth of the system.

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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

‘Optical communication space applications’ V.K. Jain, IETE Technical Review, 11(1994)191. J.H. Franz, V.K Jain, Optical Communication: Components and Systems, Narosa Publishing House, New Delhi, 2000. ‘Trellis coded pulse position modulation for optical communication systems impaired by pulse width inaccuracies’, Kamaran Kiasaleh, Tun-Yee Yan and Meera srinivasan, Journal of Lightwave Technology, 17(1999) 13336. ‘Channel coding with multi level phase signals’, G. Ungerboeck, IEEE Transaction in Information Theory, 28(1982)55. ‘Modulation analysis for wireless infrared communication’, Hyuncheol Park, John R. Barry, Proceeding of International. Conference on Communication, (1995) 1182. ‘Some implication of Trellis coded modulation for optical direct detection channels’, Coastas N. Georghiades, IEEE Transaction of Communications, 37(1989) 481. A.Papoulis, Probability, Random Variables and Stochastic Process, McGraw-Hill, New York-1984. ‘A statistical theory of target detection by pulsed radar: Mathematical appendix’, J.I. Macrum, IRE Trans. Inform. Theory, 6(1960) 59. ‘Modulation and coding for throughput efficient optical systems rate analysis of overlapping multi-pulse position modulation’, C.N. Georghiades, IEEE Transactions on Information Theory”, 40(1994) 1313. Ezio Biglieri, Dariush Divasalar, Peter J.McLane, Marvin K. Simon, Trellis Coded Modulation, Macmillan Publishing Company, New York, 1991. ‘Cutoff rate and capacity of MPPM in noiseless photon counting channel’, T.Ohtsuki, I.Sasae,and S.Mori, IEICE Tran.77(1994) 1298. Ranjan Bose, Information Theory, Coding Theory and Cryptography, Tata McGraw-Hill, New Delhi, 2002.

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