Joint Source Coding and Higher-Dimension Modulation

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1 Jont Codng and Hgher-Dmenon Modulaton Tze C. Wong and Huck M. Kwon Electrcal Engneerng and Computer Scence Wchta State Unvert, Wchta, Kana 676, USA {tcwong; Abtract Th paper propoe a novel jont ource codng and modulaton cheme. The modulaton contellaton pont are elected accordng to ther pror mbol probablte for better bandwdth a well a better bt error rate performance. Both the anal and mulaton reult are preented to verf that the propoed cheme can acheve better performance than the conventonal djont ource codng and modulaton cheme f the modulaton and ource codng are degned jontl and effcentl. Index Term Quadrature Phae Shft Keng, Bt Error Rate, Maxmum A Poteror Probablt, Maxmum Lkelhood. I. INTRODUCTION YPICAL nput to a ource encoder have unequal Tpror probablte. For example, ome alphabet have hgher pror probablte than other alphabet n an alphabet tream. However, the output of the conventonal ource encoder ha been regarded a a bt tream wth equal pror probablt and fed nto a channel encoder. Refer to Fgure (a). Then, a modulaton cheme ha been elected ndependentl, regardle of the ource unequal pror probablte. For example, quadrature phae hft keng (QPSK) modulaton take two bt from the channel encoder output bt tream and map them nto a modulaton mbol regardle of ource alphabet pror probablte. Hence, all the mbol or contellaton pont occur equall. However, the output mbol from the ource encoder, e.g., Huffman ource encoder, have unequal probablte tpcall. Thu, the conventonal modulaton, e.g., QPSK, would not acheve optmum effcenc for the mbol tream wth unequal probablt. Th the motvaton for th paper, whch conder a jont ource codng and modulaton a hown n Fgure (b). Th paper propoe a novel jont ource codng and modulaton cheme that explot the unequal mbol probablt and ntroduce a novel dtance crteron that dfferent from the one n [], []. Th paper alo how that the propoed jont ource codng and modulaton cheme wth the maxmum a poteror probablte (MAP) rule can acheve better performance than the conventonal cheme,.e., the djont ource codng and modulaton. We wll compare performance between the conventonal QPSK and the propoed cheme, called Y-Shape modulaton, wth the new dtance crtera. There have been man paper on jont ource and channel codng, but ver few on jont ource codng and modulaton, e.g., []. However, there are man dfference between [] and the propoed cheme n th paper. One of the man dfference that the one n [] cannot be extendable for M-ar PSK wth M 4, wherea the propoed cheme can be applcable for an M-ar modulaton. On another note, n tpcal communcaton tem, a ource encoder followed b a channel encoder and then b modulaton. However, the ource encoder followed b modulaton n the propoed cheme. Hence, to ue a channel codng, a non-bnar mbol channel encoder, e.g., a q-ar low-dent part check encoder, can be emploed after the jont ource codng and modulaton. However, the anal requre M-dm ntegral, whch not mple. In th paper, that dffcult wll be olved. In the future, performance of the jont ource codng and modulaton ncludng channel codng wll be preented. Th paper focue on the jont ource codng and modulaton onl. II. SYSTEM MODEL A. Y-3 Modulaton n Two Dmenon Y-3 modulaton contan three mbol. An alphabet tream wth unequal probablt can be mapped nto a bnar bt tream b ung the Huffman ource codng algorthm. Table how an example of the ource encoder, whch map a ource alphabet tream of three alphabet { abc,,} nto a bt tream of output bt {,,} or a contellaton pont tream of contellaton { S, S, S }, wth probablte {.5,.5,.5}, repectvel. The entrop of the output mbol from the ource encoder can be computed a [3] H( X) = log log.5 bt. = 4 4 () And the bandwdth effcenc of the jont ource codng and modulaton /9/$. 9 IEEE

2 η = + + =.5 bt/ Hz. () 4 4 Hence, the propoed cheme acheve the bet bandwdth effcenc. And the gnal contellaton can be wrtten a π π Sm = E co m, E n m (3) 3 3 where m =,,. σ = 3x+ ln σ = 3x ln 3 =. (5) E, E, ( E,) Fgure. (a) Conventonal djont ource codng and modulaton, and (b) propoed jont ource codng. Table. Output mbol generated b the Huffman ource encoder lted wth ther pror probablte. Fgure. Propoed gnal contellaton of Y-3 for the propoed jont ource codng and modulaton cheme. Refer to Fgure 3 for the optmum decon boundar equaton. Modulaton Contellaton Pont Output Bt Pror Probablt S.5 S.5 S.5 Fgure how the propoed gnal contellaton wth equal mbol energ for the propoed jont ource codng and modulaton cheme. Snce S, S, and S have dfferent probablt, we appl the maxmum a poteror probablt (MAP) rule ntead of the maxmum lkelhood (ML) to fnd the optmum decon regon. Then, the decon boundar lne between two mbol S, and S j, hould atf S ) S x S jx x+ S S j + σ ln = (4) j ) becaue t ue the MAP rule, where S x and S are the x and coordnate of mbol S, repectvel, σ the noe varance, and P(S ) the pror probablt of modulaton mbol, =,,, j. From equaton (4), we can derve the decon boundare for the Y-3 contellaton n Fgure a Th paper can conder other ource encoder, but the Huffman ource codng one of the bet. The concluon n th paper can be modfed lghtl to be applcable for other ource codng. Fgure 3. Optmum decon boundar equaton baed on the MAP for the propoed gnal contellaton of Y-3 n Fgure. Note that the crong pont of three optmum decon boundar lne far to the left of the orgn becaue three contellaton pont have unequal probablt. Th can be the caue for the mproved performance, compared to the conventonal cheme becaue the crong pont of the conventonal djont ource codng and modulaton cheme equal to the orgn, and there no room for mprovement. B. Y-hape and Z-hape n N-dmenon In [4], a look-up table wa ued for a hgh-dmenonal cae of up to 3 dmenon. However, th paper propoe a formula-baed method for an n-dmenonal gnal contellaton degn. In the n-dmenon cae, we could form a ( n + ) n

3 matrx ung the -th gnal contellaton S n order to determne the coordnate of each mbol where S = ( S S S 3 S n), =,,, n+. (6) If the conventonal Gram Schmdt procedure ued to fnd the n-dmenonal gnal contellaton, t take too much calculaton and tme n determnng the locaton of the contellaton pont for large n, e.g., n 4. So, th paper propoe a new technque for degnng n-dmenonal gnal contellaton atfng n+ n Sj = (7) and Sj = E = (8) = j= where the mbol energ wa normalzed to n (8). So the overall ( n + ) n contellaton matrx can be wrtten a S S n S S S S. (9) 3 3 n n S S S33 S nn n n n Smlar to the Y-3 propoed for two dmenon, the decon boundar lne between two mbol S, and S j, hould atf S ) S x S ln jx x + + S x S jx x n n n + σ =. j ) () In four dmenon, we have two dfferent branche, whch are Y-5 and Z-5. Ther properte are lted n Table : Table. Output mbol generated b the Huffman ource encoder lted wth ther pror probablte for Y-5 and Z-5. Modulaton Contellaton Pont Output Bt of Y-5 Pror Prob. of Y-5 Output Bt of Z-5 Pror Prob. of Z-5 S.5.5 S.5.5 S.5.5 S S III. PROBABILITY OF ERROR Proak derve the probablt of the par we mbol error n term of the mnmum dtance between mbol and j, and gnal-to-noe rato [5] a d ( ) P ( ) j SER = Pj E = S j S kq () N o where Q(α) the tal probablt,.e., the ntegral of the normal Gauan dent functon from α to the nfnt. So, the mbol error probablt of the QPSK cae ( ) SER Q SNR () becaue the mnmum dtance of QPSK between two contellaton pont E = E and k =. In the cae of Y-3, frt we determne the condtonal correct decon probablt of each mbol [5], [6]. For example, the correct decon probablt for gven mbol S σ ( x, ) x> ln and 3 E PCS ( ) = P. (3) σ σ 3x ln< < 3x+ B applng normal dtrbuton to the equaton above, a well a P( C ) and P( C ), we can obtan a general equaton a t P( C ) = ln e c 3SNR π 3 3SNR ln (4) Q 3t + a dt ln Q 3t + + b where a b c Ung equaton (4), we can fnd the average mbol error rate a Pe = P( C S ) + P( C S ) + P( C S ) 4 4. (5) For the BER, we fnd the condtonal probablt of mbol error that a mbol mapped to another mbol a ) = e π (6) ln Q t + 3SNRdt b

4 t ) = e π (7) ln Q t 3SNRdt In order to mplf the calculaton for other mbol, nce we know that the dtance between two mbol equal, we mpl rotate other mbol to the poton of S. Then, ln t ) = e π (8) ln Q t + 3SNRdt ) = e ln π (9) Q t 3SNRdt 3 ) = e ln π () ln Q t + 3SNRdt and ln t ) = e π. () Q t 3SNRdt 3 Then, the average BER can be wrtten a Total#of error bt BER = () Total# of tranmtted bt P S + P S + ) + ) 4 + ) + ) 4 =. (3).5 Th can be compared to the BER of conventonal QPSK a BER = SER. (4) IV. ANALYSIS AND SIMULATION RESULTS Fgure 4 how the theoretcal and mulaton reult for the propoed jont ource codng and modulaton wth Y-3 contellaton. It oberved that the propoed Y-3 ha lower BER than the conventonal BPSK or QPSK, from 4 db to hgh SNR, about.3 db better than BPSK at BER equal to -5. It alo oberved that the anal reult agree wth mulaton reult. Fgure 5 compare the BER reult between the propoed Y-4 and the conventonal BPSK or QPSK [7]. It alo oberved that the propoed Y-4 ha lower BER than the conventonal BPSK or QPSK from 7 db to hgh SNR, about. db better than BPSK at BER equal to -5. Refer to Table 3. Table 3. Comparon of conventonal djont ource codng and QPSK/BPSK modulaton, and the propoed jont ource codng and Y-Shape modulaton. QPSK Y-4 Y-3 BPSK No. of Smbol Bandwdth Eff. η.75.5 Entrop.75.5 Mnmum Dt. d mn (E ) (.67E ) ( ) E Decon Rule ML MAP MAP ML E b/n BER= db 9.4 db 9.3 db 9.6 db V. CONCLUSION From Table 3, we can oberve the advantage of the propoed jont ource codng and modulaton wth Y-3 and Y-4 contellaton over the conventonal djont degn. The propoed cheme can be better than the conventonal BPSK n both bandwdth effcenc a well a energ,.e., BER. For hgher modulaton, e.g., M larger than 4, we alo oberve that the propoed cheme can be better than QPSK n both ene. In the future, non-bnar channel codng wll be ncluded n the propoed cheme, uch a non-bnar low-dent part check code. VI. ACKNOWLEDGEMENT Th work wa partl ponored b the Arm Reearch Offce under DEPSCoR ARO Grant W9-NF-8--56, and b NASA under EPSCoR CAN Grant NNX8AV84A. The vew and concluon n th document are thoe of the author and hould not be nterpreted a repreentng the offcal polce, ether expreed or mpled, of the ARO, NASA, or the U.S. government. REFERENCES [] Shvratna Gr Srnvaan and Maheh K. Varana, Contellaton Degn wth Unequal Pror and New Dtance Crtra for the Low SNR Noncoherent Ralegh Fadng Channel, 5 Conference on Informaton Scence and Stem, the John Hopkn Unvert, March 6-8, 5.

5 [] Ranjan K. Mallk and George K. Karagannd, Equal-Gan Combnng wth Unequal Energ Contellaton, IEEE Tran. Wrele Communcaton, vol. 6, no. 3, March 7. [3] Thoma M. Cover and Jo A. Thoma, Element of Informaton Theor Second Edton. [4] A. K. Khandan and P. Kabal, Shapng of Multdmenonal Sgnal Contellaton Ung a Lookup Table, IEEE Tran. Informaton Theor, vol. 4, no. 6, November 994. [5] John G. Proak, Dgtal Communcaton, Fourth Edton. [6] Athanao Papoul and S. Unnkrhna Plla, Probablt, Random Varable and Stochatc Procee, Fourth Edton. [7] John G. Proak, Maoud Saleh, and Gerhard Bauch, Contemporar Communcaton Stem ung MATLAB and Smulnk, Second Edton. Fgure 4. Smulaton and theoretcal BER/SER veru bt energ-to-thermal noe power pectral dent rato, E b /N, n db for the propoed jont ource codng and modulaton of Y-3 contellaton. Fgure 5. Smulated BER/SER veru bt energ-to-thermal noe power pectral dent rato, E b /N, n db for the propoed jont ource codng and modulaton of Y-4 contellaton.