End-to-End BER Analysis of Space Shift Keying in Decode-and-Forward Cooperative Relaying

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1 3 IEEE Wirele Communication and Networking Conference (WCNC: PHY End-to-End BER Analyi of Space Shift Keying in Decode-and-Forwa Cooperative Relaying Pritam Som and A. Chockalingam Department of ECE, Indian Intitute of Science, Bangalore 56, India Abtract Space hift keying (SSK i a pecial cae of patial modulation (SM, which i a relatively new modulation technique that i getting recognized to be attractive in multi-antenna communication. Our new contribution in thi paper i an analytical derivation of exact cloed-form expreion for the endto-end bit error rate (BER performance of SSK in decode-andforwa (DF cooperative relaying. An incremental relaying (IR cheme with election combining (SC at the detination i conidered. In SSK, ince the information i carried by the tranmit antenna index, traditional election combining method baed on intantaneou SNR can not be directly ued. To overcome thi problem, we propoe to do election between direct and relayed path baed on the Euclidean ditance between column of the channel matrix. With thi election metric, an exact analytical expreion for the end-to-end BER i derived in cloed-form. Analytical reult are hown to match with imulation reult. Keywo: Space hift keying, decode-and-forwa, cooperative relaying, incremental relaying, election combining, BER analyi. I. INTRODUCTION Multi-antenna communication technique are increaingly getting adopted in emerging wirele tanda and ytem. An iue with multi-antenna communication i the need to have multiple radio frequency (RF chain in communication terminal, which reult in increaed complexity/cot and iue related to inter-antenna ynchronization. Spatial modulation (SM i a relatively a new modulation technique which allow the ue of le number of tranmit RF chain than the number of tranmit antenna, while achieving high pectral efficiencie [. Space hift keying (SSK i a pecial cae of SM. In SSK,n t m tranmit antenna and one tranmit RF chain are ued for ignaling. In each channel ue, a group of m information bit are ued to chooe one among n t m tranmit antenna. A ignal, ay +, which i known to the receiver, i tranmitted on thi choen antenna. The remaining n t antenna remain ilent. By doing o, the problem of detection at the receiver become one of merely finding out which antenna i tranmitting. Alo, the pectral efficiency i m bpcu. Analytical evaluation of the performance of SSK i of interet. In thi paper, our focu i on the bit error rate (BER analyi of SSK in cooperative relaying. Work in the recent pat have analyzed the performance of SSK and SM on fading channel [-[8. Several of thee performance analyi work conider point-to-point fading channel, and only few of them conider relay channel. For example, BER analyi of SSK in variou point-to-point fading cenario have been reported in [-[6; analyi for i.i.d. * Thi work wa upported in part by a gift from the Cico Univerity Reearch Program, a corporate advied fund of Silicon Valley Community Foundation. Rayleigh fading MIMO in [, Nakagami-m fading MISO in [3,[4, Rician fading MIMO in [5, and Rayleigh fading MIMO with imperfect CSI in [6. Likewie, BER analye of SM in point-to-point fading have been reported in [7,[8; analyi for i.i.d. Rayleigh fading MIMO in [7, and correlated Rayleigh and Nakagami-m fading MIMO in [8. We note that the reported BER analyi work on relay channel are fewer compared to thoe on point-to-point channel. For example, BER analyi of SSK in dual-hop amplify-andforwa (AF relaying i reported in [9. Our contribution in thi paper make a new addition to the BER analyi literature of SSK with relaying. In particular, we derive exact cloed-form expreion for the end-to-end BER of SSK in decode-and-forwa (DF cooperative relaying, which, to our knowledge, ha not been reported o far. In dual-hop relaying (like the cheme conidered in [9, there are two link, one from ource to relay (S-to-R and another from relay to detination (R-to-D. In cooperative relaying [, on the other hand, a direct link from ource to detination (S-to-D i preent, in addition to the S-to-R and R- to-d link. We conider cooperative relaying, which i more general than dual-hop relaying. In cooperative relaying, the combining of S-to-D and R-to-D ignal at the detination can be done uing one of maximal-ratio combining (MRC, equal-gain-combining (EGC, and election combining (SC. We conider decode-and-forwa incremental relaying [ with election combining at the detination in thi paper. A difficulty that arie in doing election combining with SSK i that traditional election combining method baed on intantaneou SNR [ can not be directly ued in SSK. Thi i becaue, unlike in traditional modulation method, information i carried in the antenna index in SSK. We overcome thi problem by doing election baed on the Euclidean ditance between column of the channel matrix of the R-to-D and the S-to-D link. We ue a imilar metric for incremental relaying a well. For thi ytem, we derive exact cloed-form expreion for the end-to-end BER and validate the analytical reult through imulation. II. SYSTEM MODEL Conider a cooperative relaying ytem coniting of a ource, relay, and detination a hown Fig.. The ource and relay are equipped with n tranmit antenna. Tranmiion from the ource and relay ue SSK. The mapping from data bit to SSK ignal for tranmiion i hown in Table I. The relay and detination are equipped n r and n d receive antenna, repectively, n r,n d. The communication happen in two phae. In the firt phae, the ource (S tranmit u /3/$3. 3 IEEE 3465

2 data Fig.. Source H r n n r Relay Cooperative relaying with SSK. H n H n d Detination ing SSK, which i hea by both the relay (R and detination (D. The received ignal vector at R and D, repectively, are Data bit SSK tx. ignal vector,x Antenna tatu [, T Ant. #: +, Ant. #: OFF [, T Ant. #: OFF, Ant. #: + TABLE I DATA BIT TO SSK SIGNAL MAPPING FOR n. y r H r x+w r ( y H x+w, ( wherexi the SSK ignal vector tranmitted from S (defined in Table I, H r and H are the n r n S-to-R channel matrix andn d n S-to-D channel matrix, repectively. Each element ofh r andh are modeled a i.i.d. CN(,σ r and CN(,σ, repectively. w r C nr and w C n d are additive noie vector at R and D, repectively. Incremental Relaying (IR: In the econd phae, the relay check whether a certain metric (η, which ignifie the quality of the S-to-D channel, i below a pre-determined threhold (µ. If η < µ, the relay decode the SSK ymbol tranmitted by the ource in the firt phae, and tranmit the decoded SSK ymbol to the detination. Let x r denote the SSK ignal vector tranmitted by the relay. If η µ, R doe not tranmit in the econd phae. The ignal received at D from R i y H x r +w, (3 where H C n d n i the R-to-D channel matrix whoe entrie are modeled a i.i.d. CN(,σ. w C n d i the additive noie vector. We aume that the relay ue maximum likelihood (ML detection rule [: x r arg min ǫ S n y r H r ǫ, (4 where S n i the SSK ymbol alphabet correponding to n tranmit antenna. Selection Combining at the Detination: At the end of the two phae of tranmiion, D procee either the received ample y and y (if η < µ, or only y (if η µ. In the later cae, D perform ML detection ony. In the former cae, election combining i performed among the two received vector y and y. Thi combining i done baed on two metric, η and η. The metric η ignifie the quality of the R-to-D channel. In thi election combining, only one among y and y i elected for receiver proceing, i.e., D procee y for ML detection if η > η, and procee y otherwie. The element of the additive noie vector in all the three channel are modeled a i.i.d. CN(,σ. To take into account path lo, we define σr d α r, σ dα, and σ dα, where d r, d r and d r are the ditance between S-to-R, S-to-D and R-to-D, repectively, and α i the path lo exponent. Selection Metric: For the purpoe of incremental relaying and election combining in SSK, we define the following metric: η h σ, and η h σ. Here, h i andh i denote theith column ofh andh, repectively. The intuition behind chooing the parameter η in uch a way i driven by the fact that the BER of the S-to-D channel ( conditioned on H i Q h σ [, and likewie for chooing η. III. END-TO-END BER ANALYSIS The average end-to-end bit error probability i given by [ P(E P(η µp(e η µ+p(η < µ P(E r η < µ P(E E r,η < µ+p(er c η < µp(e Er c,η < µ, (5 where E i the end-to-end error event, E r i the error event in the S-to-R link, E c r i the complement of the event E r, P(η < µ P(η µ, and P(E c r η < µ P(E r η < µ. In the following, we derive cloed-form expreion for the variou probabilitie in (5. P(η µ: The event η µ can be written a η µ/, where σ and η h. Hence, P(η µ P(η µ/. η i a gamma random variable with parameter {n d,σ. Therefore, P(η µ µ/ n d j nd exp( σ (σ n d (n d! d j! ( µ σ j exp( µ where (6 follow from the gamma ditribution CDF. σ (6, P(E η µ: When η µ, the relay doe not tranmit, and an error occur due to event E (the error event in the S-to-D link. Therefore, P(E η µ P(E η µ. The conditional PDF of η given η µ i given by f η ( η µ nd exp( σ (σ n d (n d!p(η µ, if µ, otherwie. (7 The conditional error probability on S-to-D link i [ P(E H Q ( η. (8 3466

3 Uing (7 and (8, we can write P(E η µ Q( µ µ P(η µ Q( f η ( η µ d n d j u j exp( u ω µ (σ j j! du, (9 π where ω σ σ +, and (9 follow after few tep involving changing oer of integral and uing the CDF of gamma ditribution. Subtituting u withz ω in the integral in (9, we can write u j exp ( u du ω j+ I (j( µ, ( µ ω ω wherei (j (α α zj exp( z dz. For j,i( (α πq(α. When j i any poitive integer, I (j (α can be computed uing the following lemma. α zj e z Lemma : IfI (j (α dz, then for any poitive integer j, the following relation hold j I (j (α α k (j!! (k!! eα +(j!!i ( (α, ( k where (j!! (j.(j Proof: Proof can be hown by induction, which i omitted due to page limit. Uing Lemma in (, (9 can be implified a P(E η µ Q( { ωq ( µ µ P(η µ ω n d j ω j+ (j!! (σ j j! [ j ( µ e ωk µ ( ω µ +Q. ( π(k!! ω k P(E r η < µ: Since E r i independent of η, we have P(E r η < µ P(E r, and P(E r can be derived a P(E r [ n r δ j (j! δ, (3 (4σr j (j! whereδ σ r +σ. See Appendix I for the derivation of (3. r P(E E r,η < µ: P(E E r,η < µ i the average probability of the bit error at D, given R tranmit the incorrectly decoded x, i.e., x r x. In election combining, the received vector y i choen for proceing at the detination if η > η, i.e., η > η, where η h. In thi cae, the error at the detination i due to the error event in R-to-D link (denoted bye. Therefore, we can write j P(E E r,η < µ,h,h P(E H, for η > η P(E E r,h, for η η <, (4 where P(E E r,h can be obtained a P(E E r,h Q( η. (5 See Appendix II for the derivation of (5. Subtituting (8, (5 in (4, and ubequently on averaging over the denitie ofη andη,p(e E r,η <µ,h,h can be written a P(E E r,η < µ P(η < µ + { f η (rf η (d {{ A Q( f η (rf η (d {{ A Q( rf η (rf η (d {{ A. (6 Expreion for the term A and A in (6 are derived in Appendice III and IV, repectively. Expreion for the A term i derived a follow. A n d j n d j [ nd j r nd exp( r σ (σ n d (nd! drf η (d j e σ (σ j j! λ j+n d ( j+n d j σ j σn d λ j+n d ( j+n d j σ j σn d n d e σ d (σ n d (nd! j+n d e λ d (λ j+n d (j +nd! (7 [ j+n d µ i e µ λ, (8 (λ i i! where λ σ σ, and (7 and (8 follow from the CDF σ +σ of gamma ditribution. Subtituting the expreion of A, A anda from (38, (44 and (8 repectively in (6, we get P(E E r,η < µ. P(E Er,η c < µ: P(E Er,η c < µ i the conditional probability of end-to-end bit error, given that the relay tranmit the correctly decoded x, i.e.,x r x. Therefore, i P(E E c r,η < µ,h,h Q( η Q( η, for η > η, for η η <, (9 which, on averaging over the denitie of η and η, can be written a { P(E Er,η c <µ P(η < µ Q( f η (rf η (d + {{ A Q( rf η (rf η (d {{ A. ( Subtituting the expreion ofa anda from (38 and (44, repectively, in (, we get P(E E c r,η < µ. Finally, ubtituting the expreion for the probabilitie in (5, we get a cloed-form exact expreion for the end-to-end BER. IV. NUMERICAL RESULTS In thi ection, we preent numerical reult of the BER performance obtained by computing the analytical BER expreion derived in the previou ection. We alo preent the 3467

4 BER 3 4 n r n d, d r d d No relay DF IRSC, imulation, µ 6 db DF IRSC, analyi, µ 6 db DF IRSC, imulation, µ db DF IRSC, analyi, µ db BER 3 4 SNR6 db, imulation SNR6 db, analyi SNR9 db, imulation SNR9 db, analyi SNR db, imulation SNR db, analyi SNR5 db, imulation SNR5 db, analyi DF IRSC, n r n d, d r d d, SNR (db Fig.. BER of SSK in DF IR-SC cooperative relaying cheme for n r n d, d r d d for different threhold value (µ 6 db, db. Analyi and imulation. imulated BER performance to validate the analyi. We refer to the conidered ytem a DF IR-SC (decode-and-forwa, incremental relaying-election combining cooperative relaying cheme. The path lo exponent α i taken to be 3 in all the plot. In Fig., we plot the BER of SSK in DF IR-SC cheme a a function of SNR, obtained through both analyi and imulation, for n n r n d, inter-node ditance d r d d, and threhold value µ 6 db, db. The performance with no relaying (ytem with S-to-D link only i alo plotted for comparion. It can be een thati the relaying cheme achieve better performance compared to the no-relaying cheme, and ii the comparion between the analytical and imulation reult how an exact match, thu validating the analyi. Figure 3 how the BER plot a a function of the threhold µ, for different value of SNR (6 db, 9 db, db, and 5 db andn n r n d,d r d d. The plot how that i the optimum threhold value that minimize BER vary for varying SNR, and ii analyi and imulation reult match in thi figure a well. In Fig. 4, we plot the analytical BER veru SNR plot for DF IR-SC at optimum threhold value, i,e., at µ opt (optimum µ for n r n d,4, and d r d d. µ opt value and the correponding BER for a given SNR are obtained by numerically evaluating the BER expreion at the given SNR by varying the threhold value in. db tep. Alo,c /SNR,c /SNR 4, c 3 /SNR 8 line are plotted (c,c,c 3 are contant. It can be een that the BER plot for no relaying withn d and 4 run parallel to the c /SNR and c /SNR 4 line at high SNR indicating nd and 4th oer diverity. With relaying, on the other hand, the correponding BER plot with n r n d and 4 run parallel to thec /SNR 4 andc 3 /SNR 8 line at high SNR indicating 4th and 8th oer diverity. Thi how the cooperative diverity advantage of DF IR-SC in SSK. V. CONCLUSIONS We analyzed the end-to-end BER performance of SSK in DF incremental relaying with election combining. Since the information in SSK i carried in the antenna index, we propoed to do election between direct and relayed path baed on Threhold µ (db Fig. 3. BER veru µ plot for SSK in DF IR-SC cooperative relaying cheme for n r n d, d r d d for different SNR value (SNR 6 db, 9 db, db, and 5 db. Analyi and imulation. BER 5 5 No relay, n d DF IRSC, µ opt, n r n d No relay, n d 4 DF IRSC, µ opt, n r n d 4 c/snr c/snr 4 c3/snr 8 d r d d c/snr 4 c/snr c3/snr SNR (db Fig. 4. BER veru SNR plot for SSK in DF IR-SC at optimum threhold value (µ opt forn r n d,4; d r d d. Analyi. the Euclidean ditance between column of the channel matrix. We ued a imilar metric for incremental relaying. We derived exact expreion for the end-to-end BER in cloedform. Analytical reult were validated through imulation. APPENDIX I DERIVATION OF (3 Expreion for (3 i derived a follow. P(E r e u u / n r [ n r j n r j du nr exp( σr π (σr nr (n r! d nd exp( σr (σr nr (n r! deu j u j exp( u σr j!(σr j u j exp( u δ j!(σ r j π du j!(σ r j δ π du e u π du ( u j exp( u du. πδ ( δ 3468

5 ( follow from the definition of the CDF of gamma ditribution. Uing the expreion of the jth central moment of normal r.v. (i.e.,δ j (j! j! j in (, we get (3. We have APPENDIX II DERIVATION OF (5 P(E E r,h P(E c E r,h, (3 where E c i the event of correct detection at D. Denoting x [, T, and x [, T, we can write by the law of total probability P(E c E r,h i P(Ec E r,h,x r x i. (4 We firt compute P(E c E r,h,x r x a follow. P(E c E r,h,x r x P( y H x > y H x x rx P (R{(h h H w > h h. (5 R{(h h H w i ditributed a CN(, h h σ. Hence, uing (5, we can write ( P(E E c r,h,x r x Q h. (6 σ Proceeding imilarly, we can derive ( P(E E c r,h,x r x Q h. (7 σ Hence, uing (6, (7 and (4, we get ( P(E E c r,h Q h. (8 σ Uing (3 and (8, we arrive at (5. APPENDIX III DERIVATION OF A Expreion fora i derived a follow. A Q( r nd exp( r σ (σ n d (nd! drf η (d Q( Q( [ n d j j e σ (σ j j! n d e σ (σ n d (nd! d {{ A n d j f η (d (9 Q ( j+n d e λ (σ j (σ n d j!(nd! d {{ A j. (3 where λ σ σ. We firt derive A σ j a follow. +σ A j B j B j [ µ e u u j+nd e λ du π (λ j+n d (j +nd! d j+nd e λ (λ j+n d (j +nd! deu π du {{ A j,a j+nd e λ (λ j+n d (j +nd! deu {{ A j,b µ where B j λj+n d ( j+n d σ j σn d j π du, (3. Uing the the CDF of gamma ditribution in the inner integral of A j,a in (3 a before, we can write µ A j,a [ [ e u π du µ e u du π j+n d i du π µ e u u i e u τ (λ i i! π du j+n d i (λ i i! u i e u τ du u i e u τ du, (3 π µ π where τ λ λ+. Uing the expreion of ith central moment of normal r.v. ( i.e.,τ i (i! i!, we can write i u i e u τ du τ π τ i+ u i e u τ du πτ (i! i!i+. (33 Subtituting u with z u τ in the integral µ u i e u τ du in (3, we get u i e u τ du τ i+ z i e z dz µ µ τ ( τ i+ I (i µ. (34 τ Uing the derivation of (33 and (34 in (3, we get the expreion of A j,a a A j,a j+n Q( d { τ i+ (i! µ (λ i i! i i! i+i(i ( µ τ. π (35 Next, from the gamma ditribution CDF, we can write A j,b a A j,b Q( [ j+n d µ l e µ λ µ (λ l. (36 l! l The integral A can be derived imilarly a A j, and the correponding final expreion i 3469

6 A n d Q( [ ω t+ (t! µ (σ t t! t π t! t+ I(t( µ ω +Q( [ n d µ q e µ σ µ (σ. (37 q q! q Subtituting the derivation ofa j,a anda j,b from (35 and (36, repectively, in (3, we get the expreion of A j. Subequently, combining the the expreion of A in (37 and the expreion of A j thu obtained, we get A a A n d Q( [ ω t+ (t! µ (σ t t! t π t! t+ I(t( µ ω +Q( [ n d µ q e µ σ n d [ µ (σ B j q q! Q( µ q j+n d τ i+ { (i! I (λ i i! i! i+ i Q( { µ j+n d l j (i( µ τ π µ l e µ λ. (38 (λ l l! APPENDIX IV DERIVATION OF A We conider the inner integral ofa, i.e., Q( rf η (r. Simplifying the above integral following the tep that involve interchange of the oer of the integral and ue of CDF of gamma ditribution, we can writea a A n d Q ( j+n d e λ j (σ j (σ n d j!(nd! d {{ A n d i u i exp( u du ζ (σ i i! π f η (d, (39 {{ A where ζ σ σ +. Comparing (3 and (39, it can be een that A n d j A j. Hence, A can be derived from (35 and (36. To derive A in (39, we firt ubtitute u with ζz. The inner integral of A can be implified a u i exp( u du ζ (σ i i! π ζ i+ (σ i i! π I(i(. (4 ζ Expanding I (i ( ζ uing lemma in (4, A become A H Q( ζ f η (d {{ A p E p n d + p k J k k e ζf η (d, {{ A,k (4 whereh n d ζ i+ (i!,e (4σ i i (i! p ζp+ ( p p,j (4σ i π k k A in (4 can be derived in a imilar way aa j, and can be written a ζ k (k!!. A n d µ Q( ζ q [ µ +Q( ζ ζ q χ q + (σ q q! n d q µ q e µ σ (σ q q! π { (q! I ( (q µ q ζχ! q +, (4 where χ σ σ +ζ. The ubtitution of with z v in A,k in (4, where v σ +ζ, give ζσ A,k Γ i( vµ,n d+k (σ n d(n d!v n d +k, (43 where Γ i (m,n m ez z n dz i the incomplete gamma function. Subtituting the derivation of A and A,k from (4 and (43, repectively, in (4, we get the expreion of A. Subequently combining the the expreion of A derived from (3, (35 and (36, and the expreion ofa thu obtained, and ubtituting them in (39, we get A a n d A j [ j+n d B j Q( µ t ( µ I (t τ Q( { µ π i k τ t+ { (t! (λ t t! t! t+ j+n d l µ l e µ λ (λ l l! n d i HA + E i J k A,k, (44 where A and A,k i given in (4 and (43, repectively. REFERENCES [ R. Y. Meleh, H. Haa, S. Sinanovi, C. W. Ahn, and S. Yun, Spatial modulation, IEEE Tran. Veh. Tech., vol. 57, no. 4, pp. 8-4, Jul. 8. [ J. Jeganathan, A. Ghrayeb, L. Szczecinki, and A. Ceron, Space hift keying modulation for MIMO channel, IEEE Tran. Wirele Commun., vol. 8, no. 7, pp , Jul. 9. [3 M. D. Renzo and H. Haa, A general framework for performance analyi of pace hift keying (SSK modulation for MISO correlated Nakagami-m fading channel, IEEE Tran. Commun., vol. 58, no. 9, pp , Sep.. [4 M. D. Renzo and H. Haa, Space hift keying (SSK modulation with partial channel tate information: Optimal detector and performance analyi over fading channel, IEEE Tran. Commun., vol. 58, no., pp , Nov.. [5 M. D. Renzo and H. Haa, Space hift keying (SSK- MIMO over correlated Rician fading channel: Performance analyi and a new method for tranmit-diverity, IEEE Tran. Commun., vol. 59, no., pp. 6-9, Jan.. [6 M. D. Renzo, D. D. Leonai, F. Grazioi, and H. Haa, Space hift keying (SSK- MIMO with practical channel etimate, IEEE Tran. Commun., vol. 6, no. 4, pp. 998-, Apr.. [7 J. Jeganathan, A. Ghrayeb, and L. Szczecinki, Spatial modulation: optimal detection and performance analyi, IEEE Commun. Lett., vol., no. 8, pp , Aug. 8. [8 M. D. Renzo, D. D. Leonai, F. Grazioi, and H. Haa, Bit error probability of SM-MIMO over generalized fading channel, IEEE Tran. Veh. Tech., vol. 6, no. 3, pp. 4-4, Mar.. [9 R. Meleh, S. Ikki, and M. Alwakeel, Performance analyi of pace hift keying with amplify and forwa relaying, IEEE Commun. Lett., vol. 5, no., pp , Dec.. [ J. Laneman, D. Te, and G. Wornell, Cooperative diverity in wirele network: efficient protocol and outage behavior, IEEE Tran. Inf. Theory., vol. 5, pp , Dec. 4. [ M. D. Selvaraj and R. K. Mallik, Error analyi of the decode and forwa protocol with election combining, IEEE Tran. Wirele Comm., vol. 8, no. 6, pp , Jun