Optimal Pilot Symbols Ratio in terms of Spectrum and Energy Efficiency in Uplink CoMP Networks

Optimal Pilot Symbols Ratio in te rms of Spectrum and Ener gy Ef ficienc y in Up link CoMP Netw o rks Y uhao Zhang, Qimei Cui, and Ning W ang School of Informatio n and Communicatio n Engineer in g, Beijing University of Posts an d T elecomm unication s, B eijing, 10087 6, China Email: cuiqimei@bupt.edu. cn Abstract —In wireless networks, Spectrum Efficiency (SE) and Energy Efficiency (EE) can be a ffected by the channel estimation that needs to be well designed in practice. In th is paper , considering channel estimation err or an d non-id eal backhaul links, we optimize the pilot symbols ratio in terms of SE and EE in uplin k Coordinated Multi-p oint (CoMP) networks. Modeling the channel estimation error , we f ormulate the SE and EE maximization problems by analyzing the system capacity with imperfect channel estimation. The maximal system capacity in SE optimization and the minimal transmit p ower i n EE optimization, which both hav e the closed-form expr essions, are deriv ed by some reas onable approximations t o reduce the complexity of solving complicated equations. Simulations are carried out to validate t h e superiority of our scheme, verify the accuracy of our approximation, and show the effect of p ilot symbols ratio. Index T erms —Spectrum efficiency , Energy efficiency , Coordi- nated multi-p oint, Channel estimation err or , Pil ot symbols ratio. I . I N T RO D U C T I O N In Coordin ated Multi-poin t (CoM P) networks, Spectrum Efficiency (SE) and Energy Efficiency (EE), the impor tant indexes in wireless network s [1], [2] , can be both improved by the provided coop erative diversity gains [3], [4]. F or uplink transmission, Joint Recep tion CoMP ( JR-CoMP), the efficient uplink CoMP scheme, is ch aracterized by simultaneous recep- tion at multiple co o perative nodes with f ully data and con tr ol informa tio n exchange [5]. It is kn own that the closed-fo rm approx imation f or EE-SE trade- off is der iv ed with idealistic and realistic accuracy dem onstration on th e assump tion th at the bac k haul links are pr efect [3]. Howe ver, ideal ba ckhaul links and perfect Channel State Infor mation (CSI) ar e lim- ited and impossible in prac tica l networks [6]. T herefor e, w e mainly study the SE and EE maximization based on non-ideal backhau l links and imperfect CSI. In prac tice, on ly imp erfect and non -real-time (lo ng-term ) CSI c a n b e obtain ed and exchanged in th e network s d ue to the chan nel estimation error and the capacity limited bac khaul links. Th e se limitation s need to be well c onsidered and have already catch e d much research attentio n s. In [7 ] , con sid e ring the b ackhaul link constraint, different the o retical up lin k CoMP concepts are analyzed. And the fra m ew ork inco rporatin g these concepts is pr ovided with pr actical CoMP algorith ms. I n [8], the optimal clustering an d rate allocation scheme for JR-CoMP networks with delayed CSI feedb ack is proposed by takin g a stochastic decision appr oach. However , these works only assume no n-ideal b ackhaul link s a nd imp erfect CSI sha r ing but without the consider a tion of channel estimatio n e r ror . It is k nown that the chan nel estimatio n is usually carrie d out by p eriodic p ilot o r tra ining sym bols tr ansmission, wh e r e the estimation error o ccurs inevitably [9]–[1 1]. It is straig ht- forward that too much and too less pilo t sym bols in chan nel estimation will b oth affect the sy stem perfo rmance, so its suitable ratio ne e d s to be determined rationally . I n terms o f SE and EE, decre a sin g tr a n smit p ower and red u cing p ilot sym bols will overcome system throughpu t d eterioration , wh ich un fortu- nately leads to the degradation o f Bit Er ror Probability ( BEP) perfor mance [9]. In ord e r to fin d th e optimal trade-off, the power and spacing of the pilot symbols are op timized to maximize SE in ad aptive modulatio n OFDM n etworks with the imperfect CSI [10]. And in [11], closed-form BEP expr ession is derived fo r Filter Bank Mu lticarrier (FBM) networks, based on which the optimal power allo cation between pilot an d data symb ols is propo sed to minimize BEP . In up link CoMP networks with perfect b ackhau l links, the chan n el estimation error is modeled and the sy stem-level co mputer simulatio n is con ducted to in vestigate system and user thr oughp ut [12]. Howe ver, to the best o f our knowledge, the pilot sym bols assignment for op timal SE an d E E in JR-CoMP n etworks with non-id eal backhaul links has never bee n discussed so far . In this p aper, we investigate optimal p ilot sy mbols ratio fo r channel estimation to m aximize SE and E E in u plink CoM P networks with non-id eal an d limited backhau l links. Modeling the cha n nel estimation error, we first for mulate the SE an d EE maxim iz a tio n pro blems by der i ving th e system c a pacity . Utilizing the optimal deriv ative cond itio ns in SE maxim ization and the Lagr ange multiplier method in SE maximization , th e optimal pilot symbols r atio at e very coo perative node are derived, b a sed on which th e maxim a l system cap acity in SE optimization and the minimal transmit power in EE optimiza - tion are obtained consequently . T o av oid solving com plicated equations, we intr o duce som e rea so nable app roximate rela- tionships to g e t the closed - form expressions of these re sults. Finally , numer ic a l simulations validate th e sup eriority of our scheme and the accuracy of our approximation , and rev eal that with the incr e a se of pilot symb ols ratio, SE and EE first rise rapidly in channel estimation limited r egion, and then decrease slowly in useful informatio n limited r egion. The rest of the paper is organized as follows. System model is described in Section II. SE and EE optimization s are reso lved in Section III and Section IV. Simulations ar e provided in Section V, followed by con clusion in Section VI. I I . S Y S T E M M O D E L In u plink JR-CoMP n e tworks, a comm on single-an tenna User Equipmen t (UE ) transmits in f ormation to M co ordinate d Base Stations (BSs), which ca n p rocess the received signal cooper a tively . These M BSs are conne c te d thro ugh non -ideal and limited backh aul links ( e.g. , wireless links), which ca n not afford too much traffic loads. Ther efore, with the c o nsideration of backh a ul exchange delay an d in order to reduce backh aul overhead, only long- term CSI can be exchan g ed during long time interval among the M coordin ated BSs. Due to th e limitation, only th e non- coheren t JR-CoMP ca n work in th is situation. Mo reover , the CSI is detected and estimated at each BS with er rors, i.e. , only im perfect CSI can be ob tain ed. The flat bloc k fading is assumed to ch aracterize the com plex channel gain between BS m and the co mmon UE, denoted by h m . W is the system bandwidth an d n is the Ad ditiv e White Gaussian Noise (A WGN) with power N = W × N 0 , wh ere N 0 is th e Power Spectral Density (PSD) o f th e noise. A. Signa l Model In each time fr ame, the desired info rmation x is transmitted to th e cluster of M co operative BSs b y the UE with tr ansmit power P . The receiv ed signal at BS m can b e expressed as y m = √ P h m x + i m + n, (1) where E {| x | 2 } = 1 . i m , with the power I m , is the overall interferen ce at BS m cau sed by other UEs, using the same resource, outside the M coo rdinated BSs. Therefo re, with pe rfect syn chron iz a tio n am ong the clus- ter and c onsidering the imp erfect channel estimation , the non-co herent combine d received signal can be expressed as y = M X m =1 √ P b h m x + M X m =1 √ P e h m x + M X m =1 ( i m + n ) , (2) where b h m is the r esult o f im p erfect ch annel estimatio n, e h m is the correspo nding error, an d h m = b h m + e h m . B. Chann e l Estimation Err or It is kn own that the channel estimation is b a sed on pilot symbols detection f ollowed by data demodu lation in wireless networks [9]. The per iodicity of pilot symb ols depen ds on the channel c o herence over tim e, freque n cy and space etc. It is assumed tha t pilot sym bols are transmitted dur ing the channel coheren ce interval, i.e. , L sy mbols fra me in th is p aper . W e de n ote the ratio of the pilo t symbols for ch annel estimation by α m (0 ≤ α m ≤ 1) at BS m , th erefore , (1 − α m ) represents th e o ther part for useful infor m ation. Under block fading cha n nel, the Minimum Me a n Sq uare Er r or (MM SE) of the estimation at BS m can be expressed [ 9], as g iv en by e m = 1 1 + α m · L · S N R m , (3) where S N R m is SNR of receiv ed signal at BS m , given by S N R m = P | h m | 2 I m + N . (4) Hence, e h m and b h m then can be formulated in terms of e m , as written b y | e h m | 2 = | h m | 2 · e m , | b h m | 2 = | h m | 2 · (1 − e m ) . (5) C. System Capacity W ith imperfect chann el estimation, S m = P | b h m | 2 turns ou t to be the usefu l signal strength at BS m , while Z m = P | e h m | 2 degrades into th e in ter ference, which is treated as a noise for average perfor mance due to the random ness of chan nel estimation each time in pra ctice. Amon g the received power S m , α m · S m is dev oted to pilo t sym bols demod ulation, th us only (1 − α m ) · S m is a vailable for desired inform ation. Therefo re, using (2) and (5), the SNR for desired informatio n can be ob tained un der non- coheren t combination , as given by S N R = M P m =1 (1 − α m ) P | h m | 2 (1 − e m ) M P m =1 P | h m | 2 e m + M P m =1 I m + M · N . (6) For simplicity and withou t loss of generality , it is assumed that th e overall interference power I m is id entical at each BS for average perform ance. Then sub stituting (3) an d (4) into (6), th e SNR is th en tr ansforme d , as expressed by S N R = M P m =1 (1 − α m ) S N R m α m · L · S N R m 1+ α m · L · S N R m M P m =1 S N R m 1+ α m · L · S N R m + M . (7) According to Shannon capacity formula, the ach iev able uplink data rate C can be attained, as given by C = W · log 2 (1 + S N R ) . (8) I I I . S E O P T I M I Z A T I O N The SE is defined a s the rad io between th e achievable u p link data rate and th e system b andwidth, as given by η S = C W = log 2 (1 + S N R ) . (9) Due to mo noton icity of Log function , max imizing η S is equiv alent to maximizing S N R . Th us, given fixed tran smit power P , the SE op timization p roblem can b e formulated as max { α m } S N R, ( P1 ) s . t . 0 ≤ α m ≤ 1 , m ∈ { 1 , 2 , · · · , M } . It c a n be proved that ( P1 ) is con cave since its Hessian m a trix is negative d efinite and the f easible region is linear, which guaran tee the existence of exclusive maximum . Mo reover , using th is con cave prop erty , the optimal α m , denoted b y α ∗ m , must satisfies 0 < α ∗ m < 1 becau se S N R > 0 w h en 0 < α m < 1 an d S N R = 0 wh e n α m = 0 o r α m = 1 . Proposition 1: W ith fi xed transmit p ower P a t the com- mon UE , the ap pr oximate maximal system capa city C ∗ in non-co her ent JR-CoMP n etworks can be presented by C ∗ = 2 · W · lo g 2 − b S E + p b 2 S E + 4 · M · c S E 2 · M ! , (10) wher e b S E = M X m =1 2 · S N R m √ L · S N R m , c S E = M X m =1  L · S N R m + 2 L + 1  . Pr oof: The first derivati ve of S N R can be expr essed as ∂ S N R ∂ α m = − L · S N R 2 m  α 2 m · L · S N R m + 2 · α m − 1   M P m =1 S N R m 1+ α m · L · S N R m + M  (1 + α m · L · S N R m ) 2 + L · S N R 2 m  M P m =1 (1 − α m ) S N R m α m · L · S N R m 1+ α m · L · S N R m   M P m =1 S N R m 1+ α m · L · S N R m + M  2 (1 + α m · L · S N R m ) 2 (11) By letting (11) be zero and utilizin g (7), the optim al condition s for each α m can be obtained, as g iv en by S N R = α 2 m · L · S N R m + 2 · α m − 1 , ∀ m. (12) By resolving (12), two solutions will be derived, b ut on ly one is f easible due to α m > 0 , which is pre sen ted as α m = θ m − 1 L · S N R m , (13) where θ m = p 1 + L · S N R m ( S N R + 1) . Then substituting (1 3) into (7), th e maximal S N R must satisfy S N R = M P m =1  1 − θ m − 1 L · S N R m  S N R m θ m − 1 θ m M P m =1 S N R m θ m + M . (14) Therefo re, the max imal S N R , d enoted by S N R ∗ , can b e attained by solving (1 4). Howe ver, it is clear that the closed - form expression can not be derived and only num erical result is available becau se of the complex stru c tures o f ( 14). After computing the op timal S N R ∗ and sub stituting it into (13), α ∗ m can be obtained, as g iv en by α ∗ m = p 1 + L · S N R m ( S N R ∗ + 1) − 1 L · S N R m . (15) Howe ver, only the n umerical results are o btained in this way , so we furth er utilize some app roximate re la tio nships to acquire th e analytic and closed-form solu tions. W itho ut conside ring the arrangem e n t of the sig n aling alo ng the time and frequen cy , the total symb ols dur ing th e cha n- nel coh erence time in terval can be roughly calcu lated by L = B c · T c , where B c and T c are the chann el coh e rence on frequency and time, r espectiv ely . And the ty p ical value of L and B c are 10 3 and 370 K H z in practice , b ased on wh ich two appr oximate relation ships can be obtained u nder high data rate d emand, as wr itten b y 1 + L · S N R m ( S N R + 1) ≈ L · S N R m ( S N R + 1) , 1 L √ L · S N R m ≈ 0 . Utilizing the two approximate rela tio nships above, (14) can be tr ansforme d , as gi ven by M  √ S N R + 1  2 + b S E · √ S N R + 1 − c S E = 0 . (16) Resolving (16) an d co nsidering √ S N R + 1 > 0 , S N R ∗ can be derived, as given by S N R ∗ = − b S E + p b 2 S E + 4 · M · c S E 2 · M ! 2 − 1 . (17) Therefo re, by substituting (17) into ( 8), the maximal system capacity C ∗ can be obtained and f ormulated as (10). And th e n the optimal p ilo t symbo ls ratio α ∗ m can be calculated b y substituting ( 17) in to (15). I V . E E O P T I M I Z A T I O N In this section , we will d iscuss th e optim al ratio of pilot symbols to maximize EE with requ ired uplink data ra te , denoted by R ul . It is kn own that the total power con sumption of the network s contains transmit power and circuit power . The circuit power can be further decomp osed into static compon ent (drive h ardware) and d ynamic compo n ent (pro cess signal), which ar e denoted by P base and P c = ε · R ul , where ε is the power co nsumption fo r transm ittin g a data bit. T he EE is defin ed a s the radio betwe e n the av erage data rate and the av erage total power consump tion at a ll no des [1], as giv en by η E = R ul P total = R ul P + ε · R ul + P base , (18) which indicates th at given the r equired uplink d ata rate, i.e. , R ul , maxim izin g η E is equ i valent to minim izing th e total transmit power P . Therefo re, th e EE o ptimization pro blem can be formulated , as g iv en by min { α m } P, ( P2 ) s . t . SNR ≥ 2 R ul W − 1 , 0 ≤ α m ≤ 1 , m ∈ { 1 , 2 , · · · , M } . It can be easily proved that when the op timal solution is obtained, S N R = 2 R ul W − 1 m ust be satisfied ju st by reducing transmit power P to make it hold. Proposition 2: W ith the requir ed u p link d ata rate R ul in non-co her ent JR-CoMP ne tworks, th e app r oximate minimal transmit p o wer P ∗ of the c o mmon UE can be p r esented by P ∗ =       b E E + s b 2 E E + 4 · c E E · M P m =1 | h m | 2 σ 2 2 · M P m =1 | h m | 2 σ 2       2 , ( 1 9) wher e b E E = M X m =1 2 · q | h m | 2 σ 2 · 2 R ul W √ L , c E E = M  2 R ul W − 1  − 2 · M L . Pr oof: Lagr a n ge multip lier meth od is ad opted to prove this proposition , where the Lagrang e mu ltiplier is given by L E E = P − λ E E ·  S N R − 2 R ul W + 1  , (20) where λ E E is the Lagr ange coefficient. By letting ∂ 2 L E E ∂ α 2 m = 0 and utilizing (11), th e n e c essary co ndition for optimal α m is α 2 m · L · S N R m + 2 · α m − 1 = 2 R ul W . (21) By solv ing (21) an d considerin g α m > 0 , the op timal α m can be expressed, as given by α m = q 1 + L · S N R m 2 R ul W − 1 L · S N R m . (22) By sub stituting (22) into (7) and con sid ering S N R = 2 R ul W − 1 , the following equ ation can be established as M P m =1  1 − √ ϑ m − 1 L · S N R m  S N R m √ ϑ m − 1 √ ϑ m M P m =1 S N R m √ ϑ m + M = 2 R ul W − 1 , (23) where ϑ m = 1 + L · S N R m · 2 R ul W . Similarly , the optimal P , d enoted by P ∗ , can be ca lc u lated by solvin g (23), based on wh ich α ∗ m can be obtained according to ( 22). Howe ver, only th e n umerical results are acq u ired in this way d ue to the com plex structures o f ( 23). Like SE m a ximization in th e form er section, some approx - imate relationship s are intr o duced, as expressed by ϑ m ≫ 1 , ϑ m ≈ ϑ m − 1 , 1 L q L · | h m | 2 σ 2 · 2 R ul W ≈ 0 , based on wh ich, (23) can be transformed , as given by P M X m =1 | h m | 2 σ 2 − b E E √ P − c E E = 0 . (24) 0 5 10 15 20 25 30 35 40 The transmit power (W) 0 1 2 3 4 5 6 7 8 The achievable data rate (bps/Hz) α =0.001 α =0.15 α =0.3 POS AOS GAS TS Fig. 1. T he achie va ble data rate regi on for dif ferent transmit po wers. By solving (24) an d considering √ P > 0 , the op timal P can be formulated , as g iv en by (19). And th e n the optim al p ilot symbols r a tio α ∗ m can be ob tained by substituting (1 9) into (2 2). V . S I M U L A T I O N R E S U LT S In this sectio n, the typical scen ario, involving three macro BSs, is consid ered in simulations to validate our op timal p ilo ts assignment, where th e para m eters ar e specified in T able I. Besides ou r schem es, some other schemes are con ta in ed in simulations for comparison , as d escribed b y • Precise Optimizatio n Scheme (POS) : Obtain α ∗ m and calculate S N R ∗ , P ∗ by solving the eq uation p recisely . • Appr oximate Optimizat ion Scheme (A OS) : Obtain α ∗ m , S N R ∗ , and P ∗ by approxim ate expressions. • Genetic Algo r ithm Scheme (GAS) : Obtain α ∗ m by com- plicated genetic algo rithm in Matlab. • T ra dit io nal Scheme (T A) : Distribute the same α for each BS with out optimization. T ABLE I S I M U L AT I O N P A R A M E T E R S Parame ters V alues System bandwidt h ( W ) 10 MHz Noise po wer spectra l density ( N 0 ) – 174 dBm/Hz Cooperat i ve BSs number ( M ) 3 Distance between UE and BSs ( d 1 , d 2 , d 3 ) 200 ,250,300 m A vera ge path loss (PL) 30 + 40 log 10 d dB Maximum transmit po wer ( P max ) 46 dBm Static circuit po wer consumptio n ( P base ) 50 mW Dynamic circui t factor ( ε ) 2 mW /Mbps The channel coherenc e interv al ( L ) 10 3 In Fig.1, it is observed clea rly that POS can attain the highest achievable d ata rate for all transmit powers, indicating the validity an d superiority of o u r scheme in ter ms of SE. The achiev ab le d ata rates fo r all schemes will r ise with the increment of tran smit power P , but the growth rates all become lo wer because of the lo garithmic relationship between the data rate and the SNR. It can be seen that A OS, m uch more simple than POS, c an obtain th e same SE performan ce, wh ich will be more practical in reality . Ther efore, the approximatio n is reasonable and q uite precise. The r andomn ess of GAS also 1 2 3 4 5 6 7 8 The required spectrum efficiency (bps/Hz) 0 1 2 3 4 5 6 7 8 The energy efficiency (Mbps/W) POS AOS GAS TS α =0.001 α =0.15 α =0.3 Fig. 2. The optimal energy effic ienc y versus the required spectrum ef ficienc y . 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 α , where R ul =6 bps/Hz 0 0.5 1 1.5 Energy efficiency (Mbps/W) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 α , where P=10W 4 5 6 7 8 Spectrum efficiency (Mbps/Hz) d UE,BS =250m d UE,BS =300m d UE,BS =300m d UE,BS =250m d UE,BS =200m d UE,BS =200m Fig. 3. The ef fects of α on SE and EE respecti vely with dif ferent distanc e betwee n UE a nd BSs. can be foun d in the g reen curve, and GAS of high com p lexity can get the same SE compared with POS and A OS, implying POS and A OS are optima l. As for T S, different α will cau se different pe rforman ce, where ov erlow and overhigh α are both undesirab le, which is carefully d iscussed in Fig . 3. Fig.2 compares the EE p erform a nce betwee n different schemes with different r equired up link data rate. I t can b e seen that POS can achieve optimal EE co mpared with TS and the perfor mance o f all schemes will be impair e d with the increase of R ul due to the expone ntially incre a sing n ature of transmit power with respect to data rate growth. It is o bvious that A O S can acquired the same EE per forman ce with POS, r evealing the rationality of our a p prox im ation. And the optimality of POS and A OS can be verified indirectly by GAS tha t almost have the id e n tical EE perfo rmance. Acco rding to TS, suitab le α need s to b e chosen in terms o f EE since different α will cause d ifferent perfo r mance like SE maximization. In Fig.3, it is seen that the perfor mance first soars swiftly and then decreases gr adually for b oth SE an d EE. When α is very sm a ll, the ch annel e stima tio n is extreme ly b ad and the MMSE is h igh, w h ich cause low useful signal p ower and mu ch more severe inter ference. Theref ore, the sy stem pe rforman ce is p oor due to less correctly demo dulated data e ven if there are so m u ch reso u rce for desired data. Un der this region, called the chann el estimation limited r egion, a little growth o f α will imp rove the c hannel estimation considera bly an d obtain a perfo r mance b oost due to th e prop erty of MMSE (inverse function w .r .t α ). On the contr ary , whe n α is large en ough , the c h annel estimatio n impr oves very slow d ue to the in verse function in MMSE. If we continue to increase α , less symbols will be left an d the perfo r mance will deteriora te . This region is c alled useful informatio n limited r egion. V I . C O N C L U S I O N In this pap e r , with th e conside r ation of channel estimation error and n on-ideal bac k haul link s, the optima l pilot symbols ratio is derived to max im ize SE an d E E in uplink JR-CoMP networks. The maximal system capacity in SE optimization and the minimal transm it power in EE optimization ar e also obtained by solving complicated eq u ations. In order to redu ce the comp lexity , some reasonable app r oximation s are in tro- duced to ob tain the closed-for m expressions of the se resu lts. Simulation reveals the supe riority of our scheme, the accuracy of our approxim ations, and th e effect of pilot sym bols r atio. A C K N OW L E D G E M E N T The work was sup ported by National Nature Science Foun- dation of China Pro ject (Grant No.6 14710 58), Ho ng K ong, Macao and T aiwan Science and T echnolog y Coop eration Projects (20 14DFT10 3 20, 201 6YFE012 2900) , th e 111 Pro ject of China (B160 06) and Beijing Training Project fo r The Leading T alents in S&T ( No. Z141101 0015 14026). R E F E R E N C E S [1] Q. Cui, T . Y uan, and W . Ni, “Energy-e f ficient two-w ay relaying under non-idea l po wer amplifiers, ” IEEE T rans. V eh. T ech nol. , vol. 66, no. 2, pp. 1257-127 0, 2017. [2] M. Cai, and J. N. Laneman, “W ideband distribut ed spectrum sharing with multichannel immedia te multip le acce ss, ” Analo g Inte gr . Circ . and Signal Pr ocess , Springer , https:/ /arxi v .org/pdf/1702.02695.pdf. [3] O. Onireti, F . Heliot, and M. A. Im ran, “On the energy ef ficiency-spe ctral ef ficienc y trade-of f in the uplink of CoMP system, ” IEEE T rans. 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