arXiv:1201.4793v1 [cs.IT] 23 Jan 2012 u htol e riigplt r eddt e reliable get are knowledge. to channel TOSD–SSK imperfect needed with and trans even are that SSK preserved pointed and of pilots detection, is receive–diversity data training and for it few estimates Furthermore, estimation channel only enough scheme. channel that Alamouti to out the robust than more a robust errors is as an is knowledge, TOSD–SSK channel ( modulation imperfect distributed SSK to systems identically that single–antenna and channels, independent fading anal over Rayleigh The shown, simulatio is pilots. Carlo it Monte training through substantiated and is distrib derivation correlations, fading spatial inversion receive–antennas, used fading be Gil–Pelaez and can and transmit– the general very arbitrary is with model Gauss analytical along The complex theorem. (RVs) conditional Variables in quadratic–forms fo Random exploits of framework The theory mismatched modulations. a the TOSD–SSK of and (ABEP) SSK A both Probability the Error compute sin Bit to age a framework analytical SSK develop closed–form of We integral version transmit–diversity. improved an providing is K modulation Time–Orthogonal–Signal–Des which Shift ii) modulation, Space SSK and i) (TOSD–) modulation: modulation; wh space (SSK) technologies, receiv of ing transmission blocks the two building at the on knowledge are attention channel our focus imperfect We with wi systems (MIMO) less Multiple–Input–Multiple–Output for modulation hf eig(S)mdlto,sailmdlto (SM), modulation spatial modulation, (SSK) sp transmit–diversity. design, keying MIMO single–RF shift mismatc analysis, performance systems, receiver, (MIMO) multiple–input–multiple–output igo U)(–al [email protected]). Sco 3JL, (e–mail: EH9 (UK) Edinburgh, Kingdom Road, Mayfield Buil Building, King’s Bell (IDCOM), Graham Communications Digital for Institute o eerhDW,vaG rnh 8 uloIdsraed P di Industriale Nucleo 18, Gronchi [email protected]). (e–mail: G. Italy via L’Aquila, Cen DEWS, Engineering, Research Information for and Electrical of Department u,3reJlo–ui,912GfsrYet EE (Paris CEDEX Gif–sur–Yvette 91192 [email protected]). Joliot–Curie, mail: rue 3 Sud, 2011. December USA, TX, Houston, (GLOBECOM), Conference pi 3 01 coe ,21;adNvme 6 2011. 16, November and 2011; 9, October 18, 2011; December 23, received April Manuscript Society. Communications essses hc srciigagoigatnindeto due attention growing a receiving is which systems, less S RNATOSO COMMUNICATIONS ON TRANSACTIONS iia betIetfirXX.XXXX/TCOMM.XXXX.XX.XXXXXX. Identifier Object Digital .Ha swt h nvriyo dnug,Sho fEngine of School Edinburgh, of University The with is Haas H. .D enri n .Gais r ihTeUiest fL’A of University The with are Graziosi F. and Paris– Leonardis Univ – De SUPELEC D. – CNRS 8506 UMR L2S, with is Renzo Di M. hspprwspeetdi ata h EEGoa Communica Global IEEE the at part in presented was paper This ae prvdb .Znla h dtrfrWrls System Wireless for Editor the Zanella, A. by approved Paper ne Terms Index Abstract pc hf eig(S– IOwt Practical with MIMO (SSK–) Keying Shift Space AEmdlto 1 sanvldgtlmdlto con- wire- modulation (MIMO) digital Multiple–Input–Multiple–Output novel for a cept is [1] modulation PACE ac iRenzo, Di Marco nti ae,w td h efrac fspace of performance the study we paper, this In — mefc hne nweg,“massive” knowledge, channel Imperfect — .I I. NTRODUCTION ebr IEEE Member, hne Estimates Channel ai eLoads ai Graziosi, Fabio Leonardis, De Dario , e fExcellence of ter ig,Alexander dings, Haas, 00 revised 2010; fteIEEE the of s ,Fac (e– France ), ln,United tland, l,67100 ile, s and ns, utions, that d ebr IEEE Member, ytical i.i.d.) ering, quila, mit– tions gle– ver- hed ace ian ign ey- ich for re- er. r s ueia iuain.I 4,teatoshv tde the studied knowledge. have channel non–ideal authors with on modulation the based SSK [4], of only ABEP In and th simulations. insufficient in are numerical available they are However, knowledge literature. channel matter. imperfect this with on light lation shed to to is sensitive paper more this m is of The systems. contribution modulation conventional space than errors that estimation channel argued imperfec partic often In receiver. the is the it at realistic to channel in wireless robustness the modulation of its knowledge SSK is of environments adoption propagation ma the the principle, about [13 working criticism and inherent its [10], to [9], due However, [2]–[4], [17]. complexity has modulati similar conventional receiver with than schemes the performance modulati the better if space provide are (P–CSI), that, can Information CIRs shown State have Channel the these Perfect B among results that distance the Recent the shown that on points. and been depends (BEP) diagram, has Probability spatial–constellation Error link. it the wireless [7], of points transmit–to–receive and the any [4] at on In decoded Imp [7]: Channel be (CIR) different modulation can a Response sees transmitter data receiver the the for since by destination channel sent wireless messages location–specific the radiat the the antennas exploits of other modulation property the SSK all [4]. power while switch is no transmission which data transmit–antenna, for single on a of index the into been has principle [12]. modulation in SSK introduced the MIMO on unified a based Furthermore, space architecture [8]–[11]. in improved g proposed transmit–diversity Recently, been a have achieve [4]. can modulation that systems schemes low–complexity modulation MIMO very a for m in modulation concept data results SSK for which particular, diagram ulation, constella- In spatial–constellation the a [7]. only [1], of exploits diagr modulation role “spatial–constellation data transmit–antenn the so–called for the (the plays of diagram which positions tion antenna–array, spatial informa- the the the in of into conven- part bits to encode tion they respect schemes: with modulation unique tional them makes which work principle, fundamental th same all the (SSK) another, share Keying one technologies transmission from Shift different Space Although [2], forms and [4]. (IGCH) various modulation [3], in Hopping (SM) Modulation literature Channel Spatial the Information–Guided in modu- as known space such is The spectrall principle [2]–[7]. and lation implementations low–complexity MIMO realizing efficient of possibility the oersac ok ntepromneo pc modu- space of performance the on works research Some mapped are bits information of blocks modulation, SSK In ebr IEEE Member, n Harald and , am”) ular, ulse od- ese ing ain ain jor y– on on ed ]– as it e e 1 t , TRANSACTIONS ON COMMUNICATIONS 2

However, there are four limitations in this paper: i) the ABEP transmit– and receive–antennas, fading distributions, fading is obtained only through Monte Carlo simulations, which spatial correlations, and training pilots for channel estimation. is not very much insightful; ii) the arguments in [4] are It is shown that the mismatched detector of SSK and TOSD– applicable only to Gaussian fading channels and do not take SSK modulations can be cast in terms of a quadratic–form in into account the cross–product between channel estimation complex Gaussian Random Variables (RVs) when conditioning error and Additive White Gaussian Noise (AWGN) at the upon fading channel statistics, and that the ABEP can be receiver; iii) it is unclear from [4] how the ABEP changes computed by exploiting the Gil–Pelaez inversion theorem [23]; with the pilot symbols used by the channel estimator; and ii) over independent and identically distributed (i.i.d.) Rayleigh iv) the robustness/weakness of SSK modulation with respect fading channels, we show that SSK modulation is superior to conventional modulation schemes is not analyzed. In [18], to Quadrature (QAM), regardless of we have studied the performance of SSK modulation when the number of training pulses, if the spectral efficiency is the receiver does not exploit for data detection the knowledge greater than 2 bpcu (bits per channel use) and the receiver of the phase of the channel gains (semi–blind receiver). It has at least two antennas; iii) in the same fading channel, we is shown that semi–blind receivers are much worse than show that TOSD–SSK modulation is superior, regardless of coherent detection schemes, and, thus, that the assessment of the number of antennas at the receiver and training pulses, to the performance of coherent detection with imperfect channel the Alamouti scheme with QAM if the spectral efficiency is knowledge is a crucial aspect for SSK modulation. A very greater than 2 bpcu. Also, unlike the P–CSI setup, TOSD– interesting study has been recently conducted in [19], where SSK modulation can outperform the Alamouti scheme if the the authors have compared the performance of SM and V– spectral efficiency is 2 bpcu, just one pilot pulse for channel BLAST (Vertical Bell Laboratories Layered Space–Time) [20] estimation is used, and the detector is equipped with at least schemes with practical channel estimates. It is shown that the two antennas; iv) still over i.i.d. Rayleigh fading, we show claimed sensitivity of space modulation to channel estimation that, compared to the P–CSI scenario, SSK and TOSD–SSK errors is simply a misconception and that, on the contrary, modulations have a Signal–to–Noise–Ratio (SNR) penalty of SM is more robust than V–BLAST to imperfections on the approximately 3dB and 2dB when only one pilot pulse can be channel estimates, and that less training is, in general, needed. used for channel estimation, respectively. Also, single–antenna However, the study in [19] is conducted only through Monte and Alamouti schemes have a SNR penalty of approximately Carlo simulations, which do not give too much insights for 3dB for QAM; and v) we verify that transmit– and receive– performance analysis and system optimization. Finally, in [10] diversity of SSK and TOSD–SSK modulations are preserved the authors have proposed a Differential Space–Time Shift even for a mismatched detector. Keying (DSTSK) scheme, which is based on the Cayley uni- The remainder of this paper is organized as follows. In tary transform theory. The DSTSK scheme requires no channel Section II, the system model is introduced. In Section III and estimation at the receiver, but incurs in a 3dB performance Section IV, SSK and TOSD–SSK modulations are described loss with respect to coherent detection. Furthermore, it can be and the analytical frameworks to compute the ABEP with applied to only real–valued signal constellations. Unlike [10], imperfect channel knowledge are developed, respectively. In which avoids channel estimation, we are interested in studying Section V, the spectral efficiency of TOSD–SSK modulation the training overhead that is needed for channel estimation with time–orthogonal shaping filters is studied. In Section and to achieve close–to–optimal performance with coherent VI, numerical results are shown to substantiate the analytical detection. derivation, and to compare SSK and TOSD–SSK modulations Motivated by these considerations, this paper is aimed at with state–of–the–art single–antenna and Alamouti schemes. developing a very general analytical framework to assess Finally, Section VII concludes this paper. the performance of space modulation with coherent detection and practical channel estimates. In particular, we focus our II. SYSTEM MODEL attention on two transmission technologies, which are the We consider a generic Nt Nr MIMO system, with Nt × building blocks of space modulation: i) Space Shift Key- and Nr being the number of antennas at the transmitter and at ing (SSK) modulation [4]; and ii) Time–Orthogonal–Signal– the receiver, respectively. SSK and TOSD–SSK modulations Design (TOSD–) SSK modulation, which is an improved ver- work as follows [4], [8], [11]: i) the transmitter encodes blocks sion of SSK modulation providing transmit–diversity [8], [11]. of log2 (Nt) data bits into the index of a single transmit– Our theoretical and numerical results corroborate the findings antenna, which is switched on for data transmission while all in [19], and highlight three important outcomes: i) SSK the other antennas are kept silent; and ii) the receiver solves modulation is as robust as single–antenna systems to imperfect an Nt–hypothesis detection problem to estimate the transmit– channel knowledge; ii) TOSD–SSK modulation is more robust antenna that is not idle, which results in the estimation of the to channel estimation errors than the Alamouti scheme [21]; unique sequence of bits emitted by the encoder. With respect to and iii) only few training pilots are needed to get reliable SSK modulation [4], in TOSD–SSK modulation [11] the t–th enough channel estimates for data detection. More precisely, transmit–antenna, when active, radiates a distinct pulse wave- we provide the following contributions: i) we develop a single– form wt ( ) for t =1, 2,...,Nt, and the waveforms across the · + integral closed–form analytical framework to compute the 1 ∞ antennas are time–orthogonal, i.e. , wt1 (ξ) wt∗2 (ξ) dξ = Average BEP (ABEP) of a mismatched detector [22] for SSK −∞ 1 and TOSD–SSK modulations, which can be used for arbitrary ( )∗ denotes complex–conjugate. R · TRANSACTIONS ON COMMUNICATIONS 3

+ if and ∞ if . In is the complex channel gain with 0 t1 = t2 wt1 (ξ) wt∗2 (ξ) dξ = 1 t1 = t2 αt,r = βt,r exp(jϕt,r) 6 −∞ other words, SSK modulation is a special case of TOSD–SSK βt,r and ϕt,r denoting the channel envelope and phase, R modulation with wt (ξ)= w0 (ξ) for t =1, 2,...,Nt. respectively, and τt,r is the propagation time–delay. In [11], we have analytically proved that the diversity order The time–delays τt,r are assumed to be known at the • of SSK modulation is Nr, while the diversity order of TOSD– receiver, i.e., perfect time–synchronization is considered. SSK modulation is , which results in a transmit–diversity Furthermore, we consider τ τ ... τ , 2Nr 1,1 ∼= 1,2 ∼= ∼= Nt,Nr equal to 2 and a receive–diversity equal to Nr. Thus, TOSD– which is a realistic assumption when the distance between SSK modulation provides a full–diversity–achieving (i.e., the the transmitter and the receiver is much larger than the diversity gain is NtNr) system if Nt = 2. This scheme spacing between the transmit– and receive–antennas [7]. has been recently generalized in [24] to achieve arbitrary Due to the these assumptions, the propagation delays can transmit–diversity. It is worth emphasizing that in TOSD–SSK be neglected in the remainder of this paper. modulation a single–antenna is active for data transmission and that the information bits are still encoded into the index C. Channel Estimation of the transmit–antenna, and are not encoded into the impulse (time) response, wt ( ), of the shaping filter. In other words, Let Ep and Np be the energy transmitted for each pilot pulse TOSD–SSK modulation· is different from conventional Single– and the number of pilot pulses used for channel estimation, Input–Single–Output (SISO) schemes with Orthogonal Pulse respectively. Similar to [27] and [28], we assume that channel Shape Modulation (O–PSM) [25], which are unable to achieve estimation is performed by using a Maximum–Likelihood transmit–diversity as only a single wireless link is exploited (ML) detector, and by observing Np pilot pulses that are for communication [11]. Also, TOSD–SSK modulation is transmitted before the modulated data. During the transmission different from conventional transmit–diversity schemes [26], of one block of pilot–plus–data symbols, the wireless channel and requires no extra time–slots for transmit–diversity. Further is assumed to be constant, i.e. a quasi–static channel model is details are available in [11] and are here omitted to avoid considered. With these assumptions, the estimates of channel repetitions. gains αt,r (t =1, 2,...,Nt, r =1, 2,...,Nr) can be written Throughout this paper, the block of information bits en- as follows: coded into the index of the t–th transmit–antenna is called αˆt,r = βˆt,r exp(jϕˆt,r)= αt,r + εt,r (1) “message”, and it is denoted by mt for t =1, 2,...,Nt. The Nt messages are assumed to be equiprobable. Moreover, the where αˆt,r, βˆt,r, and ϕˆt,r are the estimates of αt,r, βt,r, and related transmitted signal is denoted by st ( ). It is implicitly ϕt,r, respectively, at the output of the channel estimation unit, · assumed with this notation that, if mt is transmitted, the analog and εt,r is the additive channel estimation error, which can be signal st ( ) is emitted by the t–th transmit–antenna while the shown to be complex Gaussian distributed with zero–mean and · 2 other antennas radiate no power. variance σε = N0/(EpNp) per dimension [27], [28], where N0 denotes the power spectral density per dimension of the AWGN at the receiver. The channel estimation errors, ε , are A. Notation t,r statistically independent and identically distributed, as well as Main notation is as follows. i) We adopt a complex– statistically independent of the channel gains and the AWGN envelope signal representation. ii) j = √ 1 is the imagi- at the receiver. + − nary unit. iii) (x y) (u) = ∞ x (ξ) y (u ξ) dξ is the ⊗ − convolution of signals x ( ) and−∞y ( ). iv) 2 is the square R D. Mismatched ML–Optimum Detector absolute value. v) E is· the expectation· operator|·| computed over channel fading{·} statistics. vi) Re and Im are real For data detection, we consider the so–called mismatched and imaginary part operators, respectively.{·} vii) Pr{·} denotes ML–optimum receiver according to the definition given in + {·} probability. viii) Q (u) = 1 √2π ∞ exp ξ2 2 dξ is [22]. In particular, a detector with mismatched metric estimates u − the Q–function. ix) δ ( ) and δ , are Dirac and Kronecker the complex channel gains as in (1), and uses them in the same ·  · ·  R   delta functions, respectively. x) MX (s) = E exp(sX) and metric that would be applied if the channels were perfectly { } ΨX (ν) = E exp(jνX) are Moment Generating Function known. To avoid repetitions in the analysis of SSK and TOSD– (MGF) and Characteristic{ } Function (CF) of RV X, respec- SSK modulations, the mismatched detector is here described tively. xi) denotes “is proportional to”. by assuming arbitrary shaping filters. ∝ The mismatched ML–optimum detector can be obtained as follows. Let mq with q = 1, 2,...,Nt be the transmitted B. Channel Model message. The signal received after propagation through the We consider a general frequency–flat slowly–varying chan- wireless fading channel and impinging upon the r–th receive– nel model with generically correlated and non–identically antenna can be written as follows: distributed fading gains. In particular (t = 1, 2,...,Nt, r = zr (ξ)=˜sq,r (ξ)+ ηr (ξ) if mq is sent (2) 1, 2,...,Nr):

ht,r (ξ) = αt,rδ (ξ τt,r) is the channel impulse re- where: i) s˜q,r (ξ) = (sq hq,r) (ξ) = αq,rsq (ξ) = • − ⊗ sponse of the transmit–to–receive wireless link from βq,r exp(jϕq,r) sq (ξ) for q = 1, 2,...,Nt and r = the t–th transmit–antenna to the r–th receive–antenna. 1, 2,...,Nr; ii) sq (ξ) = √Emwq (ξ) for q = 1, 2,...,Nt, TRANSACTIONS ON COMMUNICATIONS 4

Nr Nr 2 1 Dˆ m (mt)= zr (ξ) sˆt,r (ξ) dξ Re zr (ξ)ˆs∗ (ξ) dξ sˆt,r (ξ)ˆs∗ (ξ) dξ (4) q − | − | ∝ t,r − 2 t,r r=1 Tm r=1 Tm Tm X Z  X  Z  Z 

N 2 r E η (ξ) E E Dˆ (m )= m α w (ξ)+ r m α w (ξ)+ m ε w (ξ) dξ (5) mq t  q,r 0 t,r 0 t,r 0  − Tm " N0 √N0 # − " N0 N0 # r=1 Z r r r  X

  ˆ ˆ (e) mˆ = argmax Dmq (mt) arg min Dmq (mt) mt for t=1,2,...,Nt ∝ mt for t=1,2,...,Nt n o n o 2 Nr (6) η˜0,r Em Em = arg min (αt,r αq,r)+ εt,r mt for t=1,2,...,Nt  √N0 − " N0 − N0 #  r=1 r r  X

  (a) (b) where Em is the average energy transmitted by each an- where = comes from [31, Eq. (4) and Eq. (5)], and = tenna that emits a non–zero signal; and iii) ηr ( ) is the is the asymptotically–tight union–bound recently introduced · complex AWGN at the input of the r–th receive–antenna for in [7, Eq. (35)]. Furthermore, NH (t, q) is the Hamming r = 1, 2,...,Nr, which has power spectral density N0 per distance between the bit–to–antenna–index mappings of mt dimension. Across the receive–antennas, the noises ηr ( ) are and mq; and APEP (mq mt)=E PEP(mq mt) = · → { → } statistically independent. E Pr mq mt is the Average Pairwise Error Probability { { → }} In particular, (2) is a general Nt–hypothesis detection prob- (APEP), i.e., the probability of estimating mt when, instead, lem [29, Sec. 7.1], [30, Sec. 4.2, pp. 257] in AWGN, when mq is transmitted, under the assumption that mt and mq are conditioning upon fading channel statistics. Accordingly, the the only two messages possibly being transmitted. mismatched ML–optimum detector with imperfect CSI at the Let us note that (7) simplifies significantly when receiver is as follows: APEP (mq mt) = APEP for t = 1, 2,...,Nt and → 0 ˆ q =1, 2,...,Nt (e.g., for i.i.d. fading). In this case, the ABEP mˆ = argmax Dmq (mt) (3) mt for t=1,2,...,Nt in (7) becomes: n o ˆ Nt Nt where mˆ is the estimated message and Dmq (mt) is the mis- APEP0 (a) Nt ABEP NH (t, q) = APEP0 (8) matched decision metric [7], [11], which is shown in (4) on top ≤ Nt log (Nt) 2 2 q=1 t=1 of this page, where sˆt,r (ξ)=ˆαt,rst (ξ) = (αt,r + εt,r) st (ξ) X X (a) and Tm is the symbol period. Nt Nt where = comes from the identity q=1 t=1 NH (t, q) = 2 III. SSK MODULATION Nt 2 log2 (Nt), which can be derived via direct inspection for all possible bit–to–antenna–indexP mappings.P A. Decision Metrics   In SSK modulation, the decision metric in (4) can be re– written from (1) and (2) as shown in (5) on top of this page, C. Computation of PEPs where we have taken into account that for SSK modulation the Let us start by computing the PEPs, i.e., the pairwise shaping filters are all equal to w0 ( ), and we have introduced probabilities in (7) when conditioning upon fading channel the scaling factor 1/N , which does· not affect (3). 0 statistics. From (6), PEP(mq mt) is as follows: From (5), and after some algebra, the maximization problem → ˆ (e) ˆ (e) in (3) reduces to (6) shown on top of this page, where PEP(mq mt)=Pr Dm (mt) < Dm (mq) (9) ˆ (e) → q q η˜0,r = ηr (ξ) w∗ (ξ) dξ, and Dmq (mt) is statistically Tm 0 n o ˆ where: equivalent to Dmq (mt). In particular, (6) can be thought as a R N 2 e r mismatched detector in which: i) first, pulse–matched filtering ˆ ( ) η˜0,r Em Em Dmq (mt)= (αt,r αq,r)+ εt,r √N N0 N0 is performed; and ii) then, ML–optimum decoding is applied r=1 0 − −  N 2 to the resulting signal. e r hq q i  ˆ ( ) P η˜0,r Em  Dmq (mq)= εq,r  √N N0 r=1 0 − B. ABEP q (10)  P The ABEP of the detector in (6) can be computed in closed– By introducing the notation (r =1, 2 ,...,Nr): form as follows: η˜0,r Em Em Xr = (αt,r αq,r)+ εt,r √N N0 N0 Nt Nt 0 − − (a) NH (t, q) (11) ABEP = E Pr mˆ = m m  η˜0,r hqEm q i t q Yr = √ N εq,r Nt log (Nt) { | }  N0 0 (q=1 t=1 2 ) − X X q N N (7) the PEP in (9) can be re–written in the general form (with (b) t t  NH (t, q) , , and ): E Pr mq mt A =1 B = 1 C =0 ≤ Nt log (Nt) { { → }} − q=1 t=1 2 X X APEP(mq mt) PEP(mq mt)=Pr D< 0 (12) → → { } | {z } TRANSACTIONS ON COMMUNICATIONS 5

Nr 2 (vavb) vavb ν gaγt,q + jνgbγt,q ΨD ( ν αt,q)= exp − = Υ (ν)exp ∆ (ν) γt,q (15) Nr Nr | (ν + jva) (ν jvb) ( (ν + jva) (ν jvb) ) { } − − 

Nt Nt + 1 1 1 ∞ Im Υ (ν) Mγt,q (∆ (ν)) ABEP NH (t, q) dν (19) ≤ Nt log (Nt) 2 − π ν 2 q=1 t=1 " 0  !# X X Z where: where ΨD (ν) = E ΨD (ν αt,q) is the CF of RV D { | } Nr averaged over all fading channel statistics. It can be computed 2 2 D = A Xr + B Yr + CXrY ∗ + C∗X∗Yr (13) from (15), as follows: | | | | r r r=1 (a) X   Ψ (ν)=E Υ (ν)exp ∆ (ν) γ = Υ (ν) M (∆ (ν)) From [23, Sec. III], we notice that, when conditioning upon D t,q γt,q { { }} (18) fading channel statistics, the RV D in (13) is a quadratic–form (a) in complex Gaussian RVs. In fact, AWGN at the receiver input where Mγt,q ( ) is the MGF of RV γt,q, and = comes from · and channel estimation error are Gaussian distributed RVs. the definition of MGF. Furthermore, they are mutually independent among themselves In conclusion, the ABEP of SSK modulation over arbitrary and across the Nr receive–antennas. Literature on quadratic– fading channels and with practical channel estimates can be forms in complex Gaussian RVs is very rich, and during the computed in closed–form from (7), (17), and (18), as shown last decades many different techniques have been developed in (19) on top of this page. for their analysis (see, e.g., [23] and [32] for a survey). The formula in (19) provides a very simple analytical tool Furthermore, effective methods for the computation of the PEP for performance assessment of SSK modulation with channel over generalized fading channels have been proposed, e.g., estimation errors, and allows us to estimate the number of pilot [33]–[35], and simple analytical frameworks for some special pulses, Np, and the fraction of energy, rpm, to be allocated to fading scenarios are available in [36, Ch. 9]. In this paper, we each pilot pulse to get the desired performance. In particular, propose to use the Gil–Pelaez inversion theorem [37]. (19) needs only the MGF of RV γt,q to be computed. This Accordingly, by using [37] the PEP can be computed as latter MGF is the building block for computing the ABEP follows: with P–CSI, and it has been recently computed in closed– + form for a number of MIMO setups and fading conditions. 1 1 ∞ Im ΨD ( ν αt,q) PEP(mq mt)= { | }dν In particular: i) it is known in closed–form for arbitrary → 2 − π 0 ν correlated Nakagami–m fading channels and Nr =1 [7]; ii) it Z π/2 1 1 Im ΨD (tan (ξ) αt,q) can be derived from [7] for independent Nakagami–m fading = { | }dξ 2 − π sin (ξ)cos(ξ) channels and arbitrary N [38]; iii) it can be derived from Z0 r (14) [7] for Nakagami–m fading channels and arbitrary Nr when the channel gains are correlated at the transmitter–side but where ΨD ( αt,q) is the CF of RV D when conditioning upon ·| Nr are independent at the receiver–side [38]; and iv) it is known the channel gains, and α = α , α is a short–hand t,q t,r q,r r=1 in closed–form for arbitrary correlated Rician fading channels to denote all the channel gains{ in (13). } and arbitrary N [11]. For example, for i.i.d. Rayleigh fading The conditional CF, Ψ ( α ), of RV D is given by r D t,q channels, the ABEP in (19) reduces, from (8), to: [23, Eq. (2) and Eq. (3)] shown·| in (15) on top of this + page, where: i) γ¯=Em/N0; ii) rpm = Ep/Em; iii) ga = Nt Nt ∞ Υ (ν) 1 1 ABEP Im dν 2¯γ 1 + (N r )− ; iv) g =γ ¯; and: Nr p pm b ≤ 4 − 2π 0 ( ν (1 2Ω0∆ (ν)) ) Z − h i (20) 1 2 1 2 − where Ω =E αt,r is the mean square value of the i.i.d. va = vb = (1/2) (Nprpm)− + 2 (Nprpm)− 0 | | r channel gains.  Nhr i n o  α 2 Finally, we conclude this section with three general com-  γt,q = γ ( t,q)= αq,r αt,r  r=1 | − | ments about (19): i) the integrand function is, in general, well–  2 1 1  ∆ (ν)= vavb ν Pga + jνgb (ν + jva)− (ν jvb)− − − behaved when ν 0 for typical MGFs used in wireless Nr Nr Nr →  Υ (ν) = (vavb) (ν + jva)− (ν jvb)− communication problems. Thus, the numerical computation of   −  (16) the integral does not provide any critical issues. The interested  reader might check this out in (20), where it can be shown that D. Computation of APEPs the integrand function tends to a finite value when ν 0; ii) → The APEP can be computed from (14) by removing the since the ABEP depends on the MGF of RV γt,q, from [11] conditioning over the fading channel: and [39] we conclude that the diversity order of the system is given by Nr, which is the same as the P–CSI scenario. We APEP (mq mt)=E PEP(mq mt) will verify this statement in Section VI with some numerical → { + → } 1 1 ∞ Im ΨD (ν) (17) examples, which will highlight that there is no loss in the = { }dν 2 − π ν diversity order with practical channel estimation; and iii) by Z0 TRANSACTIONS ON COMMUNICATIONS 6

Nr Nr E 2 ˆ ∗ m Dmq (mt)= Re (αt,r + εt,r) √Emη˜t,r 2 αˆt,r r=1 − r=1 | | (22)  Nr Nr P  P E 2  ˆ ∗ m  Dmq (mq)= Re (αq,r + εq,r) αq,rEm + √Emη˜q,r 2 αˆq,r r=1 − r=1 | |  P   P  PEP(mq mt) N→r 1 η˜q,r 1 η˜q,r 1 2 ∗ ∗ 2 (αq,r√γ¯ + εq,r√γ¯) αq,r√γ¯ + √N + 2 (αq,r√γ¯ + εq,r√γ¯) αq,r√γ¯ + √N 2 αq,r√γ¯ + εq,r√γ¯  r=1 h  0   0  − | | i   P  (24) = Pr  <   Nr ∗  1 η˜t,r 1 η˜t,r 1 2 (α γ¯ + ε γ¯)∗ + (α γ¯ + ε γ¯) α γ¯ + ε γ¯ 2 t,r√ t,r√ √N 2 t,r√ t,r√ √N 2 t,r√ t,r√  r  0 0 − | |    P=1    direct inspection, it can be shown that (19) reduces to the the PEP in (24) can be re–written (with A = 1/2, B = 0, − P–CSI lower–bound if Nprpm + . and C = 1/2) as: → ∞ PEP(mq mt)=Pr Dq

Nr compute the ABEP in (7) we are interested only in the cases Dˆ (m )= Re α αˆ∗ E δ +α ˆ∗ E η˜ where t = q, as NH (t, q)=0 if t = q. mq t q,r t,r m t,q t,r m t,r 6 r=1 From (26), the PEPs can be still computed by using the X n p o Nr Gil–Pelaez inversion theorem [37]: Em 2 αˆt,r − 2 | | + α r=1 1 1 ∞ Im ΨDt,q (ν t,q) PEP(mq mt)= | dν X (21) → 2 − π ν Z0  + α α where we have taken into account that the shaping filters, (a) 1 1 ∞ Im ΨDq (ν q)ΨDt ( ν t) = | − | dν wt ( ), are time–orthogonal to one another, and we have 2 − π ν · Z0  defined η˜t,r = ηr (ξ) wt∗ (ξ) dξ. In particular, for t = q (27) Tm 6 and t = q, the decision metric in (21) simplifies as shown in α R where ΨDt,q ( t,q) is the CF of RV Dt,q when con- (22) on top of this page. ·| Nr ditioning upon the fading gains αt,q = αt,r, αq,r , { }r=1 and ΨD ( αq) and ΨD ( αt) are the CFs of RVs Dq q ·| t ·| B. Computation of PEPs and Dt when conditioning upon the fading gains αq = Nr α Nr The PEPs, PEP(m m ), in (7) can be computed from αq,r and t = αt,r , respectively. Furthermore, q t { }r=1 { }r=1 (3) and (22): → (a) = comes from the independence of the conditional RVs Dq α ˆ ˆ and Dt, and the definition of CF, i.e., ΨDt,q ( ν t,q) = PEP(mq mt)=Pr Dmq (mq) < Dmq (mt) | → Eη,ε exp(jνDt,q) = Eη,ε exp(jνDq)exp( jνDt) = n ˆ ˆ o (23) { α } α { − } Dmq (mq) Dmq (mt) ΨDq (ν q)ΨDt ( ν t). We emphasize that Eη,ε is = Pr < the expectation| operator− | computed over AWGN and channel{·} ( N0 N0 ) estimation errors, as we are conditioning upon the channel By using the identity Re ab∗ = (1/2) ab∗ + (1/2) a∗b, gains. { } which holds for every pair of complex numbers a and b, and The last step is to compute the CFs in (27), which can be by explicitly showing the SNR γ¯=Em/N0 in (22), the PEPs obtained from [23, Eq. (2) and Eq.(3)] by using the theory of in (23) simplifies as shown in (24) on top of this page. quadratic–forms in conditional complex Gaussian RVs, as: By introducing the RVs (r = 1, 2,...,Nr): i) Xq,r = ΨD (ν αq)=Υq (ν)exp ∆q (ν) γq αq,r√γ¯ + εq,r√γ¯; ii) Yq,r = αq,r√γ¯ + η˜q,r √N0 ; iii) q | { } (28) ΨD (ν αt)=Υt (ν)exp ∆t (ν) γt Xt,r = αt,r√γ¯ + εt,r√γ¯; iv) Yt,r =η ˜t,r √N ; and: t 0    | { } Nr where we have defined: i) va = (1/4) + Nprpm + (1/2); 2 2  Dq = A Xq,r + B Yq,r + CXq,rYq,r∗ + C∗Xq,r∗ Yq,r ii) ; iii) α  r=1  | | | |  vb = (1/4) + Nprpm (1/2) γq = γ ( q) =  NPr Nr 2 − p Nr 2 (q)  2 2 αq,r ; iv) γt = γ (αt) = αt,r ; v) ga = Dt = A Xt,r + B Yt,r + CXt,rYt,r∗ + C∗Xt,r∗ Yt,r r=1 | p| r=1 | | r=1  | | | |  1 (q) (t) (t)  P (1/2)γ ¯ 1 + (Nprpm)− ; vi) g = ga = g = (1/2)¯γ;  (25) P b P − b h i TRANSACTIONS ON COMMUNICATIONS 7

Nt Nt + Im Υq (ν)Υt ( ν) Mγ(∆) ν (1) 1 1 1 ∞ − t,q ( ) ABEP NH (t, q) dν (32) ≤ Nt log (Nt)  2 − π n ν o  2 q=1 t=1 0 X X Z    + Nt Nt ∞ Υq (ν)Υt ( ν) 1 ABEP Im − dν (34) Nr Nr ≤ 4 − 2π 0 ( ν (1 Ω0∆q (ν)) (1 Ω0∆t ( ν)) ) Z − − − and: in closed–form in [29] for almost all fading channel models

2 (q) (q) 1 1 of interest in wireless communications. For example, if the ∆q (ν)= vavb ν ga + jνgb (ν + jva)− (ν jvb)−  −  − channel gains are i.i.d. Rayleigh distributed the ABEP in (32) 2 (t) (t) 1 1  ∆t (ν)= vavb ν ga + jνg (ν + jva)− (ν jvb)−  − b − reduces, from (8), to (34) shown on top of this page.  Nr  Nr Nr Υq (ν)=Υt (ν) = (vavb) (ν + jva)− (ν jvb)− Finally, similar to Section III-D, we note that: i) the  − (29)  integrand function in (32) is, for typical MGFs used in C. Computation of APEPs communication problems, well–behaved when ν 0; ii) since the ABEP in, e.g., (33) and (34) is given by the→ products of The APEP can be computed from (27) and (28) by still 2Nr MGFs, we conclude from [11] and [39] that the diversity using the Gil–Pelaez inversion theorem [37]: order of the system is 2Nr, which is the same as the P–CSI APEP (mq mt)=E PEP(mq mt) scenario [11]; and iii) (32) reduces to the P–CSI lower–bound → { → } + if Nprpm + . 1 1 ∞ Im ΨD (ν) (30) = t,q dν → ∞ 2 − π ν V. BANDWIDTH EFFICIENCY OF ORTHOGONAL SHAPING Z0  α α FILTERS DESIGN where ΨDt,q (ν)=E ΨDq ( ν q)ΨDt ( ν t) , which, for generic fading channels, is: | − | In Section IV, we have shown that TOSD–SSK modulation  provides, even in the presence of channel estimation errors and ΨD (ν)=E ΨD (ν αq)ΨD ( ν αt) t,q q | t − | with a single active antenna at the transmitter, a diversity order =E Υq (ν)exp ∆q (ν) γq Υt ( ν)exp ∆t ( ν) γt that is equal to 2N . This is achieved by using time–orthogonal {  { } − { − }} r =Υq (ν)Υt ( ν)E exp ∆q (ν) γq + ∆t ( ν) γt shaping filters at the transmitter, which is an additional design − { { − }} constraint that might not be required by SSK modulation and =Υq (ν)Υt ( ν) M (∆) (1) − γt,q (ν) conventional single– and multiple–antenna systems. Thus, for (31) a fair comparison among the various modulation schemes, (∆) it is important to assess whether the time–orthogonal con- and M (∆) (s)=E exp sγt,q (ν) is the MGF of RV γt,q (ν) straint affects the overall bandwidth efficiency of the com- (∆) γ (ν) = ∆q (ν) γq +n ∆t ( ν) γt. o munication system. More specifically, this section is aimed t,q − In conclusion, the ABEP of TOSD–SSK modulation over at understanding whether a larger transmission bandwidth is arbitrary fading channels and with practical channel estimates required for the transmission of the same number of bits in a can be computed in closed–form from (7), (30), and (31) as given signaling time–interval Tm, i.e., for a given bit/symbol shown in (32) on top of this page. or bpcu requirement. To shed light on this matter, in this Similar to SSK modulation, (32) is general and useful for section we analyze the bandwidth occupancy of commonly every MIMO setups. To be computed, a closed–form expres- used shaping filters and, as an illustrative example, a family (∆) sion of the MGF of RV γ (ν) = ∆q (ν) γq + ∆t ( ν) γt, of recently proposed spectrally–efficient orthogonal shaping t,q − which is given by the linear combination of the power–sum of filters. More specifically: i) as far state–of–the–art shaping generically correlated and distributed channel gains, is needed. filters are concerned, we consider well–known time–limited This MGF is available for various fading channel models in rectangular, half–sine, and raised–cosine prototypes [40, Sec. [29], or, e.g., it can be readily computed by exploiting the III–B]; on the other hand, ii) as far as time–orthogonal Moschopoulos method for arbitrarily correlated and distributed shaping filters are concerned, we consider waveforms built Rician fading channels, as described in [11]. In particular, upon linear combinations of Hermite polynomials [11], [25]. if all the channel gains are independent, but not necessarily The analytical expressions of time and frequency responses of identically distributed, the MGF (∆) reduces to: Mγ (ν) ( ) these letter filters are available in Appendix I for Nt =4. t,q · Three important comments are worth being made about the (∆) shaping filters that are considered in our comparative study: Mγ(∆)(ν) (s)=E exp sγt,q (ν) t,q 1) we limit our study to considering time–limited shaping =E exp s∆qn(ν) γq E expos∆t ( ν) γt { { }} { { − }} filters, which are simpler to be implemented than bandwidth– Nr Nr limited filters [40], [41]. This choice allows us to perform = M 2 (s∆ (ν)) M 2 (s∆ ( ν)) αq,r q αt,r t a fair comparison among SSK modulation and conventional " | | # · " | | − # r=1 r=1 modulation schemes. In fact, an important benefit of SSK and Y Y (33) TOSD–SSK modulations is to take advantage of multiple–

2 2 where the MGFs M αt,r ( ) and M αq,r ( ) are available antenna technology with a single Radio Frequency (RF) front | | · | | · TRANSACTIONS ON COMMUNICATIONS 8

TABLE I BANDWIDTH OF VARIOUS TIME–LIMITEDSHAPINGFILTERS.TIME AND FREQUENCY RESPONSES OF RECTANGULAR, HALF–SINE, AND RAISED–COSINE

SHAPING FILTERS ARE AVAILABLE IN THE CAPTIONS OF FIG.1.THE SHAPING FILTERS wt ( ) ARE GIVEN IN (35). LET + · P (ω)= 1/√2π ∞ p (ξ) exp ( jωξ) dξ BE THE FOURIERTRANSFORMOFAGENERICSHAPINGFILTERWITHTIMERESPONSE p ( ).THEN: I) THE − ·   R−∞ B 2 R0 P (ω) dω FRACTIONAL POWER CONTAINMENT BANDWIDTH (FPCB) ISDEFINEDAS FPCBX = min B +∞| | > X% [48, P. 15], WHICHIS % B [0,+ )  | R P (ω) 2dω  ∈ ∞ 0 | | THE BANDWIDTH B WHERE X% PERCENT OF THE ENERGY IS CONTAINED; AND II) THE BOUNDED POWER SPECTRAL DENSITY BANDWIDTH (BPSDB) 2 2 ISDEFINEDAS P WHICH IS THE BPSDBTHdB = min B log10 P (ω) < log10 P ωpeak THdB, ω > B [48, . 18], B [0,+ ) n | | |    − ∀ o ∈ ∞  2 BANDWIDTH B BEYONDWHICHTHESPECTRALDENSITYIS THdB BELOW ITS PEAK (MAXIMUM VALUE), i.e., P ωpeak . 

Fractional Power Containment Bandwidth (B/(2π) kHz)

X Rectangular Half-Sine Raised-Cosine wt ( ) % · 99% 7.61 1.18 1.41 4.97 99.995% >30 6.98 3.29 6.46 99.9999% >30 22.14 6.64 7.31 99.99999% >30 29.96 10.57 7.76 Bounded Power Spectral Density Bandwidth (B/(2π) kHz)

TH Rectangular Half-Sine Raised-Cosine wt ( ) dB · 3dB 9.59 2.28 1.85 6.35 5dB >30 8.18 4.62 7.40 6dB >30 15.13 6.64 7.85 7dB >30 28.03 9.65 8.27 10dB >30 >30 >30 9.39

end at the transmitter [3], [4], which is a research challenge time–limited and time–orthogonal shaping filters. Two ex- that is currently stimulating the development of novel MIMO amples, which allow us to jointly tuning time–duration and concepts based, e.g., on parasitic antenna architectures [42]– bandwidth and to guaranteeing low out–of–band interference, [44]. A recent survey on single–RF MIMO design is available are given in [46] and [47]. in [45]. In order to use a single–RF chain, SSK and TOSD– Let us now compare the bandwidth efficiency of the orthog- SSK modulations need shaping filters that are time–limited onal shaping filters available in Appendix I with state–of–the– and have a duration that is equal to the signaling time– art shaping filters. A qualitative and quantitative comparisons interval Tm. In fact, as remarked in [4, Section II–D], the are shown in Fig. 1 and in Table I, respectively, by using adoption of shaping filters that are not time–limited would two commonly adopted definitions of bandwidth [48]: i) the require a number of RF chains that is equal to the number Fractional Power Containment Bandwidth (FPCB) [48, p. of signaling time–intervals Tm where the filter has a non– 15]; and ii) the Bounded Power Spectral Density Bandwidth zero time response (i.e., the time–duration of the filter). Thus, (BPSDB) [48, p. 18]. The formal definition of these two bandwidth–limited shaping filters [41] would require multiple concepts of bandwidth is given in the caption of Table I. RF chains; 2) even though the orthogonal shaping filters By carefully analyzing both Table I and Fig. 1, we notice considered in the present paper and summarized in Appendix that the bandwidth efficiency of the different shaping filters I are obtained by using the algorithm proposed in [25], which depend on how stringent the criterion to define the bandwidth was introduced for Ultra Wide Band (UWB) systems, time– is. In particular, if the percentage of energy that is required to duration and bandwidth can be adequately scaled for narrow– be contained in the bandwidth (FPCB) is 99%, then the best band communication systems. For example, Fig. 1 is represen- shaping filter to use is the half–sine. On the other hand, if, tative of a narrow–band system with pulses having a practical to reduce the interference produced in adjacent transmission time–duration of milliseconds and a practical bandwidth of bands, the requirement moves from 99% to 99.99999%, then kilohertz. Thus, neither UWB nor (SS) the best shaping filters to us are those given in Appendix I. systems with orthogonal spreading codes are needed for space A similar comment applies when the BPSDB definition of modulation; and 3) the method proposed in [25] for the design bandwidth is used, but the best shaping filters are the raised– of orthogonal shaping filters guarantees that all the waveforms cosine (less stringent requirement) and the orthogonal filters in have the same time–duration and (practical) bandwidth. Thus, Appendix I (more stringent requirement). A similar trade–off unlike conventional Hermite polynomials, time–orthogonality has been shown in [40] and [48] for conventional modulation is guaranteed without bandwidth expansion. Let us emphasize schemes and shaping filters. that other methods are available in the literature to generate In other words, the shaping filters in Appendix I are TRANSACTIONS ON COMMUNICATIONS 9

coexistence capabilities might be required, the shaping filters

0 given in Appendix I might not be the best choice, as they would require a larger bandwidth. In that case, by using the algorithms in [25], [46], [47], and references therein, new −3 orthogonal pulses could be generated with the required time–

−5 duration and (practical) bandwidth. ) 2 )| ω −7 (|V(

10 VI. NUMERICAL AND SIMULATION RESULTS log

−10 In this section, we show some numerical examples in order

Rectangular to: i) study the performance of SSK and TOSD–SSK modula- Half−Sine tions in the presence of channel estimation errors; ii) compare Raised−Cosine the achievable performance with single–antenna and Alamouti W (ω) t −15 schemes; and iii) assess the accuracy of our analytical deriva- −2000 0 2000 4000 6000 8000 10000 12000 14000 16000 ω/(2π) [Hz] tion. For illustrative purposes, i.i.d Rayleigh fading channels are considered in all the analyzed scenarios. The interested Fig. 1. Examples of time–limited shaping filters commonly used in reader might find in [7], [11], and [18] numerical examples the literature (frequency responses). The time responses are as follows. i) about the performance of SSK and TOSD–SSK modulations Rectangular pulse: v (ξ) = pT (ξ), where pT (ξ) = 1 if T0/2 0 0 − ≤ for different wireless channels. Single–antenna and Alamouti ξ T0/2 and pT (ξ) = 1 elsewhere. ii) Half–sine pulse: v (ξ) = ≤ 0 schemes are chosen as state–of–the–art transmission technolo- √2sin[π (ξ + 0.5T ) /T ] p (ξ). iii) Raised–cosine pulse: v (ξ) = 0 0 T0 gies for performance comparison because they have the same 2/3 1 cos [2π (ξ + 0.5T0) /T0] pT0 (ξ). iv) The orthogonal shaping { − } 4 filtersp for Nt = 4 are given in (35) in Appendix I with t0 = 10− . T0 = diversity order and the same decoding complexity as SSK and 3 10− is the time duration of the filters. The frequency response is defined TOSD–SSK modulations, respectively. The interested reader + as V (ω) = 1/√2π ∞ v (ξ) exp ( jωξ) dξ. They are as follows. i) might find in [49, Fig. 2] the comparison with transmit–   −∞ − Rectangular pulse: V (2πωR ) = κrsinc (ωT0), where sinc (x) = 1 if x = 0 diversity Space–Time–Block–Codes (STBCs) for MIMO sys- and sinc (x) = sin(| πx) /πx| if x = 0. ii) Half–sine pulse: V (2πω) = 2 2 6 | | κ cos (πωT0) / 1 4ω T . iii) Raised–cosine pulse: V (2πω) = tems with more than two antennas at the transmitter, and in hs − 0 | | κrc [sinc (ωT0)+(1 /2)sinc(ωT0 1)+(1/2)sinc(ωT0 + 1)]. iv) The − [50, Fig. 8] the comparison with spatial MIMO orthogonal shaping filters for Nt = 4 are given in (36) in Appendix I with 4 systems with multi–user detection. In these latter cases, both t0 = 10− . κr, κhs, and κrc are constant factors that are not relevant for our analysis. STBCs and MIMO have higher decoding complexity and worse performance than space modulation. The simulation setup used in our study is as follows: i) we designed to have a very flat spectrum in the transmission band consider i.i.d Rayleigh fading with unit–power over all the to improve the energy efficiency, as well as a very fast roll–off wireless links. The related analytical framework is available, to reduce interference and enhance coexistence capabilities. by setting Ω0 =1, in (20) and (34) for SSK and TOSD–SSK This is especially useful to increase the system efficiency modulations, respectively; ii) rpm = 1 for all the analyzed since current standards require the transmitted spectrum to scenarios; iii) the bpcu of SSK and TOSD–SSK modulations occupy a well–defined spectral mask, e.g., for Wireless Local are equal to R = log2 (Nt); iv) as far single–antenna and Area Networks (WLAN) and UWB wireless systems. For this Alamouti schemes are concerned, we consider QAM with reason, the shaping filters in Appendix I have a very good constellation size M and bpcu equal to R = log2 (M); v) as energy containment and bounded energy spectrum. Finally, we mentioned in Section V, the shaping filters are obtained from emphasize that the shaping filters given in Appendix I are just [25]. For example, when Nt =4, wt1 ( ) in Appendix I is used an example of time–orthogonal filters that can be obtained with for SSK modulation, single–antenna, and· Alamouti schemes, state–of–the–art signal processing algorithms [46], [47], as while the set of four orthogonal filters in (35) is used for well as that the waveforms compared in Table I have the same TOSD–SSK modulation. Furthermore, for a fair comparison time–duration Tm, and, thus, they provide the same signaling among the modulation schemes, the same spectral efficiency rate 1/Tm. (measured in bpcu) is considered; vi) the ABEP of the P– For illustrative purposes, in this paper we choose the CSI scenario is computed by assuming an infinite number of shaping filters with the main objective to limit, as much pilot pulses; vii) the ABEP of single–antenna and Alamouti as possible, out–of–band interference in order to enhance schemes is obtained through Monte Carlo simulations only, the coexistence capabilities of our communication system, but to check that our simulator is well–tuned the numerical and to reduce interference in adjacent transmission bands. results are compared, for the P–CSI scenario, to the ABEP Thus, our criterion is based on choosing filters which, for predicted by the union–bound recently developed in [50] for the same time–duration, have a stringent energy containment single– and multi–user systems; viii) as far as single–antenna or bounded energy spectrum. For example, we assume either schemes are concerned, Em is the average energy transmitted X% > 99.9999% or THdB > 6dB in Table I. With these for each information symbol; and ix) as far as the Alamouti assumptions, the orthogonal shaping filters given in Appendix scheme is concerned, Em is the average energy transmitted I are the best choice, and are chosen to obtain the simulation for each information symbol from the two active transmit– results in Section VI. For applications where less stringent antennas, i.e., Em is equally split between the two antennas. TRANSACTIONS ON COMMUNICATIONS 10

N = 2 (1 bpcu) N = 16 (4 bpcu) t t 0 0 10 10 N = 1 p N = 3 p −1 N = 10 −1 10 p 10 P−CSI N = 1 r −2 N = 1 −2 10 r 10

ABEP ABEP N = 4 −3 −3 r 10 10

N = 2 N = 2 r N = 4 r N = 1 r p −4 −4 10 10 N = 3 p N = 10 p P−CSI −5 −5 10 10 01 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 −1 E /N [dB] E /N [dB] m 0 m 0

Fig. 2. ABEP of SSK modulation against Em/N0 for: i) Nt = 2 (1 bpcu); Fig. 4. ABEP of SSK modulation against Em/N0 for: i) Nt = 16 (4 bpcu); ii) Nr = 1, 2, 4 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP ii) Nr = 1, 2, 4 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP with no channel{ estimation} errors.{ Solid} lines show the analytical model and with no channel{ estimation} errors.{ Solid} lines show the analytical model and markers show Monte Carlo simulations. markers show Monte Carlo simulations.

N = 8 (3 bpcu) t 0 10 a SNR penalty, with respect to the P–CSI lower–bound, of approximately 3dB and 2dB when Np = 1, respectively; v) −1 10 the ABEP gets worse for increasing Nt, as a consequence of the increased size of the spatial–constellation diagram, and

−2 N = 1 gets better for increasing Nr, due to the receive–diversity gain; 10 r and vi) TOSD–SSK modulation significantly outperforms SSK N = 4 r

ABEP modulation, due to the transmit–diversity gain introduced by −3 10 the orthogonal pulse shaping design. As far as the performance comparison with single–antenna N = 1 N = 2 p r −4 and Alamouti schemes is concerned, the following conclu- 10 N = 3 p sions can be drawn (see Table II for numerical values): i) N = 10 p P−CSI SSK modulation outperforms single–antenna QAM, in all the −5 10 0 5 10 15 20 25 30 35 40 analyzed scenarios, for spectral efficiencies greater than 2 bpcu E /N [dB] m 0 and for Nr > 1. If Nr = 1, QAM always outperforms SSK modulation; ii) SSK and single–antenna QAM have Fig. 3. ABEP of SSK modulation against Em/N0 for: i) Nt = 8 (3 bpcu); almost the same robustness to channel estimation errors, with ii) Nr = 1, 2, 4 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP a SNR penalty, with respect to the P–CSI lower–bound, of with no channel{ estimation} errors.{ Solid} lines show the analytical model and markers show Monte Carlo simulations. approximately 3dB when Np =1; iii) TOSD–SSK modulation outperforms the Alamouti scheme with QAM, in all the ana- lyzed scenarios, for spectral efficiencies greater than 2 bpcu. The results are shown in Figs. 2–6 for transmission tech- In particular, unlike SSK modulation, TOSD–SSK modulation nologies with no transmit–diversity gain (SSK and QAM), and is superior to the Alamouti scheme with QAM for Nr =1 as in Figs. 7–10 for transmission technologies with transmit– well. This is due to the transmit–diversity gain of TOSD– diversity gain (TOSD–SSK and Alamouti). As far as SSK SSK modulation; iv) TOSD–SSK modulation is more robust and TOSD–SSK modulations are concerned, we observe to channel estimation errors than the Alamouti scheme with that: i) our analytical frameworks are very accurate and QAM. A clear example can be observed in Table II when asymptotically–tight for all the analyzed scenarios. In par- R = 2 bpcu and Np = 1. In fact, the Alamouti scheme is ticular, as expected, they are exact for Nt = 2; ii) there superior to TOSD–SSK modulation in the P–CSI scenario, is no loss of the diversity order in the presence of channel but TOSD–SSK modulation provides better performance if estimation errors. Only a loss of the coding gain can be Np = 1 and Nr > 1. More in general, Table II shows observed for all MIMO setups; iii) even though in space that the Alamouti scheme with QAM has a SNR penalty, modulation the information is encoded into the CIRs, the with respect to the P–CSI lower–bound, of approximately 3dB performance degradation observed when reducing the number when Np = 1, while TOSD–SSK modulation has a SNR of pilot pulses, Np, is not very high, and the ABEP is very penalty of only 2dB; and v) the performance gain of SSK and close to the P–CSI lower–bound, in the analyzed scenarios, TOSD–SSK modulations with respect to single–antenna and for Np = 10; iv) SSK and TOSD–SSK modulations have Alamouti schemes increases with Nr, because, as analytically TRANSACTIONS ON COMMUNICATIONS 11

M = 8 (3 bpcu) 0 than 2 bpcu. In all the cases, the price to be paid for this 10 performance improvement is the need of increasing the number of radiating elements Nt at the transmitter, while still retaining −1 10 a single–RF chain and avoiding inter–antenna synchroniza- tion, which are beneficial for low–complexity implementations [42]. This remark is somehow similar to [51], as far as the −2 N = 1 10 r N = 4 achievable transmit–diversity of STBCs is concerned. Finally, r it is worth emphasizing that the need of a large number ABEP −3 10 of radiating elements seems not to be a critical bottleneck for the development of the next generation cellular systems, N = 1 (Monte Carlo) p N = 2 as current research is moving towards the utilization of the −4 N = 3 (Monte Carlo) r 10 p N = 10 (Monte Carlo) millimeter–wave frequency spectrum [52]. In fact, in this band p P−CSI (Monte Carlo) compact horn antenna–arrays with 48 elements and compact P−CSI (Union−Bound) −5 10 patch antenna–arrays with more than 4 elements at the base 0 5 10 15 20 25 30 35 40 E /N [dB] station and at the mobile terminal, respectively, are currently m 0 being developed to support multi–gigabit transmission rates [53]. Furthermore, SSK and TOSD–SSK seem to be well– Fig. 5. ABEP of QAM against Em/N0 for: i) M = 8 (3 bpcu); ii) Nr = 1, 2, 4 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP suited low–complexity modulation schemes for the recently with no{ channel} estimation errors.{ Solid} lines with markers or just markers proposed “massive MIMO” paradigm [54], according to which show Monte Carlo simulations. Dashed lines show the union–bound computed from [50] with no channel estimation errors at the receiver (P–CSI scenario). unprecedent spectral efficiencies can be achieved in cellular This union–bound is shown only for a subset of curves in order to improve networks by using antenna–arrays with very large (with tens the readability of the figure, and avoid overlap among closely–spaced curves. or hundreds) active radiating elements.

M = 16 (4 bpcu) 0 10 VII. CONCLUSION

−1 In this paper, we have analyzed the performance of space 10 modulation when CSI is not perfectly known at the receiver. A very accurate and general analytical framework has been N = 1 −2 r 10 proposed, and it has been shown that, unlike common be- N = 4 r lief, SSK modulation has the same robustness to channel

ABEP estimation errors as conventional modulation schemes, while −3 10 TOSD–SSK modulation is less sensitive to channel estima- N = 1 (Monte Carlo) p tion errors than conventional modulations. Also, it has been N = 2 N = 3 (Monte Carlo) r −4 p 10 shown that few pilot pulses are needed to achieve almost the N = 10 (Monte Carlo) p same performance as the P–CSI lower–bound, and that the P−CSI (Monte Carlo) P−CSI (Union−Bound) performance gain, over state–of–the–art MIMO technologies, −5 10 0 5 10 15 20 25 30 35 40 promised by space modulation is retained even with imperfect E /N [dB] m 0 channel knowledge. These results confirm the usefulness of space modulation in practical operating conditions, and, in Fig. 6. ABEP of QAM against Em/N0 for: i) M = 16 (4 bpcu); ii) particular, the notable performance advantage of TOSD–SSK Nr = 1, 2, 4 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP with modulation, which provides transmit–diversity and is more no channel{ estimation} errors.{ Solid lines} with markers or just markers show Monte Carlo simulations. Dashed lines show the union–bound computed from robust to channel estimation errors than conventional schemes, [50] with no channel estimation errors at the receiver (P–CSI scenario). This such as the Alamouti code. union–bound is shown only for a subset of curves in order to improve the readability of the figure, and avoid overlap among closely–spaced curves. ACKNOWLEDGMENT proved in [50], space modulation takes much better advantage We gratefully acknowledge support from the European of receive–diversity. The results shown in this section confirm Union (PITN–GA–2010–264759, GREENET project) for this that this trend is retained in the presence of channel estimation work. M. Di Renzo acknowledges support of the Laboratory errors as well. of Signals and Systems under the research project “Jeunes In conclusion, SSK modulation is as robust as single– Chercheurs”. D. De Leonardis and F. Graziosi acknowledge antenna systems to imperfect channel knowledge, and it pro- the Italian Inter–University Consortium for Telecommunica- vides better performance when the target spectral efficiency tions (CNIT) under the research grant “Space Modulation for is greater than 2 bpcu and Nr > 1. On the other hand, MIMO Systems”, and Micron Technology under the Ph.D. TOSD–SSK modulation is more robust than the Alamouti fellowships program awarded to the University of L’Aquila. scheme to imperfect channel knowledge, and it provides better H. Haas acknowledges the EPSRC under grant EP/G011788/1 performance when the target spectral efficiency is greater for partially funding this work. TRANSACTIONS ON COMMUNICATIONS 12

TABLE II 4 REQUIRED Em/N0 (dB) TO GET ABEP = 10− FORALLSCENARIOSEXCEPT SSK MODULATION AND SINGLE–ANTENNA QAM IF Nr = 1, FOR 2 WHICH THE Em/N0 (dB) TO GET ABEP = 10− IS SHOWN.FOR SSK MODULATION AND SINGLE–ANTENNA QAM, EACH ROW SHOWS THE Em/N0 (dB) FOR Nr = 1 / Nr = 2 / Nr = 4.FOR SSK MODULATION AND THE ALAMOUTI SCHEME WITH QAM, EACH ROW SHOWS THE Em/N0 (dB) FOR Nr = 1 / Nr = 2.THE VALUES HAVE AN ERROR APPROXIMATELY EQUAL TO 0.1dB. ±

SSK

Rate Np =1 Np =3 Np = 10 P CSI − 1 bpcu 22.9 / 25.3 / 16.2 21.1 / 23.5 / 14.5 20.3 / 22.7 / 13.6 19.9 / 22.3 / 13.2 2 bpcu 26 / 26.8 / 17 24.2 / 25.1 / 15.4 23.4 / 24.3 / 14.5 23 / 23.8 / 14 3 bpcu 29 / 28.4 / 17.9 27.3 / 26.6 / 16.2 26.4 / 25.8 / 15.3 26 / 25.4 / 14.9 4 bpcu 32 / 29.9 / 18.7 30.3 / 28.1 / 17 29.5 / 27.3 / 16.2 29 / 26.9 / 15.7 Single–Antenna QAM

Rate Np =1 Np =3 Np = 10 P CSI − 1 bpcu 19.8 / 22.3 / 13.1 18.2 / 20.6 / 11.5 17.2 / 19.7 / 10.6 16.8 / 19.3 / 10.1 2 bpcu 22.7 / 25.2 / 16.2 21.1 / 23.5 / 14.5 20.3 / 22.7 / 13.6 19.9 / 22.2 / 13.2 3 bpcu 27.5 / 29.9 / 21.1 25.6 / 28 / 19.1 24.9 / 27.4 / 18.2 24.6 / 27 / 18.1 4 bpcu 29.6 / 32 / 23.3 27.8 / 30.1 / 21.3 27.1 / 29.6 / 20.5 26.8 / 29.1 / 20.3 TOSD–SSK

Rate Np =1 Np =3 Np = 10 P CSI − 1 bpcu 27.2 / 18.2 26 / 16.9 25.5 / 16.4 25.3 / 16.2 2 bpcu 28.7 / 19 27.5 / 17.8 27 / 17.3 26.8 / 17 3 bpcu 30.2 / 19.8 29 / 18.6 28.5 / 18.2 28.4 / 17.8 4 bpcu 31.9 / 20.7 30.5 / 19.4 30.1 / 18.9 29.9 / 18.7 Alamouti QAM

Rate Np =1 Np =3 Np = 10 P CSI − 1 bpcu 25.3 / 16.2 23.5 / 14.5 22.8 / 13.5 22.3 / 13.2 2 bpcu 28.4 / 19.3 26.5 / 17.5 25.7 / 16.6 25.4 / 16.3 3 bpcu 32.9 / 24 31.4 / 22.2 30.4 / 21.3 30 / 21 4 bpcu 35.2 / 26.2 33.3 / 24.3 32.6 / 23.5 32.3 / 23.3

4 √4 2√2 11 √4+2√2 2 wt (ξ)= p (ξ)+ − p (ξ)+ p (ξ)+ p (ξ)+ p (ξ) 1 − √165 1 4 2 30 3 − 4 4 − √110 5    q    4  √4 2√2  11 √4+2√2  2   −  wt2 (ξ)= √ p1 (ξ)+ 4 p2 (ξ)+ 30 p3 (ξ)+ 4 p4 (ξ)+ √ p5 (ξ)  − 165 − − 110    q   (35)   3  √4+2√2   √4 2√2 2    wt (ξ)= p (ξ)+ p (ξ) + (0) p (ξ)+ − p (ξ)+ p (ξ)  3 22 1 − 4 2 3 4 4 − √11 5 q      3  √4+2√2 √4 2√2  2   −  wt4 (ξ)= 22 p1 (ξ)+ 4 p2 (ξ) + (0) p3 (ξ)+ 4 p4 (ξ)+ √ p5 (ξ)  − − 11  q          4 √4 2√2 11 √4+2√2 2 Wt (ω)= P (ω)+ − P (ω)+ P (v)+ P (ω)+ P (ω) 1 − √165 1 4 2 30 3 − 4 4 − √110 5    q    4  √4 2√2  11 √4+2√2  2   −  Wt2 (ω)= √ P1 (ω)+ 4 P2 (ω)+ 30 P3 (ω)+ 4 P4 (ω)+ √ P5 (ω)  − 165 − − 110    q   (36)   3  √4+2√2   √4 2√2 2    Wt (ω)= P (ω)+ P (ω)+(0) P (ω)+ − P (ω)+ P (ω)  3 22 1 − 4 2 3 4 4 − √11 5 q      3  √4+2√2 √4 2√2  2   −  Wt4 (ω)= 22 P1 (ω)+ 4 P2 (ω) + (0) P3 (ω)+ 4 P4 (ω)+ √ P5 (ω)  − − 11  q          TRANSACTIONS ON COMMUNICATIONS 13

N = 2 (1 bpcu) N = 8 (3 bpcu) t t 0 0 10 10

−1 −1 10 10

−2 −2 10 10

ABEP N = 2 ABEP N = 2 −3 r −3 r 10 10 N = 1 r N = 1 r N = 1 N = 1 p p −4 −4 10 N = 3 10 N = 3 p p N = 10 N = 10 p p P−CSI P−CSI −5 −5 10 10 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 E /N [dB] E /N [dB] m 0 m 0

Fig. 7. ABEP of TOSD–SSK modulation against Em/N0 for: i) Nt = 2 Fig. 9. ABEP of TOSD–SSK modulation against Em/N0 for: i) Nt = 8 (1 bpcu); ii) Nr = 1, 2 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes (3 bpcu); ii) Nr = 1, 2 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes the ABEP with no channel{ } estimation errors.{ Solid} lines show the analytical the ABEP with no channel{ } estimation errors.{ Solid} lines show the analytical model and markers show Monte Carlo simulations. model and markers show Monte Carlo simulations.

N = 16 (4 bpcu) t M = 16 (4 bpcu) 0 0 10 10

−1 −1 10 10

−2 −2 N = 2 10 10 r

N = 2 ABEP ABEP r N = 1 −3 −3 r 10 10 N = 1 N = 1 (Monte Carlo) r p N = 1 p N = 3 (Monte Carlo) −4 −4 p 10 10 N = 3 N = 10 (Monte Carlo) p p N = 10 p P−CSI (Monte Carlo) P−CSI P−CSI (Union−Bound) −5 01 −5 10 −1 10 0 5 10 15 20 25 30 35 40 −101 0 5 10 15 20 25 30 35 40 E /N [dB] E /N [dB] m 0 m 0

Fig. 8. ABEP of TOSD–SSK modulation against Em/N0 for: i) Nt = 16 Fig. 10. ABEP of Alamouti scheme with QAM against Em/N0 for: i) (4 bpcu); ii) Nr = 1, 2 ; iii) Np = 1, 3, 10 ; and iv) P–CSI denotes M = 16 (4 bpcu); ii) Nr = 1, 2 ; iii) Np = 1, 3, 10 ; and iv) P–CSI the ABEP with no channel{ } estimation errors.{ Solid} lines show the analytical denotes the ABEP with no channel{ estimation} errors.{ Solid li}nes with markers model and markers show Monte Carlo simulations. or just markers show Monte Carlo simulations. Dashed lines show the union– bound computed from [50] with no channel estimation errors at the receiver (P–CSI scenario). This union–bound is shown only for a subset of curves in order to improve the readability of the figure, and avoid overlap among APPENDIX I closely–spaced curves. ORTHOGONAL SHAPING FILTERS FOR Nt =4

In this appendix, we show an example of orthogonal shaping filters that can be used for TOSD–SSK modulation. Without tom of the previous page too, where we have defined: loss of generality we consider the case study with Nt =4, but 2 1 1 ξ the procedure can be generalized to larger antenna–arrays. p1 (ξ)= exp t √√π − 2 0     2 More specifically, we consider the procedure described 2ξ 1 ξ  p2 (ξ)= exp in [25], which allows us to generate orthogonal shaping  √2√π − 2 t0    2   2 filters with the same time–duration and bandwidth. Similar  4ξ 2 1  ξ   p3 (ξ)= − exp (37) techniques are available in [46], [47]. From [25], we can  √8√π − 2 t0    3   2 obtain the four orthogonal impulse (time) responses shown 8ξ 12ξ 1 ξ p4 (ξ)= − exp in (35) at the bottom of the previous page, as well as the √48√π − 2 t0    4 2   2 four related frequency responses (Fourier transform) P (ω)=  16ξ 48ξ +12  1 ξ +  p5 (ξ)= − exp √ ∞  2 t0 1 2π p (ξ)exp( jωξ) dξ shown in (36) at the bot-  √384√π − −∞ −           R  TRANSACTIONS ON COMMUNICATIONS 14 and: [18] M. Di Renzo and H. Haas, “Space shift keying (SSK) modulation with partial channel state information: Optimal detector and performance t0 1 2 analysis over fading channels”, IEEE Trans. Commun., vol. 58, no. 11, P1 (ω)= exp (t0ω) √√π − 2 pp. 3196–3210, Nov. 2010.   2  h i [19] M. M. Ulla Faiz, S. Al–Ghadhban, and A. Zerguine, “Recursive least– 2jt0ω 1 2  P2 (ω)= exp (t0ω) squares adaptive channel estimation for spatial modulation systems”,  √2√π − 2  IEEE Malaysia Int. Conf. Commun., pp. 1–4, Dec. 2009.   3 2 h i  2t0 4t0ω 1 2 [20] P. 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[42] A. Kalis, A. G. Kanatas, and C. B. Papadias, “A novel approach to Dario De Leonardis was born in Atri, Italy, in 1984. MIMO transmission using a single RF front end”, IEEE J. Select. Areas He received the Bachelor and the Master degrees Commun., vol. 26, no. 6, pp. 972–980, Aug. 2008. in Telecommunications Engineering from the De- [43] O. N. Alrabadi, C. B. Papadias, A. Kalis, and R. Prasad, “A universal partment of Electrical and Information Engineering, encoding scheme for MIMO transmission using a single active element University of L’Aquila, Italy, in October 2006 and for PSK modulation schemes”, IEEE Trans. Wireless Commun., vol. 8, in May 2009, respectively. no. 9, pp. 5133–5143, Sep. 2009. From September 2008 to March 2009, he was [44] O. N. Alrabadi, C. Divarathne, P. Tragas, A. Kalis, N. Marchetti, C. a Visiting ERASMUS student at the Telecommuni- B. Papadias, and R. Prasad, “Spatial multiplexing with a single radio: cations Technological Center of Catalonia (CTTC), Proof–of–concept experiments in an indoor environment with a 2.6 GHz Barcelona, Spain, where he conducted research for prototypes”, IEEE Commun. Lett., vol. 15, no. 2, pp. 178–180, Feb. his Master thesis project on low–complexity receiver 2011. design for ultra wide band wireless systems. In April 2010, he received a [45] A. Mohammadi and F. M. Ghannouchi, “Single RF front–end MIMO personal research grant from the National Inter–University Consortium for transceivers”, IEEE Commun. Mag., vol. 49, no. 12, pp. 104–109, Dec. Telecommunications (CNIT), Italy, to conduct research on space modulation 2011. for multiple–input–multiple–output wireless systems, and he was affiliated [46] B. Parr, B.–L. Cho, K. Wallace, and Z. Ding, “A novel ultra–wideband with the Department of Electrical and Information Engineering and the Center pulse design algorithm”, IEEE Commun. Lett., vol. 7, no. 5, pp. 219– of Excellence for Research DEWS, University of L’Aquila, Italy. Since 221, May 2003. December 2010, he has been a Ph.D. candidate in the same institution. His [47] X. Wu, Z. Tian, T. N. Davidson, and G. B. Giannakis, “Optimal main research interests are in the area of wireless communications. waveform design for UWB radios”, IEEE Trans. Signal Processing, vol. 54, no. 6, pp. 2009–2021, June 2006. [48] F. Amoroso, “The bandwidth of digital data signal”, IEEE Commun. Mag., vol. 18, no. 6, pp. 13–24, Nov. 1980. [49] M. Di Renzo and H. Haas, “Transmit–diversity for spatial modulation (SM): Towards the design of high–rate spatially–modulated space–time block codes”, IEEE Int. Conf. Commun., pp. 1–6, June 2011. [50] M. Di Renzo and H. Haas, “Bit error probability of space shift keying Fabio Graziosi (S’96-M’97) was born in L’Aquila, MIMO over multiple–access independent fading channels”, IEEE Trans. Italy, in 1968. He received the Laurea degree (cum Veh. Technol., vol. 60, no. 8, pp. 3694–3711, Oct. 2011. laude) and the Ph.D. degree in electronic engineering from the University of L’Aquila, Italy, in 1993 and [51] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block PLACE coding for wireless communications: Performance results”, IEEE J. Sel. in 1997, respectively. PHOTO Since February 1997, he has been with the De- Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. HERE [52] F. Khan and J. Pi, “Millimeter–wave mobile broadband: partment of Electrical Engineering, University of Unleashing 3–300GHz spectrum”, IEEE Wireless Commun. L’Aquila, where he is currently an Associate Pro- Netw. Conf., Mar. 2011, tutorial presentation. [Online]. Available: fessor. He is a member of the Executive Com- http://www.ieee-wcnc.org/2011/tut/t1.pdf. mittee, Center of Excellence Design methodologies [53] S. Rajagopal, S. Abu–Surra, Z. Pi, and F. Khan, “Antenna array design for Embedded controllers, Wireless interconnect and for multi–Gbps mmWave mobile broadband communication”, IEEE System-on-chip (DEWS), University of L’Aquila, and the Executive Commit- Global Commun. Conf., pp. 1–6, Dec. 2011. tee, Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT). [54] T. L. Marzetta, “Noncooperative cellular wireless with unlimited num- He is also the Chairman of the Board of Directors of WEST Aquila s.r.l., bers of base station antennas”, IEEE Trans. Wireless Commun., vol. 9, a spin–off R&D company of the University of L’Aquila, and the Center of no. 11, pp. 3590–3600, Nov. 2010. Excellence DEWS. He is involved in major national and European research programs in the field of wireless systems and he has been a reviewer for major technical journals and international conferences in communications. He also serves as Technical Program Committee (TPC) member and Session Chairman of several international conferences in communications. His current research interests are mainly focused on wireless communication systems with emphasis on wireless sensor networks, ultra wide band communication Marco Di Renzo (SM’05–AM’07–M’09) was born techniques, cognitive radio, and cooperative communications. in L’Aquila, Italy, in 1978. He received the Laurea (cum laude) and the Ph.D. degrees in Electrical and Information Engineering from the Department of Electrical and Information Engineering, University of L’Aquila, Italy, in April 2003 and in January 2007, respectively. From August 2002 to January 2008, he was with the Center of Excellence for Research DEWS, Uni- Harald Haas (SM’98–AM’00–M’03) holds the versity of L’Aquila, Italy. From February 2008 to Chair of Mobile Communications in the Institute for April 2009, he was a Research Associate with the Digital Communications (IDCOM) at the University Telecommunications Technological Center of Catalonia (CTTC), Barcelona, of Edinburgh. His main research interests are in the Spain. From May 2009 to December 2009, he was an EPSRC Research Fellow areas of wireless system design and analysis as well with the Institute for Digital Communications (IDCOM), The University of as digital signal processing, with a particular focus Edinburgh, Edinburgh, United Kingdom (UK). on interference coordination in wireless networks, Since January 2010, he has been a Tenured Researcher (“Charg´e de spatial modulation and optical wireless communica- Recherche Titulaire”) with the French National Center for Scientific Research tion. (CNRS), as well as a research staff member of the Laboratory of Signals and Professor Haas holds more than 15 patents. He has Systems (L2S), a joint research laboratory of the CNRS, the Ecole´ Sup´erieure published more than 50 journal papers including a d’Electricit´e(SUP´ ELEC),´ and the University of Paris–Sud XI, Paris, France. Science Article and more than 140 peer–reviewed conference papers. Nine of His main research interests are in the area of wireless communications theory, his papers are invited papers. He has co–authored a book entitled “Next Gen- signal processing, and information theory. eration Mobile Access Technologies: Implementing TDD” with Cambridge Dr. Di Renzo is the recipient of the special mention for the outstanding five– University Press. Since 2007, he has been a Regular High Level Visiting year (1997–2003) academic career, University of L’Aquila, Italy; the THALES Scientist supported by the Chinese “111 program” at Beijing University of Communications fellowship for doctoral studies (2003–2006), University of Posts and Telecommunications (BUPT). He was an invited speaker at the TED L’Aquila, Italy; and the Torres Quevedo award for his research on ultra wide Global conference 2011. He has been shortlisted for the World Technology band systems and cooperative localization for wireless networks (2008–2009), Award for communications technology (individual) 2011. Ministry of Science and Innovation, Spain.