A Joint Code-Frequency Index Modulation for Low-complexity, High Spectral and Energy Efficiency Communications

IEEE TRANSA CTIONS ON COMMUNICA TIONS 1 A Joint Code-Frequenc y Inde x Modulation for Lo w-c omple xity , High Spectral and En er gy Ef ficienc y Commu nications Minh Au, Member , IEEE, Georges Kaddoum, Me m be r , IEEE, Francois Gagnon, Senior Me mber , IEEE and Ebrahim Soujeri, Senior Member , IEEE, Abstract A relativ ely simple low complexity multiuser communica tion system based on simultaneou s code and freq uency index mo dulation (CFIM) is pro posed in this paper . T h e propo sed ar chitecture reduc e s the emitted energy at th e tran smitter as well as the p eak-to-average-power ratio (P APR) o f o rthogon al frequen cy-division multiplexing (OFDM)- ba sed schemes fu nctions withou t relegating data rate. In the scheme we introd u ce her e, we implement a joint code- frequency- ind ex modulation (CFIM) in order to en hance sp ectral and energy efficiencies. After intro ducing an d analysing the structure w ith respect to latter me trics, we d e riv e closed-for m expression s of th e bit er ror rate ( BER) perfo rmance over Rayleigh fading channels and we validate the outco me by simulation results. Simu lation are used verify the analyses and s how t hat, in terms of BER, the p roposed CF IM outperfo rms the existing in dex modulatio n (IM) based systems such as spatial modu lation (SM), OFDM-IM and OFDM schemes significantly . T o better exhibit th e particu larities o f the propo sed schem e, P APR, com p lexity , spectral efficiency (SE) and energy efficiency (EE) are thoro ughly examined . Results ind icate a h igh SE while ensuring an elev ated re liability compare d to the afo r ementione d systems. In addition , the con cept is extended to sy n chron o us multiu ser communication n etworks, w h ere full functionality is o btained. W ith the characteristics demonstrated in th is work, the p roposed system w ould constitute an exceptional nominee for Internet of Th ings ( IoT) application s where low-comp lexity , lo w-power consum ption an d high d ata rate are param ount. M. Au, G. Kaddoum, F . Gagnon and E. Soujeri are with the Department of Electrical Engineering, ´ Ecole de T echnologie Sup ´ erieure, Montr ´ eal, QC, H3C 1K3 CA e-mail: minh.au@lacime.etsmtl.ca. Manuscript receiv ed Nov ember 20, 2017; revised December 26, 2017. IEEE TRANSA CTIONS ON COMMUNICA TIONS 2 Index T erms Code-freq uency index-mo dulation, spectral efficiency , energy efficiency , low-complexity , I o T , mul- tiuser co mmunica tio n. I . I N T R O D U C T I O N Facing a tremendou s d emand for higher data rate commun i cation systems associ ated with an increasing need in the number of devices and gadgets for mobile Internet and Internet of Thi n gs (IoT) appl i cations, is the challenge that needs t o be address ed i n the next generation of wi reless systems. Indeed, they are required to achieve high sp ectral efficienc y (SE) while maintaining massive connectivity . For som e emerging d e vices such as d ron es and tiny sensors in intellig ent vehicle technolog i es, ultra reli abl e and extremely low-latency are absolut e requirements . W ithin th e de velopment of future wireless networks, the concept of IoT has em er ged as the most prom ising technology that supports and enables many devices t o gain Int ernet connectivity . This emerging technology has onl y become av ailable via major advances i n and av ailabil ity of tiny , cheap and i n telligent sensors that are cost-eff ectiv e and easily deployable [1]. Guaranteeing battery l ife sa ving by h a ving a low energy con s umption shoul d be another note worthy feature of these IoT devices. A lar ge number of researches hav e dev oted th emselves to fulfil t hese requirements over the last decade [2]–[4]. Recently , the concept of index mod ulation (IM), which utilizes the indices of some trans- mission entities to carry e xtra informati on b i ts, has been proposed as a competitive alternative in digital mod ulation techniques for the future wireless communications [5]. These IM-based systems aim to in crease data rate and energy efficienc y (EE) in emerging wireless comm unication devices. In this directio n, research in th e last decade has seriously consi d ered the usage of certain parameters such as sp ace, polarity , code and frequency as m odulation indices to enable the transmi ssion of extra bit s per symbol withou t demandin g l ar ger bandwidth o r higher power . In th is vein, the s patial modu lation (SM) technique which is based on transiti o ning between multiple antennas and us i ng the in dex of the acti ve antenna as a measure to carry e xtra in- formation is an area of gro wing interest where fruitful results hav e been provided. [6]–[10]. Howe ver , this technique requires additio n al antenn ae to achie ve m ore data rate. So, it is not implementable in senso r networks. Other examples of index-based mod u lation s y s tems that in volve orthogonal frequency-di vi sion multiplexing (OFDM) include subcarrier-inde x modu lation (SIM)-OFDM [11] and enhanced subcarrier-index modul ation (ESIM)-OFDM [12]. Howe ver IEEE TRANSA CTIONS ON COMMUNICA TIONS 3 these s y s tems are primitive and stay s hort of con ventional OFDM systems in t erms of ove rall performance. A relatively stronger but more complex in dex-based modul ati on system that in volves OFDM is called OFDM with Index Mo dulation (OFDM-IM) and is p rop osed in [13]. In this approach, the subcarriers are grouped into many blocks where a subdivision of subcarriers contained in each block is t u rned on depending on indexing bits, where the activated subcarriers t ransmit a symbol each. This OFDM-based IM techniqu e utilises maximum likelihood (ML) detectio n at the recei ver which i s indeed too complex for IoT and wi reless sensor de v i ces because lo w complexity and low power consumpti on say the final word in these tiny gadgets. As the IM modulatio n schemes menti oned above are either hardware-demanding or too com- plex, very low-complexity and ener gy effi cient IM techniques have been prop osed m o re recently [14]–[17]. In the generalized code index mod ulation (GCIM) [14]–[16], extra inform ation bits are carried by the selection of spreading codes. On the other hand, frequency index modulatio n (FIM) [17] is an OFDM system based, in this scheme extra bits are carried by the activ ati on of one s u bcarrier , wh i le the rest are nulled ou t . By choosi ng orthogonal spreading codes and frequencies, these bits are estimated based o n square law energy detectio n (SLED) at the recei ver side, yieldin g low-complexity and high EE. Howe ver , neither syst ems achiev e a h i gh SE because GCIM is designed for single-carrier commu nications, while some subcarriers are not used in FIM. Furthermore, in terms of multiuser c ommun i cation perspecti ve, gi ven that 5 G s tandards prom ise massive connectivity for billio n s of d evices [18], [19], these aforementioned s ystems migh t be limited and cost-ineffecti ve because SM requires hardwar e, i.e many antennae, while mo re bandwidth is needed in OFDM -bas ed systems i ncorporating IM. Therefore, achie ving hig h SE in FIM and GCIM is imperative. T o th e best of our knowledge, there is no performance analysis of IM systems for multiuser communications. Mo reov er , no attempt has been made regarding the benefit of having sev eral t ransmission enti ties to carry extra i nformation bits. Motiv ated by t he aforementio n ed prob l ems, this paper exploits the benefits o f having a joi n t code-frequency IM scheme that could meet the needs of 5G wireless systems to a great extent. This paper extends the idea of GCIM to mul t icarrier systems to enhance SE while m aximizing EE by reducing the peak-to-ave rage power ratio (P APR) vi a FIM. Bot h of these schemes exhibit low- complexity and can be used for m u ltiuser comm unications via spreadin g codes. The combinati on of these suit IoT applications where several sensors and devices mu s t be connected. The cont ributions of this paper are summarized as foll ows: IEEE TRANSA CTIONS ON COMMUNICA TIONS 4 • The j oint frequency and spreading code indexing is used as an information -bearing un it to increase both SE and EE w i thout adding extra h ardware com plexity t o the system. • A l ow- complexity detection meth od b ased on SLED is proposed. • Closed-form mathem atical expressions for the probability of bit error of CFIM are derive d. The performance of CFIM is compared to other IM-based approaches such as SM, OFDM- IM as well as con ventional methods such as OFDM. Results obtained show that CFIM outperforms these schemes. • Under t he frame work of m u lticarrier sy s tems, CFIM schem e is analysed i n synchronous transmissio n for the first t ime. The remainder of th i s paper is organized as follows. Section II provides the system m odel of CFIM including the transmitter and recei ver structu res. Closed form expressions of the probability of bit error are analy zed in Sec tion III. SE, EE and comp l exity of the syst em are di scussed in Section IV . CFIM is extended to s y n chronous mu l tiuser comm unications in Section V . Section VI presents computer simulat i on results concerning SE, EE and com p lexity in comparison to SM, OFDM-IM and con ventional OFDM systems. Conclusions follow in Section VII. I I . S Y S T E M M O D E L In t h is section, we present t he CFIM transmitt er and receiv er architectures. A. The T ransmitter A t ypical architecture of the CFIM transmi tter with K × N subcarriers is depicted in Fig. 1. A sequence b of p T bits is equally separated into K blocks. Since each CFIM block has the same processing procedure, we consid er the k th block for simplici t y hereafter . In this blo ck, the sequence b k = [ b p 1 k , b p 2 k , b p 3 k ] leaves the transm itter in chun ks or sub-blo cks of p bits where each sub-block consists o f 3 sub-chunks such that p = p 1 + p 2 + p 3 . In this format, log 2 ( M ) = p 1 denotes the number of modulated bits that m ap into an M − ary signal constel l ation to produce a symbol s k ∈ S , where S is a set of M − ary si g nals. Further , log 2 ( N c ) = p 2 is the num ber of mapped bits required to select the spreading code among N c codes, and lo g 2 ( N ) = p 3 is the n umber of mapped bits needed to activ ate one subcarrier . In this paper , we consider MPSK modulation only for i t s h i gher energy ef ficiency . The symbo l s k is multip l ied by a spreading code c J k k of length L in the time domain. Th i s is indexed by J k ∈ J k = { 1 · · · N c } and indicates t he selection of a code in a p o o l that contain s N c IEEE TRANSA CTIONS ON COMMUNICA TIONS 5 Bit Splitter Code Index Selection M -ary Modul ator F requency Index Selection Code Index Selection M -ary Modul ator F requency Index Selection b 1 b K b p 1 1 b p 2 1 b p 3 1 b p 1 K b p 2 K b p 3 K X ( I 1 ;J 1 ) X ( I K ;J K ) 9 > > = > > ; X 1 9 > > = > > ; X K x 1 x N T S/P b IFFT D/A x ( t ) I 1 c J 1 1 s 1 Spreader I K c J K K s K Spreader Fig. 1. CFI M architecture of the tr ansmitter of them . The signal is then trans m itted via the sub carrier index ed b y I k ∈ I k = { 1 · · · N } out of N av ailabl e subcarrier indices. The spreading cod e c J k k ∈ C L is selected from a predefined codebook in the blo ck k denoted as C k =  c 1 k , · · · c N c k  . Since part of the transm itted bits are con veyed by the code and the frequency indices, and to hav e a fair comparison with other con ventional systems, we define an equi valent system bit ener gy E bs which represents the effecti ve energy cons u med per transm itted bi t . This equiva lent system bit energy is related to t he phy sically mo d ulated bit energy by the relationship: E bs = p p 1 E b . (1) Thus, the transm itted symbol has an ener gy per coded bit E s = E [ s k s ∗ k ] = p 1 E bs . At th e subcarrier indexed b y I k , t he spread signal X ( I k ) k is giv en by: X ( I k ) k = h X ( I k ) k , 0 , · · · , X ( I k ) k ,L − 1 i T = h s k c J k k , 0 , · · · , s k c J k k ,L − 1 i T (2) while the inactiv ate subcarriers ∀ i 6 = I k , X ( i ) k are zero vectors of L elements. Due to their excellent correlation properties, orthogonal spreading codes such as W alsh-Hadamard or Zadoff- Chu may be used. In this work, we employ W alsh-Hadamard codes. Under such a condition , we hav e c J k k ·  c J ′ k k  ∗ = L − 1 X l =0 c J k k ,l  c J ′ k k ,l  ∗ =    1 i f J k = J ′ k 0 ot herwise. (3) IEEE TRANSA CTIONS ON COMMUNICA TIONS 6 where ( · ) ∗ is the complex conjugat e. After obtain ing and concatenating X k for all k , we have a vector X X =    X (1) , 1 ,l · · · , X ( N ) 1 ,l | {z } X 1 , · · · , X (1) K,l , · · · , X ( N ) K,l | {z } X K    T , (4) where X ( i ) k ,l for all k is giv en by X ( i ) k ,l =    s k c J k k ,l if i = I k 0 otherwise. (5) In addit i on, the orthogonality between the subcarriers h olds is assumed i n order to combat the intersymbol interference. This is given b y ∆ f = 1 /T N ≥ 1 /T c where T N is the duration of a CFIM s ymbol, and T c is the chi p in t erv al of the spreading code. Not e that direct spread spectrum is applied to each active s ubcarrier . Thu s , t h e spreading op eration i s performed over tim e. Thi s process is simil ar to mult icarrier direct-sequence code division mult iple access (MC-DS-CDMA) systems. Afterwards, the remaining procedures are t he same as th o se classical OFDM, where N T is the length after performing the fast Fourier transform (FFT). So, the complete CFIM transm itter that incorporates all blo cks would be s tated as: x ( t ) = 1 √ T N K X k =1 N X i =1 L − 1 X l =0 X ( i ) k ,l p ( t − l T c ) e j 2 π f i,k t , (6) where p ( t ) is assumed t o be a rectangul ar signaling pulse shifted in time given by p ( t ) =    1 i f 0 ≤ t ≤ T N 0 ot herwise. (7) The subcarrier f i,k is given by f i,k = N ( k − 1) T N + i T N . (8) Note that the active subcarrier index i s given by f I k ,k , with i = I k . As a matter of fact, the insertion and removal of cyclic guard prefix is no t expressed in our mathematical equ at i ons for sake of simplicit y . Fig. 2 illustrates the case where b p 2 k = [1 , 0] and b p 3 k = [0 , 1] which leads to the s el ecti on of the spreading code c 3 , and the selecti on of the second subcarrier , i.e. I k = 2 out of 4 av ailable subcarriers in the block. IEEE TRANSA CTIONS ON COMMUNICA TIONS 7 Co de Idx 00 01 10 11 00 01 10 11 c 1 c 2 c 3 c 4 f 1 f 2 f 3 f 4 F req Idx. Fig. 2. An illustration of the CFIM system wi th 4 subcarriers and 4 codes where the transmitter has index ed t he message 10 and 01 . T hen encodes the rest of the message via t he spreading code c 3 , t ransmits the encoded message via the subcarrier f 2 only . B. The Receiver A t ypical architecture of the CFIM recei ver with K × N subcarriers is depi ct ed in Fig. 3. In this paper , we assume that th e chip interval T c is larger than th e delay spread of the multipath fading channel, s o that a flat Rayleigh fading channel is assumed. Moreover , an additive white Gaussian nois e (A WGN) is also considered, In this condition, the spread receive d s ignal denoted by Y ( i ) k ,l at the i th subcarrier for all l = 0 · · · L − 1 in the k th block is given by Y ( i ) k ,l =    s k c J k k ,l h ( i ) k + Z ( i ) k ,l if i = I k Z ( i ) ′ k ,l otherwise, (9) where h ( i ) k ∼ C N (0 , 1) i s the FFT output prod u ced by the Rayleigh fading channel effec t, Z ( i ) k ,l and Z ( i ) ′ k ,l are independent complex A WGN, with zero mean and variance N 0 / 2 . In order to estimate the transmit ted message in the k th block with low-complexity decodi ng, we may first consid er a matrix of N × L elements d eno t ed by Y k such that Y k =      Y (1) k , 0 · · · Y (1) k ,L − 1 . . . . . . Y ( N ) k , 0 · · · Y ( N ) k ,L − 1      (10) Therefore, the recei ver u s es the a matrix c k =  c 1 k , · · · , c N c k  T of N c × L elements that contains those spreading codes in t h e codebook C k to despread the matrix Y k . So, the output of the despreader would s imply b e ˆ X k = Y k c H k , (11) IEEE TRANSA CTIONS ON COMMUNICA TIONS 8  Y 1 c H 1 c H K SLED ζ 1 = j Y 1 c H 1 j 2  arg max i;j f ζ 1 g arg max i;j f ζ K g ( ^ I 1 ; ^ J 1 ) Select Select ( ^ I K ; ^ J K ) M -ary Demodulator M -ary Demodulator Demapp er Sort and Merge ^ b ^ b p 1 1 ^ b p 2 1 ^ b p 3 1 ^ b p 1 K ^ b p 2 K ^ b p 3 K S/P y 1 y N T FFT G 1 G K r ( t ) A/D Despreader  Despreader Y K  ζ K = j Y K c H K j 2 Fig. 3. CFI M architecture of the receiver where ( · ) H is the Hermit ian transpose of the matri x. Then, a SLED is used to estim ate both the activ e subcarrier and t he selected code indices in the k th block. Since the spreading codes as well as t he subcarriers are mutually orth o gonal, the outp u t variables in the matrix ˆ X k are fed to SLED forming an N × N c matrix ζ k of decision v ariables such that for i = 1 , · · · , N and j = 1 , · · · , N c we have ζ ( i,j ) k =   ˆ X ( i,j ) k   2 =      s k h ( i ) k + η ( i,j ) k   2 H 1   η ( i,j ) ′ k   2 H 0 , (12) where H 1 and H 0 are two hypot heses wi th which H 1 indicates the presence of the symbo l s k , and H 0 is the alternative that there is no si gnal. The additive noise component η ( i,j ) k is give n by η ( i,j ) k = L − 1 X l =0 Z ( i ) k ,l  c j k ,l  ∗ , (13) and η ( i,j ) ′ k is written in th e same manner as the latter with Z ( i ) ′ k ,l . Since the A WGN components Z ( i ) k ,l and Z ( i ) ′ k ,l are mutually independent, η ( i,j ) k and η ( i,j ) ′ k are also independent. In order to esti mate the indices actually i nv olved in transmission i.e ( I k , J k ) , the SLED chooses the arguments of the maximum value of the matrix ζ k such that ( ˆ I k , ˆ J k ) = argmax i ∈I k ,j ∈J k n ζ ( i,j ) k o . (14) IEEE TRANSA CTIONS ON COMMUNICA TIONS 9 Therefore, the decision variable m ight land in H 1 if ( ˆ I k , ˆ J k ) = ( I k , J k ) . By estimatin g the activ e subcarrier and the selected code in d ices ( ˆ I k , ˆ J k ) , the receiv er can extract the p 3 + p 2 mapped bits. The receiver then dem odulates the correspond i ng branch outp u t usi ng a con ventional M − ary demodulator to extract the remain ing p 1 modulated bits. In this p aper , we assume that the channel coef ficients are perfectly estimated by the recei ver . Thus , an equalization is performed at the recei ver by dividing the recei ved symbol ˆ X ( i,j ) k by G k = 1 /h ( i ) k . By virtue of SLED, CFIM exhibit s a lo w -com plexity structure in comparison t o other IM schemes. When compared to OFDM-IM for inst ance, our method does not require sharing an indexing look up table between comm unicating parti es. I I I . P E R F O R M A N C E A NA L Y S I S In thi s sectio n, we inv estigate the performance of th e CFIM sy s tem in term s of th e probabilit y of error and revie w the system gains obtained by concurrently mappin g information bits into subcarrier and spreading code indices. A. Pr o bability of Bit Error of the Syst em In t he CFIM scheme, the t ransmitted data in ever y block can be divided into three parts; t wo blocks to represent t he mapped bits and a singl e bl ock to identify the modu l ate bits. The former two blocks determine the combination of subcarrier and spreading code selected, whil e the latt er block contains the remain ing data. In such structure, t h e probabil ity of b i t error o f the CFIM system consists of the probabilit y of b i t error of the mapped bi ts P map and the probabil i ty of bit error of the modul ated bits P mod . Subsequently , the prob abi lity of bit error of th e system could be described as P CFIM = p 1 p P mod + p 2 + p 3 p P map . (15) The p robability of errors P mod and P map are respectiv ely weight ed by the number of modu lated bits and the number of m apped bits divided b y the total nu mber of bit s in volved in transmiss ion. Moreover , these probabilities of error are linked to t h e probability of erroneous detection of the code-frequenc y index P ed pair . Indeed, P mod depends on the correct estimation of the selected indices and t he probabilit y of bit error o f the M − ary modu l ation P b when the i n dices are correctly estimated. Thereupon, errors happen in two dif ferent manners. The first case is when the selected indices are correctly est imated but an error in t he demodul at i on process takes IEEE TRANSA CTIONS ON COMMUNICA TIONS 10 place. The second case is when an err or befalls in the assessment of the indices chosen at transmissio n , and the mod ulated bits are t herefore estimated using inaccurate code-frequency indices. In thi s case, the recei ver has no choi ce but t o guess the modulated bits. In fact, the probability of bit error will be simply equal to 1 / 2 . Consequently , t he BER probability of the modulated bits would be expressed as: P mod = P b (1 − P ed ) + 1 2 P ed . (16) In determ i ning the P map , the erroneous detectio n o f the code-frequency ind ex pair can cause a wrong estimation of the combination of mapped mod u lated bits. Each wrong combination can hav e a diffe rent numb er of in correct bits com pared to the correct combi n ation, i.e the actually t ransmitted one. Since, we hav e assumed that the frequency and the spreading codes are mutually orth o gonal, the detecti o n of t he i n dices is sim ply equiv alent to a noncoherent N × N c - ary orth o gonal system upon w h ich operates t h e SLED. Therefore, the probability o f index in error P ed is con verted i nto the corresponding BER prob ability of mapped bi ts as P map = 2 ( p 2 + p 3 − 1) 2 p 2 + p 3 − 1 P ed . (17) B. Pr o bability of Err oneous Detection of the Code-F r equ ency Index Since a SLED is used for estimatin g the pair of ind i ces, we shall determi n e the probabi lity of erroneously detecting the pair o f the code-frequency indices P ed . T o do s o, we assu m e an equiprobable selection of th e active s u bcarrier and the spreading code. Moreover , for the sake of clarity , we focus on a single block, i.e the k th block such that ζ ( i,j ) k of (12) simpli fies to ζ ( i,j ) ∀ i ∈ I and ∀ j ∈ J . Moreover , Since the frequency as well as the spreading codes are mutually orthogonal, we can write t he matrix ζ as a vector o f N N c elements, which is d eno ted as vec ( ζ ) =  ζ (1 , 1) , · · · , ζ ( N ,N c )  =  ζ (1) , · · · , ζ ( N N c )  . Denoting t he selected index at the transmitter as ν , the probabili ty of index error P ed conditioned on the selected index µ ∈ { 1 , · · · , N N c } would be P ed = P  ζ ( ν ) < max  ζ ( µ )  | ν  , for 1 ≤ µ µ 6 = ν ≤ N N c . (18) As deduced from in equation (18), an error in the estimation of the selected index will occur if the decision variable max  ζ ( µ )  is larger than ζ ( ν ) . In this condition , i t is easy that P ed is IEEE TRANSA CTIONS ON COMMUNICA TIONS 11 P m = 1 2 π    π (1 − (2 m − 1) / M ) Z 0 exp  − p 1 γ b sin 2 [(2 m − 1) π / M ] sin 2 θ  dθ − π (1 − (2 m +1) / M ) Z 0 exp  − p 1 γ b sin 2 [(2 m + 1) π / M ] sin 2 θ  dθ    (22) equiv alent to the probabi lity of detection of a noncoherent N × N c -ary orthog o nal system over the Rayleigh channel. Th i s i s given by [20]: P ed = N N c − 1 X µ =0 ( − 1) µ 1 + µ + µ ¯ γ c  N N c − 1 µ  , (19) where ¯ γ c = E [ | h | 2 ] E s N 0 is the ave rage signal-to -no ise ratio (SNR) per symbo l . C. Pr obab i lity of Bit E r r or of Modul a ted Bits When the subcarrier and spreading code indices are correctly estimated, then we must consider the presence of the channel coeffi cient h when formu l ating the av erage bit error probability P b of the modulated bi t , wh i ch becomes condition al on the received power and may be drafted as P b = Z + ∞ 0 P b | γ b P ( γ b ) dγ b , (20) where γ b = | h | 2 E bs N 0 is the i nstantaneous SNR per bit. P ( γ b ) is the condit ional probability distribution fun ct i on of γ b giv en the correct index estimation . P b | γ b is the condition al probability of bit error in A WGN channels [21 ]. Specifically , for M-PSK modulation using Gray code, we may use the clo se expression in [22] t o d erive P b | γ b : P b | γ b ≈ 1 p 1 M / 2 X u =1 w ′ m P m , (21) where w ′ m = w m + w M − m , w ′ M / 2 = w M / 2 , w m is the Hamming weight of bits assigned t o th e symbol m , and P m is given i n equati on (22) as stated in [22]. In order to comp ute t he probabilit y of bit error P b , t he probabilit y d istribution of γ b needs to be deri ved. It should be noted that t he random va riable γ b is obtained knowing that the index estimation is correct. As a result, if the index esti m ation is always correct, i.e. P ed = 0 , th en γ b is chi -square distri buted with two degrees o f freedom. This is due to the fact that h ∼ C N (0 , 1) . In t h is case, P b in equation (20) is equivalent to t he probabi lity of bit error of th e con ventional MPSK in Rayleigh fading channels. Howe ver , in the low-SNR regime, SLED selects the index IEEE TRANSA CTIONS ON COMMUNICA TIONS 12 that has the greatest SNR. Consequently , γ b is no longer chi-square distributed with two degrees of freedom. In such a condition, the conditional probability distribution functi on P ( γ b ) is not a vailable. Neve rtheless, an empirical distribution may be obtained. By computi ng P b using equation s (21) and (22), t he BER probabil ity of the modulat ed bits P mod can be obtain ed via equation (16). W it h P map in equation (17), the comprehensiv e probability of bit error of the proposed P CFIM scheme is presented i n equat i on (15). I V . S P E C T R A L E FFI C I E N C Y , E N E R G Y E FFI C I E N C Y A N D C O M P L E X I T Y A N A L Y S E S In th is s ection, we consider the spectral and th e energy efficienc y as well as the complexity analyses for the propo s ed CFIM system. A. Spectral Efficiency The CFIM system has a tot al of K N subcarriers, but uses K subcarriers to transmit K p bits at ev ery transmi ssion i nstant. In fact o n e out of N subcarriers is activated in each block, so the spectral effic iency is simpl y indicated as ξ CFIM = p N = p 1 + p 2 + p 3 2 p 2 . (23) It can be seen th at increasing p 2 e.g th e number of su b carriers would reduce the spectral ef ficiency , while increasing the mo d ulation order and t he number of spreading codes bot h enhance the spectral effic iency . Howe ver , these would impact the overall performance of the syst em. B. Ener g y Efficiency In CFIM systems, only K p 1 bits from the t otal K p are di rectly modulat ed using M − ary modulation , wh ereas K ( p 2 + p 3 ) b its are con veyed in the selectio n of codes and subcarriers. Considering that each m odulated bit requires an energy o f E b to be transm itted, th en mapp i ng to index subcarriers and spreading cod es should reduce t he tot al required transmiss i on energy . Consequently , the percentage o f energy saving per block in th e proposed system is g iv en by E sa ving =  1 − p 1 p  E b % = 1 − 1 1 + p 2 + p 3 p 1 ! E b % . (24) Under this condition, the energy sa ving depends on the rati o of the number of sub carriers and spreading codes in volve d in indexing t o t h e mo d ulation o rder of the system. Clearly , an IEEE TRANSA CTIONS ON COMMUNICA TIONS 13 augmentation in the number of subcarriers o r spreading codes result s i n m ore energy saving i n the CFIM system . Howe ver , increasing the num b er of subcarriers reduces th e spectral efficienc y . Hence, ind exing via s preading codes is asso ci ated with less cost in terms of energy and spectral ef ficiency . A drawback o f multi carrier sy s tems is their hi g h peak to a verage power ratio (P APR). This impacts the p erformance by distorting the si gnal i nduced by the nonlinearity of hig h power amplifier (HP A) [23]–[25]. This cou ld i n turn deteriorate the s p ectral and ener gy effic iency of the system . Reducing P APR leads to a significant power sa ving, which improves the ener gy ef ficiency performance. In fact, a high P APR value appears when a number of subcarriers in a giv en OFDM system are out of phase with each oth er . Thus , a high P APR value o ccurs when a lar ge num ber of sub carriers is activ ated. Since the propos ed s cheme acti vates a s i ngle frequency in each block, P APR is reduced. The P APR i s d efined as P APR = max 0 ≤ t ≤ T N | x ( t ) | 2 1 /T N R T N 0 | x ( t ) | 2 dt . (25) Using equation (6), the maximu m expected P APR can be determined. In con ventional OFDM, assuming that all sym bols are equal, the peak value is give n by max [ x ( t ) x ∗ ( t )] = E s T N K 2 N 2 L 2 , (26) while the mean square value of the sign al is E [ x ( t ) x ∗ ( t )] = E s T N K N L. (27 ) In t his condi t ion the maximum expected P APR of the OFDM is K N L . In CFIM, since a singl e subcarrier per block is activ ated, it is easy to determi ne the maxi m um expected P APR as K L . Therefore, the CFI M can reduce the P APR by 1 / N compared to con ventional O FDM . Since N = 2 p 2 , the P APR will signi ficantly reduce as the number of bit s p 2 increases. C. System Complexity The com plexity of the CFIM syst em can be ev aluated by the number of operations required to accompl ish transmissio n . Since the CFIM uses IFFT and FFT operations with a length of N T , the computational complexity will be O FFT/IFFT ∼ O (2 N T log 2 ( N T )) . For the sake of clarity , th is will be omi tted in this paper . IEEE TRANSA CTIONS ON COMMUNICA TIONS 14 The number of spreading codes and n o t its length is used for i ndexing. Considering th at each spreading code cont ains L elements , and the transmitter selects one o ut of N c codes in t he k th block, the resulti ng numb er of o p erations is L on th e transmitter side. The receiver forms the matrix Y k ∈ C N × L as d escribed in equati on (10) after t he FFT operation. In order to estimate the transmitted informatio n, the receiver despreads the received signal using a matrix c k of N c × L elements as in equation (11). As such, the computational complexity of this operatio n becom es O spread/de spread ∼ O (2 N LN c − N N c + L ) . Since there are K blocks, the overall complexity that incorporates t he transm ission and reception t urns i nto K O spread/de spread . The SLED mul t iplies the vector ˆ X k by its compl ex conjugate. Th erefore, the compu t ational complexity is simply O SLED ∼ O ( N N c K ) . Regarding the mo dulation and the demo d ulation, we assume t hat a modulator con verts a stream of p 1 bits into an M − ary s y mbol by com puting an inner product between the bit st ream and the vector [ 2 p 1 − 1 , · · · , 1] . The demodulat o r wil l con vert the symbol into a bit s tream by computing p 1 Euclidean divisions. As a result, the computational complexity of a modulator and a demodulator wil l be O Mod/Demod ∼ O (3 p 1 − 1) per activ e s ubcarrier . In total, the com plexity of the CFIM sy s tem would be expressed as O CFIM = O SLED + K ( O spread/de spread + O Mod/Demod ) . (28) In order to have a fair comparison between the propos ed system with other multi-carrier systems in th e state of the art, we ass u me t hat d irect spread spectrum com munications are used in the following sy s tems. In the con ventional DS-OFDM scheme, all subcarriers are acti vated during the transmission, and a s preading code is used in each subcarrier . Since there is one spreading code p er subcarrier , the complexity o f the spreading and desp reading operations is giv en by O spread/de spread ∼ O (2 L − 1 + L ) per subcarrier . Assuming that there is K different symbols that are di stributed over K N subcarriers, the computational complexity will b e O DS-OFDM = K N ( O spread/de spread + O Mod/Demod ) . (29) Aiming at comparin g th e CFIM sy stem to system s which share th e same modulati on concept but with a different recei ver design, w e comp are the complexity of CFIM t o that of DS-OFDM-IM. In the latter , on l y g ou t of N subcarriers are activ ated. Howe ver , th e compl exity of the maxim um likelihood detector is O MLD ∼ O ( 2 p 2 M g ) per acti ve subcarrier where g = p 2 / log 2 ( M ) , [13]. Since g spreading codes are used per block, the comp u t ational complexity is g L per active IEEE TRANSA CTIONS ON COMMUNICA TIONS 15 subcarrier . At th e receiv er , despreading of the received sign al is required. Since t here are g activ ated su bcarriers out of N , there exists N choose g tim es g ! possib l e combination . Un d er this condition the comput ational comp l exity that in corporates sp reading and depsreading o p erations would be O spread/de spread ∼  N g  g !( 2 L − 1) + g L . As suming t hat there are K different blocks the total computatio nal complexity of t he system becomes O OFDM-IM-DS-CDMA = K ( g O spread/de spread + O Mod/Demod ) + O MLD . (30) V . E X T E N S I O N T O S Y N C H RO N O U S M U LT I U S E R C O M M U N I C A T I O N S In t h is section, we extend the proposed sys tem to synchronous mul tiuser comm unications. A. Index Modul ation f o r Mu ltiuser Commun ications u sing CFIM Assuming that a codebook C k in the k th block can be split into a group of U codebook s of N c orthogonal spreading codes each. In this condit ion, CFIM can be extend to multiuser communication s where it operates i n the same frequency band with no mult iuser interference (MUI). Fig. 4 il lustrates the case where three users hav e two spreading codes each and four subcarriers are av ailable for transmis sion. Co de Idx 00 01 10 11 c 1 c 2 c 3 c 4 f 1 f 2 f 3 f 4 User1 z }| { 0 1 User2 z }| { 0 1 u 1 u 2 c 5 c 6 u 3 User3 z }| { 0 1 F req. Idx Fig. 4. The CF IM system with 4 subcarriers and 2 codes per user . In the i llustration, 00 has been i ndex ed for User 1 and User 2, and thus f 1 has been selected, while f 3 has been selected for User 3 . Regarding the spreading code selection, c 1 , c 4 and c 5 hav e been selected respectiv ely for User 1 , 2 and 3 . B. Downlink T ransmission In downlink transmi s sion, a base station prepares one message for each us er using U CFIM modulators and transmits the overa ll signal where they are expected to b e receiv ed by U IEEE TRANSA CTIONS ON COMMUNICA TIONS 16 CFIM 1 CFIM 2 CFIM U b 1 b 2 b U User 1 User 2 User U X 1 X 2 X U OFDM Modulator Modulator Modulator Modulator Y CFIM 1 Detector CFIM 2 Detector CFIM U Detector User 1 User 2 User U ^ b 1 ^ b 2 ^ b U Fig. 5. Multiuser CFIM in downlink scenario independent recei vers. In sy n chronous transmission, MUI can be a voided by using orthog o nal spreading codes. Fig. 5 i llustrates the multi user CFIM i n d ownlink scenario. Focusing on the k th block, the spread receiv ed signal at the i th subcarrier for all l = 0 · · · L is giv en by Y ( i ) k ,l =      h k N u,i P u =1 s u k c J k,u k ,l + Z ( i ) k ,l if i = I k ,u Z ( i ) ′ k ,l otherwise, (31) where h k is the f ading channel coef ficient, N u,i is the total n u mber of users whose combined signal is transm i tted b y the i th subcarrier . More, s u k , c J k,u k ,l for l = 0 · · · L − 1 are respecti vely the modulated symbol and the s preading code selected by the u th user . Note that ( I k ,u , J k ,u ) is the code-frequency index pair selected by the u th user . Z ( i ) k ,l and Z ( i ) ′ k ,l are t he addit ive n o ise component i n the i th subcarrier . In s ynchronous downlink scenario, si nce orthog o nal spreading codes are used, the d etecti on of the transmitted signal for t he u th recei ver is the same as a si ngle CFIM recei ver as described in Section II-B. C. Uplink T ransmission In uplink transmiss ion, the users transmit U messages to the base s t ation using CFIM trans- mitters. The base s tation has U CFIM receive rs to decode those m essages as d epi cted i n Fig. 6. Since we hav e assumed synchron o us transmissio n, at t he k th block, and at the i th subcarrier for all l = 0 · · · L the spread receiv ed s ignal at the b ase stati on i s g iven by Y ( i ) k ,l =      N u,i P u =1 h k ,u s u k c J k,u k ,l + Z ( i ) k ,l if i = I k ,u Z ( i ) ′ k ,l otherwise, (32) where h k ,u is the fading channel coefficient o f the uth user . In synchron ous transmi ssion, each recei ver decodes their message as described in section II-B. IEEE TRANSA CTIONS ON COMMUNICA TIONS 17 CFIM 1 CFIM 2 CFIM U b 1 b 2 b U User 1 User 2 User U T ransmitter T ransmitter T ransmitter CFIM 2 Receiver CFIM 1 CFIM U Receiver Receiver OFDM Demodulator ^ b 1 ^ b 2 ^ b U Base station Fig. 6. Multiuser CFIM in uplink scenario V I . N U M E R I C A L R E S U L T S In thi s section, we stu d y the o btained analy t ical and simulat i on resul ts for the proposed CFIM system and sho w that t hese results are in g ood agreement. Then we compare the performance of CFIM to other i ndex-based schemes like SM and OFDM-IM system s . W e also st u dy the complexity , spectral and energy efficienc y for the propos ed system. Finally , CFIM for multius er communication i n sy nchronous transm ission is analyzed. In this paper , we have used a W alsh- Hadamard matrix with v arious size for spreading operations. Moreover , we have omit ted t he cyclic prefix for sake of s implicity . A. P erformance of CFIM T o ha ve a better s ight into the problem, we scrutini ze for va rious scenarios the parameters that affe ct the performance of the proposed CFIM in thi s subsection. As the o u t going bit s in the system constit ute of three main parts t hat are related to t he modulat i on order M , subcarriers N and spreading codes N c , we shall consider t h e influence of each of these elements o n the system performance. W e start by plottin g th e bit error rate (BER) performance of the CFIM scheme for various M , N and N c in Fig . 7. This ov erall performance is extracted from equation (15). W e witness in this figure that analytical and sim ulation result s for t h e CFIM s y s tem are in correspondence and great harmo ny , which approves the sureness of our method. The modulation order M having the m ost dest ructiv e in fluence on the BER performance is natural, as increasing M reduces the Euclidean distance between the transmitted symbol s and narro ws down d ecis i on zones at t he recei ver . IEEE TRANSA CTIONS ON COMMUNICA TIONS 18 0 5 10 15 20 25 30 35 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 7. Performance of CFIM over Rayleigh fading channels with a spreading factor of L = 32 . Furthermore, Fig. 7 also s hows that ou r system is desi gn-flexible and adapt able to acquire sev eral forms in terms of M , N and N c . W e observe that the CFIM system using M = 2 , N = 4 , N c = 2 plott ed as red curve and the CFIM using M = 4 , N = 2 , N c = 2 plott ed as blue curve transmit both 4 b i ts per transmiss ion. Howe ver , the first one exhibits a better performance, because the modulated part transm its one sin g le bit per symbol while the other transm its two bits per sym b ol. The BER performance of CFIM for a fixed mod ulation order M = 2 and v arious N and N c is sho wn in Fig. 8. as depicted in this figure, the parameters N and N c hav e a marginal influence on th e BER performance compared to t h e parameter M . In fact, th e propos ed system exhibits the same property as FIM systems. The BER performance would in d eed deteriorate if the contribution made by the ener gy contain ed in the mapped bits were not there. This is because higher values of N and N c would simpl y challenge the receiver to choose a correct code-frequency index pair from withi n a larger set and would deteriorate the BER performance. But mapped bits balance the performance by virtu e of their energy cont ribution. It should be noted that unl ike FIM sys t ems, the proposed sys t em can improve the reliabilit y while enhancing SE. Indeed, in FIM schemes only one subcarrier out of N is activa ted. T o get additional mapped bit s, this system n eeds to increase the numb er of subcarriers a vailable N , which requires larger band width expansion, whi ch reduces SE. Fortunately , the propos ed CFIM compensates the spectral efficienc y loss by in d exing via spreading codes. IEEE TRANSA CTIONS ON COMMUNICA TIONS 19 0 5 10 15 20 25 30 35 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 8. Performance of CFIM with a modulation order of M = 2 , a spreading factor of L = 32 and v arious number of spreading codes and subcarriers. B. P erformance Compariso n with FIM, OFDM-IM and SM Systems W e compare the p erformance of our prop o s ed w ork to the performance of two other in d ex modulation based schemes, nam ely t he FIM, the o rt hogonal frequency division m ultiplexing with index modulation (OFDM-IM) and the spati al modulati o n (SM) techniques. The FIM s ystem the mapped bits are indexe d by the activation of one out of N subcarriers [17]. The concepts of OFDM-IM and SM are very well explained in [13] and [6]–[9]. For sim plicity , we consider a comparison with th ese index modulati on schemes for the case of d i spatching 4 bits per transm ission where the number of subcarriers i s l imited to N = 4 , and the num ber of transmi tting antennae for SM is also li mited to N t = 2 . In this case, since N = 4 , the CFIM uses a modulati o n order of M = 2 , and the number of spreading codes av ailable is N c = 2 . In OFDM-IM , w e acti vate g = 2 su b carriers out to N and a m odulation order of M = 2 is used. In SM, since N t = 2 , a modulat i on order of M = 8 and a single receiving antenna N r = 1 are considered. In the last two systems, m axi mum-likelihood is imp lemented i n the receiv er to ensure a higher reliabili ty . For the FIM system , a mod ulation o rd er of M = 4 is employed. In addition to these, a com parison with a si ngle carrier system t hat employs 16 -PSK modulation is considered. Fig. 9 shows the performance of the proposed system in compariso n t o ot h er index modulati o n based schemes. It can be seen that CFIM outperforms SM, FIM and single carrier 16 -PSK IEEE TRANSA CTIONS ON COMMUNICA TIONS 20 0 5 10 15 20 25 30 35 E bs / N 0 10 -4 10 -3 10 -2 10 -1 10 0 Av erage B ER CFIM M = 2 , N = 4, N c = 2 FIM M = 4, N = 4 SM M = 2, N t = 8, N r = 1 OFD M-IM M = 2, (4 ,2) 16-PS K Fig. 9. Performance of CF IM in comparison to FIM, SM, OFDM-IM, OFDM and 16 -PSK schemes f or 4 bits per transmission systems. Alth o ugh OFDM-IM outperforms ot h er index modulatio n based sy s tems at E b / N 0 = 2 5 dB, CFIM exhibits a bett er BER performance in the lo w SNR regime. In IoT appli cations as well as i n wireless sensor networks, com m unication takes p lace more often v i a transmissi on at the lo w SNR regime, i .e the SNR would typically range from 0 to 25 dB. As a result, CF IM suggests an ideal solution for this type o f communicatio n n etworks, where low transm i ssion power con s umption is an absolu t e requirement. When t he num ber of subcarriers is limited but we wi s h to enhance the spectral efficienc y , the modulation order needs to be increased in con ventional OFDM. In such a condition, the BER performance degrades. The p roposed system is mo re flexible so that it can be designed in diffe rent fashions, i.e in a way to optimize the sp ectral ef ficiency whi l e having h i gh reliabilit y without any additional bandwidth. In Fig. 10, we il lustrate this by plot ting the performance of CFIM vs. con ventional OFDM schemes with spectral efficiencies of 3 bps/Hz and 4 bps/Hz. In either cases, w i th a total nu m ber of four subcarriers, it can be seen that CFIM achie ves b etter performances compared to OFDM. In fact, the propos ed s chem e activ ates two subcarriers out of four and the number of transmi tted bits is maxim ized with the help of spreading codes selecti o n. This allows for m inimizing the modulation order that guarantees high er reliability . Unli ke SM and OFDM-IM that use ML detection, su ch a scheme achieves a n o ticeable p erformance despit e its sim p licity and ease. The proposed system overcomes SM as it does not require neither space nor heavy hardware and it IEEE TRANSA CTIONS ON COMMUNICA TIONS 21 0 5 10 15 20 25 30 35 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 10. Performance comparison of CFIM to con ventional OFDM with spectral ef ficiencies of 3 bps/Hz and 4 bps/Hz. Number of blocks i s K = 2 and number of subcarriers per block is N = 2 . does not totally depend on channel coef ficients to perform. It also overcomes OFDM-IM which is far too complex for IoT applications . C. Complexity , Ener g y and Spectral Efficiency Despite its s implicity , CFIM can achieve better spectral ef ficiency while maintaini ng hi gher reliability compared to ot h er index m odulation based systems. Mo reov er , CFIM as well as FIM schemes acti vate on e subcarrier out of N sub carriers in a bl ock, yielding l ow- complexity and high energy ef ficiency com munication systems. Therefore, these two criteria are paramount for IoT devices as well as for low-cost wireless sens o rs. 1) S ystem Complexity: The computati o nal complexity of these systems i s cons idered for e valuation here. Since, CFIM can be applied to mult i user communications , we app l y direct s p read spectrum to other comm unication schemes in order to hav e a f air comparison. As a result , th e complexity analysis of these systems may be discuss ed by considering DS-FI M, conv ent ional DS-OFDM and D S-OFDM-IM. It should be n oted t hat we ha ve om itted the comp l exity of the IFFT/FFT part because the number of operations i s the s am e for all the aforementioned s y stems. In fact, t he complexity of the CF IM d epends on the n umber of spreading codes inv olved in indexing. This is because on th e receiv er side, N c despreading operations are needed on each subcarrier to estimate the mapped bit. Moreover , the SLED has N × N c decision variables, which IEEE TRANSA CTIONS ON COMMUNICA TIONS 22 50 100 150 200 250 10 2 10 3 10 4 10 5 10 6 10 7 Fig. 11. Complex ity comparison of the proposed CFIM to FIM, OF DM and OFDM-IM( N , 2 ) systems using a modulation order of M = 2 and a spreading factor of L = 32 in a single block, e.g K = 1 . results in more operations when N c is increased. Therefore, to offset better spectral efficienc y and high reliabi l ity comm unication, CFIM may incur more com p lexity on the receiver side. Fig. 11 depi ct s t h e complexity of the CFIM in comparis o n wit h other index m odulation based systems according to equatio n s (28 ), (29) and (30). Note that, the comp l exity of FIM is similar to equation (28) wit h N c = 1 . By fixing the modul ation order to M = 2 and increasing the number of subcarriers av ail abl e, FIM exhibits the lowest complexity b ecause o n ly o n e subcarrier is activ ated for t h e transmi ssion, and one singl e uniq u e spreading code is used for the spreading and the despreading operation. In DS-OFDM, assuming that all subcarriers are activated, more o perations is needed to estim at e the transmitted inform at i on, which results in higher com plexity . On the other hand, CFIM syst ems exhibit higher compl exity as the number of spreading codes in volved in indexing increases. An augmented system compl exity is the price to pay to get a higher number o f bits per transmission while maintaining reliabi lity . Nevertheless, t he comp lexity is reasonable for a small number of spreading codes, i.e N c = 2 . Furthermore, it can be seen that the proposed sy stem is clearly the winner compared to OFDM- IM for large N as the latt er employs maximum-li kelihoo d detectio n to estimate the inform ation, with a complexity of O MLD ∼ O (2 p 2 M g ) . T h is induces further complexity when the number of activ ated subcarriers g is h igher . IEEE TRANSA CTIONS ON COMMUNICA TIONS 23 T ABLE I C O M PA R I S O N C F I M , D S - F I M , D S - O F D M A N D D S - O F D M - I M S Y S T E M S U S I N G A M O D U L A T I O N O R D E R O F M = 2 , N = 4 S U B C A R R I E R S , A N D S P R E A D I N G FAC T O R L = 32 System Complexity Spectral Eff. Energy saving CFIM( 4 , 2 ) 546 1 bps/Hz 75% CFIM( 4 , 8 ) 2082 1 . 5 bps/Hz 83 . 3% CFIM( 4 , 32 ) 8226 2 bps/Hz 87 . 5% DS-FIM 290 0 . 7 5 bps/Hz 66 . 7% DS-OFDM 388 1 bps/Hz 0% DS-OFDM-IM( 4 , 2 ) 841 1 bps/Hz 50% 2) E n er gy Efficiency: In order to complete the analysis, we ha ve ev aluated the energy ef- ficiency of the proposed CFIM system in comparison with the aforementio n ed syst ems. Since modulated bits are on l y t ransmitted via the channel, one can sav e s ignificant amount of energy by mapping more bits . According to equatio n (24 ), th e system can achieve better energy effi ciency by mapping more informatio n bi ts. Ho wev er , sa vi ng ener gy comes at the cost o f dim inished spectral ef ficiency in FIM syst em . This is due to the fac t that only on e single subcarrier is activ ated out of N . The proposed CFIM system can save a great amount o f ener g y while m aintaining a desired spectral ef ficiency . This is made possibl e by indexing via spreading codes. Indeed, t his does not require neither additional bandwi dth nor energy E b to transmit extra b its in the system. T able I compares our CFIM system with DS-FIM, DS-OFDM and DS-OFDM-IM in terms of complexity , spectral ef ficiency and energy sa ving. By using a mod u lation order of M = 2 and N = 4 subcarriers in all syst ems, CFIM is superior in t erms of ener gy saving. No t e that increasing the number of spreading codes saves ener gy , enhances s pectral effi ciency , but increases complexity . Furthermore, b y comparing CFIM( 4 , 2 ), DS-OFDM and DS-OFDM( 4 , 2 ) at equal spectral conditions, CFIM exhibits less complexity compared to DS-OFDM( 4 , 2 ) and saves more ener gy compared to the two aforementioned systems . IEEE TRANSA CTIONS ON COMMUNICA TIONS 24 By comparing the proposed system in terms of energy saving with different spectral efficien- cies, it can be seen that CFIM systems are s u p erior to OFDM-IM and FIM in terms o f energy saving as depicted in Fig 12. Again, increasing the number of s p reading codes N c improves significantly the energy ef ficiency . Furthermo re, it can b e seen th at for 1 bps/Hz, CFIM( 4 , 4 ) hav e 100% energy saving. In fact, i t means t hat one can transm i t mapped bits with no addi t ional ener gy for mod u lated bit s. M o reover , i mproving spectral effic iency results i n less energy saving, because the modulati on order M needs to be higher . As depicted i n Fig. 13, i ncreasing th e number of subcarriers does not achieve a better energy efficienc y . 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 12. Energy efficienc y analysis vs. spectral efficienc y wi t h N = 4 subcarriers In Fig. 14, w e hav e ev aluated the com p lementary cumulative distrib ution function (CCDF ) of the P APR in the proposed s y stem in comparison with OFDM and OFDM-IM( 4 , 2 ) syst ems. It should be noted that th ese sy s tems transmit 52 b i ts per transmission and t h e FF T lengt h is N T = 64 . It can be seen t hat t h e CFIM has the lowest P APR reducti o n compared to these two systems. This i s due to th e fact that only one subcarrier per block is activ ated, which reduces the probabili ty of having a high value of P APR. D. P erformance of CFIM in Multiuser Communications Since spreading codes are used in the proposed syst em , it is easily extendible to the concept of m ultiuser comm unications. In s y nchronous transm ission, each user transmit s or recei ves the information in the s am e time slot. As such, orthogonal spreading codes may be implemented IEEE TRANSA CTIONS ON COMMUNICA TIONS 25 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 13. Energy efficienc y vs spectral efficienc y of C FIM with various number of subcarriers and N c = 4 spreading codes. 0 2 4 6 8 10 12 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 14. CCDFs of the P APR of the proposed CFIM( 4 , 2 ) in comparison with con ventional OFDM and OFDM-IM( 4 , 2 ) using a modulation order of M = 2 , number of blocks K = 13 , the number of subcarriers per block is N = 4 and the F F T length is N T = 64 in order t o m itigate m ultiuser i nterferences (MUI). Thus, assum ing that each d e vice uses CFIM systems, in both downlink and up link transmissi on s cenario, the BER performance i s equal to equation (15), i. e. this is equ al t o t he performance of a single user CFIM transmissi on scenario. Furthermore, in a single-user scenario, we ha ve seen that CFIM can enhance t he spectral ef ficiency by increasing the n umber o f spreading N c . Howev er , there is a trade-off bet w een the IEEE TRANSA CTIONS ON COMMUNICA TIONS 26 number of users i n the scenario and the spectral efficienc y , because the number of orthogonal codes i s a limi ted resource. Given the number of users and th e size of a W alsh-Hadamard m atrix, the maximal achiev able spectral ef ficiency per user can be determined via equation (23). Fig. 15 depicts the s pectral effi ciency using a modul ation order of M = 2 with a tot al o f N = 4 subcarriers. It can be seen that CFIM system can significantly enhance the spectral ef ficiency compared to th e con ventional OFDM, wi t h a l arge number of spreading codes av ailable. It should be noted that t he maximal achiev able spectral ef ficiency decreases as the nu m ber of users increases. No t ably , above 32 users with a size of 64 × 64 W alsh-Hadamard matrix, it is mo re ef ficient to use the con ventional OFDM rather than CFIM systems to achie ve higher spectral effic iency . 5 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 3.5 4 Fig. 15. S pectral efficiency in multiuser CFIM using a modulation order of M = 2 with a total of N = 4 subcarriers, and using W alsh-Hadamard matrices of v arious size. V I I . C O N C L U S I O N S In this paper , we ha ve proposed a low-complexity index modulation based system th at can significantly enhance the spectral and the energy efficiencies whi l e mai n t aining reliabi l ity . This scheme i s b ased on a join t index mo d ulation: code and frequency index. The code i ndex enhances the SE while the frequency ind ex ensures a better EE by reducing h igh P APR values. The proposed scheme suits IoT applications and can be e x tended to mul t iuser com munications, as notably a remarkable performance has been shown in synchronous transmissi on. IEEE TRANSA CTIONS ON COMMUNICA TIONS 27 The in troduced system may be built on OFDM platforms as this l atter is abundant and easy to program. In short, the system is divided into K blocks t h at operate N sub carriers and N c spreading codes each, and use M − ary keying. At the transmitt er side, eac h block contains an outgoing bit s t ream that is divided into three parts. The bits in the first part m odulate an M − ary symbol that is actually transmitted t h rough the channel. The bits in the second part are used to acti vate a subcarrier and the bits i n the t hird p art choose a spreading code. The M − ary symbol generated by the bits in th e first part is first s pread by the s preading code and then transmitted via the activ ated subcarrier . Note that a single su bcarrier out of N is activ ated per block and a spreading code out of N c is chosen. The remaining procedures are the s ame as those of classical OFDM. FFT i s performed first at the receiver , then the signal is despread and square-law en velope detecti o n is applied to estimate the code and frequency indices in order to recov er the mapped bits, foll owe d by a con ventional M − ary demodulati on process. The acquired closed-form terms of the BER performance ov er fading channels is exa mined and confirmed by comp u ter simulati on. Moreover , analys es regarding complexity , SE and EE and complexity analyses hav e p erformed, where ou r finding s indicate a drop in the peak to avera ge power ratio. This P APR cutback shows t he s u itability of the propos ed approach to sensor-based IoT application s, where the portrait of bo t h p owe r and complexity should be preserved at s m all values. The modul ation architecture presented in t his work satisfies the requirements of 5 G - based wireless systems as it minimizes P APR and power cons umption at the t ransmitter and demonstrates a satis fying overall performance in t erms o f reliability and hig h SE. Some comm u nication system s operate through asynchronou s transmi ssion. Under such a condition, performance seve rely degrades if orthogon al spreading codes such as W alsh-Hadam ard or Zado ff-Chu sequences are used due to strong MUI. Therefore, using Go ld codes wi ll const i tute an ideal alternativ e for such scenarios . 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