arXiv:2012.04527v1 [eess.SP] 8 Dec 2020 ytm h rpsdfruaini ple orealistic OFDM to each OFD applied considered of the b of is comparisons systems. rate assessments through performance formulation numerical data effectiveness on its the proposed showing examples, computing The for system. useful block ratio, derived signal-to-interferenc are the data interference calculating three of allow not expressions kinds the or different expressio is three two of Theoretical of study signal. of powers length received the case the the that in particular interfering on is means the restrictions to which interferen samples, constrained no response, t the has impulse CP works, and previous channel matrix Unlike of signal channel usef characterized. received is and new number defined that the impulse is matrix the both power channel channel calculating equivalent than the intercarrier for new of higher and A order with characterized. intersymbol is the formulated when The response jointly considers caused convolution. being transmitte interference, It systems, channel the OFDM in systems. the performed windowed OFDM procedure of different overlap-and-add framework proposed (CP the seven prefix The cyclic CP-OFDM. and/or encompasses conventional OFD suffix windowed the i cyclic (called add besides windowing receiver may with the which allow in systems), those and and/or including parameters transmitter six the schemes, interference on various syste relies intercarrier studying (OFDM) formulation and proposed multiplexing The intersymbol frequency-division of orthogonal analysis the wOD) OAOD,cci rfi C) ylcsfx(CS), (SINR). suffix cyclic ratio (CP), signal-to-interference-plus-noise prefix cyclic multiple WOLA-OFDM, frequency-division (w-OFDM), orthogonal windowed (OFDM), I EA-TDU) ULue,30 evn egu (e-mail: Belgium Leuven, 3001 Leuven, [email protected]). KU (ESAT-STADIUS), Secur Luxembourg. Luxembourg, for of Centre University Interdisciplinary (SnT), the Trust with also is Martins DLPl) eea nvriyo i eJnio(FJ,21 (UFRJ), Janeiro (e-mails: de Brazil Rio Janeiro/RJ, Computer of and de University Electronics Federal of Department (DEL/Poli), the and (PEE/Coppe) el enlyCmnccoe,EceaPltenc Superi Polit´ecnica Escuela (Madrid), Henares Comunicaciones, de Alcal´a [email protected]). y 28871 Alcal´a, Se˜nal de Universidad la de is popular most [1]–[ the MCM (OFDM) but subcarrier. multiplexing ways, frequency-division per several orthogonal coefficient in implemented one be using can effect by whose equalized channels, flat- be of set a into partitioned FMSses nfidFruainadAnalysis and Formulation Unified Systems: OFDM .Moe swt h eateto lcrclEngineering Electrical of Department the with is Moonen M. ..MrisadPSR ii r ihteEetia Engine Electrical the with are Diniz P.S.R. and Martins W.A. .Cu-oda n .Gr´aG r ihteDprmn fT of Department the with are G. Garc´ıa F. and Cruz-Rold´an F. ne Terms Index Abstract UTCRIRMDLTO MM systems, effectiv are (MCM) channels communication frequency-selective MULTICARRIER N Auie arxfruaini rsne for presented is formulation matrix unified —A nesmo n necrirItreec in Interference Intercarrier and Intersymbol ennoCruz–Rold´an, Fernando Otooa rqec-iiinmultiplexing frequency-division —Orthogonal asoGr´aG,Mr Moonen, Marc G., Garc´ıa Fausto .I I. NTRODUCTION { wallace.martins,diniz eirMme,IEEE, Member, Senior } stur.r.W.A. @smt.ufrj.br). t,Rlaiiyand Reliability ity, pi (e-mail: Spain rn Program ering e-plus-noise 4-7,Rio 941-972, Engineering sfrthe for ns rd la de or These . elw IEEE Fellow, can s ased xing eor´ıa ms. 4]. ely his M M ce ul in ), n r s s oto neie O pcrlcmoet,ie,t reduce to i.e., t components, helps spectral which [15]), OOB (e.g. undesired time-domai (Tx) control transmitter comprise the They in [15]–[25]. windowing [5]–[7], [2], systems communication wireline widely and [10]–[14]). been see wireless has (e.g., several standards w-OFDM in higher reasons, deployed these a For efficiency, convent adjacen CP-OFDM. to compared and/or smooth spectral rejection, emission (ACI) interference (OOB) channel better out-of-band its side-lobe the reduces achieves of conventional to reduction w-OFDM and domain, as Due time levels efficiency the (CP-OFDM). in reduce windowing, spectral transitions to OFDM the same as from cyclic-prefix so the resulting other overhead achieving each the time-domain with addition, the overlap In parts windowing. windowed time-domain or pulse-shaping compatibl solutions. fully OFDM-based filter-bank not existing or the are OFDM with which wavelet [7]–[9], as service, (FBMC) MBB such OFDM multicarrier the Radio that solutions on was impact New other negative OFDM selection unlike severe 5G this a vehicular in for have windowed not reason 3GPP or would One them, by 1. (IoT), Phase selected Among things (NR) been of has (V2X). Internet (w-OFDM) (mMTC), everything the communications as to machine-type studie known extensively massive also were services as new such providing for [6] [5], tandhg fcec o oiebodad(B)service downlink (MBB) broadband the mobile for is has efficiency it high OFDM where LTE/LTE-A, attained of access. scheme modulation multiple layer physical support t to compati and systems, ability offsets, (MIMO) timing multiple-output multiple-input to with insensitivity relative frequency-se against fading, effectiveness and simplicity offers alc .Martins, A. Wallace ti osbet n nteltrtr ifrn w-OFDM different literature the in find to possible is It includes that OFDM of variation a is w-OFDM The waveforms or schemes alternative 5G, of context the In FMi ae ntedsrt ore rnfr DT and (DFT) transform Fourier discrete the on based is OFDM al .R Diniz, R. S. Paulo , WOLA-OFDM C CPwrx-OFDM CPwtx-OFDM CPW-OFDM wrx-OFDM wtx-OFDM AATRSISO H CONSIDERED THE OF HARACTERISTICS CP-OFDM System Windowing Tx/Rx Tx/Rx eirMme,IEEE Member, Senior Side Rx Rx Tx Tx ✗ AL I TABLE elw IEEE Fellow, CP ✓ ✓ ✓ ✓ ✓ ✓ ✓ CS ✓ ✓ ✓ ✓ ✗ ✗ ✗ OFDM xml of Example Reference SYSTEMS , [19] [18] [17] [16] [15] [7] [2] lective bility ional he d, s. o n e 1 t 2

TABLE II TYPICAL PARAMETERS USED IN THE CONSIDERED OFDM SYSTEMS, FOR A GIVEN ν-ORDERCHANNELIMPULSERESPONSE.

Parameter Description CP-OFDM wtx-OFDM wrx-OFDM WOLA-OFDM CPW-OFDM CPwtx-OFDM CPwrx-OFDM δ δ µ CP length µ ≥ ν µ−β ≥ ν µ− 2 ≥ ν µ−β−δ ≥ ν µ − β − 2 ≥ ν µ − 2β ≥ ν µ − δ ≥ ν δ µ β Tx window tail 0 β < µ 0 β < µ − δ β < µ − 2 β < 2 0 δ δ δ Rx window tail 0 0 2 ≤ µ δ ≤ µ − β 2 ≤ µ − β 0 δ ≤ µ δ δ ρ CS length 0 β 2 β β + 2 0 0 δ δ γ Rx removed samples µ µ µ − 2 µ − δ µ − 2 µ − β µ − δ δ δ κ Rx circular shift 0 0 0 2 0 β 2

spectral leakage. Some systems also include a time-domain µ N ρ windowing in the receiver (Rx) to increase the OOB rejection and to reduce the power of interfering signals [2], [16], [20], [25], which can increase the signal-to-interference-plus-noise β β ratio (SINR). On the other hand, a CP is always inserted in each transmitted data vector, and in some systems, an (a) wtx and WOLA additional guard interval or cyclic suffix (CS) is also appended µ N ρ [6], [16]–[18]. In this paper, we focus our attention on the seven OFDM systems shown in Table I. The typical parameter values of each system are given in detail in Table II, and β βδ/2 the characteristics of the windows are given in Figs. 1 (b) CPW and 2, where N is the number of subcarriers. First, we include in our study conventional CP-OFDM because the µ N proposed unified formulation is also valid for the most widely standardized MCM scheme. Second, we analyze Tx and Rx time-domain w-OFDM [2], [19], [21], [22], [24], β β respectively referred to as wtx-OFDM and wrx-OFDM. Third, (c) CPwtx our study also includes the weighted overlap-and-add OFDM (WOLA-OFDM), proposed for asynchronous 5G [5], [6], Fig. 1. Tx windows: (a) wtx and WOLA, (b) CPW, and (c) CPwtx. [17], also with an additionally extended suffix scheme named

CPW-OFDM [18]. As can be seen in Table I, these transceivers μ N use independent time-domain windows in both transmitter and receiver sides, and both CP and CS are inserted into each data vector to be transmitted. Lastly, as there are standards that employ in their physical layers w-OFDM without CS, e.g., γ δ/2 δ/2 δ/2 δ/2 [11]–[13], it is necessary to study a fourth group of OFDM (a) wrx and CPW systems: they just include CP with a windowing in the Tx (referred to as CPwtx-OFDM) and a windowing in the Rx μ N (CPwrx-OFDM) [7], [23]. Thus, our study covers a wide range of w-OFDM systems that will play an important role in various communication systems over the next decade. γ δ/2 δ/2 δ/2 δ/2 OFDM systems suffer from intersymbol and intercarrier (b) WOLA and CPwrx interference (ISI and ICI) when the order of the channel impulse response (CIR) does not satisfy the conditions Fig. 2. Rx windows: (a) wrx and CPW, (b) WOLA and CPwrx. related to the CP length in Table II. In this context, the interference analysis in conventional CP-OFDM has been widely addressed. For instance, different SINR models are considered, and the ICI induced by the proposed windowing derived in [26]–[34]. For more details, we refer the reader to is obtained. [35], where the impact of highly dispersive channels on OFDM We perform an exhaustive analysis of ICI and ISI in the under finite-duration CIR with arbitrary length is shown. seven different system considered in our study. Since no Regarding w-OFDM, previous studies focusing on the analysis constraint is imposed upon the order of the CIR, our results of interference are [7], [15], [23], [25], [36]. In [7], [15], [23], is applicable to the cases where the interference is due to any both ISI and ICI are grouped under a single time-domain term, number of transmitted data blocks. The main contributions in and the systems analyzed in these papers are CPwtx-OFDM this paper can be summarized as follows: [7], [15], [23] with a unique windowing in the Tx unit, and • A unified matrix formulation for a wide range of OFDM wtx-OFDM [15]. In [25], an Rx windowing OFDM system is systems is presented. It includes the full transmission 3

chain, the overlap-and-add operation, the convolution The matrix Γ introduces µ + ρ redundant samples: with a channel of arbitrary length, and the reception 0 × − I process. This matrix formulation is a powerful tool to µ (N µ) µ Γ = IN . reduce the computational time needed to perform Monte   I 0 × − Carlo simulations. Moreover, this compact formulation ρ ρ (N ρ) based on a small set of only six parameters allows It appends a µ-length CP and, when applicable, also a ρ-length easy reconfiguration of the systems —different OFDM CS. Observe that a cyclic shift, as employed in [14], is systems are obtained by simply changing parameters in equivalent to the inclusion of a CS into each data vector. the formulation. In addition, it has excellent potential for The following block performs the windowing, and it can be use in an educational context, because it enables quickly represented as a diagonal matrix explaining several OFDM systems. tx tx • Theoretical closed-form expressions of interference and V(N+µ+ρ) = diag v1×(N+µ+ρ) , noise for the case of an insufficient length of redundant obtained with a tapering windown function, definedo as samples (i.e. CP) are obtained. The interference is tx tr tf identified in the frequency domain, where the symbol is v1×(N+µ+ρ) = v1×β 11×(N+µ+ρ−2β) v1×β . reconstructed, and classified into three different classes. tr  tf  This classification helps to study which one is most The vectors v1×β and v1×β have as entries the rise and fall harmful to the system’s performance. samples of the window tails, respectively. • Interference and noise powers are derived to obtain the After the or windowing, there is a β-samples SINR, and hence the data rate and the bit-error rate. The overlap-and-add operation between successive symbols. This results are also useful for bit-loading, adaptive CP and operation is jointly formulated with the channel convolution. power/subcarrier allocation algorithms [3]. The rest of this paper is organized as follows. In Section II, we B. Channel present the unified system model, considering seven different The signal xs is convolved with the transmission channel, OFDM systems and adopting a unified matrix formulation defined as h = h0 h1 hν , and becomes covering all of them. Then, three types of interference are contaminated by noise. In general,··· the number of transmitted calculated in Section III. In addition, theoretical expressions data vectors that affect the first N + δ +γ samples of the for both interference and noise powers are derived, and the received data vector is M +1, with corresponding SINR is determined. Simulation results are ν + β presented in Section IV, and finally, conclusions are drawn M =∆ , (2) N + δ + γ in Section V.   The notation used in this paper is as follows. Bold-face in which represents the ceiling function. Therefore, the l-th letters indicate vectors (lower case) and matrices (upper case). received signal⌈·⌉ vector is given by T The transpose of A is denoted by A and IN represents the r N N identity matrix. The subscript is omitted whenever the y(N+δ+γ)×1 [l] × size is clear from the context. 0 and 1 denote, respectively, a M (m) s matrix of zeros or ones. = H x × [l m] (N+δ+γ)×(N+µ+ρ) · (N+µ+ρ) 1 − m=0 X II. UNIFIED SYSTEM MODEL + q(N+δ+γ)×1[l], A block diagram is shown in Fig. 3, where the transmitted where H(m) is a matrix whose entries, for 0 b N + δ + data vector in the transform domain is given by γ 1 and 0 c N + µ + ρ 1, are ≤ ≤ T − ≤ ≤ − X = X X XN− , (1) 0 1 ··· 1 0, mN0 + b c< 0, (m) ∆ − with N being the number of subcarriers. The parameters used H = hmN0+b−c, 0 mN0 + b c ν, (3)   b,c  0,≤ mN + b −c >≤ ν, in the equations are given in detail in Table II. We assume h i  0 − perfect synchronization in time and frequency, and also that where N0 = N+ µ + ρ β and q represents the channel the receiver has perfect channel-state information (CSI). noise. −

A. Transmitter C. w-OFDM Receiver The time-domain vector before the overlap-and-add block In the absence of noise, the received data vector can be is expressed in the transform domain as s tx −1 x(N+µ+ρ)×1 = V(N+µ+ρ) Γ(N+µ+ρ)×N WN XN×1. rx · · · YN×1 = WN KN PN×(N+δ) V(N+δ) − · · · Here, W 1 represents the inverse DFT matrix with the r R(N+δ)×(N+δ+γ) y N δ γ × , (4) (k,n)-th entry given by × · ( + + ) 1 r where y × is the received signal, and the matrices are − 1 2π (N+δ+γ) 1 W 1 = ej N kn, 0 k,n N 1. defined as follows. k,n N ≤ ≤ −   4

X Transmitter Receiver

X0 Y0 W-1 tx RPVrx KW X1 V Y1 xs yr

XN-2 YN-2

XN-1 YN-1 Overlap-and-add

Fig. 3. General block diagram of windowed OFDM over a channel with additive noise.

First, R represents removal of the first γ samples of the ISI received data vector: ICI XN [l-1] XN [l] 2 Y l -1 -1 N-1 [ ] R = 0 N δ ×γ I N δ . ( + ) ( + ) XN [l-1] XN [l] Y l -2 -2 N-2 [ ] ) ) ICI k The diagonal matrix representing the windowing is k XN [l-1] XN [l] 1 Y l -3 -3 N-3 [ ] rx rx V = diag v1×(N+δ) , where the tapering window inn the Rx is definedo as Subcarrier ( Subcarrier Subcarrier ( Subcarrier rx rr rf X [l-1] X [l] Y [l] v = v1×δ 11×(N−δ) v1×δ , 2 2 2 X [l-1] X l Y l where vrr and vrf have as entries the rise and fall samples 1 1[ ] 1[ ] X l X l Y l of the Rx window tails. Next, P is a matrix that represents a 0[ -1] 0[ ] 0[ ] δ-samples overlap-and-add operation: l-1 l Time l Time 0δ/2 Iδ/2 0δ/2×(N−δ) 0δ/2 Iδ/2 Transmitted Reconstructed   Data Vectors Data Vector P = 0(N−δ)×δ IN−δ 0(N−δ)×δ .   Fig. 4. Types of interference.    I 0 0 × − I 0   δ/2 δ/2 δ/2 (N δ) δ/2 δ/2  Basically, it adds the first δ samples to the last δ samples. noise rx GN×(N+δ+γ) = W K P V R. Then, a circular shift of κ samples is needed in some systems · · · · (WOLA, CPwtx, and CPwrx). This operation is formulated Note that for ν γ β, the number of data vectors affecting the reception (in≤ the− transform domain) of a single data vector with the matrix KN , defined as follows: is one. In this case, a set of N independent parallel subcarriers 0(N−κ)×κ IN−κ KN = . is obtained, each having a channel gain of Hk, defined as the Iκ 0 × −  κ (N κ)  N-point DFT of the CIR h. Thus a one-tap per subcarrier In some other systems, e.g., those that include a CS in each equalizer can be used to mitigate the phase and the amplitude transmitted data vector (wtx and CPW), this is an identity introduced by the channel. This means that Am = matrix: KN = IN . Finally, W is a DFT matrix: 0, m > 0, and A0 is a diagonal matrix with elements Hk, − 2π 0 k (N 1). For the other cases (ν>γ β), we have [W] =e j N kn, 0 k,n N 1. ≤ ≤ − − k,n ≤ ≤ − three different types of interference, as depicted in Fig. 4 [35], [37]: III. ANALYSIS OF INTERFERENCE • Type-I intercarrier interference (ICI1): corresponding to This section provides a comprehensive analysis of the the elements [A ] , i = j. 0 i,j 6 interference and its power for the considered OFDM systems. • Type-II intercarrier interference (ICI2): corresponding to Given a finite duration CIR with arbitrary length, without the elements [Am] , i = j, m> 0. i,j 6 any constraint on the length, the received data vector can be • Intersymbol interference (ISI): corresponding to the expressed in the transform domain as diagonal elements [Am]i,i, m> 0. M We now derive theoretical expressions for the powers noise Y[l]= Am X[l m]+ G q[l], (5) corresponding to the desired signal component in the received · − · m=0 data vector, as well as to the ISI, ICI, and noise. These powers X are used to compute the SINR. For this study, we assume that where the components of the data vector Xk and the noise vector qk rx (m) tx −1 Am,N×N = W K P V R H V Γ W , are zero-mean wide-sense stationary uncorrelated processes, · · · · · · · · 5 independent and identically distributed for all k, with variances 100 2 2 CP-PED200 σ and σ , respectively. We follow the same reasoning as in wtx-PED200 X n -1 10 wrx-PED200 [35]. The desired signal component in the received data vector WOLA-PED200 can be written as CPW-PED200 10-2 CPwtx-PED200 CPwrx-PED200 des Y [l]= A X[l], (6) -3 des 0 · 10 SER des -4 where A0 is a diagonal matrix with entries 10

des 10-5 A0 i,i = [A0]i,i . (7)

-6 The desired signal power (before the transform-domain 10 equalization) at subcarrier k is obtained as the (k, k)-th 10-7 s 0 5 10 15 20 25 30 35 40 45 element of the covariance matrix, i.e., Psignal (k) = [C ]k,k, SNR (dB) where (a)

s H 100 C = E Ydes[l] Ydes[l] · CP-VEH200 H H wtx-VEH200 des des wrx-VEH200 = E A X[l] X [l] A -1  0 0 10 WOLA-VEH200 · · · CPW-VEH200 desn H des Ho CPwtx-VEH200 = A0 E X[l] X [l] A0  . CPwrx-VEH200 -2 · · H · 10 = σ2 Ades Ades ,  (8) X · 0 · 0 10-3 where E is the expected-value  operator. The noise data SER vector is given{·} by 10-4

noise Ynoise[l]= G q[l], (9) · 10-5 n As a result, the noise power is given by Pnoise (k) = [C ]k,k, 10-6 where 0 5 10 15 20 25 30 35 40 45 50 SNR (dB) H Cn = σ2 Gnoise Gnoise . (10) n · · k (b) The interference component is given by  Fig. 5. SER versus SNR for different OFDM systems. (a) PED200. (b) VEH200. Y [l] = Y[l] Y [l] Y [l] int − des − noise M IV. SIMULATIONS ICI1 = A X[l]+ Am X[l m], (11) 0 · · − m=1 In order to demonstrate the applicability of the proposed X formulation, this section compares the performance of the ICI1 des where A0 = A0 A0 . Using the above, the ISI and ICI studied systems in terms of SER and achievable data rate. − i power is PISI,ICI (k)= C k,k, where It is worth noting that OOB emissions will not be taken into account here. The set of parameters used in the simulations   Ci = E Y [l] YH [l] are summarized in Table III. BPSK modulation is used as int · int H the primary mapping, the number of active subcarriers is ICI1 H ICI1 = E  A X[l] X [l] A N = 256, which is the DFT size, and the frequency spacing is 0 · · · 0   11.16071492 kHz. Two sets of 250 wireless fading channels M   H H each, according to the ITU Pedestrian A and Vehicular A + E Am X[l m] X [l m] (Am) · − · − · channels [38], [39], are used as multipath channels. They m=1 X n o have been generated with Matlab’s stdchan using the H M 2 ICI1 ICI1 H channel models itur3GPAx and itur3GVAx with a carrier = σ A A + Am (Am) . X · 0 · 0 · m=1 ! frequency fc = 2 GHz and two different sets of parameters:   X (12) (a) 4 km per hour as pedestrian velocity, Ts = 200 ns and length L = ν +1=11; (b) 100 km per hour as mobile speed, Hence, the SINR for subcarrier k is Ts = 200 ns and length L = ν +1 = 21. These channels are referred to as PED200 and VEH200, respectively. The P (k) SINR (k)= signal . (13) noise is modeled as an additive white Gaussian noise. It is PISI,ICI (k)+ Pnoise (k) assumed that the channel remains unchanged within the same simulation and perfect channel estimation is performed at the 6

TABLE III PARAMETERSUSEDINTHEEXPERIMENTS (N = 256)

Parameter CP-OFDM wtx-OFDM wrx-OFDM WOLA-OFDM CPW-OFDM CPwtx-OFDM CPwrx-OFDM β 0 8 0 8 8 8 0 δ 0 0 10 10 10 0 10 ρ 0 8 5 8 13 0 0 γ 32 32 27 22 27 24 22 κ 0 0 0 5 0 8 5

20 100 19

18.5 10-1

CP-PedA200 18 SNR=5dB wtx-PedA200 wrx-PedA200 15 -2 17.5 10 WOLA-PedA200 CPW-PedA200 Data Rate (Mbps) CPwtx-PedA200 17 CPwrx-PedA200

10-3 SER 40 41 42 43 SNR (dB) Data Rate (Mbps) 10 SNR=25dB CP-PED200 10-4 wtx-PED200 wrx-PED200 WOLA-PED200 CPW-PED200 -5 CPwtx-PED200 10 SNR=40dB CPwrx-PED200

5 0 5 10 15 20 25 30 35 40 45 10-6 0 5 10 15 20 25 30 35 SNR (dB) Length of CP (a) (a)

22 100 20 20 19.5 19 -1 10 CP-VehA200 18 18.5 SNR=5dB wtx-VehA200 18 wrx-VehA200 WOLA-VehA200 16 Data Rate (Mbps) 17.5 CPW-VehA200 -2 17 10 CPwtx-VehA200 CPwrx-VehA200 14 40 41 42 43 44 45 SNR (dB) SER

-3 Data Rate (Mbps) 12 10 CP-VEH200 SNR=25dB wtx-VEH200 wrx-VEH200 10 WOLA-VEH200 SNR=40dB CPW-VEH200 CPwtx-VEH200 10-4 8 CPwrx-VEH200

6 0 5 10 15 20 25 30 35 40 45 50 10-5 SNR (dB) 0 5 10 15 20 25 30 35 Length of CP (b) (b)

Fig. 6. Total achievable data rate versus SNR for different OFDM systems. Fig. 7. SER versus CP length for different OFDM systems. (a) PED200. (b) (a) PED200. (b) VEH200. VEH200. receiver. Perfect time and frequency synchronization is also where γ∗ is the modified SINR gap defined for a target SER assumed. as −1 2 In Fig. 5, the symbol error rate (SER) performance curves ∗ Q (SER/2) of the different OFDM systems and channels are depicted. As γ = . √2π can be seen, the results for the different systems are practically   indistinguishable for each set of channels. Thus, there is no The total achievable data rate is thus − clear advantage in terms of SER of any particular OFDM N 1 N system over the other systems. R = fs C (k), (15) N · k 0 Next, we investigate the data rate performance for a fixed X=0 CP length (µ ). As we use BPSK modulation, the data = 32 where N0 = N + µ + ρ and fs = 1/Ts. We employ the SER rate for subcarrier k is given by [40] obtained in the previous simulations to compute the values ∗ 1 SINR (k) of γ corresponding to each SNR. Fig. 6 shows the resulting C (k)= log , (14) 2 2 γ∗ data rate as a function of the SNR. In this set of experiments,   7

7.4 7.7 CP-VehA200-SNR=5 wtx-VehA200-SNR=5 CP-PedA200-SNR=5 7.6 wrx-VehA200-SNR=5 wtx-PedA200-SNR=5 WOLA-VehA200-SNR=5 7.2 wrx-PedA200-SNR=5 CPW-VehA200-SNR=5 WOLA-PedA200-SNR=5 7.5 CPwtx-VehA200-SNR=5 CPW-PedA200-SNR=5 CPwrx-VehA200-SNR=5 CPwtx-PedA200-SNR=5 CPwrx-PedA200-SNR=5 7.4 7

7.3

6.8 7.2 Data Rate (Mbps) Data Rate (Mbps) 7.1

6.6 7

6.9 6.4

6.8

6.2 6.7 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Length of CP Length of CP (a) (b)

14 14

13 13

12 12

11 11

10 10

Data Rate (Mbps) CP-PedA200-SNR=25 Data Rate (Mbps) CP-VehA200-SNR=25 wtx-PedA200-SNR=25 wtx-VehA200-SNR=25 wrx-PedA200-SNR=25 wrx-VehA200-SNR=25 9 WOLA-PedA200-SNR=25 9 WOLA-VehA200-SNR=25 CPW-PedA200-SNR=25 CPW-VehA200-SNR=25 CPwtx-PedA200-SNR=25 CPwtx-VehA200-SNR=25 CPwrx-PedA200-SNR=25 CPwrx-VehA200-SNR=25 8 8

7 7 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Length of CP Length of CP (c) (d)

20 20

18 18

16 16

14 14

12 12 Data Rate (Mbps) CP-PedA200-SNR=40 Data Rate (Mbps) CP-VehA200-SNR=40 wtx-PedA200-SNR=40 wtx-VehA200-SNR=40 wrx-PedA200-SNR=40 wrx-VehA200-SNR=40 10 WOLA-PedA200-SNR=40 10 WOLA-VehA200-SNR=40 CPW-PedA200-SNR=40 CPW-VehA200-SNR=40 CPwtx-PedA200-SNR=40 CPwtx-VehA200-SNR=40 CPwrx-PedA200-SNR=40 CPwrx-VehA200-SNR=40 8 8

6 6 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Length of CP Length of CP (e) (f)

Fig. 8. Achievable data rate for different lengths for the CP and SNR values. 8

102 CP-VEH200 of the CP length is obtained for each OFDM system (see wtx-VEH200 Fig. 7), assuming SNR = 5, 25, and 40 dB. These results are wrx-VEH200 ∗ 100 WOLA-VEH200 employed to calculate γ . Then, we obtain the achievable data CPW-VEH200 CPwtx-VEH200 rate, depicted in Fig. 8 for the PED200 and VEH200 channels. CPwrx-VEH200 10-2 In all cases, CP-OFDM outperforms the other systems, except for SNR = 40 dB, VEH200, and for smaller values of the

ICI1 CP, for which the wrx scheme shows a better performance. P -4 10 However, this improvement is not very significant in this case of insufficient redundant samples. The remaining OFDM 10-6 systems have better performance whenever the windowing is in the receiver. For both small CP lengths and low SNR

10-8 values, the systems that only have a CP outperform those that incorporate a CS. 10 15 20 25 30 35 Length of CP Finally, the formulation presented here allows analysis in the (a) transform domain of the three different interference powers that appear in each OFDM scheme. Fig. 9 shows the total CP-VEH200 power results (PICI1, PICI2, and PISI) as a function of the CP wtx-VEH200 wrx-VEH200 length, obtained in the previous experiment for the VEH200 100 WOLA-VEH200 CPW-VEH200 channel. The interference power is higher for systems whose CPwtx-VEH200 windowing is performed in the transmitter than those with CPwrx-VEH200 10-2 windowing in the receiver. Note that the CPW system has low levels of interference power, but the data rate results do ICI2 P 10-4 not outperform the other systems. This is due to the overhead involved in the inclusion of both CP and CS.

10-6 V. CONCLUSION In this paper, we presented a unified formulation 10-8 that describes conventional CP-OFDM and other six 10 15 20 25 30 35 different w-OFDM systems. The unified matrix formulation Length of CP describes the whole transmitter, including the overlap-and-add (b) windowing operation, and the operation of convolving the 102 transmitted signal with the channel, as well as the complete CP-VEH200 wtx-VEH200 receiver operation. Moreover, we derived expressions for wrx-VEH200 WOLA-VEH200 100 intersymbol interference as well as two different kinds CPW-VEH200 CPwtx-VEH200 of intercarrier interference, along with their corresponding CPwrx-VEH200 powers, besides the noise component. We developed analytical 10-2 expressions for the SINR so as to evaluate the effects of

ISI interference on the considered OFDM systems and to study P 10-4 the achievable data rate. Computer simulations were carried out with practical scenarios. Comparing the obtained results, 10-6 we observed that in terms of SER, all OFDM systems behave similarly. But in terms of data rate criteria, the OFDM systems

10-8 that only have a windowing in the receiver, or that only include a CP, outperform the others. It has also been noted that 10 15 20 25 30 35 Length of CP some systems (such as CPW-OFDM) have low interference (c) power levels, but the performance in terms of the transmission data rate is slightly lower than other systems with more Fig. 9. Total power for different CP lengths. (a) Power of type-I intercarrier interference. The reason for this can be found in the penalty interference. (b) Power of type-II intercarrier interference. (c) Power of intersymbol interference. paid for including both the prefix and suffix.

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