Intersymbol and Intercarrier Interference in OFDM Transmissions Through Highly Dispersive Channels

Intersymbol and Intercarrier Interference in OFDM Transmissions Through Highly Dispersive Channels Wallace Alves Martins Fernando Cruz–Roldan´ Universite´ du Luxembourg, Luxembourg Universidad de Alcala´ Universidade Federal do Rio de Janeiro, Brazil Alcala´ de Henares (Madrid), Spain [email protected] [email protected] Marc Moonen Paulo Sergio Ramirez Diniz Katholieke Universiteit Leuven Universidade Federal do Rio de Janeiro Leuven, Belgium Rio de Janeiro, Brazil [email protected] [email protected] Abstract—This work quantifies intersymbol and intercarrier dancy [7]–[23]. Perhaps, the first attempt to analyze systemat- interference induced by very dispersive channels in OFDM ically the harmful ISI/ICI effects are the works in [7], [8], [24] systems. The resulting achievable data rate for suboptimal OFDM in which preliminary results in terms of interference spectral transmissions is derived based on the computation of the actual signal-to-interference-plus-noise ratio for arbitrary length finite power are presented. The authors in [9] analyze ISI inter- duration channel impulse responses. Simulation results point to ference in OFDM-based high-definition television (HDTV). significant differences between data rates obtained via conven- The works in [10] address time-variant channels that induce tional formulations, for which interference is supposed to be Doppler effects. A landmark work is [11], which presents the limited to two or three blocks, versus the data rates considering interference power due to the tails of the channel impulse the actual channel dispersion. Index Terms—Orthogonal frequency-division multiplexing, response (CIR). The works [12], [13] use the ISI/ICI inter- highly dispersive channels, intersymbol interference, intercarrier ference analyses to design the front-end prefilter employed at interference, cyclic prefix, zero padding the receiver side for channel shortening in xDSL applications. The work in [14] uses the results from [11] to show it might I. INTRODUCTION be useful to allow the existence of ISI/ICI due to insufficient Since its origins in the analog [1] and digital [2] domains, guard periods, as long as the transceiver is aware of it—via the orthogonal frequency-division multiplexing (OFDM) sys- feedback mechanism of channel-state information (CSI). The tem has striven to combat interference induced by frequency- authors in [19] analyze the ISI/ICI impacts on a power-line selective channels [3]. A major breakthrough was the use communication (PLC) system, whereas [20] optimize the re- of redundant elements, such as cyclic prefix, for preserving dundancy length for a PLC system, allowing for the existence orthogonality among subcarriers at the receiver [4]. of controlled ISI/ICI. A similar kind of optimization is also Practical wireless broadband communications use increas- conducted for coherent optical communications in [21]. The ingly high sampling rates while trying to meet the demands works [22] analyze the ISI/ICI effects from a theoretical and an for high data rates. In this context, the redundancy overhead experimental viewpoints for wireless local area networks. The turns out to be a big issue due to spectral-efficiency losses. works [25]–[28] address the design/analysis of more general Furthermore, as some practical wired applications (e.g., digital transceivers or equalizers, mostly for doubly-selective channel subscriber lines, including ADSL, VDSL, and more recently, models. G.Fast) work with highly dispersive channels, it can be All of these works start from a common point—the analysis virtually impossible to append so many redundant elements of ISI/ICI for a general setup where the amount of redundant in the transmission, calling for alternative solutions, such as elements can be smaller than the delay spread of the channel— prefiltering at the receiver side to shorten the effective channel to eventually achieve different goals. We shall start from the model [5]. Besides, the use of on-channel repeaters to extend same point, but without imposing any kind of constraints system coverage [6] can induce delays which are longer than upon the order of the CIR, which is only assumed to have those the guard interval was originally designed to cope with. finite duration. This eventually implies that the proposed The aforementioned issues motivated many works to an- analysis can be applied to the cases where the interference alyze ISI/ICI in OFDM systems with insufficient redun- is due to several data blocks, not being limited to two or three blocks, as all previous works do. As mentioned before, This study was financed in part by Capes, CNPq, and Faperj, Brazilian this is especially important in delay-constrained applications research councils. The work was also partially supported by the Spanish working in highly dispersive environments where the number Ministry of Economy and Competitiveness through project TEC2015-64835- C3-1-R and by the Spanish Ministry of Education, Culture and Sports through of subcarriers cannot be set larger than the order of the CIR. Research Grant PRX17/0080. Besides, the proposed analysis unifies and generalizes many q[l] ˆ X0[l] X[l] Y[l] Xˆ [l] X0[l] IDFT Channel DFT Equalizer . . H Γ Υ . WN H( M), , H0 WN E − · · · ⊕ XN 1[l] XˆN 1[l] − − T R Fig. 1. Block diagram of an OFDM transceiver. of the aforementioned works, providing closed-form matrix The channel model is represented by a causal finite duration expressions as functions of the channel taps that are easy to impulse response (FIR) filter with coefficients h0; : : : ; hν C 2 use for different purposes—including the ones listed above. of order ν N, and an additive noise vector q[l] CN0 . In Both cyclic-prefix and zero-padding OFDM transmissions are fact, the FIR2 model can also be regarded as an overall2 channel considered and a fine distinction between two types of ICI impulse response, encompassing any pulse shaping and front- is provided. As for the application of the proposed analysis, end receive prefiltering. Thus, assuming a synchronization 2 this work focuses on providing achievable data rates for delay ∆ 0; 1;:::;N0 1 , the number of data vectors suboptimal OFDM transmissions based on the computation of that may affect2 f the reception− ofg a single data vector is M + 2, signal-to-interference-plus-noise ratios (SINRs) for arbitrary where ν length finite duration CIR. M ; (5) , N 0 II. SYSTEM MODEL in which stands for the ceiling function. Hence, the Let us consider an OFDM transceiver model as depicted in reconstructedd·e data vector can be written as Fig. 1. The data samples X [l], with k 0; 1; ;N k 2 N , f ··· − M 1 N, belong to a particular constellation C, such ^ g ⊂ C ⊂ X[l] = R H( m) T X[l m] + R q[l]; (6) as QAM or PSK, and comprise vector X[l] at the (block) · − · · − · m= 1 time index l Z. After the transmission/reception process, the X− 2 reconstructed data samples are denoted as X^ [l], with k , where H( m) is an N0 N0 matrix, in which, for 0 b; c k 2 N − × ≤ ≤ and comprise the reconstructed data vector X^ [l]. N0 1, one has − The cyclic-prefix OFDM (CP-OFDM) is described by the 0; mN0 + b c + ∆ < 0; following transmitter and receiver matrices, respectively: ∆ − H( m) = hmN0+b c+∆; 0 mN0 + b c + ∆ ν; − b;c 8 − 0;≤ mN + b −c + ∆ >≤ ν: 0µ (N µ) Iµ H < 0 − TCP , × − WN ; (1) (7) IN · Note that in: (6), up to M + 2 data vectors may contribute N ×N ΓCP 0 ^ 2C to the reconstructed vector X[l]. In fact, this number can be RCP , |E WN{z 0N µ} IN ; (2) smaller, depending on the delay ∆. Indeed, based on (5), one · · × can write ν = (M 1)N + ρ + 1, with ρ being a number in N×N 0 ΥCP 0 − 2C the set 0; 1;:::;N 1 . Based on (7), if there is perfect f 0 − g where N N + µ denotes| the{z size of} the transmitted synchronization (i.e., ∆ = 0), then H(1) = 0N0 N0 and 0 , N × 2 1 data vector after appending µ < N redundant elements, WN the reconstructed data vector is affected by at most M + 1 is the normalized N N discrete Fourier transform (DFT) transmitted data vectors. In this case, the last matrix H( M) − × j 2π kn matrix with entries [WN ]kn = e− N =pN, IN is the N N has only ρ + 1 nonzero rows. This eventually implies that, × th identity matrix, 0N µ is an N µ matrix whose entries are for ∆ > ρ, what would be the (M + 2) channel matrix will N ×N × zero, and E C × is an equalizer diagonal matrix. actually be a null matrix, and again up to M +1 vectors affect An alternative2 OFDM system inserts zeros as redundancy X^ [l]. In summary, the number of data vectors contributing to and is called zero-padding OFDM (ZP-OFDM). There are the reconstructed vector is up to M +2 whenever ∆ is chosen many variants of ZP-OFDM. A common choice is the ZP- within the set 1; 2; : : : ; ρ , or otherwise up to M + 1. f g OFDM-OLA (overlap-and-add), with: Considering the scenarios in which the redundancy is long enough, i.e. 1 ν µ, one has M = 1. In that case, the ≤ ≤ IN H received data vector at time l might be affected only by the TZP , WN ; (3) 0µ N · transmitted data vectors at times l + 1, l, and l 1, for a × − N ×N nonzero delay ∆. As mentioned before, when ∆ = 0, one has ΓZP 0 2C H(1) = 0N0 N0 , and matrices Υ and Γ are able to eliminate | {z } Iµ × RZP , E WN IN : (4) the interference induced by the transmitted data vector at l 1, · · 0(N µ) µ − − × i.e., Υ H( 1) Γ = 0N N .

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