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 ∈ 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 FIR∈ model can also be regarded as an overall∈ 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 affect∈ { the reception− of} 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 ∈ N , { ··· − M 1 N, belong to a particular constellation C, such ˆ } ⊂ 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− ∈ reconstructed data samples are denoted as Xˆ [l], with k , where H( m) is an N0 N0 matrix, in which, for 0 b, c k ∈ N − × ≤ ≤ and comprise the reconstructed data vector Xˆ [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  − 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 ˆ ∈C 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 − ∈C the set 0, 1,...,N 1 . Based on (7), if there is perfect { 0 − } where N N + µ denotes| the{z size of} the transmitted synchronization (i.e., ∆ = 0), then H(1) = 0N0 N0 and 0 , N × ∈ 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 /√N, 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 alternative∈ OFDM system inserts zeros as redundancy Xˆ [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. { } 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 ∈C 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 . In addition, one has that Υ H0 Γ N×N · − · × · · ΥZP 0 ∈C 2In fact, ∆ could be any natural number, but we assume ∆ ∈ 1 | {z } In principle, there is no point in using µ ≥ N in practice. {0, 1,...,N0 − 1} for the sake of simplicity. is a right-circulant matrix of dimension N N that can be elements BICI1 , i = j, where BICI1 Bdes,ICI1 × i,j 6 , − diagonalized by the DFT matrix. As a result, Bdes. Note that BICI1 = 0.   i,i Type-2 ICI (ICI ): This is the interference among dif- Xˆ [l] = E D X[l] + R q[l], (8) • 2 · · · ferent subcarriers of the data vector transmitted at time where the diagonal matrix D is l m, with m 1, 1, 2 ...,M , in the considered data − ∈ {− } vector transmitted at time l. The elements AISI,ICI2 , h m i,j D , diag √NWN , (9) i = j, contribute to this interference. · 0(N ν 1) 1 6     − − ×  Finally, the contribution of noise q[l] to the reconstructed data T in which h , [ h0 h1 hν ] . As can be noted, there are vector Xˆ [l] depends on matrix Gnoise. no ISI or ICI when ν ···µ. IV.SINRA NALYSIS AND APPLICATIONS The equalizer E for this≤ transceiver can be defined in several ways, where the most popular ones are the zero-forcing (ZF) This section presents an application of the previous ISI/ICI and the minimum mean-squared error (MMSE) equalizers, analysis for conducting an SINR analysis of OFDM systems, 1 1 H H 1 − which can then be employed in many applications, including with EZF , D− or EMMSE , D D D + SNR IN , where SNR stands for signal-to-noise· ratio.· In the latter case, computing achievable data rates. We shall start with the signals the transmitted symbols and environment noise are wide-
before the multiplication by the equalizer matriz E. Based on sense stationary (WSS), mutually independent, white random (10) and (11), one has sequences. M As mentioned before, some previous works have analyzed Y[l] Bdes,ICI1 X[l] + AISI,ICI2 X[l m] , · m · − the case where ν > µ, but with the restriction of having ν m= 1 mX=0− N . Next section presents an analysis for general ν. ≤ 6 0 + Gnoise q[l], (13) · III.ISI/ICIANALYSIS des Ydes[l] , B X[l]. (14) This section focuses on OFDM transceivers with insufficient · Now, assuming the transmitted symbols and the noise signal number of redundant elements, i.e., ν > µ. In this case, the are WSS, mutually independent, white random sequences with received data vector at time l can be affected by the transmitted zero means and variances σ2 and σ2 , respectively, then one data vectors at times l + 1, l, l 1, . . . , l M, and matrices X Q can compute the covariance matrices of the desired signal Υ and Γ cannot eliminate all ISI/ICI.− − (C ), of the noise component (C ), and of ISI/ICI (C ). One can rewrite eq. (6) as s n i Indeed, one has M H 2 des des H ˆ H Cs = E Ydes[l] Y [l] = σ B (B ) , (15) X[l] = E WN Υ H( m) Γ WN X[l m] des X · · · − · · · − · · · m= 1 ISI,ICI H mX=0− A 2  6 m Ci + Cn =E (Y[l] Ydes[l]) (Y[l] Ydes[l]) H − · − +E WN | Υ H0 Γ {zW X[l] +}E WN Υ q[l]. · · · · · N · · · ·  M noise 2 ICI1 ICI1 H ISI,ICI2 ISI,ICI2 H Bdes,ICI1 G = σ B (B ) + A (A ) X  · m · m  (10) m= 1 | {z } | {z }  mX=0−  The desirable signal of (10) is  6  Ci ˆ des Xdes[l] = E B X[l], (11) + σ2 Gnoise (Gnoise)H . (16) · · |Q · · {z } des in which B is a diagonal matrix with elements Cn des des,ICI1 (12) Hence, the SINR related to the kth subcarrier is given as B i,i , B i,i . | {z } [C ] The difference between (10) and (11) defines the ISI and SINR(k) = s kk . (17) [Ci]kk + [Cn]kk ICI. Now, the products Υ H( m) Γ, cannot be expressed as N N right-circulant matrices,· − and· therefore they cannot be When QAM is used and error probability is measured in diagonalized× using DFTs. Based on [24], the interference can terms of symbol error rate (SER), the achievable data rate for be classified into three different types: the kth subcarrier is ISI: This is the interference from the data vector trans- SINR (k) • C (k) = log 1 + , (18) mitted at time l m, with m 1, 1, 2,...,M , in the 2 Γ   considered data− vector transmitted∈ {− at time on the} same l in which the SNR gap Γ for a target SER is Γ = γdm γc+Γm subcarrier. All diagonal elements ISI,ICI2 contribute − Am i,i (in decibels), where γdm is a design margin, γc is the coding to this interference. gain, and Type-1 ICI (ICI ): This is the interference among differ- • 1 2 1 1 SER ent subcarriers belonging to the considered data vector Γ = Q− , (19) m 3 4 transmitted at time l. It appears as a consequence of the    ×10-3 14 4 MSSNR-Conventional MSSNR-Actual 12 EIGAP-Conventional 3 EIGAP-Actual 10 2 8

1 6 Amplitude Data rate [Mbps] 0 4

-1 2 512 samples 0 -2 5 10 15 20 25 30 0 500 1000 1500 2000 2500 TEQ length Time index (a) Fig. 2. Response of the CSA loop 4 in series with a high-pass filter. 18 MSSNR-Conventional MSSNR-Actual 16 EIGAP-Conventional EIGAP-Actual with Q( ) being the tail distribution function of the normal 14 distribution.· By disregarding the signal correlation among the transceiver 12 sub-channels, the achievable data rate of a suboptimal system 10 can finally be obtained as 8 N Data rate [Mbps] R = fs C (k), (20) 6 · N0 · k X∈N 4 where is the underlying sampling rate. fs 2 25 30 35 40 45 CP length V. SIMULATION RESULTS (b) The SINR calculation is an important task for several Fig. 3. Achievable data rates as a function of (a) TEQ and (b) CP lengths. applications, including the design of time-domain equalizers (TEQs) and the calculation of achievable data rates. In this context, this section exemplifies two key points: the influence task is conducted considering: the first 512 samples in Fig. 2 that the limitation of CIR length has on TEQ design, and the (Conventional), or the entire response in Fig. 2 (Actual). Once differences of calculating the achievable data rate considering the TEQs have been designed, the overall impulse responses the interference limited to two or three blocks (Conventional), (OIRs) are obtained by convolving the response in Fig. 2 versus the one considering the actual dispersion (Actual). with the TEQ response. From these OIRs, the achievable data A classic simulation setup, based on the widely used carrier- rates are obtained with the interference limited to two or three serving area (CSA) downstream loops [12], [13], is employed blocks (Conventional) or to a larger number of blocks (Actual). to achieve this goal. In the experiments, the CSA loops are in Although there are several sources of DSL noise, the achiev- series with a fifth-order Chebyshev Type-I high-pass filter with able data rates are computed by considering only additive cut-on frequency at 4.8 kHz, which filters out the telephone white Gaussian noise (AWGN) at -140 dBm/Hz, assuming an voiceband signal [29]. For downstream, the IDFT and DFT input signal power of 23 dBm/Hz [13]. All powers are defined have size N = 512, which is also the maximum CIR length with respect to a 100-Ω resistor. The following parameters in the Conventional analysis for obtaining the SINR. However, were chosen to compute the data rate for each subcarrier k: 7 the convolution of any CSA loop with the high-pass filter an SNR gap Γm = 9.8 dB (for a SER = 10− ), a noise margin has significant samples beyond the 512-sample index. Fig. 2 γdm = 6 dB, and a coding gain γc = 4.2 dB. The sampling depicts an example of the convolution of CSA loop 4 with rate is f = 2.208 MHz. The active tones are 7, 8,..., 256 , s { } the high-pass filter. The discarded samples beyond the time and the TEQ time-offset has been optimized over the values index 512 correspond to 21.18% of the effective CIR energy. 2, 3,..., 50 . { } The Actual analysis via the proposed formulation considers Figs. 3(a) and 3(b) respectively depict the achievable data all samples in Fig. 2. rates as a function of the TEQ lengths (for a fixed CP To use relatively small CP lengths, one can employ TEQs, length of µ = 32) and as a function of the CP length which can be designed via classic techniques, such as those (for an optimized TEQ length in the range of values from proposed in [30] (MSSNR) and [31] (EIGAP). The design 2 to the CP length of each experiment). The goal of these simulations is not comparing the design techniques of [30], [13] K. Vanbleu, G. Ysebaert, G. Cuypers, M. Moonen, and K. V. Acker, [31], but showing the differences in the results of the TEQ “Bitrate-maximizing time-domain equalizer design for DMT-based sys- tems,” IEEE Transactions on Communications, vol. 52, no. 6, pp. 871– designs or in the calculation of data rates. As can be seen in 876, June 2004. Figs. 3(a) and 3(b), the results obtained differ for virtually all [14]E.Z ochmann,¨ S. Pratschner, S. Schwarz, and M. Rupp, “Limited the design parameters considered. These differences highlight feedback in OFDM systems for combating ISI/ICI caused by insufficient cyclic prefix length,” in Proceeding of the 48th Asilomar Conference on the importance of using a correct formulation for the above Signals, Systems and Computers, Nov. 2014, pp. 988–992. calculations. [15] T. Pham, T. Le-Ngoc, G. K. Woodward, and P. A. 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