A Low-Complexity Turbo Equalizer for OFDM Communication ... · 100 IEICE TRANS. COMMUN., VOL.E89–B, NO.1 JANUARY 2006 PAPER A Low-Complexity Turbo Equalizer for OFDM Communication

100IEICE TRANS. COMMUN., VOL.E89–B, NO.1 JANUARY 2006

PAPER

A Low-Complexity Turbo Equalizer for OFDM CommunicationSystems

Alexander N. LOZHKIN†a), Nonmember, Mitsuhiro AZUMA†, and Tomohiko TANIGUCHI†, Members

SUMMARY With the growing demand for mobile communications,multicarrier (MC) schemes are receiving an increasing amount of atten-tion, primarily because they handle frequency selective channels better thanordinary single-carrier schemes. However, despite offering several advan-tages, MC systems have certain weak points. One is their high sensitivityto interchannel interference (ICI). The influence of Doppler shift and ICIare the focus of this paper. Newly proposed B3G/4G systems are devel-oped for data transmission rates higher than those of the IEEE 801.11. It isthen necessary that the bandwidth of the subcarrier be small. Moreover, fora higher carrier frequency and mobile speed, the influence of the Dopplershift will be large; therefore, the influence of ICI becomes severer. Us-ing a Markov chain approach, we synthesized a turbo equalizer (TE) thatminimizes ICI when interference affects the arbitrary number M of adja-cent subchannels. This approach shows the complexity of the proposedalgorithm exhibits linear growth with respect to M and independence withrespect to the total number of subchannels in the multicarrier system. Theproposed ICI cancellation scheme can also be effective in the case of multi-ple Doppler frequency offsets. This makes the proposed approach attractivefor practical implementations.key words: OFDM, DMT, multicarrier system, interchannel interference,turbo-equalizer, frequency offset

1. Introduction

Orthogonal-frequency division multiplexing (OFDM) hasemerged as an attractive and powerful alternative to con-ventional modulation schemes in the recent past due to itsvarious advantages in lessening the severe effect of fre-quency selective fading. Therefore, OFDM is currently be-ing adopted and tested for many standards, including ter-restrial digital broadcasting (DAB and DVB) in Europe, andhigh-speed modems over Digital Subscriber Lines in the US.It has also been proposed for broadband indoor wirelesssystems including the IEEE 802.11, MMAC and HIPER-LAN/2. An OFDM system operating over a wireless com-munication channel effectively forms a number of parallelfrequency nonselective fading channels thereby reducing in-tersymbol interference (ISI) and obviating the need for com-plex equalization thus greatly simplifying channel estima-tion/equalization task. Moreover, OFDM is bandwidth effi-cient since the spectra of the neighboring subchannels over-lap, yet channels can still be separated through the use of or-thogonality. Furthermore, its structure also allows efficienthardware implementations using FFT and polyphase filter-ing. Despite the progress made in combating ISI, OFDM-

Manuscript received December 8, 2004.Manuscript revised May 13, 2005.†The authors are with YRP Center, Fujitsu Laboratories LTD.,

Yokosuka-shi, 239-0847 Japan.a) E-mail: [email protected]

DOI: 10.1093/ietcom/e89–b.1.100

based systems still have a number of drawbacks when com-pared to single-carrier systems. The main drawback is theirhigh sensitivity to ICI, which can be caused by a numberof factors including frequency deviation in the receiver andDoppler shift. ICI is the main limiting factor in multicarrier(MC) communication systems, which require a high rate ofdata transmission over mobile channels.

The influence of Doppler shift and ICI are the focus ofthis paper. The complete analysis of Doppler spread per-turbations in OFDM systems was presented in References[1]. It has been shown in [1] that the difference in Dopplerfrequencies affecting different channels paths leads to theloss of orthogonality, since a common Doppler shift canbe viewed as a frequency offset and can be corrected [2].Newly proposed B3G/4G systems are developed for datatransmission rates higher than those of the IEEE801.11a.It is then necessary that the bandwidth of the subcarrier besmall. Moreover, for a higher carrier frequency and mobilespeed, the influence of the Doppler shift will be large andtherefore, the influence of ICI becomes severer.

With some modifications, ideas developed in this papercan be applied for multiple accesses of the OFDMA systemuplink, where the separation of different users is achievedby exploiting the orthogonality of the subcarriers. In suchsystems, frequency offsets are present among the users andsimple adjusting of the receiver oscillator with PLL [2] toone carrier frequency offset, as in a single user system, can-not restore the orthogonality among subcarriers. In order tocounteract frequency offsets in OFDMA, carrier frequencyoffsets are estimated at the base-station receiver and feed-back to user terminal. The feedback approach is efficient,but requires an additional network signaling and does notwork in situations in which the mobile terminal does notnecessarily have access to downlinks or where the downlinksuffers from a different Doppler spread (frequency divisionduplex (FDD), e.g.) [1].

In a practical mobile radio channel, time-variant mul-tipath propagation causes Doppler frequency spread. Thereceived signal on each subcarrier can then be considered asa linear combination of signals received via different pathswith different Doppler frequencies. In the proposed method,the transmission channel is modeled by a combination ofmultiple Doppler-shifted propagation paths and their fre-quency offset values are estimated using pilot symbols, ICIis then cancelled by implementation of the proposed turbo-equalizer that operates without requiring any feedback in-formation.

Copyright c© 2006 The Institute of Electronics, Information and Communication Engineers

LOZHKIN et al.: A LOW-COMPLEXITY TURBO EQUALIZER101

In [3] we proposed a non-linear iterative turbo signalprocessing technique based on iterative serial maximum aposteriori estimation for the case when M = 2. This tech-nique significantly mitigates the effect of ICI on the bit er-ror rate (BER) performance of an MC system. We used afinite-state discrete Markov process model to describe ICI.Because of the discrete nature of ICI, this model seems tobe more realistic than a simple Gaussian approximation [1]–[5]. The spectra of adjacent subchannels of OFDM signalsapproximately cross at the −3 dB point and the first sidelobeis high as −13 dB. Such frequency localizations that causethat contribution to ICI in practice usually comes from sev-eral adjacent channels. In fact, in order to completely elim-inate the ICI affect, turbo processing may have to consideras many subchannels with ICI as possible in order to satisfythe physical model. In cases where ICI appears due to thefrequency offset or Doppler shifts, the number of subcarriersor subchannels M affected by ICI is equal to the total num-ber of subchannels N., i.e. all subchannels are subject to themutual ICI. The main motivation of this paper is develop-ing turbo algorithms with better ICI suppression proposedearly in [3]. In order to complete this task it is necessaryto increase the number of sources ICI (M > 2) taken intoaccount, and, at the same time, to keep the complexity ofimplementation as low as possible.

This paper represents an extension of the previousstudy [3], and it intends to achieve two main goals. First,the earlier proposed method is extended to cases when thesubchannel of interest is affected by interchannel interfer-ence from M > 2 adjacent subchannels. The second goalis to show that under special conditions, the proposed algo-rithm’s complexity exhibits linear growth with respect to Mand independence with respect to the total number of sub-channels in the multicarrier system. This makes the pro-posed approach attractive for practical implementations.

The paper is organized as follows: In Sect. 3, we definethe key concepts and briefly describe the synthesis and fea-tures of the previously proposed turbo-receiver for OFDMor Discrete Multitone (DMT)-based systems in the presenceof ICI caused by a frequency offset. Generally speaking, ICIin OFDM/(DMT)-based systems can be caused by many dif-ferent factors (e.g., insufficient guard-interval length, non-linear distortions in the amplifier [5], [6], or other factors,as a result of which the subcarriers lose mutual orthogonal-ity), not only by the frequency offset. All results presentedhere apply directly to the above cases. In the discussion, weassume that ICI is caused by a frequency offset. This as-sumption allows us to keep the simple and direct ICI modeldefined in the previous study [3].

In Sect. 4, we show that it is possible to significantlysimplify the previously proposed algorithm without any sig-nificant loss in the BER performance. In this section, wealso derive a linearization for the proposed algorithm. InSect. 5, we derive the structure of the linearized turbo re-ceiver for signals with ICI for cases when the subchannelof interest is affected by interchannel interference from Madjacent subchannels. Section 6 summarizes the results and

establishes common ground between the proposed schemeand the classical Decision Feedback Equalizer (DFE) ap-proach [4]. In Sect. 7, we discuss improvements in the BERperformance that result from implementing the proposedscheme and present simulation results. Here, we describea DMT-based communication system and outline possibleimprovements in its performance as a result of implement-ing the proposed signal post-processing algorithm comparedwith the DFE approach. Finally, Sect. 8 presents our conclu-sions.

2. Inter-Channel Interference in OFDM

The main disadvantage of OFDM-based systems is theirsensitivity to ICI. In an ideal transmission channel with nosignal distortion, a DMT system would guarantee error-freerecovery of transmitted symbols by the receiver and wouldcreate conditions close to “perfect reconstruction” [5]. Thatis, perfect reconstruction conditions are satisfied in the caseof a DMT system. Note that perfect reconstruction usuallyrequires a substantial overlap of adjacent subchannels.

In most practical applications, the orthogonality be-tween the subchannels is destroyed by the receiver when-ever the amplitude, phase, or frequency is distorted by thetransmission channel or the front-end of the receiver. Inmost cases, this causes unacceptable performance degrada-tion [6]. Here, we restrict our discussion to cases in whichthe deterioration of BER performance can be defined mostlywith respect to ICI and other noise signals. During equalizersynthesis we assumed that ICI exists in the general form asshown in Fig. 4. Thus, ICI is characterized only by the sub-channels’ mutual leakage coefficients. During simulationwithout any loss of generality, we assumed that ICI in DMTsystem occurs due to a frequency offset. This assumptionallowed us to use a simple model of ICI in a DMT-basedsystem.

Muschallik [7] showed that a frequency offset due tofrequency differences in local oscillators between the trans-mitter and receiver also disturbs the orthogonality of re-ceiver carriers, causing the demodulated signal of one car-rier to be distributed to other carriers as ICI. Such a fre-quency offset can be generated by inaccuracies in the tun-ing oscillator or by an instantaneous Doppler shift in rapidlytime-varying channels and will degrade the BER of the de-modulated signals.

A number of methods have been developed to reducethis ICI sensitivity to a frequency offset [1], [6]–[10], in-cluding the windowing of the transmitted signal [7], the useof ICI self-cancellation schemes [8], [9] and the use of thechannel matrix inversion for ICI cancellation [10]. Althoughall these approaches are effective in many practical applica-tions (HDTV, WLAN, WiFi, WiMAX), minimizing the ef-fect of ICI in this way is inherently suboptimal because in-terference contains information about the symbols that havebeen transmitted. Moreover, most of them are inefficient inthe multiple Doppler spread environment.

The optimal symbol-wise reception of signals gives us


a starting point for increasing the reliability of data trans-mission. The structure of the maximum likelihood sequencedetector for M asynchronous signals in the presence of ICIand additive white Gaussian noise has been derived and de-scribed by Zhenhua at al. [11]. This receiver consists of abank of M single-user matched filters followed by a Viterbidecision algorithm. If the input is binary, the Viterbi algo-rithm has a computational complexity in the order of 2M perbit decision. Thus, the optimal receiver is too complex toimplement, even for a moderate number of users or chan-nels. On the other hand, the performance of the conven-tional receiver quickly becomes interference-limited as thenumber of users or channels increases [1], [4], [11]. Hence,we need a suboptimal receiver or equalizer whose perfor-mance approaches that of the optimal receiver [3] but whosecomplexity is significantly lower.

3. System Configuration and ICI Model

In this study, we assume a DMT-based communication sys-tem. We also assume that the guard interval length exceedsthe maximal channel delay spread, thus the ISI level is neg-ligible compared to that of ICI and other noise signals. Itis assumed that an OFDM signal represented in (1) is trans-mitted

s(t) =

+∞∑

m=−∞g(t − m · TS )·

N∑n=0

Dn(m) exp{ j2π · fC · t}

· exp{ j2π · n · ∆ f (t − m · TS )} (1)

where, N is the number of subcarriers (subchannels), TS isthe symbol length, ∆ f is intercarrier frequency space, fCis the lowest carrier frequency, Dn (m) is the data symbol onthe subcarrier (subchannel) whose frequency fn = fC+n ·∆ fin the m-th OFDM symbol.

In a practical mobile radio channel, time-variant multi-path propagation causes Doppler frequency spread. There-fore we assumed that the transmission channel consists ofNP differently Doppler-shifted propagation paths with inci-dent angles γi. In fact, difference in incident angles γi foreach transmission path defines the difference in the Dopplershifts [10]. Thus, the received OFDM signal can be writtenas:

y(t) =NP∑i=1

Ai · s(t − τi) exp{ j2π∆ fi · (t − τi)} + n(t) (2)

where, Ai, τi and ∆ fi are attenuation, relative delay andDoppler shift of the i-th path respectively; n (t) is additiveGaussian noise.

Figures 1 and 2 illustrate this suggestion about thephysical model, where δi = ∆ fi/∆ f is the normalized fre-quency offset of the i-th path.

As shown in [10], after down converting and samplingthe received signal (2), the block of samples are transformed

Fig. 1 ICI physical model.

Fig. 2 Multipath and multi Doppler spectrum.

by FFT to generate signal yl (t) on l-th subcarrier or, simi-larly, the data symbol D∗l (k) that corresponds to each sub-carrier (subchannel). Thus, yl (t) and D∗l (k) are expressedas:

yl(t) = D∗l (k) = αl,l(k) · Dl(k)

+

N−1∑n=0n�l

αl,n(k) · Dn(k) + nl(k) (3)

where nl (k) is additive noise that corresponds to the l-th car-rier in the k-th OFDM symbol and αl,n (k) is the transferfunction from symbol Dn (k) to the l-th carrier.

αl,n(k) =NP∑i=1

1N

sin(π(n − l + δi))

sin(π(n−l+δi)

N

)

· exp

{− jπ(N − 1)(n − l + δi)

N

}

· exp{ j · 2πδi · ∆ f · TS · k} · Ai

· exp{− j · 2π( fn + δi · ∆ f )τ}Here, the coefficient αl,n (k) is the transfer function

from symbol Dn (k) to the l-th subcarrier. If l � n, αl,n (k)represents the influence of ICI. For simplicity, we startwith only two adjacent channels (one placed below andthe other placed above the channel of interest) as shownin Fig. 3. Thus we assumed that the influences of compo-nents αl,l±2 (k), αl,l±3 (k), etc. are insignificant. Although thismeans that there is some loss of generality, this restrictionreflects the fact that in practice, the main contribution to ICIusually comes from adjacent channels and any contributionto ICI from channels further from the channel of interestcan be considered negligible and treated as Gaussian noise.


Fig. 3 ICI model. M=2.

Fig. 4 General model of ICI.

In the following sections, we will extend this model to amore general situation, when M adjacent subchannels intro-duce ICI into the subchannel of interest (Fig. 4). For illustra-tion purposes, in Fig. 4, we split the M subchannels into twogroups with M/2 subchannels in each group. However, gen-erally speaking, the number of subchannels in each groupcan be different. In this model, the case shown in Fig. 3 canbe referred to as a case with M= 2 when only the ICI fromthe two adjacent subchannels is taken into account. Withoutany loss of generality, we assume that ICI in DMT systemsoccurs due to a difference in Doppler shifts ∆ fi. We alsoassume that the sampling frequency at the receiver is track-ing the correct value. This assumption allowed us to use asimple ICI model in the DMT-based system.

The models shown in Figs. 1–4 are useful for under-standing the physical processes that cause ICI and will beused throughout this study. With the model defined, theproblem then is to determine the value (or its sign in thecase of binary transmission) of the transmitted informationsymbol, Di, for a given received signal in a specific subchan-nel.

4. The Receiver and the Mathematics of Probability

In this section, we define some of the key notions and brieflypresent some of the results obtained in the previous study[3] for the case when M= 2. Here, we assume that binaryinformation is transmitted through subchannels by means ofsignals Di (k, j), i = −1, 0, +1; j = 0, 1, whose durationis T . In this notation index, i defines the number of eachsubchannel with respect to the subchannel of interest, whileindex j is determined by the sign of information symbol Di

in the corresponding subchannel i{j = 0 if Di = +1j = 1 if Di = −1

(4)

For simplicity, we omit time index k and index j de-pendences for signals Di (k, j) i.e. Di (k, j) → Di and as-sume that the information signals are opposite in signsi.e. D−1 (k, 0) = −D−1 (k, 1), D0 (k, 0) = −D0 (k, 1), andD+1 (k, 0) = −D+1 (k, 1) and that the same signals are used inthe lower and upper subchannels as well as in the subchan-nel of interest for the transmission of information symbols.The latter assumption reflects the fact that all subchannelsare presumed to be identical, with no differences (in ampli-tude, shape, energy, etc.) between the information signals.

From Fig. 3, signals in the subchannel of interest (sub-channel number zero), affected by ICI from the lower andupper subchannels can be represented as a linear combi-nation of signals transmitted in the lower and upper sub-channels and in the subchannel of interest with αi j co-efficients corresponding to the cross-channel leakage i.e.D0 + α−10 · D−1 + α+10 · D+1.

For example, in the k-th OFDM symbol for informationsymbol D0 = +1 in the subchannel of interest we have:

S 0 = 1 + α−10 + α+10,D0 = +1,D−1 = +1,D+1 = +1S 1 = 1 + α−10 − α+10,D0 = +1,D−1 = +1,D+1 = −1S 2 = 1 − α−10 + α+10,D0 = +1,D−1 = −1,D+1 = +1S 3 = 1 − α−10 − α+10,D0 = +1,D−1 = −1,D+1 = −1

(5)

In this case, for information symbol D0 = −1, the sig-nals in the subchannels are paired and are opposite in sign:

S 4=−1 + α−10 + α+10=−S 3,D0=−1,D−1=+1,D+1=+1S 5=−1 + α−10 − α+10=−S 2,D0=−1,D−1=+1,D+1=−1S 6=−1 − α−10 + α+10=−S 1,D0=−1,D−1=−1,D+1=+1S 7=−1 − α−10 − α+10=−S 0,D0=−1,D−1=−1,D+1=−1

(6)

From Eqs. (5) and (6), instead of a set of two transmit-ted signals (4), after the introduction of ICI, we have a setof eight signals S j, j = 0, . . . , 7 at each subchannel receiverinput. The possible number of signals S j, j = 0, . . . , 7 isdetermined by the number of symbols in the subchannel ofinterest, D0, and it depends on the symbols in adjacent sub-channels D−1 and D+1.

Let us assume that a maximum a posteriori probabilitysymbol-wise algorithm (MAP) [12] is employed for making


decisions about the sign of the information symbol trans-mitted in the subchannel of interest. The motivation for thisapproach is that the MAP-based algorithm provides betterresults than maximum likelihood estimation [12]. This ismostly because this algorithm takes into account the a pri-ori knowledge about receiving signals. In the previous study[3], we treated the received symbols to be estimated as de-terministic parameters; we also used the a priori knowledgeabout the symbols in the adjacent subchannels and treatedthem as stochastic parameters, but limited them to a finiteset of discrete values. The a posteriori probability of recep-tion of an information symbol in the subchannel of interest,P (D0/y0(t)), is obtained as the sum of the a posteriori prob-abilities of reception of signals that correspond to the trans-mitted information symbol D0 in the subchannel of interest.From Eqs. (5) and (6), the a posteriori probability of recep-tion of signals in the subchannel of interest, P (D0/y0(t)),can be written as

P(D0 = +1/y0(t)) = k · [P(S 0/y0(t)) + P(S 1/y0(t))+P(S 2/y0(t)) + P(S 3/y0(t))]

P(D0 = −1/y0(t)) = k · [P(S 4/y0(t)) + P(S 5/y0(t))+P(S 6/y0(t)) + P(S 7/y0(t))]

(7)

where P(S j/y0 (t)

)is the a posteriori probability of receiv-

ing signal S j given y0 (t), and y0 (t) = S j+n0 (t) is the receiv-ing signal representing an additive mixture of a sequence ofsignals with ICI and white Gaussian noise n0 (t) with spec-tral power density N0.

In the maximum a posteriori probability receiver, thesign of the received information symbol D is determined ac-cording to the result of comparing the a posteriori probabili-ties received information symbol D with an appropriate signor their logarithms with a threshold [12]. Therefore, for thesubchannel of interest,

P(D0 = +1/y0(t))P(D0 = −1/y0(t))

>/<1

or its logarithms

ln P(D0 = +1/y0(t)) − ln P(D0 = +1/y0(t))>/<0 (8)

In the previous study, we derived a decision rule forsigns of the received information symbol in the subchannelof interest. Here, we present the same results expressed ex-plicitly in terms of cross-channel leakage coefficients α+10

and α−10. For further detail, please refer to the previouswork [3].

∆ ln P0 = ∆ ln P (D0) = ln P (D0 = +1/y0 (t))

− ln P (D0 = +1/y0 (t)) =2

N0

∫ T

02 · y0(t)dt

+0.5 ln cosh

{1/2

{∆ ln P+1+

2N0

∫ T

02α+10y0(t)dt−∆E01

}}

− ln cosh

{1/2

{∆ ln P+1+

2N0

∫ T

02α+10y0(t)dt+∆E01

}}

+0.5 ln cosh

{1/2

{∆ ln P+1 +

2N0

∫ T

02α+10y0(t)dt−∆E23

}}

− ln cosh

{1/2

{∆ ln P+1 +

2N0

∫ T

02α+10y0(t)dt + ∆E23

}}

+ ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt − ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E23

}}

− ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt + ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E23

}}

(9)

In order to preserve the succession, we employed thesame notation in (9) as in the previous study [3]. Here,∆ ln Pi = ln P (Di = +1/yi (t)) − ln P (Di = +1/yi (t)), i =−1, 0,+1 denotes the difference in the a posteriori probabil-ity logarithms; the energy of signal S j (t) is denoted as E j,

so E j =∫ T

0S 2

j (t) dt. Therefore:

∆E01 ≡ E0 − E1

N0; ∆E23 ≡ E2 − E3

N0

and ∆EΣ ≡ (E0 + E1) − (E2 + E3)2

(10)

The term ∆ ln Pi represents a soft decision in the i-thsubchannel. Here, the decision rule makes a hard-bit de-cision, i.e. the decision about the transmitted signal’s sign.The decision rule for the subchannel of interest is graphi-cally depicted in Fig. 5.

The non-linear transfer function

F(x,∆E) = ln cosh

{x − ∆E

2

}− ln cosh

{x + ∆E

2

}

(11)

used in (9) can be described as a limiter with a linear re-gion. Notice that the level of limits depends on ∆E and theSNR ratio defined in (10) (precisely speaking, it dependson the Gaussian noise spectral power density N0). Figure 6shows the transfer function of a non-linear unit, F (x,∆E),for different parameter ∆E values and Fig. 7 its graphicalrepresentation.

Figure 7 shows a graphical representation of F (x,∆E)as a soft limiter, which will be used throughout the followingdiscussion. As can be seen from (9) and Fig. 5, despite theobvious simplification decision algorithm compared to theprevious decision rule presented in [3] as a result of usingexplicit expressions, the proposed receiver still looks com-plicated. Let us see now how we can simplify it even further.


Fig. 5 Turbo-receiver based on Ref. [3], M =2.

Fig. 6 F (x,∆E).

Fig. 7 Graphical representation of F (x,∆E).

Fig. 8 Hierarchical representation of turbo receiver, M = 2.

5. Hierarchical Turbo Equalizer Structure

First note that the proposed receiver has a hierarchicalstructure. The turbo receiver shown in Fig. 5 can be de-composed into different levels as shown in Fig. 8. Thetwo blocks marked as “Level 0” in Fig. 8 correspond tothe first four non-linear items in (9) followed by linear

item 2N0

∫ T

02 · y0 (t) dt and are expressed asF (x,∆E01) and

F (x,∆E23); the blocks marked as “Level 1” correspondto the two items under functional log cosh{•}, and theblock marked as “Main Pass” corresponds to the Main Passthrough the first linear item, 2

N0

∫ T

02 · y0 (t) dt, in (9). Each

rectangular block marked as the certain “Level” in Fig. 8 has


a transfer function shown in Fig. 6 and an internal structureshown in Fig. 7. The bold line marks inter-level connectionsin Fig. 8, which are also shown in Fig. 5. As can be seenfrom Fig. 8 and Eqs. (9) and (10), the receiver for M = 2consists of the above three levels: the Main Pass (the maincontribution), level 0, and level 1.

Let us try to find more details about parameters for non-linear blocks at the different levels of turbo equalizer shownin Fig. 8. At level zero, there are two non-linear blocks withtransient functions F (x,∆E01) and F (x,∆E23). Inputs ofthese blocks are the same, and they are coming from the out-put of the correlator with α+10 (cross-channel leakage fromthe upper subchannel). Parameters ∆E01 and ∆E23 corre-spond to the energy difference between the signals with op-posite signs that are affected by ICI from the same uppersubchannel (the couples of signals S 0, S 1 are different fromS 2, S 3 in information symbol D+1 = ±1).

∆E01 ≡ E0 − E1

N0; S 0 = D0 + α−10 · D−1 + α+10 · D+1;

S 1 = D0 + α−10 · D−1 − α+10 · D+1 (12)

∆E23 ≡ E2 − E3

N0; S 2 = D0 − α−10 · D−1 + α+10 · D+1;

S 3 = D0 − α−10 · D−1 − α+10 · D+1 (13)

Similary, at level one, there is only one non-linear blockwith a transient function F(x,∆EΣ), where

∆EΣ ≡ (E0+E1)−(E2+E3)2

=(E0−E2)+(E1−E3)

2

=∆E02 + ∆E13

2(14)

S 0 = D0 + α−10 · D−1 + α+10 · D+1;

S 2 = D0 − α−10 · D−1 + α+10 · D+1 (15)

S 1 = D0 + α−10 · D−1 − α+10 · D+1;

S 3 = D0 − α−10 · D−1 − α+10 · D+1 (16)

The input signal for the non-linear block at level one

is coming from the output of the correlator with α−10

(cross-channel leakage from the lower subchannel). FromEqs. (14)–(16), it is easy to see that parameter ∆EΣ is thesum of the energy differences between the signals with op-posite signs that are affected by ICI from the same lowersubchannel (the couples of signals S 0, S 2 are different fromS 1, S 3 in information symbol D−1 = ±1).

From the above discussion, we can conclude that eachlevel of the turbo equalizer corresponds to a single path fromthe ICI source (Fig. 4). Therefore it seems reasonable to as-sume that if we increased the number of ICI sources, for ex-ample, by one, from M= 2 to M= 3, the new turbo equalizerwill have the structure of the turbo equalizer for M = 2 (inorder to process all M= 2 ICI sources), plus an extra levelfor processing an additional source of ICI. In the next Sec-tion, we will discuss a possible extension of the proposedtechnique for M > 2.

6. Further Algorithm Development

So far we have restricted the discussion for cases when onlytwo adjacent subchannels introduce ICI into the subchannelof interest (M = 2). Figure 9 shows the proposed turbo-equalizer structure for cases when three adjacent subchan-nels affect the subchannel of interest.

In Fig. 9, this is shown as M= 3. As expected, the newturbo equalizer, in addition to having the structure of theturbo equalizer developed for M = 2 (Fig. 8), uses an ad-ditional level for processing an additional ICI source. InFig. 8, the turbo equalizer structure is shown without anymathematical prove. A direct approach, similar to the onedescribed in the earlier work [3] for M = 3 was used to de-rive this structure; however, this approach is too complicatedto be presented here, and at the same time, it does not pro-vide any new solutions. In Appendix A, we present a simpli-fied explanation and definitions related to the turbo equalizershown in Fig. 8.

In Appendix A, we use a mathematical induction

Fig. 9 Hierarchical decomposition of turbo receiver, M=3.


method to show that the number of levels in the turbo equal-izer exhibits linear growth with respect to M and indepen-dence with respect to the total number of subchannels in themulticarrier system.

6.1 Inter-Level Connections

The inter-level connections in Figs. 8 and 9 make the turboequalizer algorithm very complicated. Such connections(shown in Figs. 5, 8, and 9 by the bold lines) start at thelower level, go through all intermediates levels, and end atthe upper level (level number M). As can be seen fromFigs. 8 and 9, the inter-level connections themselves do notuse any likelihood functions calculated at the correlator’soutputs. Instead, they use signals already processed byother non-linear blocks and modify parameters ∆E of thenon-linear blocks at the upper level as shown in Fig. 8 andFig. 9 (Eqs. (9) and (17)). Let’s see what happens if we sim-plify receiver structure by removing inter-level connectionsin Figs. 8 and 9.

F(x) = FDirect

= ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt − ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E23

}}

− ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt + ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E23

}}

(17)

F(x) = FSimpl

= ln cosh

{1/2

(∆ ln P−1+

2N0

∫ T

02α−10y0(t)dt−∆EΣ

)}

− ln cosh

{1/2

(∆ ln P−1+

2N0

∫ T

02α−10y0(t)dt+∆EΣ

)}(18)

Here, Eq. (17) applies to the non-linear block at levelone shown in Fig. 8, or to the two last items in (9), thatis, when all inter-level connections are taken into account.Equation (18) shows the same calculations with all inter-level connections omitted. Figure 10 shows the plots basedon Eqs. (17) and (18) for a high SNR ratio (SNR=10) wherex ≡ 2

N0

∫ T

02α+10y0 (t) dt and α+10 = α−10 = 0.25.

As can be seen from Fig. 10, signals from level zeroslightly modify the non-linear block’s transfer function(gray plots in Fig. 10); however, they do not affect slope and

Fig. 10 Direct and simplified calculations.

limiting level. Thus, the linear part and the limiting level aredefined mostly by parameter ∆EΣ. Therefore, it is reason-able to assume that the impact of the inter-level connectionson the total BER performance is not so strong. In order toprove it, we remove all inter-level feeds to the non-linearblock at level one. We use Eq. (18) instead of Eq. (17) todetermine the BER performance of the newly obtained sim-plified turbo equalizer. The corresponding simulation con-ditions for the proposed turbo equalizer algorithm are shownin Table 1.

Thus, during simulation the two last non-linear itemsin (9) were reduced to:

ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt − ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt − ∆E23

}}

≈ ln cosh

{1/2

(∆ ln P−1+

2N0

∫ T

02α−10y0(t)dt−∆EΣ

)}(19)

and

ln cosh

1/2

∆ ln P−1 +

{2

N0

∫ T

02α−10y0(t)dt + ∆EΣ

}

+ ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E01

}}

− ln cosh{1/2

{∆ ln P+1

+ 2N0

∫ T

02α+10y0(t)dt + ∆E23

}}

≈ ln cosh

{1/2

(∆ ln P−1+

2N0

∫ T

02α−10y0(t)dt+∆EΣ

)}(20)

Figure 11 shows the BER performance as a function of2Eb/No (a doubled average received signal energy per bit-to-background noise power spectrum density ratio) and thefrequency offset is normalized to the inter-subchannel space


Table 1 Simulation setup.

Fig. 11 BER performance.

δ = 0.25 as a parameter for a conventional DMT-based re-ceiver [14] using 64 subcarriers with QPSK modulation anda Gaussian noise (AWGN) channel (points marked DMT),and the same DMT receiver using the proposed turbo post-processing algorithm (DMT+TE) with two turbo-iterations(M= 2). As a reference, the simulation results for ICI-freetransmission (δ = 0.0) with the matched filter receiver arealso shown in Fig. 11. Points corresponding to the directcalculations based on Eq. (9) are denoted as “DMT+TE.”Results corresponding to the simplified calculation basedon Eqs. (19) and (20) are not shown in Fig. 11 because theyare completely the same as the results obtained with Eq. (9).Only some insignificant degradation in the low 2Eb/No area(2Eb/No < 3, . . . , 5 dB) was found by analyzing the numeri-cal data. In coded OFDM transmission (COFDM), the BERfor the effective use of forward-error-correction codes (con-volutional, for example) should be in the order of 10−3 orlower; in some coding schemes it can be as high as10−2.As can be seen in the BER plots in Fig. 11, the proposedscheme calculated with Eqs. (19) and (20) provides the nec-essary BER for the following implementation of the forwarderror correction. Therefore, inter-level connections can beomitted in most practical cases.

Despite the linear growth of the number of levels withrespect to M, which enables keeping the complexity of theturbo equalizer the same as that of the Decision FeedbackEqualizer (DFE) or similar to that of the Viterbi decoder, thenumber of non-linear blocks at each level still grows expo-nentially as 2M−1. For example, when M = 2, the number ofnon-linear blocks at level zero is two; however, when M= 3,the number of non-linear blocks at the same level increasesto 4. Obviously, such exponential grow of the turbo equal-izer complexity makes its practical implementation very dif-ficult. In this Section, we will show how this problem can

Fig. 12 Approximation.

be overcome while preserving the linear complexity growth.As can be seen in Figs. 8 and 9, despite the exponential

growth of the number of non-linear blocks in the levels, thedecision rule uses the sum of non-linear blocks outputs fromeach level. This is good news. Figure 12 shows the sum ofoutputs of two non-linear blocks as the following function:

FΣ(x,∆E1,∆E2) = F(x,∆E1) + F(x,∆E2) (21)

LimΣ = |∆E1| + |∆E2| (22)

This sum can be considered as the output of the non-linear blocks at level zero in Fig. 8 or at level one in Fig. 9,with the appropriate substitution of ∆E. As can be seenfrom Fig. 12, the resulting curve for FΣ (x,∆E1,∆E2) stillhas the shape of the same limiter with a linear area. More-over, the limiting level, LimΣ, of the resulting function,FΣ (x,∆E1,∆E2), equals the sum of the limiting levels ofeach item (Fig. 12). Thus, we can easily replace the sumof the two non-linear blocks with a single non-linear blockwith limiting level LimΣ defined as the sum of the individualitems’ limiting levels (21), (22).

This approach becomes more obvious if we apply abroken-line approximation to each item shown in Fig. 12. Itis well known in mathematics that the sum of several linearfunctions is also a linear function. Therefore we can extendthis notion to the sum of an arbitrary number of non-linearblocks.

It was found experimentally by authors during sim-ulations that the following broken-line approximation ofF (x,∆E) gives the best results (does not cause any BERdegradation) for most practical 2Eb/No values (2Eb/No >5 dB)

F(x) =

{if |x| < 0.5 · Scale · LimΣ; F(x) = −x·LimΣ

0.5·Scaleelse F(x) = −LimΣ · sign(x)

(23)


where parameter Scale is defined as

Scale =

{if LimΣ > 4; Scale = 1else Scale = 4/LimΣ

(24)

The broken-line approximation of the resulting func-tion, FΣ (x,∆E1,∆E2), defined in (21) is shown in Fig. 12. Itwas calculated for 2Eb/No=10 dB according to the approxi-mations in Eqs. (22)–(24).

The average BER for a system using a broken-line ap-proximation for non-linear blocks is also shown in Fig. 13(points marked “Approx.”). As can be seen, there is no BERdegradation at all. Instead, there is a small improvement inBER in the high-2Eb/No area. This insignificant improve-ment is due to the fact that during the direct calculationsbased on Eq. (11), cosh {◦} is calculated as a difference oftwo exponential functions and the parameters of these expo-nential functions grow significantly with N0 → 0:

∆EΣ ≡ (E0 + E1) − (E2 + E3)2

→ ∞,∆E1 ≡ E0 − E1

N0→ ∞,

∆E2 ≡ E2 − E3

N0→ ∞ when N0 → 0

Therefore, this exponential overflowing, or the errorsof calculation in the case of a huge function’s arguments,slightly affect the BER results.

Let us now define the limiting levels for the sum ofnon-linear blocks for the turbo equalizer schemes shown inFigs. 8 and 9. When M=2 (Fig. 8), the limiting levels of non-linear blocks at level zero and level one are (see AppendixB):

∆ELev0 = (∆E01 + ∆E23)/2 = 4 · α+10/N0 (25)

∆ELev1 = ∆EΣ = (∆E02 + ∆E13)/2 = 4 · α−10/N0 (26)

Here in (25), we include a scaling factor of 1/2 followedby the summing circuit in Fig. 8. Similar results were ob-tained for the turbo equalizer shown in Fig. 9 for M= 3:

∆ELev0 = (∆E01 + ∆E23 + ∆E45 + ∆E67)/4

= 4 · α+20/N0 (27)

Fig. 13 Turbo equalizer, general case of ICI.

∆ELev1 = ((∆E02 + ∆E13)/2 + (∆E46 + ∆E57)/2)/2

= 4 · α+10/N0 (28)

∆ELev2 = (∆E04 + ∆E15 + ∆E26 + ∆E37)/4

= 4 · α−10/N0 (29)

Equations (27)–(29) show that after scaling, output sig-nals at the non-linear blocks depend only on the cross-channel leakage coefficients αi j and SNR. Now we areready to derive a turbo equalizer structure for cases whenM sources of ICI affect the subchannel of interest.

7. The General CASE of ICI and Comparison withDFE

In this section, we will present a turbo equalizer for the gen-eral case shown in Fig. 4, when M sources of ICI affect thesubchannel of interest. Figure 13 shows the structure of alow-complexity turbo equalizer. The proposed turbo equal-izer employs M subchannels in addition to the main path(Main Pass). Each subchannel corresponds to a single pathof ICI shown in Fig. 4. In order to keep the complexity lin-ear, we replace the sum of non-linear blocks (Figs. 8 and 9)with a single non-linear block whose total limiting level isequal to the sum of the limiting levels of individual blocks.Thus, in Fig. 13, each subchannel is incorporated with a sin-gle non-linear block. As shown in Appendix B, the limitinglevel of non-linear blocks in the subchannels is defined onlyby the cross-channel leakage coefficient, αi j, and summingfactor 2M. Finally, in order to keep a partial subchannel am-plitude balance, an additional multiplication with a scalingfactor of 2−M is introduced.

By analyzing the structure shown in Fig. 13, it is easyto find certain similarities with the DFE [12], [13] approachproposed in reference [13] to combat ICI. This similaritybecomes more pronounced if we assume that leakage coef-ficients αi j are relatively small, i.e. αi j � 1.0. This assump-tion reflects the fact that in practice for well-designed sys-tems, the contribution form ICI (undesirable signal contribu-tion) is significantly lower than the contribution of the useful(desirable) signal. Thus, for the leakage coefficients αi j inorder of magnitude 0.1 or less, all correlators with the refer-ence signals αi j can be omitted with only SNR penalty lessthan 0.1 dB. Such a modified turbo equalizer with omittedcorrelators and a DFE-based equalizer are shown in Figs. 14and 15, respectively.

Removing ICI components from the input signal is per-formance by passing soft-decision from the adjacent sub-channels through the hard limiter circuit with the followingweighting as shown in Fig. 15.

Complexity of equalizer is the essential criteria for itspractical implementation. Table 2 below shows the estima-tion of the number of operations per subchannel with param-eter M for the proposed algorithm and classical DFE. QPSKconstellation is assumed (More details are in Appendix C).

Here we assumed that the binary shifting by M bits per-forms the scaling (multiplication) by 1/2M and the look-up-table (LUT) or linear approximation (23), (24) employs for


the non-linear transfer function (11) calculations. As canbe seen from Table 2, both approaches have about the samecomplexity.

We’d like to stress that despite the above-mentionedsimilarities in structure between the proposed turbo equal-izer and the DFE, there is a significant difference in opera-tion. The main difference between these two approaches isthat in the case of the turbo equalizer, the operational pointof the non-linear block is placed mostly in the linear area(Figs. 6 and 12). Only high-amplitude noise componentscan move the operating point from the linear to the limit-ing area. This reflects the fact that the linear area and lim-iting levels depend directly on the Gaussian noise spectral

Fig. 14 DFE-like presentation of turbo equalizer.

Fig. 15 DFE [12].

Table 2 Relative complexity.

power density, N0. Thus, when the Gaussian noise spectralpower density, N0, decreases, the limiting level and linear ar-eas in Fig. 12 increase and vice versa. In contrast, the DFEis specifically designed to work in the limiting area in or-der to re-construct the transmitted signal. The transfer func-tion (the weight) for the DFE is static and does not dependon SNR. Therefore for most practical implementations, theDFE must have a high SNR for proper operation [12], [13].From this, we conclude that the proposed turbo equalizeroperates in a more intelligent way because the non-linearlevels are adjusted according to the SNR value.

Finally, the denoted similarity in the structures does notautomatically suppose the similarity in the operations’ prin-ciples. The compensation technique is behind that of theDFE. In the turbo-equalizer, the proposed structure approx-imately calculates a posteriori probabilities of transmittedinformation symbol.

8. Noise Immunity and Simulation Results

Figure 16 shows the average BER for these two approachesfor an AWGN channel (δ=0.25) as a function of 2Eb/No, andM equals the number of ICI sources taken into account dur-ing signal processing as a parameter. The simulation condi-tions, number of subchannels, modulation scheme, numberof turbo-iterations, and corresponding channel conditionsare shown in Table 1. Additional cross-channel leakage co-efficients αi j were selected according to Table 3.

Fig. 16 Turbo equalizer vs. DFE.


Table 3 Leakage coefficients αi j.

As can be seen from Fig. 16, the proposed turbo equal-izer provides better BER performance than does the DFEat the same level of complexity. The BER performance ofthe DFE is approximately constant when the SNR valuesare high (2Eb/No > 15 dB). There is no additional improve-ment in BER even when the SNR is constantly increased.This reflects the fact that when the estimations in the adja-cent channels are correct, only the Gaussian noise and otheruncompensated ICI define the BER performance.

In the previous work [3], we showed that in order tocompletely eliminate the effect of ICI, the equalizer musttake into account as many subchannels with ICI as possi-ble in order to satisfy the physical model. In cases whereICI appears due to frequency offsets, the number of sub-channels affected by ICI, M, is equal to the total number ofsubchannels, N, i.e. all subchannels are subject to mutualICI. In contrast to the DFE, the turbo equalizer in the sec-ond and all following steps calculates a new estimation forthe logarithm a posteriori probability in the subchannel ofinterest, which is also used in the adjacent subchannels, evenfor a limited M (M =2) [3]. Thus the results obtained in theadjacent subchannels are used during logarithm a posteri-ori probability calculations in all other adjacent subchannels(30). Even if there are no direct relation between the chan-nels placed higher and those placed lower than the channelof interest, there is a chain, or indirect relationship, betweenthe results obtained for all channels, even if M is still limitedto 2. In this way, the output of one-subchannel receiver isused as a priori information by another subchannel, i.e.

ln P(D−2) = F{ln P(D−3), ln P(D−1)}ln P(D−1) = F{ln P(D−2), ln P(D0)}ln P(D−0) = F{ln P(D−1), ln P(D+1)}ln P(D+1) = F{ln P(D0), ln P(D+2)}ln P(D+2) = F{ln P(D1), ln P(D+3)}

(30)

Indirect relationship between the results obtained for allchannels (30).

where operand F in (30) denotes the processing algorithm(9) or its possible modifications (19) and (20).

This chain structure of improvements is shown in (30)for two adjacent subchannels. This is the main reason whythe proposed turbo equalizer significantly outperforms theDFE at the same level of complexity.

The BER performance achievable under a Dopplerspread channel was also evaluated by computer simulation.Doppler spread 300 Hz was selected during simulations. As-

suming 5 GHz carrier frequency, the corresponding travel-ing speed of a mobile terminal is about 65 km/h. In the sim-ulation, we used a multipath channel derived from a typi-cal channel profile of urban, non-hilly terrain with six rays,or paths, and an exponentially decaying power delay pro-file. For each path, Raleigh-distributed random attenuationwas assumed. The adopted channel model is similar to thevehicle test channel, A, in the recommendation of ITU-RM1225. There are 6-path fading channels with time delaysof 0, 1, 2, 3, 5 and 8 sample intervals, each path experienc-ing independent fading with an average power of {0, −2, −9,−10, −15 −20} dB, respectively.

Observe that different paths having different Dopplerfrequencies, which break the orthogonality between sub-channels, cause the ICI. During simulation we assumedthat the path with maximal amplitude had the Doppler shiftwhich equals zero, i.e. a carrier’s PLL is locked onto thesignal at the strongest path.

In the simulation, we also assumed that the transmis-sion channel’s parameters were not changed by the singleDMT symbol duration. The configuration of the simulationunder frequency-selective fading is shown in Table 4.

We assumed here the case of the multiple Doppler fre-quency offsets. The normalized frequency offset of the di-rect path equals 0 and other paths have Doppler shifts equalto 0.15, 0.10, −0.10, 0.08 and 0.6 correspondingly (see Ta-ble 4).

The length of the guard interval was selected to be 8%of OFDM symbol duration that significantly exceeds thechannel’s spread maximum delay value. Thus, no ICI orISI is introduced by multipath channel [4], [14] and there-fore all ICI in the DMT communication system are causedonly by the differences in the Doppler shifts δi.

Figures 17 and 18 show the BER performance of a con-ventional DMT-based receiver [14] using QPSK modulation(points indicated in Figs. 17 and 18 as DMT STD) and ofthe same DMT receiver equipped with the proposed turboequalizer (points marked as DMT+TE) for the simulationparameters shown in Table 3. In Fig. 17, “Static” denotesa multipath channel that does not change with time, i.e. astatic multipath channel when signals from all paths resultin interference that does not change in time. Meanwhile theDoppler shifts for individual paths are not equal to zero andselected from Table 3.

Figure 17 shows the BER performance for the samesystem as a function of the number of turbo iterations (TI).Experimentally, we found that an increase in the number ofturbo iterations to more than three (points denoted in Fig. 17as 3TI and 4TI), which complicated the calculations and in-creased the decoding latency, did not significantly improveBER performance. Increase in the number of turbo iterationby 1 (from 3 to 4) yields an SNR gain of less than 0.2 dB.This demonstrates good convergence of the proposed algo-rithm. For example, in order to achieve the high-gain per-formances, the standard turbo decoders have to employ atleast eight turbo iterations, or more.

Plots in Fig. 17 show substantial improvement in BER


Table 4 Simulation setup.

Fig. 17 BER turbo equalizer QPSK, static mobile channel. MultipleDoppler frequency offsets.

when the number of subchannels M taken into account isincreasing. In fact, in order to completely eliminate the ICIaffect, a turbo equalizer may consider as many subchannelswith ICI as possible in order to satisfy to the physical modelshown in Fig. 4. In cases where ICI appears due to the dif-ferent Doppler shifts, the number of subchannels affectedby ICI M is equal to the total number of subchannels N,i.e. all subchannels are subject to the mutual ICI. However,the main contribution to ICI comes from adjacent subchan-nels and any contribution to ICI from subchannels furtherfrom the subchannel of interest can be considered negligi-ble. Experimentally, we found that for maximum normal-ized Doppler shift equal to 0.15, an increase in the numberof subchannels M taking into account no more than 6 didnot significantly (less than 0.5 dB) improve the BER perfor-mance. Similar results have been reported in [3].

As shown in Figs. 17 and 18, the proposed turbo post-processing scheme works well in the multipath and multi-Doppler radio channel. However, as can be find seen in

Fig. 18 BER turbo equalizer QPSK, mobile channel, Doppler shift =300 Hz. Multiple Doppler frequency offsets.

Figs. 16 and 17, the effectiveness of the proposed methodin the case of the multipath channel is lower than in the caseof AWGN channel. This is mostly due to the frequencyselective fading environment, when signals in the differ-ent subchannels (signals at the different subcarriers) havedifferent amplitudes. As it was discovered in [3], the re-ported turbo algorithm provides better performances whenthe mutual coupling (leakage) between adjacent subchan-nels does not exceed 30% of signal’s amplitude in the sub-channel of interest. In this case there is no any significantBER degradation due to ICI with respect to the case of ICI-free transmission. In the case of AWGN channel (Fig. 16)when ICI is caused only by a single non-zero frequency off-set, these leakage coefficients are not exceeding 0.3; there-fore, the proposed scheme works very efficiently. In the caseof Raleigh fading when the different subchannels have dif-ferent amplitudes due to frequency selective fading and dif-ferent Doppler shifts, leakage coefficients can exceed 0.3.That makes the proposed algorithm less efficient.

One possible solution that can improve the situation in


Fig. 19 BER turbo equalizer QAM-16, static mobile channel. MultipleDoppler frequency offsets.

the case of frequency selective fading is increasing the num-ber of subchannels in the system in order to get a strong cor-relation between signals’ amplitudes for the adjacent sub-channels. Because of high correlation, the signal’s ampli-tudes into adjacent subchannels will remain at the same orapproximately the same level, and, therefore, effect fromimplementation of the proposed turbo equalizer for ICI sup-pression will be increased.

QPSK constellation was selected during simulation.However our study could not have been completed withoutdiscussion about implementation of the proposed methodinto M-QAM constellations, which the most OFDM com-munication systems have adopted [5], [14]. Because of thespace limitation here we only show QAM-16 BER plots(Fig. 19) similar to plots shown in Fig. 17 and obtained forthe QPSK case. As can be seen in Fig. 19, the proposedturbo equalizer also shows better performance than DFE inthe case of quadrature amplitude modulation. Some ideasabout possible ways of extending the proposed algorithm forQAM-16 constellation case and structure of QAM-16 turboequalizer have been presented by authors recently in [15]and more results are under development. It has been foundthat when we are moving from QPSK to QAM-16 constel-lation, the complexity of QAM-16 turbo equalizer increasesapproximately twice, i.e. turbo equalizer complexity againexhibits linear growth respect to the number of constellationpoints.

Points corresponding to BER obtained with actualchannel estimation are marked in Figs. 17 and 18 as “Es-tim.CSI.” In the proposed turbo-scheme, pilots’ symbols (P)are used to estimate the channel parameters. Pilot symbolsare inserted among the data symbols as shown in Fig. 20. We

Fig. 20 Data and pilot symbols.

insert a pilot symbol block every m = 6 OFDM blocks. Theinsertion of such pilot symbols guarantee that the transmis-sion channel conditions do not change significantly betweentwo consecutive channel states estimations [16]. An alter-native possible pilot signal insertion method using scatteredpilot symbols is described in [10].

We implemented the channel transfer function estima-tion method based on minimizing the mean square error pro-posed in [10] for DFE approach. According to this method,the parameters of channel transfer function are estimated soas to minimize the mean square error of equation, where

∑P

means that summation is performed as long as Dl (k) is apilot symbol.

E(NP) =∑PK

|y(k, l) − α∗l,l(k) · Dl(k)|2

+∑PK−1

|y(k − 1, l) − α∗l,l(k − 1) · Dl(k)|2 (31)

In Eq. (31), the received symbols and locally generatedreplicas of the received symbols against pilot symbols arecompared and channel parameters such as δi are estimatedto minimize this criterion. In the case when Np is more thanone, we assumed that the influence to the MMSE from eachpath is approximately independent. Thus, the influence ofeach path, estimated with (31) can be removed from the to-tal E (NP). After this, there remains the influence of otherpaths in the MMSE. By removing the influence of the firstestimated path E (1) from the MMSE, the following crite-rion, E (2) is obtained. This procedure has to be repeatedcontinuously for NP times for all paths. More details are in[10]. As shown in Figs. 17 and 18, BER degradation dur-ing actual channel estimation comparing with the case witha perfect known channel is less than 1–1.5 dB for all rangesof SNRs employed in such type of systems.

9. Conclusion

We investigated the effect of ICI on two adjacent subchan-nels in a multicarrier communication system. The perfor-mance of a conventional matched filter receiver degradesrapidly as the coupling between adjacent subchannels in-creases. This increases the BER in uncoded and coded sys-tems.

Further improvement in the DMT performance was ob-tained by increasing the number of adjacent subchannels forICI cancellation. In order to keep the turbo equalizer com-plexity as low as possible, a linear approximation was intro-duced for non-linear blocks. The proposed turbo equalizeralgorithm’s complexity exhibited linear growth with respectto M and independence with respect to the total number of


subchannels in the multicarrier system. The internal struc-ture and level of complexity of the proposed turbo equalizerare approximately the same as those of the DFE. However,the BER performance of the proposed scheme obtained bycomputer simulation was much better than that of the stan-dard DFE.

In the multipath and multiple Doppler shifts environ-ments, the proposed turbo equalizer provides probabili-ties of error10−3or less that cannot be reached in the stan-dard DMT-based systems. This makes possible the suc-cessful implementation of different forward error correctionschemes like a convolutional code or the turbo code.

The implementation of the turbo algorithm is indepen-dent of transmitted signals, providing complete standardOFDM (DMT) reception compatibility. This makes the pro-posed scheme attractive alternative for practical implemen-tations in the newly developed B3G/4G mobile communica-tion systems, i.e. WiMAX MAN, where the influence of ICIcaused by multiple Doppler shifts becomes severer.

Acknowledgments

This work was supported, in part, by the National Institute ofInformation and Communications Technology (NICT, for-mer TAO) under contracted research entitled “The researchand development of advanced radio signal processing tech-nology for mobile communication systems.”

Also we would like to thank Dr. Takeshi TAKANOfrom Fujitsu Lab., Ltd. and Professor Yoshihiko AKAIWAfrom Kyushu University for their comments on themanuscript and helpful discussion. We also want to expressour gratitude to the management of Fujitsu Laboratories andto colleagues for their encouragement and support.

References

[1] P. Roberson and S. Kaiser, “Analysis of Doppler spread perturba-tions in OFDM systems,” ETT, vol.11, no.6, pp.585–592, Nov.-Dec.2000.

[2] P.H. Moose, “A technique for orthogonal frequency division multi-plexing frequency offset correction,” IEEE Trans. Commun., vol.42,no.19, pp.2908–2914, Oct. 1994.

[3] A.N. Lozhkin, “Turbo receiver for OFDM signals with interchannelinterference,” IEICE Trans. Commun., vol.E86-B, no.8, pp.2395–2413, Aug. 2003.

[4] D. Froney, Jr. and M.V. Eyuboglu, “Combined equalization and cod-ing using precoding,” IEEE Commun. Mag., vol.29, no.12, pp.25–34, Dec. 1991.

[5] G. Cherubini, E. Eleftheriou, S. Olcer, and J.M. Cioffi, “Filter bankmodulation techniques for very high speed digital subscriber line,”IEEE Commun. Mag., vol.38, no.5, pp.98–104, May 2000.

[6] X. Cai and G.B. Giannakis, “Bounding performance and suppress-ing ICI in wireless mobile OFDM,” IEEE Trans. Commun., vol.51,pp.2047–2056, Dec. 2003.

[7] C. Muschallik, “Improving an OFDM reception using an adaptiveNyquist windowing,” IEEE Trans. Consum. Electron., vol.42, no.3,pp.259–269, Aug. 1996.

[8] Y. Zhao and S.G. Haggman, “Intercarrier interference self-cancellation scheme for OFDM mobile communication systems,”IEEE Trans. Commun., vol.49, no.7, pp.1185–1190, July 2001.

[9] J. Armstrong, “Analysis of new and existing methods of reducing

intercarrier interference due to carrier frequency offset in OFDM,”IEEE Trans. Commun., vol.47, no.3, pp.365–369, March 1999.

[10] M. Nakamura, M. Itami, K. Itoh, and A.H. Aghvami, “ICI cancel-lation technique based on estimation delay and Doppler profile inOFDM reception,” ITE Journal, Dec. 2002.

[11] X. Zhenhua, G.K. Rushforth, and R.T. Short, “Multiuser signal de-tection using sequential decoding,” IEEE Trans. Commun., vol.38,no.5, pp.578–583, May 1990.

[12] Y. Akaiwa, Introduction to Digital Mobile Communication, Wiley,New York, 1997.

[13] E. Viterbo and K. Fazel, “How to combat long echoes in OFDMtransmission schemes: Subchannel equalization or more powerfulchannel coding,” Proc. IEEE Globecom’95, pp.2069–2074, Singa-pore, Nov. 1995.

[14] J.M. Cioffi, “A multicarrier primer,” in ANSI T1E1.4 CommitteeContribution, no.91-157, Boca Raton, FL, Nov. 1991.

[15] A.N. Lozhkin, “Turbo receiver for QAM signals with interchan-nel interference,” Proc. 9th International OFDM-Workshop (In-OWo’04), pp.261–265, Dresden, Germany, Sept. 2004.

[16] J.K. Cavers, “An analysis of pilot symbol assisted modulation forRayleigh fading channels,” IEEE Trans. Veh. Technol., vol.40,no.11, pp.686–693, Nov. 1991.

Appendix A

Figure A· 1 shows a model of a four-subchannels mutual ICI(M =3).

There are three sources of ICI for the subchannel ofinterest as shown in Fig. A· 1. From Fig. A· 1, signals in thesubchannel of interest affected by the ICI from the lower andupper subchannels can be represented as a linear combina-tion of signals transmitted in the lower and upper subchan-nels and in the subchannel of interest with coefficients αi j

corresponding to the cross-channel leakage (Fig. A· 1). Forinformation symbol D0 = +1 in the subchannel of interestwe have:

S 0 = 1 + α−10 + α+10 + α+20,D0 = +1,D−1 = +1,D+1 = +1,D+2 = +1

S 1 = 1 + α−10 + α+10 − α+20,D0 = +1,D−1 = +1,D+1 = +1,D+2 = −1

S 2 = 1 + α−10 − α+10 + α+20,D0 = +1,D−1 = +1,D+1 = −1,D+2 = +1

S 3 = 1 + α−10 − α+10 − α+20,D0 = +1,D−1 = +1,D+1 = −1,D+2 = −1

(A· 1)

Fig. A· 1 Four-subchannels mutual ICI model.


S 4 = 1 − α−10 + α+10 + α+20,D0 = +1,D−1 = −1,D+1 = +1,D+2 = +1

S 5 = 1 − α−10 + α+10 − α+20,D0 = +1,D−1 = −1,D+1 = +1,D+2 = −1

S 6 = 1 − α−10 − α+10 + α+20,D0 = +1,D−1 = −1,D+1 = −1,D+2 = +1

S 7 = 1 − α−10 − α+10 − α+20,D0 = +1,D−1 = −1,D+1 = −1,D+2 = −1

(A· 2)

And for information symbol D0 = −1 in the subchannel ofinterest, we have:

S 8 = −1 + α−10 + α+10 + α+20,D0 = −1,D−1 = +1,D+1 = +1,D+2 = +1

S 9 = −1 + α−10 + α+10 − α+20,D0 = −1,D−1 = +1,D+1 = +1,D+2 = −1

S 10 = −1 + α−10 − α+10 + α+20,D0 = −1,D−1 = +1,D+1 = −1,D+2 = +1

S 11 = −1 + α−10 − α+10 − α+20,D0 = −1,D−1 = +1,D+1 = −1,D+2 = −1

(A· 3)

S 12 = −1 − α−10 + α+10 + α+20,D0 = −1,D−1 = −1,D+1 = +1,D+2 = +1

S 13 = −1 − α−10 + α+10 − α+20,D0 = −1,D−1 = −1,D+1 = +1,D+2 = −1

S 14 = −1 − α−10 − α+10 + α+20,D0 = −1,D−1 = −1,D+1 = −1,D+2 = +1

S 15 = −1 − α−10 − α+10 − α+20,D0 = −1,D−1 = −1,D+1 = −1,D+2 = −1

(A· 4)

From Eqs. (A· 1)–(A· 4), instead of a set of two trans-mitted signals (4), after the introduction of ICI, we havea set of sixteen signals S j, j = 0, . . . , 15 at each sub-channel receiver input. The possible number of signals S j,j = 0, . . . , 15 is determined by the number of symbols in thesubchannel of interest, D0, and it depends on the symbols inthe adjacent subchannels D−1, D+1, and D+2.

Let us present the newly developed turbo equalizer hi-erarchically, in a way similar to the turbo equalizer shown inFig. 10, when M = 3. It should be a three-level structure inorder to satisfy the following condition: one level for eachICI path.

Let us assume that the first ICI source is informationsymbol D+2 that belongs to subchannel number +2. Thereare eight couples or groups of signals in (A· 1)–(A· 3) thatare different in information symbol D+2 = ±1. These are S 0,S 1; S 2, S 3; S 4, S 5; and S 6, S 7. Therefore “Level 0” of thenewly developed turbo equalizer consists of four non-linealblocks with transient functions F (x,∆E01), F (x,∆E23),F (x,∆E45), and F (x,∆E67). In order to maintain the sameoutput level after summing of non-linear blocks outputs,(level balance), the values equal to the inverse of the num-ber of non-linear blocks is a scale factor. Thus, for levelzero with four non-linear blocks, a scaling coefficient of 1/4is used.

The second source of ICI is information symbol D+1

that belongs to subchannel number +1. There are alsoeight couples or groups of signals in (A· 1)–(A· 3) thatare different in information symbol D+1 = ±1. These

are S 0, S 2; S 1, S 3; S 4, S 6; and S 5, S 7. Accordingto the structure proposed for level two, these signals aregrouped by two—the first group consists of S 0, S 2 and S 1,S 3, while the second group consists of S 4, S 6 and S 5, S 7.Therefore, the second level in the proposed turbo equalizerconsists of two non-lineal blocks with transient functionsF (x, (∆E02 + ∆E13)/2) and F (x, (∆E46 + ∆E57)/2). Thescaling factor equals 1/2 for level two. Here again, the firstnon-linear block, F (x, (∆E02 + ∆E13)/2), is the same as inFig. 10.

Finally, the last source of ICI is information sym-bol D−1 that belongs to subchannel number −1. Simi-larly to levels zero and one, there are also eight couplesof signals in (A· 1)–(A· 4) that are different in informa-tion symbol D−1 = ±1. These are S 0, S 4; S 1, S 5; S 2,S 6; and S 3, S 7. According to the structure proposed forlevel three, these signals represent only one group. There-fore the second level in the proposed turbo equalizer con-sists of only one non-lineal block with transient functionF (x, (∆E04 + ∆E15 + ∆E26 + ∆E37)/4). The scaling factorequals 1 for level 3. The hierarchically constructed struc-ture of the turbo equalizer for M = 3 is shown in Fig. 11.We obtained the same turbo equalizer structure by straightmathematical synthesis.

The proposed discussion can be extended for an arbi-trary M. Thus, the total number of signals after introduc-tion of ICI equals 2M. Therefore, at level zero, there are2M−1non-linear blocks combining couples of signals, andthe scaling factor for level zero is 1/2M−1. Further, at levelone, there are 2M−2non-linear blocks combining quadrupletsof signals, and the scaling factor for level zero is 1/2M−2.This continues until finally at level M we get a single non-linear block combining M/2 signals, and the scaling factorequals 1, which can obviously be omitted, but is still shownin Figs. 10 and 11 for the purposes of illustration.

Appendix B

When M = 2, from (2) and (3), the energy of signal S j (t) is

defined as E j =∫ T

0S 2

j (t)dt. Here, in order to simplify theexpressions, we assume that T = 1.

E0 = (1 + α−10 + α+10)2

E1 = (1 + α−10 − α+10)2

E2 = (1 − α−10 + α+10)2

E3 = (1 − α−10 − α+10)2

(A· 5)

Therefore, from (A· 5)

∆E01 = (1 + α−10 + α+10)2 − (1 + α−10 − α+10)2

= 4 · (1 + α−10) · α+10 (A· 6)

∆E23 = (1 − α−10 + α+10)2 − (1 − α−10 − α+10)2

= 4 · (1 + α−10) · α+10 (A· 7)

∆E02 = (1 + α−10 + α+10)2 − (1 − α−10 + α+10)2

= 4 · (1 + α+10) · α−10 (A· 8)

∆E13 = (1 + α−10 − α+10)2 − (1 − α−10 − α+10)2


= 4 · (1 − α+10) · α−10 (A· 9)

Finally the sum of (A· 6)+(A· 7) and (A· 8)+(A· 9) with 1/N0

scaling coefficient leads to (A· 10) and (A· 11), which areEqs. (25) and (26).

∆ELev0 = (∆E01 + ∆E23)/2 = 4 · α+10/N0 (A· 10)

∆ELev1 = ∆EΣ = (∆E02 + ∆E13)/2 = 4 · α−10/N0 (A· 11)

Appendix C

In this appendix we extend the proposed method into thequadrature modulation case. With the turbo equalizer struc-ture already defined, let’s consider two possible constella-tion schemes most widely utilized in OFDM transmission:BPSK and QPSK. When the system model and the struc-ture of turbo equalizer are fixed, the remained question ishow to define leakage coefficients αi j properly. Let’s startwith Eq. (3) where, for simplicity, we have omitted the noisecomponent.

D∗l (k) = αl,l(k) · Dl(k) +N−1∑n=0n�l

αl,n(k) · Dn(k) (A· 12)

Representing Eq. (A· 12) with a more expanded form of theICI components linear combination leads to:

D∗l (k) ≡ Dl(k) +1αl,l+

N−1∑n=0n�l

αl,n(k) · Dn(k)

= Dl(k) +αl,0

αl,lD0(k) +

αl,1

αl,lD1(k) + . . .

+αl,N−2

αl,lDN−2(k) +

αl,N−1

αl,lDN−1(k) (A· 13)

Note that in (A· 12), (A· 13) all coefficients αi j are com-plex numbers; meanwhile, Dl (k) can be any real or complexnumber depending on the type of the constellation imple-mented.BPSK Constellation:A BPSK signal takes one of two fixed phases (0, π) corre-sponding to the digitally modulating signal Dl. Thus BPSKsignal waveform consists only of the real part (non-complexwaveform) [12]. Therefore Dl (k) is a real number and thesystem model shown in Fig. 3 can be expressed as:

D∗l (k) = Re

Dl(k) +αl,0

αl,lD0(k) +

αl,1

αl,lD1(k) + . . .

+αl,N−2

αl,lDN−2(k) +

αl,N−1

αl,lDN−1

= Dl(k) + Re

αl,0

αl,l

D0(k) + Re

αl,1

αl,l

D1(k) + . . .

+Re

αl,N−2

αl,l

DN−2(k) + Re

αl,N−1

αl,l

DN−1

With the leakage coefficients in Figs. 3 and 4 defined as realnumbers as αi j ≡ Re

{αi j

αii

}.

QPSK Constellation:A QPSK signal phase can have one of four possible phases.Information symbol Dl (k) is a complex number and inde-pendent binary messages Re{Dl (k)} and Im{Dl (k)} modu-late the two-quadrature components of the carrier [12]. Withsome modifications (A· 13) leads to:

Re{D∗l (k)} = Re{Dl(k)} + Re

αl,0

αl,lD0(k) +

αl,1

αl,lD1(k)

+ . . . +αl,N−2

αl,lDN−2(k) +

αl,N−1

αl,lDN−1

(A· 14)

Im{D∗l (k)} = Im{Dl(k)} + Im

αl,0

αl,lD0(k) +

αl,1

αl,lD1(k)

+ . . . +αl,N−2

αl,lDN−2(k) +

αl,N−1

αl,lDN−1

(A· 15)

For simplicity, we will consider single first complexleakages coefficients in (A· 15) and (A· 16) with complex in-formation symbol D0 (k). Complex multiplication leads to:

Re

αl,0

αl,lD0(k)

= Re

{αl,0

αl,l

}· Re{D0(k)}

−Im

{αl,0

αl,l

}· Im{D0(k)} (A· 16)

Im

αl,0

αl,lD0(k)

= Im

{αl,0

αl,l

}· Re{D0(k)}

+Re

{αl,0

αl,l

}· Im{D0(k)} (A· 17)

Equations (A· 16) and (A· 17) show that for QPSK con-stellation, in addition to ICI, another type of interferencearises. This is cross quadrature interference. Similar resultscan be obtained for all other ICI components in (A· 14) and(A· 15). With the assumption that bits in Re and Im partsof information symbol Dl (k) are uncorrelated, the system

Fig. A· 2 Cross quadrature interference for QPSK.


models shown in Figs. 3 and 4 remain valid with only a dou-bled number of the leakage coefficients. These coefficientsstill are the real numbers (non-complex values) and definedas: α1i j ≡ Re

{αi j

αii

}and α2i j ≡ −Im

{αi, j

αi, i

}. Figure A· 2 shows

ICI model for QPSK constellation with Eqs. (A· 14)–(A· 16).Thus, in order for the system model to remain valid, it isnecessary to replace each interference path for each infor-mation symbol (source of ICI) shown in Figs. 3 and 4 withtwo ICI paths (for real and imaginary sources of ICI sepa-rately) as shown in Fig. A· 2. Therefore, for the quadraturemodulations like QPSK or M-QAM due to cross quadratureinterference, the total number of ICI sources increases to 2Mwith respect to M for the BPSK case. However, the internalstructure of the turbo equalizer remains the same (with thesubstitution of 2M) as shown in Figs. 13 and 14.

Alexander N. Lozhkin was born inMoscow, Russia on July 9, 1963. He re-ceived Radio-Engineering qualification in 1986,MS EE degrees from Moscow Aviation Insti-tute, Moscow, and Dr. Eng. Degree from Kyu-shu University, Fukuoka in 2000, respectively.Since 2000 he has been with Fujitsu Laborato-ries Ltd., Yokosuka YRP Center. He is currentlya researcher in Mobile Access group. His cur-rent research interests are Hi-speed mobile com-munication systems for B3G/4G and digital sig-

nal processing.

Mitsuhiro Azuma is currently a Director ofR&D Strategy and Planning Office, Fujitsu Lab-oratories Limited. He received his M.S. degreein Information Sciences and Electronics fromthe University of Tsukuba in 1981. From 1991to 1993, he worked as a visiting researcher atBell Communications Research (currently Tel-cordia), New Jersey, USA. He has been ac-tive in the field of network operation and man-agement including wireless communication net-works. He was a Vice Chairman of IEICE TM

Committee in Japan from 2001 to 2002. He was a delegate from Japan at-tending ITU-R Working Party 8F, where he involved with the calculationmethodology of spectrum demand for the systems beyond IMT-2000. Heis a co-translator of “Data Networks,” Ohmsha, 1987. He is a member ofIEEE.

Tomohiko Taniguchi is currently a Di-rector of Mobile Access Laboratory at FujitsuLaboratories Limited. He received his B.E. de-gree in electrical engineering from the Univer-sity of Tokyo in 1982. In 1987, he spent a year atStanford University, as a Visiting Scholar. From1996 to 2000, he worked for Fujitsu Laborato-ries of America, Sunnyvale, CA, U.S.A. He hasbeen active in the field of Signal Processing formore than 20 years. He was a member of SpeechTechnical Committee, a member of Multimedia

Signal Processing Technical Committee (IEEE SP Society), a Vice Chairof Signal Processing for Communications Electronics Technical Commit-tee (IEEE COMSOC), and on the Editorial Board of EURASIP Journal onApplied Signal Processing. He served as a Signal Processing SymposiumChair for ICC’02, Globecom’02, Globecom’03, and ICC’05. He receivedthe Young Engineer Award from the IEICE in 1985, meritorious patentaward in 1995 and 1998. He is a Fellow of IEEE.

Documents

A Low-Complexity Turbo Equalizer for OFDM Communication ... · 100 IEICE TRANS. COMMUN., VOL.E89–B, NO.1 JANUARY 2006 PAPER A Low-Complexity Turbo Equalizer for OFDM Communication