impulse noise removal ofdm

4602 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 11, NOVEMBER 2013

BER Reduction of OFDM BasedBroadband Communication Systems overMultipath Channels with Impulsive Noise

M. Mirahmadi, Member, IEEE, A. Al-Dweik, Senior Member, IEEE, and A. Shami, Senior Member, IEEE

AbstractThis paper presents an efficient technique to jointlymitigate the severe bit error rate (BER) performance degradationcaused by impulsive noise (IN) and multipath fading in broad-band transmission systems. The proposed system is based ona low complexity interleaving process applied after the inversefast Fourier transform (IFFT) in orthogonal frequency divisionmultiplexing (OFDM) systems, hence it is denoted as time-domaininterleaving (TDI). The proposed TDI introduces both time andfrequency diversity, which can be used to effectively combatimpairments such as IN and frequency-selective fading. Inaddition to its substantial BER reduction capability, the TDI doesnot degrade the spectral efficiency and has low computationalcomplexity. In frequency-selective fading channels, the BER ofthe proposed system is mathematically equal to that of Walsh-Hadamard precoded OFDM systems [1]. In presence of IN,analytical and simulation results show that TDI can remarkablyreduce the level of the error floors that are commonly observed.Specifically, TDI can achieve a BER of 105 for less than 1 dBdifference from the IN-free case.

Index TermsOFDM, interleaving, diversity, precoding, fad-ing, impulsive noise, Walsh-Hadamard.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) is a multicarrier modulation technique thatemploys orthogonal subcarriers. Due to the unique featuresof OFDM, it is currently adopted in several wirelesscommunication standards such as digital audio broadcasting(DAB) [2], digital video broadcasting-terrestrial (DVB-T) [3],worldwide interoperability for microwave access (WiMAX)technologies [4] and the 4G LTE-Advanced [5].

Besides its superiority for wireless transmission, OFDMhas rendered itself as the dominant modulation technique formany wired technologies and standards such as the secondgeneration digital video transmission over cable (DVB-C2)[6], the asymmetric digital subscriber line (ADSL) [7] andhome networking over power line communications (PLC) [8].For such applications, the main advantage of using OFDM is

Manuscript received March 26, 2013; revised August 21 and September28, 2013. The editor coordinating the review of this paper and approving itfor publication was A. Tonello.

M. Mirahmadi and A. Shami are with the Department of Electrical andComputer Engineering, Western University, London, Ontario, Canada (e-mail:{mmirahma, ashami2}@uwo.ca).

A. Al-Dweik is with the Department of Electrical and Computer Engineer-ing, Khalifa University, Abu Dhabi, UAE (e-mail: [email protected]). He isalso with the School of Engineering, University of Guelph, Guelph, Ontario,Canada (e-mail: [email protected]).

This work was conducted when A. Al-Dweik was a Visiting Professor atWestern University, London, ON, Canada.

Digital Object Identifier 10.1109/TCOMM.2013.102313.130220

the bandwidth efficiency and reduced system complexity as aresult of using the fast Fourier transform (FFT) and its inverseversion (IFFT).

However, for both wired and wireless transmission, thefrequency-selective fading remained one of the major factorsthat can cause severe bit error performance degradation toOFDM systems. Therefore, tremendous efforts have beendevoted in the literature to alleviate the effect of fading usingvarious techniques. Among many others, precoded OFDM [9]and modulation diversity techniques [10]- [11] have demon-strated significant robustness in severe frequency-selectivechannel.

In addition to the multipath propagation phenomenon,which is a major source of disturbance for several commu-nication systems, impulsive noise (IN) is another significantsource of disturbance for particular OFDM based applicationssuch as DVB-T [12][14], PLC [15] and ADSL systems [16].Unlike multipath fading, adopting OFDM for IN channelsmight impact the performance negatively [17], because animpulsive burst may destroy all subcarriers within the OFDMsymbol due to the averaging process of the FFT. Hence, usingOFDM for IN channels must be accompanied with effectiveIN mitigation techniques [18]. In the literature, the commonIN mitigation techniques used for single carrier systems suchas clipping and blanking are extended to OFDM systems [15].Approaches that are designed specifically for OFDM systemsare reported in the literature as well. For example, Abedlkefi etal. [19] exploited the pilots embedded within OFDM signalsto detect and correct the samples contaminated by IN. Zhidkov[17] proposed a frequency domain IN cancellation techniquefor DVB-T systems where the IN is estimated and subtractedafter the FFT.

In general, most of the aforementioned techniques sufferfrom limited performance improvement, particularly in heav-ily distributed IN channels. Moreover, most of the systemsreported in the literature assume that the IN bursts have ashort duration that is equal to the OFDM sample period, andhence, all IN samples can be considered uncorrelated [15][20]. Although such assumption is pivotal to enable analyticalperformance evaluation, it does not capture the bursty natureof the IN, which is confirmed by channel measurements forvarious applications [16], [21], [22], particularly for broadbandcommunications where the OFDM sample duration is veryshort [8]. For example, the OFDM symbol duration as definedby the IEEE 1901 PLC standard [8] is only 5.12 s. Hence,even very narrow bursts can affect several consecutive OFDM

0090-6778/13$31.00 c 2013 IEEE

MIRAHMADI et al.: BER REDUCTION OF OFDM BASED BROADBAND COMMUNICATION SYSTEMS OVER MULTIPATH CHANNELS WITH IMPULSIVE . . . 4603

samples [21], or even the entire OFDM symbol. Consequently,most of the techniques that are designed based on the as-sumption of uncorrelated samples might not be effective inthe presence of IN bursts.

In [23], Al-Dweik et al. proposed a new OFDM symbolstructure to combat IN by interleaving the time-domain sam-ples after the IFFT process. The technique, namely Time-Domain Interleaving (TDI), was then extended in [24] toinclude multipath fading channels. However, the results in [23]and [24] were entirely obtained via Monte Carlo simulationusing the assumption of ideal IN samples detection and blank-ing. It is worth noting that both TDI and the system reported in[11] interleave a block of OFDM symbols before transmission,however the two systems are fundamentally different. Thesystem described in [11] jointly modulates all symbols in theblock and then interleaves the modulated symbols, whereas theTDI modulates the information bits conventionally, applies theIFFT and then interleaves the block.

This paper presents a novel technique, based on TDI [23], inconjunction with a two-level threshold-based blanking schemeto combat the adverse effects of multipath propagation aswell as IN for OFDM based communications systems. Unliketraditional interleaved single carrier and OFDM systems wherethe information symbols are spread over a larger number oftransmission blocks [25], the TDI system interleaves the timedomain samples after the IFFT which are composed of amixture of all information symbols. This results in a significantimprovement in uncoded BER that can never be achieved withconventional interleaving. To the best of our knowledge, theproposed TDI has never been considered in the literature. Inaddition, unlike the traditional IN mitigation techniques, theuse of the proposed blanking scheme in conjunction with TDIenables the proposed system to efficiently combat IN evenin frequency-selective channels. Closed-form formulae arederived for the signal-to-interference and noise ratio (SINR)in frequency-selective fading channels using zero forcing (ZF)and minimum mean squared error (MMSE) equalizers. Analyt-ical and simulation results reveal that the proposed TDI systemoffers a remarkable BER performance improvement and caneffectively and jointly combat the IN bursts and multipathfading. It is worth noting that in frequency-selective fadingchannels without IN, TDI can achieve the same BER as Walsh-Hadamard precoded OFDM systems [1], [9]. In addition, itsignificantly outperforms such systems in the presence of IN.

The rest of this paper is organized as follows. The systemand channel models are presented in Section II. The proposedsystem model is described in Section III. Section IV presentsthe symbol blanking scheme suggested to mitigate the INeffects. The system performance analysis in fading channelsusing ZF and MMSE equalizers are provided in Sections Vand VI, respectively. The numerical results are given in SectionVII, and finally Section VIII concludes the paper.Notations : In what follows unless otherwise specified,

uppercase boldface and blackboard letters, such as H andH, will denote N N matrices, whereas lowercase boldfaceletters, such as x, will denote row or column vectors withN elements. Samples or data symbols from the th OFDMsymbol will be denoted using lower case letters such as y.Symbols with a hat such as x will denote the estimate of x,

and symbols with breve x denotes an interleaved sequence.Moreover, x N , Nc, U and B will denote that randomvariable x follows the normal, complex normal, uniform orBernoulli distribution, respectively. The expectation, Hermi-tian transpose and conjugation are denoted as E [], []H and[].

II. OFDM SYSTEM AND CHANNEL MODELSA. OFDM System Model

Consider an OFDM system with N subcarriers modulatedby a sequence of N complex data symbols d = [d0, d1, ....,dN1]T . The data symbols are selected uniformly from a par-ticular constellation such as quadrature amplitude modulation(QAM). The modulation process can be implemented effi-ciently using N -points IFFT, where the IFFT output during theth OFDM block x() = FHd(). Matrix F is the normalizedFFT matrix whose elements are defined as Fi,k = eik, = 1/

N , = j2/N , j =

1 and {i, k} = 0, 1, ...,N1 denote the row and column numbers, respectively. SinceF is a unitary matrix, then FH = F1. Consequently, the nthsample in x can be expressed as

xn() = N1

i=0di()e

in, n = 0, 1, ..., N 1. (1)

To eliminate the inter-symbol-interference (ISI) between con-secutive OFDM symbols in frequency-selective multipath fad-ing channels, a cyclic prefix (CP) of length N samples no lessthan the normalized channel delay spread (Lh) is formed bycopying the last N samples of x and appending them at thefront of x to compose the transmission symbol with a totallength Nt = N + N samples and a duration of Tt seconds.Hence, the complex baseband symbol transmitted during theth signaling period can be expressed as

x() = [xNN (), xNN+1(),..., xN1(),

x0() , x1(), . . . , xN1()]T . (2)Consequently, the ith transmitted sample xi() = xiN(),i = 0, 1, ..., Nt1, where i i mod N . Then, the sequencex is upsampled, filtered and up-converted to a radio frequencycentered at fc.

At the receiver frontend, the received signal is down-converted to baseband and sampled at a rate Ts = Tt/Nt.In this work, we assume that the channel is composed ofLh +1 independent multipath components each of which hasa gain hm and delay m Ts, where m {0, 1,..., N}.The channel taps are assumed to be constant over N OFDMsymbols, which corresponds to quasi-static multipath fadingchannels. The received sequence y = H()x()+z(), wherethe channel matrix H is an Nt Nt Toeplitz matrix withh0 on the principal diagonal and h1,..., hLh on the minordiagonals, respectively [27] and z() represents the overalladditive noise that includes AWGN and IN. For notationssimplicity, we omit the subscript in the remaining parts ofthis section. Given that Lh < N , the nth sample of y can beexpressed as yn =

Lhi=0 hi xniN+zn. Subsequently, the

receiver should identify and extract the sequence y = [yN ,yN+1, . . . , yN+N1] and discard the first N CP samples [y0,y1, . . . , yN1]. Therefore,

yn =Lh

i=0hi xni + zn+N , 0 n N 1. (3)


In vector notation, the sequence y can be expressed asy = Hx+ z (4)

where z = [zN , zN+1, , zN+N1]T , and the channel matrixH is circulant [27], [28]. It is worth noting that the quasi-staticchannel model, and hence the proposed system can be adoptedfor several broadband OFDM systems where the channelchanges at a rate lower than the interleaved block length.Examples for such system include the DVB-T [12]- [14],PLC [15] and ADSL systems [16]. For OFDM systems wherethe channel might change faster than the interleaver depth,higher complexity solutions such as time-domain equalization,or interleaver depth optimization might be invoked.

B. Impulsive Noise ModelThe IN is usually characterized by bursts of one or more

short pulses whose amplitude g, duration TI , and time ofoccurrence are all random parameters. The main models usedfor the amplitude distribution are the Middleton class-A [29],exponential [21] and Gaussian [30]. The common distributionsof the bursts arrival process are partitioned Markov chains(PMC) [21], Gated Bernoulli-Gaussian (GBG) [30], and Pois-son [22]. The burst width distribution is usually modeled usingthe PMC or as the sum of two lognormal random variables[16].

To enable analytical performance evaluation over IN chan-nels, the GBG model is widely adopted in the literature wherethe IN is modeled as a sequence of independent, Bernoullidistributed bursts with equal and fixed width, which is equalto the duration of a single time-domain OFDM sample [18],[19], [30]. Hence, the overall noise samples at the receiverfrontend zn = wn+ngn, where the AWGN wn Nc

(0, 2w

)and 2w = E

[|wn|2] = N0/2. The IN component ngn ismodeled as a gated Gaussian process, and n B {0, 1}where p = P{n = 1}, and gn Nc

(0, 2g

)where 2g 2w

[18].In this work, the GBG model is adopted to represent the IN,

which is modeled as a sequence of independent bursts eachof which consists of 1 N + N pulses. In general,the burst width is a random variable whose distributioncan be modeled as a uniform or lognormal distribution, or byusing PMC [16]. Therefore, the overall received noise z =w + bg, where z = [z0, z1,..., zN+N1]. The vector b isused to specify the position as well as the width of the INburst with respect to the OFDM symbol. Given that n0 is theindex of the first OFDM sample that is affected by a noiseburst, then the elements of b are

bn =

{1, n0 n < n0 + 0, otherwise , (5)

where n0 U[0, N + N ]. Therefore, the location of

the noise burst will be random and uniformly distributed overthe OFDM symbol period. Thus, any IN burst can affect amaximum of one OFDM symbol.

III. PROPOSED SYSTEM MODEL

The proposed system is similar to the conventional OFDMsystem described in Section II. However as depicted in Fig.

Fig. 1. The proposed TDI system in fading channels.

1, after the IFFT, N OFDM symbols are interleaved using asimple row/column interleaver,

write

read

x0(0) x0(1) x0(N 1)x1(0) x1(1) x1(N 1).

.

.

.

.

.

xN1(0) xN1(1) xN1(N 1). (6)

It is worth noting that using the block interleaver representedin (6) simplifies the analysis, but the delay of a pair ofsuch interleaver/deinterleaver is equal to 2N OFDM symbols.Alternatively, convolutional interleaving can offer the sameBER performance with only half of that delay [32, p. 467].

The interleaver produces N interleaved symbols{x(0), x(1), ..., x(N 1)} where

x() =[x(0) x(1) x(N 1)

]T. (7)

The CP samples are formed by copying the last N samplesof x and appending them at the front of x to compose thetransmission symbol. Hence, the complex baseband symboltransmitted during the th signaling period can be expressedas

x() = [x(N N), x(N N + 1),..., x(N 1),x(0) , x(1), . . . , x(N 1)]T . (8)

Consequently, the ith transmitted sample isxi() = x(

i N), i = 0, 1, ..., Nt 1. The

remaining transmission and reception processes aresimilar to conventional OFDM systems. Therefore,the received sequence y() =H()x()+z(), whereyn =

Lhi=0 hi x(

n i N) + zn. Subsequently, after

discarding the first N CP samples, the remaining samplescan be expressed as

yn =

Lhi=0

hi x(n i) + zn+N , 0 n N 1. (9)

In vector notation y = Hx+ z. To simplify the discussion,assume initially that the channel matrix H is the identitymatrix. Therefore, the received N samples, after removing theCP, can be written as

y() = x() +w() + ()b()g(). (10)By noting that the deinterleaving process interleaves thevectors w(), b() and g(), the deinterleaving process yields

y() = x() + w() + ()b ()g() (11)where w and g are the interleaved AWGN and IN vectors.The burst and sample gating factors and b have differentproperties from the original and b. For example, giventhat (0) = 1 and = Nt, then () = b0() = 1 {1, 2, ...N}, i.e., the first sample of every OFDM symbol


after deinterleaving will be affected by an IN pulse, while allother samples in all symbols will be IN free. Consequently,the FFT output can be expressed as

rk() = N1

n=0yn()e

nk, k = 0, 1, ..., N 1 (12)

which can be reduced tork() = dk() + k() + uk() (13)

where the FFT of the AWGN k Nc(0, 2w

). The last term

in (13) represents the FFT of the INuk() = ()

N1n=0

bn()gn()enk

where uk Nc(0, 2u

), and

2u = 2()2g

N1n=0

bn()

= 2()2g() (14)where = is the number of nonzero elements in the vectorb(), i.e., the Hamming weight of b().

It can be concluded from (13) and (14) that the deinterleav-ing process spreads the IN burst over most OFDM symbolswithin the deinterleaved block. The FFT process applied afterthe deinterleaving averages the IN pulses over all subcarrierswithin a given OFDM symbol, which may cause the loss of upto OFDM symbols because 2g 2w [18], [21], [38]. It isalso worth noting that the interleaving process described in (6)does not affect the peak-to-average power ratio (PAPR) of thetransmitted signal because the interleaver changes the order ofthe transmitted samples without changing their values.

IV. TDI SYMBOL BLANKINGAn efficient solution to mitigate the effects of impulse noise

is to apply blanking [15], [17], where the received sampleswith high amplitudes are set to zero. The contaminated sam-ples are detected and suppressed by comparing the receivedsamples values with a particular threshold T1. Therefore, theoutput of the blanking nonlinearity is qn = yn |yn| T1 and0 otherwise. We refer to this approach as sample-by-sampleblanking. After blanking and deinterleaving, the nth sampleof the FFT input qn = yn |yn| T1 and 0 otherwise,where yn = xn + wn + bn gn. Therefore, the FFT outputs() = Fq() and the kth subcarrier can be written as

sk() = N1

n=0qn()e

nk, k = 0, 1, ..., N 1

=

nC yn()enk (15)

where C = {n {0, 1, ..., N 1}|qn = 0}. The average errorprobability can be expressed as

Pe =

C P (e|C)P (C). (16)The sample blanking threshold T1 should be selected to

minimize the BER, and it is a function of several variablessuch as signal-to-noise ratio (SNR), signal-to-IN ratio (SIR)and [15]. It is worth noting that the blanking process isnot ideal in the sense that it will not necessarily blank all INsamples, and may blank some information samples, which isdue to the high peak-to-average-power-ratio (PAPR) probleminherent in OFDM systems [31]. In addition, the sample-by-sample blanking is not feasible in frequency-selective channelsdue to the vast amplitude fluctuations that a signal mayexperience in such channels [19].

In our study we observed that despite the fact that detectingcorrupted samples is very challenging, the corrupted symbolscan be easily and accurately identified by introducing anadditional threshold T2 for the blanking process, where T2denotes the number of samples with amplitudes larger thanT1. In each symbol, if the number of samples with |yn| T1exceeds T2, the entire symbol is considered corrupted and isblanked subsequently. i.e.,

q =

{0 y n 1{|yn|>T1} > T2y otherwise (17)

This approach is called symbol blanking for the rest of thepaper. It is worth mentioning that the time diversity providedby the TDI interleaving mechanism enables us to recoverthe blanked symbol and allows symbol blanking without anysignificant loss.

It is worth noting the symbol blanking substantially relaxesthe system sensitivity to T1, which is difficult to estimateaccurately in conventional OFDM systems [20], [39].

Based on the widely used assumption that 2g 2w [18],[21], [38], the symbol blanking process is expected to behighly accurate. By assuming a perfect burst detection process,the average probability of error over a block of N symbolscan be expressed as

Pe =N1

=0P (e|)P () (18)

where denotes the total number of IN bursts per block of NOFDM symbols, which has a binomial PDF

P ( = i) =

(N

i

)pi(1 p)Ni (19)

where p is the probability of the occurrence of IN duringeach symbol period. The exact value of p can be measured orextracted from the selected IN model.

It is worth mentioning that the analyses in the next sectionsare valid for any IN model that considers IN bursts longer thanthe sample period. Since symbol blanking technique blocksthe corrupted symbols, the exact amplitude of the IN is not acontributing factor in the performance, as long as the symbolsthat are hit by IN can be correctly detected. To ensure thatthis condition remains true, the thresholds T1 and T2 have tobe carefully adjusted.

V. SYSTEM PERFORMANCE USING ZF EQUALIZERAs it can be noted from (18), the probability of error is

computed as the average of all possible symbol blankingscenarios. A particular case of interest is the one where = 0,which corresponds to an IN-free frequency-selective fadingchannel. Therefore, we first consider the case of = 0, thenthe results are generalized for > 0. The TDI system blockdiagram is given in Fig. 1 for notation clarification.

A. No Samples Blanked, = 0The receiver design can be performed by noting that H =

FHHF [40], where

H = diag ([H0, H1, , HN1]) , Hk =Lhm=0

hmemk .

Thus (4) can be written asy = FHHFx+w. (20)


Therefore, the ZF equalizer output can be written as s =FHH1Fy = x+Vw, where V FHH1F. However, sinceH1 is a diagonal matrix,

H1 = diag([

1

H0,

1

H1, , 1

HN1

])(21)

and matrix V is circulant, the first row of V, using the Matlabnotations, is given by

V(0, :)=2

N1k=0

1HkN1

k=0ekHk.

.

.N1k=0

e(N1)kHk

T

. (22)

It can be noted from (22) that each element in V is composedof a mixture of all elements of H1. Specifically, each rowin V consists of the FFT of the vector

[1H0

,1H1

, , 1HN1].

Consequently, the term Vw is given by

=

N1

i=0 wiiN1i=0 wii1

.

.

.N1i=0 wii+1

(23)

where i =N1

k=0ekiHk

. After deinterleaving, the thOFDM symbol s()= x() +(), where x() = FHd()and is formed by equalizing and deinterleaving the additivenoise,

() =

N1

i=0 wi(0)+i(0)N1i=0 wi(1)+i(1)

.

.

.N1i=0 wi(N 1)+i(N 1)

. (24)

The FFT is then applied to extract the decision variablesr() = d() + F().

As an example, for the case of = 0 , the noise componentcan be described as

F(0) = 2

N1

j,i=0 wi(j)i(j)N1j,i=0 wi(j)i(j)e

j.

.

.N1j,i=0 wi(j)i(j)e

(N1)j

. (25)

Since all samples are identically distributed, without loss ofgenerality, we consider the first subcarrier of the first OFDMsymbol, which is given by

r0(0) = d0(0) + 2N1j=0

N1i=0

wi(j)i(j). (26)

Substituting the value of i and noting that Hk() = Hk for the considered quasi-static channel, then (26) can bewritten as

r0(0) = d0(0) + 2N1j=0

N1i=0

N1k=0

eki

Hkwi(j). (27)

By comparing (27) to the standard OFDM, which has

rstd0 = d0 +2

H0

N1i=0

wi. (28)

It can be noted that the channel frequency response at each

subcarrier in the TDI system is a mixture of the channelfrequency responses Hi of the entire interleaving block length,which is caused by the deinterleaving and the last FFToperations.

The conditional SNR for a given channel matrix H can beexpressed as

SNR |H =E{|d0|2

}E

{2N1i=0 N1j=0 wi(j)i2} . (29)

Since E {wi(j)wv(k)} = 0 for j = k or i = v, thedenominator of (29) can be simplified as follows,

E

2

N1i=0

N1j=0

wi(j)i

2

= 4N1i=0

N1i=0

ii

N1j=0

N1j=0

E {wi(j)wi (j)}

= 22w

N1i=0

|i|2 . (30)

It is worth noting that (30) can be obtained using Parsevalstheorem as well,

E

2w

N1i=0

N1j=0

zi(j)i

2 = 22w

N1k=0

1

|Hk|2. (31)

Therefore, the SNR using a ZF equalizer is given by

SNR |H = 2d

22wN1

k=01

|Hk|2(32)

where E{|d0|2

}= 2d . By denoting 1 = 2

N1k=0

1|Hk|2 ,

thenSNR |H =

1= (33)

where 1 = 1/ and = 2d/2w. It is interesting to notethat (32) and [1, Eq. (61)] are identical, and (32) is similarto [34, Eq. 17] where the block length M = N , [9, Eq. 24]for the case of Li = 1, and with [42, Eq. 2] for tk,i = 1.Furthermore, it can be noted from (32) that if any subcarriergoes through a deep fade, i.e. k, |Hk|2 0, then 1/ |Hk|2 , and hence SNR 0. Therefore, ZF equalizer is expectedto offer poor BER in frequency-selective fading channels evenwithout IN.

Given that the PDF f() is known, the BER can beexpressed as

PZF () =

0

Q () f() d. (34)

Note that is composed of the sum of N correlated randomvariables each of which has the form |Hk|2. If the channelfrequency response parameters {H0,..., HN1} are i.i.d. RV,then evaluating f() might be feasible for particular sce-narios. For example, if we assume that Hi Nc(0, 1), then|Hi|2 has a Chi-square 2 PDF with two degrees of freedom,and |Hi|2 has an inverse-2 PDF. Since the inverse2 PDFcan be expressed in terms of the inverse Gamma PDF, thePDF of the sum

i |Hi|2 can be evaluated as described

in [33]. However, since Hk are correlated, evaluating f()


analytically is difficult. Hence, semi-analytical solutions canbe incorporated [9], [34], [42]. In addition, the work of Wangand Giannakis [35] states that systems with instantaneousSNR similar to (33) will not have diversity gain if lim 0+f() = c > 0. Moreover, McCloud [34] demonstratedthat f() is bounded away from zero as 0+. Thisresult proves that the proposed interleaved system with ZFequalizer has no diversity gain. Nevertheless, it provides aBER advantage in high SNRs that is very challenging toquantify analytically. Consequently, Monte-Carlo simulationscan be invoked to evaluate the system performance. Besides,it can be clearly noticed from (32) that SNR |H 0 if anyof the subcarriers goes into a deep fade.

B. Samples Blanked, > 0In the process of OFDM symbol blanking, a sequence of

blanked and non-blanked symbols is generated. Therefore, theinput of the deinterleaver can be written as

s() =

{0 y(), blanked symbolx() + (), otherwise

where () is defined in (23). The deinterleaving processrearranges the N OFDM symbols back to their original orderbefore interleaving, which yields s() = BFHd()+B(),where B is a matrix that specifies the blanking process. If nosymbol is blanked, then B is the identity matrix. Each of thesamples to be blanked is nulled by setting the correspondingmain diagonal element to zero. The received signal is thenachieved by computing the FFT of s,

r() = FBFHd() + FB(). (35)Using the Appendix A, the first term in (35), v FBFHd()can be calculated for a particular B. Thus, the kth sample ofv is given by,

vk(|) = dk + 2N1i=0i=k

disin

[2(k i)]

sin [2(i k)] ej N (+1)(ik)

(36)where = NN and = N tr(B) represents the number ofblanked symbols and tr() denotes the trace.

The second term in (35), can be calculated for the = 0case as follows,

FB(0) = 3

N1j,i,k=0 bj

wi(j)Hk(j)

e(ik+0 j)N1j,i,k=0 bj

wi(j)Hk(j)

e(ik+j)

.

.

.N1j,i,k=0 bj

wi(j)Hk(j)

e(ik+Nj)

where bj = B(j, j). The assumption that = 0 is used fornotational simplicity. Following the same approach describedin previous subsection we obtain

rk = dk + 2N1i=0i=k

disin

[2(k i)]

sin [2(i k)] ej N (+1)(ik)

+ 3N1j=0

bj

N1i=0

wi(j)

N1k=0

eik

Hk(j)(37)

where the second and third terms in (37) represent the ICI andadditive noise, respectively. As depicted in the Appendix A,

the noise variance is given by

2w = E

3

N1j=0

bj

N1i=0

wi(j)

N1k=0

ewik

Hk(j)

2

= 22w

N1k=0

1

|Hk|2(38)

2w = 22w represents the variance of the AWGN given that

samples are blanked, and the ICI variance is

2ICI,k = 42d

N1i=0i=k

sin2[2(k i)]

sin2 [2(i k)] . (39)

Therefore, the SINR for the kth subcarrier can be calculatedas

SINR |H,,k = 2 2d

2w + 2ICI,k

. (40)

Since all subcarriers will experience the same SINR, thesubcarrier index k can be set to zero and dropped from (41)without loss of generality. Substituting (39) into (41) andsimplifying the results gives

SINR |H, =[2

N1i=0

1

|Hi|2+

4

2

N1i=1

sin2(2i

)sin2 (2i)

]1.

(41)The BER can be calculated by substituting (41) into the wellknown QPSK error formula,

P (e|,H) = Q(

SINR |H,)

. (42)Therefore,

P (e|H) =N1=0

Q

(SINR |H,

)(N

i

)p(1p)N. (43)

The conditional BER given in (41) can be used to compute theaverage BER semi-analytically by generating a large numberof realizations for H.

VI. SYSTEM PERFORMANCE IN FADING CHANNELS(MMSE)

A. No Samples are Blanked, = 0

The typical approach to mitigate the performance degra-dation of ZF equalizers is to use MMSE equalizers, wherethe output signal of such equalizers can be calculated ass = [HH+I]1Hy, where = 1 for optimum errorperformance. Considering that fact that H = FHHF andnoting that the input samples to the equalizer are interleaved,the equalizer output can be written as

s = FH [HH+I]1HFy. (44)By noting that y = FHHFx+w, then (44) is simplified to

s = FH[|H|2+I

]1|H|2Fx

+ FH[|H|2 +I

]1HFw. (45)

Because H,[|H|2+I

]1|H|2 and

[|H|2+I

]1H

are all NN diagonal matrices, the equalizer output s can


be written ass = FHFx+ FH Fw (46)

wherek =

|Hk|2|Hk|2 +

, k =Hk

|Hk|2 + (47)

It is clear that when = 0, then FHF = IN andFH F = H1, which implies that the MMSE equalizerbecomes ZF equalizer. For > 0, s will have some sort ofinter-symbol interference (ISI) because A is not a diagonalmatrix. Consequently, a compromise has to be made betweenthe degradation resulted from the ISI and noise-enhancementproblem. Obviously, setting = 1/ yields the conventionalMMSE equalizer, which is widely used in the literature [36].

The FFT output for the proposed system can be derivedusing Appendix B. The kth sample of the FFT output is givenby

rk() = 2N1m=0

N1i=0

xm(i)iekm

+ 2N1m=0

N1i=0

wi(m)iekm

=

N1i=0

idk(i) + 2N1i=0

N1m=0

iwi(m)ekm.

(48)Since all the elements of r are identically distributed, forsimplicity, we consider the first FFT-pin output of the firstOFDM symbol,

r0(0) = d0(0)0 +

N1i=1

id0(i) + 2N1i=0

N1m=0

iwi(m).

(49)The first term in (49) shows the desired information symbol,whereas the ISI is characterized by the second term and finally,the third term gives the noise component of the received signal.The SINR for a given channel matrix can be expressed as

SINR |H,k=0 = E{|d0(0)0|2

}

E

N1i=1

id0(i) + 2N1i=0

N1m=0

iwi(m)

21

.

(50)Since the noise and ISI terms are independent, it is possibleto separate the denominator terms and rewrite (50) as

SINR |H,k=0 = E{|d0(0)0|2

}(E

{N1i=1 id0(i)2}

+E

{2N1i=0 N1m=0 iwi(m)2})

. (51)

The expected value of the desired signal can be reduced toE{|d0(0)0|2

}= 2E

{|d0(0)|2

}|0|2

= 22d |0|2 . (52)The denominator of (51) is composed of two parts, 1 and 2.

Since E {d0(i)d0(i)} = 0 for i = i, and E{|d0(i)|2

}= 2d,

then 1 and 2 are reduced to,

1 E

N1i=1

id0(i)

2 = 22d

N1i=1

|i|2 . (53)

And similarly,

2 4E

N1i=0

N1m=0

iwi(m)

2 = 22w

N1i=0

i2 .(54)

Following the same approach used for the ZF equalizer, andinserting (53) and (54) into (51) gives

SINR |H,k=0 = 2d |0|2

2dN1

i=1 |i|2 + 2wN1

i=0

i2 . (55)By substituting the values of 0, i and i into (55), usingthe fact that

N1i=0

ei(kn)HkHn

= 0, k = n, and noting thatall subcarriers will experience the same SINR, then SINR atthe FFT output can be expressed as

SINR |H = N1

k=0 kN1k=0

k|Hk|2

. (56)

The SINR given in (56) is equal to SINR of the WHT-OFDMsystem [9], [1, Eq. (68)]. Consequently, the two systems willhave equal BER in fading channels. The BER of the proposedsystem with MMSE can be calculated semi-analytically as

PMMSE() =1

L

Li=1

Q(

SINR |Hi)

(57)

where Hi is the ith realization of H and L is the total numberof realizations.

From the definition of k (47), it is evident that unlessthe subcarrier k is in deep fade, is very close to 1. Thus,the summation

N1k=0 k in (56) is generally comparable to

2d22w

= N in (32). On the other hand, the denominatorin (32) is dominated by the subcarriers in the deep fade.i.e., if any of the subcarriers Hk 0, the summationN1

k=01

|Hk|2 , resulting in very low SNR. This effect hasbeen evaded in MMSE since

N1k=0

k|Hk|2 is bounded away

from infinity. This leads to substantially higher SNR than TDIwith ZF equalizer and diversity advantages in the operationalSNRs [37].

B. Samples Blanked, > 0Similar to the ZF case, some OFDM symbols are blanked if

they are declared as contaminated by the IN bursts. Therefore,the equalizer output can be written as

s() =

{0 y, blanked symbol() + (), otherwise (58)

where and are defined in (83), and (81), respectively.The deinterleaver output s() = B()+B() and the FFToutput r() = Fs() = FB() + FB(), where

rk() = 2N1n=0

bn

N1i=0

xn(i)iN enk

vk()


+ 2N1n=0

bn

N1i=0

wi(n)iN enk

uk()

. (59)

The first term in (59) vk() can be written asvk() =

2N1

i=0iN

N1n=0

bnxn(i)enk

= N1

i=0iN

( dk(i)

+ 2N1

n=0n=k

dn(i)sin

[2(k n)]

sin [2(n k)] e2 (+1)(nk)

).

(60)To simplify the calculations, we consider 0(0), which is equalto

0(0) = 0d0(0) +

N1i=1

id0(i)

+ 3N1i=0

i

N1n=1

dn(i)sin

[2n

]sin [2n]

ejN (+1)n. (61)

In (61), the first term includes the desired signal and thesecond term is the interference caused by the MMSE equalizer,whose power is denoted as 2I,MMSE . The last term is theinterference caused by losing the blanked samples fromthe OFDM symbol, which has a power of 2I,Blank. Theequalization and blanking interference power can be calculatedas

2I,MMSE = E

N1i=1

id0(i)

2

= 222d

N1i=1

|i|2 . (62)

and

2I,Blank = E

{3N1i=0

i

N1k=1

dk(i)

sin[2k

]sin [2k]

e2 (+1)k

2

= 62dN1

i=0|i|2 , (63)

where N1

k=1

sin2(2k)sin2(2k)

. The second term in (59),uk(), describes the noise component of the received signal,

uk() = 2N1n=0

bn

N1i=0

wi(n)iN enk

= 2N1i=0

iN

N1n=0

bnwi(n)enk.

The variance of the noise component is

2u = E

2

N1i=0

iN

N1n=0

bnwi(n)enk

2

= 22w

N1i=0

i2 . (64)

Note that the second summation in (64) is just the FFT ofthe noise samples with samples missing. Consequently, theconditional SINR is calculated as

SINR |H, = 22 20

2d

2w + 2I,MMSE +

2I,Blank

. (65)

After some mathematical manipulations (65) can be writtenas

SINR |H, =

k k1

k

k|Hk|2 +

2

2

k 2k

k k

,

k = 0, 1, ...N 1. (66)The BER can be computed semi-analytically as described in(43).

In practical OFDM systems, accurate channel estimation,symbol timing and frequency synchronization are necessaryto detect the information symbols. Therefore, every trans-mitted data frame is usually preceded by a preamble of afew OFDM training symbols. Moreover, some subcarriersare also reserved as pilots to assist channel estimation andsynchronization. The number and distribution of the pilotsdepend on the nature of the channel [3], [4]. For the TDI,the received signal described in (20) clearly implies that thechannel matrix H should be estimated and compensated beforethe deinterleaving process. Consequently, if typical channelestimation and synchronization techniques are to be used, thetraining OFDM symbols within the frame preamble shouldnot be interleaved. Moreover, for various OFDM systems thatdoes not involve mobility such as PLC, the channel can beconsidered relatively fixed for the entire frame period [43]. Ifthe channel or synchronization parameters need to be updatedmore frequently, then time-domain [44] or blind techniquescan be invoked [45].

VII. NUMERICAL RESULTS

Monte Carlo simulations are used to evaluate the per-formance of the proposed system over frequency-selectivemultipath fading channels. The OFDM system considered inthis paper has N = 128 and N = 16. All data symbolsare QPSK modulated. The channel is assumed to be time-invariant throughout the duration of each interleaving block.In addition, full channel state information and perfect synchro-nization are assumed throughout this work. Each simulationrun consists of 2.56 106 independent OFDM symbols.The multipath fading channel model considered in this workcorresponds to a Rayleigh frequency-selective channel withnormalized delays of [0, 1, 2, 3, 4] samples and average gains[0.35, 0.25, 0.18, 0.12, 0.1]. The IN is modeled as GBG pro-cess, where the bursts position is uniformly distributed withinthe OFDM symbol period. Unless it is specified otherwise,the burst gating factor probability p = 0.01, the burst width = (N + N)/2, the signal-to-IN ratio SIR = 20 dB andSNR .

To demonstrate the effect of TDI on the decision variablesr at the FFT output, the magnitude of the noise term in eachsubcarrier is presented for a particular channel realizationas shown in Fig. 2. The aggregate noise and interference(AGNI) output at the kth subcarrier is computed as |rk dk|2.The AGNI is selected because it clearly reveals the noise


30 40 50 60 70 80Subcarrier No.

Agg

rega

ted

Out

put N

oise

and

Inte

rfere

nce

TDIZFTDIMMSEOFDMCh. Res.

Fig. 2. Noise enhancement of the TDI and OFDM systems with ZF andMMSE equalizers for SNR = 15 dB.

enhancement problem caused by the ZF equalizer. As it canbe observed from the figure, the channel has two nulls atsubcarriers 50 to 56 and 64 to 69. As expected, the division bythe relevant Hk in conventional OFDM will cause severe noiseenhancement for the subcarriers with deep fading as shown inFig. 2. The ZF equalizer and the last FFT mixes the channelfrequency response (CFR) of all subcarriers, which mainlywill deteriorate the BER performance as the enhanced noiseis distributed over all subcarriers. Finally, the MMSE does notsuffer from any noticeable noise enhancement problem due tothe bias added to Hk before the division process, which provesthat the CFR mixing process is advantageous and will causeBER improvement.

Fig. 3 presents the effective SINR of the TDI and OFDMsystems versus the SNR , which measures the SNR degra-dation caused by both, fading and blanking. The immunityof the TDI to fading can be observed from the = 0 casewhere the SINR of the TDI is about 5 dB higher than theconventional OFDM. It is evident that OFDM suffers fromlosing some samples, where > 0, more than TDI whichleads to the increase in the SINR gap to approximately 7 and7.5 dB for = 2 and 1, respectively. For 3, the effectiveSINR becomes very low, which decreases the SINR differenceto about 4 dB. In other words, despite OFDM that is degradedeven with = 1, TDI can successfully resist the adverse effectof losing samples up to = 3, where its slope becomes equalto that of OFDM.

The BER performance of IN-free ( = 0) TDI, WHT-OFDM and conventional OFDM systems using ZF and MMSEequalizers is given in Fig. 4. As it can be noted from thefigure, the lowest BER is achieved by the TDI/WHT usingMMSE, whereas the TDI/WHT ZF gives the worst BER.The OFDM inherent fading-resistance gives it a mediocreperformance. Since OFDM transmits the signal in orthogonalnarrow frequency band channels, both equalizers give the same

0 5 10 15 20 25 30 352

0

2

4

6

8

10

12

14

16

SNR (dB)

Effe

ctiv

e SI

NR

(dB)

= 0

= 1

= 2

= 3

TDI

OFDM

Fig. 3. Effective SINR of TDI and OFDM for different values of .

0 10 20 30 40107

106

105

104

103

102

101

100

SNR (dB)

BER

TDIZF Theo.TDIZF Sim.TDIMMSE Theo.TDIMMSE Sim.OFDMZF Sim.OFDMMMSE Sim.WHTMMSE Sim.WHTZF Sim.

Fig. 4. BER of the TDI and conventional OFDM using ZF and MMSEequalizers.

BER; a result which is expected. Moreover, the simulationresults in this figure corroborate with the analysis. In addition,the BER curves of WHT-OFDM with MMSE and TDI-MMSEmatch, which indicates that TDI-MMSE and WHT-OFDMusing MMSE provide equal diversity advantages. As expected,the ZF suffers from noise enhancement problem which can beobserved in all ZF equalizers.

Fig. 5 shows the BER versus T1 using different values ofT2 and SNR. It can be noted from the figure that there isno unique optimum (T1, T2) that should be used. Moreover,once T2 is selected, T1 can be selected without a majorconcern about its accuracy due to the BER plateau shown inFig. 5. Therefore, the thresholds selection process is relaxed


1 2 3 4 5106

105

104

103

102

101

100

Normalized Threshold (T1)

BER

SNR = 15 dBSNR = 30 dB.

.

Analytic

T2 =5T2 = 20

Fig. 5. BER versus the threshold T1 for different values of T2 and SNR.

as compared to the sample-by-sample blanking. Besides, itis interesting to note that analytical BER can be achievedregardless of the SNR.

Fig. 6 shows the joint effect of IN and fading, and comparesthe BER of TDI, OFDM and WHT-OFDM using ZF equal-ization. The performance of OFDM without IN is shown inthe figure as the baseline. The blind blanking is performedby comparing the received sample to the optimum thresholds,while non-blind is performed by assuming that the receiverhas perfect IN state information (NSI). In general, OFDMsystems offer poor performance in IN channels even whenblanking is invoked and the perfect NSI is available. Thesimulation results show that OFDM exhibits an error floorgreater than 103, which persists even with NSI. Similar tothe conventional OFDM, the WHT-OFDM system does notoffer any noticeable immunity against IN. Although the TDI-ZF does not offer BER improvement in frequency-selectivefading channels, it managed to mitigate the IN reasonably,where the BER converges to the OFDM BER with = 0.Furthermore, the BER of blind symbol blanking applied forTDI matches the BER of ideal blanking with NSI. In all othercases, a severe BER degradation is observed.

The BER of TDI-MMSE is compared to OFDM and WHT-OFDM in Fig. 7, where the results clearly show that theTDI significantly outperforms the other considered systems. Inaddition to the high robustness in frequency-selective fadingchannels, the TDI can effectively combat the degradationcaused by IN bursts with less than 1 dB degradation from theIN-free case at BER of 106. Both, the OFDM and WHT-OFDM exhibit high errors at SNR of about 35 dB. However,the WHT-OFDM slightly outperforms the conventional OFDMdue to its robustness to fading. Comparing the BER for theblind and NSI blanking in TDI shows that blind symbolblanking can be performed with high accuracy. This figure alsoshows the agreement between semi-analytical and simulationresults.

5 10 15 20 25 30 35 40

104

103

102

101

SNR (dB)

BER

OFDM, Sam., BlindOFDM, Sam., NSIOFDM, = 0TDI, Sym., BlindWHTOFDM, NSITDITheory

Fig. 6. BER using ZF equalizer in the presence of IN.

0 5 10 15 20 25 30 35106

105

104

103

102

101

SNR (dB)

BER

OFDM, BlindOFDM, NSITDI, TheoryTDI, Sym., BlindTDI, = 0WHTOFDM, NSI

Fig. 7. BER using MMSE equalizer in the presence of IN.

The effect of the IN bursts arrival probability is given inFig. 8. It can be noted from the figure that the TDI BERdegradation for p 0.02 is negligible, while an error floor isobserved at BER 104 for p = 0.05. It is worth notingthat P ( 1)|p=0.02 0.92, which corresponds to a severeIN channel where every block of N OFDM symbols will bemainly hit by one or more IN bursts. For the p = 0.05 and0.1 cases, P ( 3)|p=0.05 0.96 and P ( 3)|p=0.1 1, respectively. Therefore, the TDI can offer a superior BERperformance even in severe IN channels where p = 0.05. Thep = 0.1 case is too extreme, and hence it will be difficult tohave reliable communications in such scenarios.


0 10 20 30 40106

105

104

103

102

101

SNR (dB)

BER

p = 0.1p = 0.05p = 0.02p = 0.01p = 0.005p = 0

Fig. 8. BER of the TDI for different gating probability p.

VIII. CONCLUSION

This work presented a novel technique for joint mitiga-tion of impulsive noise (IN) and multipath fading effects inbroadband communication systems with OFDM modulation.The proposed system, namely TDI, is composed of the timedomain interleaving that interleaves the samples of severalOFDM symbols after the IFFT and deinterleaves them beforethe final FFT stage, and a two-stage blanking to mitigatethe IN. The performance of the TDI in presence of IN hasbeen analyzed in multipath fading channels where ZF andMMSE equalizers are employed to compensate the frequency-selectivity effects. The performance of the proposed systemis evaluated in terms of BER where closed-form formulaeare derived for the SINR using ZF and MMSE equalizers.The obtained analytical and simulation results show that theTDI can effectively combat the effects of heavily distributedIN and the frequency-selectivity of the channel. Our analysesshow that the effective SINR of TDI is mathematically equalto that of Walsh-Hadamard precoded OFDM (WHT-OFDM)system, which proves that equal frequency diversity gain isobtained [1]. Moreover, the TDI has time diversity, which waseffectively used to reduce the BER degradation caused by theIN.

APPENDIX ATHE DISTURBANCE CAUSED BY BLANKING

Assume that x = F1{d}. Therefore,

xn = N1i=0

diein. (67)

Given that samples of x are zeroed, calculating s = F{y},where y =Bx, gives

sk| = N1n=0

ynenk =

N1n=

xnenk (68)

Inserting (68) in (67), gives

sk| = 2N1n=

N1i=0

dieinenk

= dk + 2N1n=

N1i=0i=k

dien(ik) (69)

where = NN . With the change of the summations order,it can be rewritten as

= dk + 2N1i=0i=k

di

N1n=

en(ik). (70)

The last summation can be simplified using the identityN1n=

en(ik) =e2 j (kiN+kN)

2 e2 j (ki+k)2e2 j i2 e2 j k2

and taking e 2 j kN as a common factor, which results inN1n=

en(ik) =e2 j(ik) e2 j(ik)2

e2 j(ik)2 1 . (71)

Note that e2 j(ik) = 1. ThusN1n=

en(ik) =1 e2 j(ik)2e2 j(ik)2 1 (72)

Using Euler formula, and after some algebraic manipulations,we have

1 e2 jm=2 sinm (sinm j cosm)where m = (ik)N . Similarly, letting

(ik)N = u, gives

e2 ju 1 = 2 sinu [ sinu+ j cosu] . Therefore,N1n=

en(ik) =sinm

sinu

sinm j cosm sinu+ j cosu

= sinmsinu

ej(m+u)

=sin

[2(k i)]

sin [2(i k)] ej2(+1)(ik)

Finally,

sk| = dk + 2N1i=0i=k

disin

[2(k i)]

sin [2(i k)] ej2(+1)(ik)

(73)The second term corresponds with the error caused by zeroingsome samples of x and its mean is

E

2

N1i=0i=k

disin

[2(k i)]

sin [2(i k)] ej2(+1)(ik)

= 0.

(74)The variance of this error can be computed as

2k = E

2N1i=0i=k

disin

[2(k i)]

sin [2(i k)] ej2(+1)(ik)

2

= 4E

N1i=0i=k

N1v=0v =k

E {didv}sin[N (k i)

]sin[N (i k)

]


sin[N (k v)

]sin

[N (v k)

] ej N (+1)(vi)}

(75)

Assuming that the data samples di are independent andidentically distributed with E {didk} = 0 i = k and variance2d, then

2k = 4N1i=0i=k

E {didi }sin

[2(k i)]

sin [2(i k)]sin[2(k i)]

sin [2(i k)]

=2dN2

N1i=0i=k

sin2[2(k i)]

sin2 [2(i k)] (76)

APPENDIX BTDI-MMSE PROCESSING

For the case of MMSE equalizer without blanking, theequalizer output s can be written as

s = FHFx+ FH Fw (77)where k and k are defined in (47). Let A FHF andA FH F where A and A are circulant matrices with theirfirst rows defined as

A(0, :)=2

N1k=0 kN1k=0 ke

kN1k=0 ke

2k.

.

.N1k=0 ke

(N1)k

T

(78)

and

A(0, :)=2

N1k=0 kN1k=0 ke

kN1k=0 ke

2k.

.

.N1k=0 ke

(N1)k

T

. (79)

Thus, the MMSE equalizer output in (46) can be reduced tos = Ax+ Aw. (80)

Following the same approach used for the ZF equalizer, wedefine Aw similar to except that 1/Hk is replacedby k. Therefore,

() = 2

N1

i,k=0 wi()keki0N1

i,k=0 wi()keki1

.

.

.N1i,k=0 wi()ke

kiN+1

. (81)

By denoting i = N1

k=0 keki

, which is the is the FFTof ( = F{}), then (81) can be written as

() =

N1

i=0 wi()iN1i=0 wi()i1

.

.

.N1i=0 wi()iN+1

. (82)

The symbol index is added in (81) to denote the noise sampletaken at th OFDM symbol, which is needed to represent the

deinterleaved noise. The first term in (80) can be written as

() Ax =2

N1

i,k=0 x(i)keki0N1

i,k=0 x(i)keki1N

.

.

.N1i,k=0 x(i)ke

kiN+1N

T

.

(83)By denoting i =

N1k=0 ke

ki, which is the FFT of

( = F{}), then (83) can be written as

() =

N1

i=0 x(i)i0N1i=0 x(i)i1N

.

.

.N1i=0 x(i)iN+1N

. (84)

The MMSE output iss() = () + (), (85)

where

sn() =

N1i=0

x(i)in + N1i=0

wi()in. (86)

The next step is to deinterleave s(), which can be calculatedby substituting (81) and (83) into (85). The result is the desireddeinterleaved signal s() = () +(), where

sn() =

N1i=0

xn(i)i + N1i=0

wi(n)i. (87)

The final step is to apply the FFT to get the N decisionvariables r() = Fs() = F()+F(), which are thenapplied to the demodulator to get the information symbols.After some algebraic manipulations, F() and F() canbe written as

F() = 2

N1

m,i=0 xm(i)iN1m,i=0 xm(i)ie

m.

.

.N1m,i=0 xm(i)ie

(N1)m

(88)

and

F()=2

N1

m,i=0 wi(m)iN1m,i=0 wi(m)ie

m.

.

.N1m,i=0 wi(m)ie

(N1)m

. (89)

Therefore, the FFT kth pin output is given by

rk() = 2N1m=0

N1i=0

xm(i)iekm

+ 2N1m=0

N1i=0

wi(m)iekm

=

N1i=0

idk(i) + 2N1i=0

N1m=0

iwi(m)ekm.

(90)

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Maysam Mirahmadi (M09) received the B.Sc.and M.Sc. degrees in Electrical Engineering fromAmirkabir University of Technology (Tehran Poly-technic), Tehran, Iran, in 2004 and 2007, respec-tively. From 2007 to 2009, he was a Senior Designerat Kavoshkam R&D Group, Tehran, Iran. In 2013,he received the Ph.D. degree from the Universityof Western Ontario. Currently he is working as aresearch scientist at Canada Research and Develop-ment Center (CRDC), IBM Canada Ltd. His currentresearch interests include wireless communications,

optimized computations and high performance computing.


Arafat Al-Dweik (S9M01SM04) received theM.S. and Ph.D. degrees in electrical engineeringfrom Cleveland State University, Cleveland, OH,USA in 1998 and 2001, respectively. From 2003to 2013 he was an Assistant and then AssociateProfessor at the Department of Electrical and Com-puter Engineering at Khalifa University, UAE, andhe recently joined the School of Engineering atUniversity of Guelph, Guelph, ON, Canada, andhe holds an Adjunct Research Professor position atWestern University, London, ON, Canada. Dr. Al-

Dweik has several years of industrial experience in the USA, recipient ofthe Fulbright Scholarship, and has been awarded several awards and researchgrants. He is also a Senior Member of the IEEE and Associate Editor of theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. The main researchinterests of Dr. Al-Dweik include Wireless Communications, synchronizationtechniques, OFDM technology, modeling and simulation of communicationsystems, error control coding, and spread spectrum systems.

Abdallah Shami (M03SM09) received the B.E.degree in Electrical and Computer Engineering fromthe Lebanese University, Beirut, Lebanon in 1997,and the Ph.D. Degree in Electrical Engineeringfrom the Graduate School and University Center,City University of New York, New York, NY inSeptember 2002. In September 2002, he joined theDepartment of Electrical Engineering at LakeheadUniversity, Thunder Bay, ON, Canada as an Assis-tant Professor. Since July 2004, he has been withWestern University, Canada where he is currently

an Associate Professor in the Department of Electrical and Computer Engi-neering. His current research interests are in the area of network optimization,cloud computing, and wireless networks. Dr. Shami is an Editor for IEEECOMMUNICATIONS SURVEYS AND TUTORIALS and has served on the edi-torial board of IEEE COMMUNICATIONS LETTERS (2008-2013). Dr. Shamihas chaired key symposia for IEEE GLOBECOM, IEEE ICC, IEEE ICNC,and ICCIT. Dr. Shami is a Senior Member of IEEE.

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