Reconstruction of the Samples Corrupted withImpulse Noise in Multicarrier Systems

Josko Radic and Nikola RozicDepartment of ElectronicsUniversity of Split, Croatia{radic, rozic}@fesb.hr

Abstract— This paper addresses estimation of the signal sam-ples corrupted with impulse noise in multicarrier systems. Themain contribution in our work is application of the well knownBussgang theorem for developing the explicit expression forestimation of the signal samples that are corrupted with impulsenoise. In the presented analysis, estimation measure based onminimizing mean square error (MMSE), has been considered.Simulation results show that MMSE measure obtained by theproposed method reduces symbol error rate (SER) for the givennumber of iterations in comparison with the standard approachwithout estimation. Simulation results show very good agreementwith the results of the theoretic analysis.

Index Terms— Multicarrier systems, impulse noise, MMSEestimation.

I. INTRODUCTION

D IGITAL data transmitting, based on orthogonal carriermodulation is relatively old. Actually, it is a special

case of parallel transmission realized by frequency divisionmultiplex (FDM). In classical FDM, channel separation isbased on frequency bands non-overlapping. Each subchannelis modulated with different carrier frequency. Resulting spec-trum is superposition of subchannels spectrum that are non-overlapped. The main disadvantage of this scheme is in poorspectrum efficiency. An improvement of the described FDMscheme can be achieved by choosing the orthogonal set ofcarriers. In that case, spectrum overlapping is permitted withthe minimal interference between the adjacent channels, whilethe channel separation is maintained. Recently, an orthogonalfrequency division multiplexing technologies (OFDM) is be-coming very popular in wireless communication systems [1].Today it is implemented in DVB-T and WiMAX standards [2]and [3]. Implementation is relatively simple by applying IDFTand DFT in transmitter and receiver respectively [4].

Performance analysis of the communication system basedon parallel transmission has been reported in [5]. The moti-vation for implementing a system for parallel transmission isin efficient utilization of the transmission media that results ineasier equalization, and in an increased immunity on impulsenoise [6]. Impulse noise immunity comes from extended sym-bol duration, since the impulse noise is usually of very shortduration compared to symbol duration in parallel transmission.

Impulse noise immunity of the orthogonally multiplexedQAM system has been reported in [7]. However, in [8]comparison of the multicarrier modulation system (MMC) andsingle carrier (SC) is performed and the simulation results

show better performance of the SC system than the MMC forhigher impulse rates and power.

Two approaches are most usual in impulse noise suppres-sion. In the case of the strong noise impulses, in comparisonwith the OFDM signal, impulse peaks can be detected inthe receiver in time domain. This method requires optimumthreshold calculation [9]. Samples detected as an impulse noisecan be replaced with zero values or can be clipped withnonlinear element. However, if the impulse noise is maskedin OFDM signal then the impulse noise can be detected inestimated noise signal under the assumption that the impulsenoise level is above the Gaussian noise.

An iterative approach, where M largest samples of theestimated noise are subtracted from the received signal, isproposed in [10]. This method is based on assumption thatthe largest amplitudes in the noise estimation are probablyimpulse noise samples.

In addition, an iterative method for decoding complexnumber codes in impulse noise environment is described in[11]. Two estimators, in time and frequency, domain exchangeinformation on impulse noise between themselves. Both esti-mators use only partial information for estimation like in turbocoding.

An algorithm for impulse noise compensation in frequencydomain is proposed in [12]. In this method author proposesimpulse noise detection based on threshold that is calculatedtaking into account estimation of the noise variance. Optimumthreshold calculation for impulse noise detection is reported in[13].

Most of papers are focused on impulse noise detection butnot on the optimization of the samples affected by the impulsenoise. The primary goal of this paper is to estimate the originalvalue of the signal samples that are corrupted by impulsenoise, under the assumption of the known impulse noisepositions and parameters of the impulse noise. Criterion forthe optimization is the minimum mean square error MMSE.

The paper is organized as follows. In section II, the systemand channel noise model are introduced. In section III, atheoretical analysis of the proposed model is given. Resultsand discussion of the theoretic analysis and simulation arepresented in section IV. Section V concludes the paper.

978-1-4244-2948-6/09/$25.00 ©2009 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Fig. 1. Block scheme of system for impulse noise suppression.

II. SYSTEM AND CHANNEL MODELING

Received baseband signal after sampling in time domainduring the OFDM symbol interval can be expressed as:

x

(kTs

N

)=

N−1∑n=0

X[n]ej 2πnkN , n, k = 0, 1, . . . , N − 1

(1)where Ts denotes OFDM symbol duration, N is length ofthe OFDM frame, X = [X[0],X[1], . . . ,X[N − 1]] rep-resents baseband information simbols. Synchronization andsampling are assumed as ideal. Coefficients X[n] are complex-valued constellation points, representing data to be trans-mitted. Values of the data carriers depend on modulationtype (MPSK, MQAM e.t.c.), and are elements of the setX = {X0,X1, . . . , XM−1}, where M is number of symbolsin constellation diagram. It is assumed for data carriers thatEs = E [X∗[n]X[n]], where E [·] denotes expectation operatorEs is energy per symbol, and Eb = Es/ log2 M is theenergy per bit. After performing a DFT in receiver over theOFDM frame (symbol) and adding the noise, received signalin frequency domain is:

Y [n] = X[n] + W [n] + I[n] n = 0, 1, . . . , N − 1 (2)

where Y [n], W [n] and I[n] denote samples of the received sig-nals Y = [Y [0], Y [1], . . . , Y [N − 1]], aditive white Gaussiannoise (AWGN) samples W = [W [0],W [1], . . . ,W [N − 1]]and impulse noise samples I = [I[0], I[1], . . . , I[N−1]]. Noisesamples W [n] are complex independent zero mean Gaussiandistributed, of variance σ2

W = No, where No/2 denotes doubleside spectral noise density.

Usual approach for impulse noise modeling in time domainis based on Middleton or Bernoulli-Gaussian model [14] and[8]. In this paper we use Bernoulli-Gaussian impulse noisemodel. Vector of impulse noise samples in frequency domainis DFT of the impulse noise samples in time domain, i.e. i =[i[0], i[1], . . . , i[N−1]], I = DFT{i}. Bernoulli-Gaussian noiseprocess is defined as:

i[n] = b[n]u[n] (3)

where b[n] denotes independent identically distributed samplesfrom Bernoulli distribution i.e. b[n] ∈ B = {0, 1}, andPr{b[n] = 1} = p. u[n] denotes complex zero mean indepen-dent identically distributed samples from Gaussian distributionvariance of σ2

u. Bernoulli Gaussian noise is applicable formodeling noise generated with hairdryer [12].

III. THEORETICAL ANALYSIS

This section explains how to reconstruct samples of theoriginal signal x[n] if received signal y[n] is corrupted by

Fig. 2. Constelation diagram for 4–QAM modulation (left), and shape ofthe function fXm

N for the case when the real component of the value X[n]is equal to a (symbols at first end forth quadrant) (right).

the impulse noise (Fig. 1). We use minimum mean squareerror (MMSE) as a criterion for optimal reconstruction. In[15] it is shown that the MMSE calculation is equivalent tothe conditional expectation i.e. estimated value is E [x[n]|x[n]]where x[n] represents hard decisions in time domain and nis element of the set QI that contains indexes of the samplesthat are affected by the impulse noise. In OFDM demodulator,received signal after transforming in the frequency domain bymeans of DFT, is hard decided with the limiter that operateindependently in real and imaginary component. Signal at theoutput of the hard limiter for 4–QAM signal can be representedas:

X[n] = fN (X[n] + W [n] + I[n])

= X[n] + fXm

N (W [n] + I[n])(4)

Where fN denotes transfer function of the hard limiter, andfXm

N is hard limiter that acts only on the noise but its transferfunction depends on the values of the signal X[n] and thenotation in the exponent (Xm) comes from that fact. FunctionfXm

N is shifted function fN and shift depends on the value ofthe element X[n]. It is important to note that function fXm

Nshould be defined independently for the real and imaginarypart of the noise W [n] + I[n] based on symbol value X[n].Fig. 2 represents shape of the function fXm

N for the case whenthe real component of the value X[n] is equal to a (symbolsat first end forth quadrant). By applying Bussgang theorem onthe nonlinear function fXm

N in (4), output of the hard limitercan be expressed as:

X[n] = X[n] + αXm(W [n] + I[n]) + D[n] (5)

where αXmdenotes scaling constant, and D[n] is additive

noise component. It is interesting to note that the covariancebetween the signal G[n] = W [n] + I[n] and D[n] is zero i.eE [G∗[n]D[n]] = 0. However, E [X∗[n]D[n]] �= 0. ConstantαXm

can be calculated based on distribution of the signalW [n] + I[n] and transfer function fXm

N [16].

αXm=

1σ2

Gc

∫ ∞

−∞xpGc

(x)fXm

N (x)dx (6)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

where σ2Gc

= (σ2W +σ2

I )/2 = (N0+NIσ2u)/2 denotes variance

of the sum of the Gaussian and impulse noise G[n] per com-plex dimension and NI = |QI | is number of the impulse noisesamples. It is worth to note that random variables G[n] andI[n] are Gaussian and independent, so the sum is also Gaussiandistributed. pGc

(x) is distribution of the real and imaginarycomponent of the noise G[n]. Coefficients αXm

have to becalculated for all symbols from the constellation diagram,independently for the real and imaginary components. In the4–QAM system αXm

= α, ∀Xm ∈ X are same for allsymbols Xm and α follows directly from the (6) (Fig. 2):

α = − 2a

σ2Gc

∫ −a

−∞

x√2πσ2

Gc

e− x2

2σ2Gc dx =

2a√2πσ2

Gc

e− a2

2σ2Gc

(7)Where a =

√Es/2 denotes absolute value of the real and

imaginary part of the values that represent symbols in 4–QAMconstellation diagram. Since, X[n] and D[n] are correlated itis not easy to estimate x[n] based on MMSE criterion, i.e.calculate E [x[n]|x[n]]. However, D[n] can be written as:

D[n] = βX[n] + Z[n] (8)

Where β denotes scaling constant and Z[n] denotes additivenoise component. From (8) follows:

cov(X[n],D[n]) = cov(X[n], βX[n] + Z[n])

= βσ2X + cov(X[n], Z[n])

(9)

for the case where cov(X[n], Z[n]) = 0, β can be calculatedas:

β =cov(X[n],D[n])

σ2X

(10)

from (4), cov(X[n],D[n]) can be written as:

cov(X[n],D[n]) = cov(X[n],−E[n] − αG[n])= −cov(X[n], E[n])

(11)

where E[n] = X[n] − X[n] denotes error vector. X[n] andG[n] are independent, so cov(X[n], G[n]) = 0. It is easy tocalculate cov(X[n],D[n]):

cov(X[n], E[n]) = E [X∗[n]E[n]]

=M−1∑m=0

M−1∑k=0k �=m

X∗m[n]E(k→m)[n]pX(Xm)pE(Xk,Xm)

=M−1∑m=0

M−1∑k=0k �=m

X∗m[n](Xk[n] − Xm[n])pX(Xm)pE(Xk,Xm)

(12)

where E(k→m) denotes error vector if symbol Xk is decidedas Xm, m and k are indexes of all possible symbols andpE(Xk,Xm) is probability that symbol Xm is chosen giventhat symbol Xk was transmitted. Error vector consists of thereal and imaginary components (phase and co-phase) that areindependent. In 4–QAM modulation it is easy to calculate

TABLE I

ERROR VECTORS PARAMETERS FOR REAL COMPONENT AND

CORRESPONDING PROBABILITIES IN 4–QAM.

k Ek pE(Ek)

0. −√2Es

pEI2

,pEQ

21. 0 1 − pEI , 1 − pEQ2.

√2Es

pEI2

,pEQ

2

probability of error from the noise distribution pGc(x) that

is Gaussian with variance σ2Gc

:

pEI = pEQ =∫ ∞√

Es2

pGc(x)dx

=12

erfc

(√Es

4σ2Gc

) (13)

where pEI and pEQ denotes probability of error for the realand imaginary components and Es = σ2

X = E [|X[n]|2] isenergy per symbol. Function pE(Xk,Xm) follows from thepEI and pEQ depending on the error vector and correspondingprobabilities. From (4) and (8) after IDFT hard decision intime domain is obtained:

x[n] = (1 + β)x[n] + αg[n] + z[n] (14)

where g[n] = w[n]+i[n] denotes superposition of the Gaussianand impulse noise in time domain, and cov(x[n], z[n]) = 0.Variance of the process z[n] has to be known for calculationof the MMSE estimation of the x[n]. Substituting (8) in (4)gives:

αG[n] + Z[n] = −E[n] − βX[n] (15)

Having in mind that E [X∗[n]E[n]] = −βE [|X[n]|2] (Eq. (10)and (11)), variance of the process Z[n] folow:

E [|Z[n]|2] = −β2E [|X[n]|2]+E [|E[n]|2]−α2E [|G[n]|2] (16)

It is easy to show that cov(G[n], Z[n]) = 0. From (4)follow that var(E[n]) = α2var(G[n]) + var(D[n]). Alsovar(D[n]) = β2var(X[n]) + var(Z[n]) and directly followthat cov(G[n], Z[n]) = 0. Variance of the process Z[n] intime domain is E [|z[n]|2] = E [|Z[n]|2]/N . Variance of theprocess E[n] can be obtained by multiplying variance percomplex dimensions with 2, since the errors are independent.Let E = {E0, . . . , ENE−1} be the set of the all possible errorvectors per complex dimensions, including zero vector (noerror), where NE = |E| denotes number of error vectors.Also, let pE(Ek) denotes probability of occurrence error Ek.Variance of the process E[n] is:

E [|E[n]|2] = 2NE−1∑k=0

E2kpE(Ek) (17)

It is easy to calculate E [|E[n]|2] in the case of 4–QAMmodulation (Table I):

E [|E[n]|2] = 4EspEI = 4EspEQ (18)

From (14), (16) and (18) MMSE estimation of the samples

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

8 9 10 11 12 13 14 15 16−10

−5

0

5

10

15

20

Es/N

0 [dB]

SN

R [d

B]

s−niSim.s−iSim.ns−niSim.ns−iSim.

Fig. 3. SNR vs. Es/N0 for the Γ = 100 and 8 noise impulses at thepositions with impulse noise and at the positions without impulse noise.

that are corrupted with impulse noise can be calculated as [15]:

E [x[n]|x[n]] =

=(1 + β)E [|x[n]|2]

(1 + β)2E [|x[n]|2] + α2E [|g[n]|2] + E [|z[n]|2] x[n]

=(1 + β)σ2

x

(1 + 2β + 4pEI )σ2x + α2(1 − NI

N )σ2u

x[n], ∀n ∈ QI

(19)

where σ2x = E [|X[n]|2]/N = Es/N . Here we assume that

αg[n] + z[n] is Gaussian distributed. Also, it is assumedthat variance of the process z[n] at the positions that arecorrupted by impulse noise i.e. n ∈ QI only slightly changesin comparison with the variance calculated in (16). Estimatedsamples are substituted instead of samples that are corruptedwith impulse noise, and new decision is obtained at the outputof the hard limiter. Procedure can be iteratively repeated.

IV. SIMULATION RESULTS

In this section we present and analyze simulation results.In all simulations we assume 64 carriers in OFDM frame

4 5 6 7 8 9 10 11 12−15

−10

−5

0

5

10

15

20

Es/N

0

SN

R [d

B]

s−niSim.s−iSim.ns−niSim.ns−iSim.

Fig. 4. SNR vs. Es/N0 for the Γ = 50 and 4 noise impulses at the positionswith impulse noise and at the positions without impulse noise.

4 6 8 10 12 14 16−5

0

5

10

15

Es/N

0 [dB]

SN

R [d

B]

Γ = 100, 8 imp., no−optSimulationΓ = 50, 4 imp., no−optSimulationSimulationΓ = 100, 8 imp., optSimulationΓ = 50, 4 imp., opt

Fig. 5. SNR vs. Es/N0 for the Γ = 100 and 8 noise impulses, as forΓ = 50 and 4 noise impulses. Comparison of the MMSE samples estimationwith the method without estimation.

and 4–QAM modulation. Figs. 3 and 4 present results of thetheoretical analysis and simulation for signal noise ratio versusEs/N0 for scenario where Γ = σ2

u/σ2w = 100 and 8 noise

impulses as for Γ = 50 and 4 noise impulses respectively.Four scenarios are considered. In first and second, SNR versusEs/N0 are displayed for signal positions that are corruptedwith impulse noise (n ∈ {QI}) and positions with only Gaus-sian noise (n /∈ {QI}), (ns-i and ns-ni curves) respectively.SNR is calculated as E [|x[n]|2]/E [|w[n] + i[n]|2], n ∈ {QI}and E [|x[n]|2]/E [|w[n]|2], n /∈ {QI} respectively. This curvesrepresent noise levels.

In third and fourth scenario, SNR for the signal x[n]− x[n]versus Es/N0 is displayed. SNR at the positions that arecorrupted with impulse noise as at the positions that arenot corrupted with impulse noise are considered (s-i and s-nicurves). SNR is calculated as E [|x[n]|2]/E [|x[n]− x[n]|2], n ∈{QI} and E [|x[n]|2]/E [|x[n]− x[n]|2], n /∈ {QI} respectively(14). These curves clearly explain the motivation for replace-ment of the received samples that are corrupted with impulsenoise y[n], n ∈ {QI} with the corresponding samples obtainedafter hard decision and transforming in the time domain i.e.x[n]. Replacement significantly increases SNR, particularly athigher Es/N0. However, replacement of the signal samplesthat are not corrupted with impulse noise y[n], n /∈ {QI}, withsamples of the hard decision in time domain i.e. x[n] resultswith SNR decreasing in comparison with the situation whensamples are not replaced. The results of computer simulationmatch the theoretic analysis very good.

Fig. 5 shows results of the theoretical analysis and sim-ulation of the SNR versus Es/N0 for situation where Γ =100 and 8 noise impulses as for Γ = 50 and 4 noiseimpulses respectively. Fig. shows SNR comparison for theoptimal (MMSE measure, Eq. (19)) and non-optimal (onlyreplacement) estimation of the samples that are corruptedwith impulse noise. SNR is calculated as E [|x[n]|2]/E [|x[n]−E [x[n]|x[n]]|2], n ∈ {QI} for optimal estimation and asE [|x[n]|2]/E [|x[n] − x[n]|2], n ∈ {QI} for non-optimal esti-mation. Theoretical and simulation results shows that at lower

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Fig. 6. SNR vs. parameter e for the Γ = 100 and 8 noise impulses, as forΓ = 50 and 4 noise impulses

Es/N0 ratio, MMSE estimation significantly increase SNR incomparison with only replacement. SNR increasing for 5 [dB]in the situation with Γ = 100 and 8 impulses as in situationwhere Γ = 50 and 4 impulses at Es/N0 = 8 and 4 [dB] re-spectively. As Es/N0 increase, MMSE estimation E [x[n]|x[n]]and x[n] converge to the same value and estimation is sameas replacement. The results of computer simulation match thetheoretic analysis very good.

Fig. 6 shows simulation results of the SNR versus parametere for situation where Γ = 100 and 8 noise impulses as forΓ = 50 and 4 noise impulses respectively. SNR is calculatedas E [|x[n]|2]/E [|x[n]− ex[n]]|2]. Simulation results show thatvalue of the parameter e coincides with the theoreticallycalculated MMSE measure that minimize MSE i.e. maximizeSNR (dashed line).

Fig. 7 shows simulation results of the symbol error rate(SER) versus number of the iteration (1–5) for situation whereΓ = 100, Es/N0 = 10 [dB] and 8 noise impulses as for Γ =50, Es/N0 = 6 [dB] and 4 noise impulses respectively. Upperbounds are also displayed, i.e. situation without impulse noise.Situation with MMSE samples estimation method is comparedwith method without estimation. Since, for the second andhigher number of the iteration it is complicate to calculateMMSE measure, we assume that the coefficient that multipliedx[n] (Eq. (19)) linearly increase with the iteration and in finaliteration it is 1. Simulation results shows that the MMSEestimation of the corrupted samples produce significantlylower SER in comparison with the method with only samplesreplacement for the same number of the iterations.

V. CONCLUSION

In this paper the developed expression for estimation of thesamples corrupted with impulse noise in multicarrier systemshas been derived. Minimum mean square error was usedas criterion for estimation. Theoretical analysis, confirmedthrough simulation results, show that the MMSE measure isgood criterion for samples estimation. The proposed analysiscan be modified and implemented for frequency selective

Fig. 7. SNR vs. es/N0 for the Γ = 100 and 8 noise impulses, as forΓ = 50 and 4 noise impulses at Es/N0 = 10 and 6 [dB] respectively.

channel and OFDM system with higher order modulation (8-QAM, 16-QAM etc.).

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.