Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 830517 7 pageshttpdxdoiorg1011552013830517
Research ArticleResidual ISI Obtained by Nonblind AdaptiveEqualizers and Fractional Noise
Monika Pinchas
Department of Electrical and Electronic Engineering Ariel University 40700 Ariel Israel
Correspondence should be addressed to Monika Pinchas monikapinchasgmailcom
Received 18 July 2013 Accepted 29 August 2013
Academic Editor Ming Li
Copyright copy 2013 Monika Pinchas This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Recently a closed-form approximated expression was derived by the same author for the achievable residual intersymbolinterference (ISI) case that depends on the step-size parameter equalizerrsquos tap length input signal statistics signal to noise ratio(SNR) and channel power and is valid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of05 le 119867 lt 1 But this expression was obtained for the blind adaptive case and cannot be applied to the nonblind adaptive versionUp to now the achievable residual ISI for the non-blind adaptive case could be obtained only via simulation In this paper wederive a closed-form approximated expression (or an upper limit) for the residual ISI obtained by non-blind adaptive equalizersvalid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of 05 le 119867 lt 1 This new obtainedexpression depends on the step-size parameter equalizerrsquos tap length input signal statistics SNR channel power and the Hurstexponent parameter Simulation results indicate that there is a high correlation between the calculated results (obtained from thenew obtained expression for the residual ISI) and those obtained from simulating the system
1 Introduction
We consider a nonblind deconvolution problem in which weobserve the output of an unknown possibly nonminimumphase linear system (single-input-single-output (SISO) FIRsystem) from which we want to recover its input (source)using an adjustable linear filter (equalizer) and trainingsymbols During transmission a source signal undergoes aconvolutive distortion between its symbols and the channelimpulse response This distortion is referred to as ISI [12] It is well known that ISI is a limiting factor in manycommunication environments where it causes an irreducibledegradation of the bit error rate thus imposing an upper limiton the data symbol rate In order to overcome the ISI probleman equalizer is implemented in those systems [1ndash12]
In this paper we consider the nonblind adaptive equalizerwhere training sequences are needed to generate the errorthat is fed into the adaptive mechanism which updates theequalizerrsquos taps [9ndash12]Thenonblind adaptive approach yieldsin most cases a better equalization performance consideringconvergence speed and equalization quality compared withthe blind adaptive version [6] In addition the blind adaptive
version has a higher computational cost compared with itsnonblind approach [6]
The equalization performance from the residual ISI pointof view depends on the channel characteristics on the addednoise on the step-size parameter used in the adaptationprocess on the equalizerrsquos tap length and on the input signalstatistics [13 14] Fast convergence speed and reaching aresidual ISI where the eye diagram is considered to be open(for the communication case) are the main requirementsfrom a blind or nonblind equalizer Fast convergence speedmay be obtained by increasing the step-size parameter Butincreasing the step-size parameter may lead to a higher resid-ual ISI which might not meet the systemrsquos requirements anymore Recently [2] a closed-form approximated expressionwas derived for the achievable residual ISI case that dependson the step-size parameter equalizerrsquos tap length input signalstatistics SNR Hurst exponent and channel power But thisexpression is valid only for the blind adaptive case and cannotbe used for the nonblind version
Up to now the achievable residual ISI for the nonblindadaptive case (for the noisy or noiseless case) could beobtained only via simulation Thus the system designer had
2 Mathematical Problems in Engineering
to spend a lot of time in simulating the whole system inorder to find the best values for the step-size parameter andequalizerrsquos tap length that meet the systemrsquos requirementsfrom the residual ISI point of view In this paper we derivea closed-form approximated expression (or an upper limit)for the residual ISI obtained by nonblind adaptive equalizersthat depends on the step-size parameter equalizerrsquos taplength input signal statistics SNR channel power and Hurstexponent parameter This expression is valid for fGn inputwhere the Hurst exponent is in the region of 05 le 119867 lt 1Please note119867 = 1 is the limit case which does not havemuchpractical sense [15ndash17] It should be pointed out that a whiteGaussian process is a special case (119867 = 05) of the fractionalGaussian noise (fGn) model [18] FGn with 119867 isin (05 1)
corresponds to the case of long-range dependency (LRD)[18] Thus the new obtained expression for the achievableresidual ISI is not only valid for the special case of whiteGaussian process but also covers those cases that correspondto the case of LRD
The paper is organized as follows After having describedthe system under consideration in Section 2 the closed-formapproximated expression (or upper limit) for the residual ISIis introduced in Section 3 In Section 4 simulation results arepresented and the conclusion is given in Section 5
2 System Description
The system under consideration is illustrated in Figure 1where we make the following assumptions
(1) The input sequence 119909[119899] belongs to a two indepen-dent quadrature carriers case constellation input withvariance 1205902
119909 where 119909
119903[119899] and 119909
119894[119899] are the real and
imaginary parts of 119909[119899] respectively and 1205902119909119903
is thevariance of 119909
119903[119899]
(2) The unknown channel ℎ[119899] is a possibly nonmini-mum phase linear time-invariant filter in which thetransfer function has no ldquodeep zerosrdquo namely thezeros lie sufficiently far from the unit circle
(3) The equalizer 119888[119899] is a tap-delay line
(4) The noise 119908[119899] consists of 119908[119899] = 119908119903[119899] + 119895119908
119894[119899]
where 119908119903[119899] and 119908
119894[119899] are the real and imagi-
nary parts of 119908[119899] respectively and 119908119903[119899] and 119908
119894[119899]
are independent Both 119908119903[119899] and 119908
119894[119899] are frac-
tional Gaussian noises (fGn) with zero mean Notethat 1205902
119908119903
= 119864[1199082
119903[119899]] 1205902
119908119894
= 119864[1199082
119894[119899]] for 119898 = 119896
119864[119908119903[119899 minus 119896]119908
119903[119899 minus 119898]] = (1205902
119908119903
2)[(|119898 minus 119896| minus 1)2119867minus
2(|119898 minus 119896|)2119867+(|119898 minus 119896| + 1)
2119867] and 119864[119908
119894[119899minus119896]119908
119894[119899minus
119898]] = (1205902
119908119894
2)[(|119898 minus 119896| minus 1)2119867
minus 2(|119898 minus 119896|)2119867
+
(|119898 minus 119896| + 1)2119867] where 119864[sdot] denotes the expectation
operator on (sdot) and119867 is the Hurst exponent
(5) The variance of 119908[119899] is denoted as 119864[119908[119899]119908lowast[119899]] =1205902
119908 where 1205902
119908= 21205902
119908119894
= 21205902
119908119903
and (sdot)lowast is the conjugateoperation on (sdot)
h[n] c[n]
w[n]
x[n] y[n] z[n]+
+
Adaptive equalizer
Figure 1 Block diagram of a baseband communication system
The transmitted sequence 119909[119899] is sent through the chan-nel ℎ[119899] and is corrupted with noise 119908[119899] Therefore theequalizerrsquos input sequence 119910[119899]may be written as
where 119901[119899] is the convolutional noise namely the residualintersymbol interference (ISI) arising from the differencebetween the ideal equalizerrsquos coefficients and those chosen inthe system and 119908[119899] = 119908[119899] lowast 119888[119899] The ISI is often used asa measure of performance in equalizersrsquo applications definedby
where 120575 is the Kronecker delta function and 120577[119899] stands forthe difference (error) between the ideal and the actual valueused for 119888[119899]
Next we turn to the adaptation mechanism of theequalizer which is based on training symbols [9ndash12 19]
119888eq [119899 + 1] = 119888eq [119899] minus 120583 (119911 [119899] minus 119909 [119899]) 119910lowast[119899] (5)
where 120583 is the step-size parameter 119888eq[119899] is the equalizervectorwhere the input vector is119910[119899] = [119910[119899] sdot sdot sdot 119910[119899minus119873+1]]119879and119873 is the equalizerrsquos tap length The operator ( )119879 denotesfor transpose of the function ( ) Please note that for thenonblind adaptive case during the training period a knowndata sequence is transmitted A replica of this sequence ismade available at the receiver in proper synchronism withthe transmitter thereby making it possible for adjustments tobe made to the equalizer coefficients in accordance with theadaptive filtering algorithm employed in the equalizer design[19]
3 Residual ISI for Fractional GaussianNoise Input
In this section a closed-form approximated expression (or anupper limit) is derived for the residual ISI valid for the fGninput case
Mathematical Problems in Engineering 3
Theorem 1 Noted the following assumptions
(1) The convolutional noise 119901[119899] is a zero mean whiteGaussian process with variance 1205902
119901= 119864[119901[119899]119901
lowast[119899]]
The real part of119901[119899] is denoted as119901119903[119899] and119864[1199012
119903[119899]] =
119898119901
(2) The source signal 119909[119899] is a rectangular QuadratureAmplitude Modulation (QAM) signal (where the realpart of 119909[119899] is independent of the imaginary part of119909[119899]) with known variance and higher moments
(3) The convolutional noise 119901[119899] and the source signal areindependent
(4) The gain between the source and equalized outputsignal is equal to one
(5) The convolutional noise 119901[119899] is independent of 119908[119899](6) The added noise is fGn with zero mean(7) The channel ℎ[119899] has real coefficients(8) The Hurst exponent is in the range of 05 le 119867 lt 1
The residual ISI expressed in dB units may be defined as
ISI = 10 log10(119898119901) minus 10 log
10(1205902
119909119903
) (6)
where119898119901is defined by
119898119901= 1205902
119908119903
(120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
times (2(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
minus120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
)
minus1
)
= 1205902
119908119903
120583 (1198731205902
119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
2 minus 120583 (1198731205902119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic(119873 minus 1)119867 (2119867 minus 1)]
(7)
and119877 is the channel length1205902119908119903
is the variance of119908119903[119899] (119908
119903[119899]
is the real part of119908[119899]) and SNR is given by SNR = 1205902119909119903
1205902
119908119903
=
1205902
1199091205902
119908
Comments It should be pointed out that assumptions (1)ndash(5)from above are precisely the same assumptions made in [214]
Proof Let us first recall the expression for the adaptationmechanism of the equalizer given in (5) Then we substitute(2) into (5) and obtain
According to [13 14] when the equalizer has convergedwe may assume that 119864[Δ(1199012
119903)] cong 0 Therefore by setting
119864[Δ(1199012
119903)] = 0 into (9) we obtain
minus 2120583119898119901119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]]
+ 1205832(119898119901+ 1205902
119908119903
)
times 119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
= 0
dArr
119898119901= 1205902
119908119903
(120583119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
times (2119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] minus 120583119864
times[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
)
minus1
)
(11)
In [13] the expression 119864[(sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898])
2
]
was approximated as (119864[sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898]])
2It should be pointed out that this approximation fits theMPSK case where QPSK is a special case of it Howeversatisfying results were obtained in [13] for the 16QAM and64QAM cases in spite of the fact that the above mentionedapproximation was applied Thus it makes sense to use thesame approximation also here for our case The expression119864[sum119898=119873minus1
119898=0119910[119899 minus 119898]119910
lowast[119899 minus 119898]]may be written as
119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] = 119873120590
Figure 2 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 10 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
Next we turn to find a closed-form approximated expressionfor1205902119908119903
The real part of119908[119899] namely119908119903[119899] may be expressed
as
119908119903 [119899] =
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896] (13)
Thus the variance of 119908119903[119899] is given by
1205902
119908119903
= 119864[
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896]
119898=119873minus1
sum
119898=0
119888119898 [119899] 119908119903 [119899 minus 119898]]
=
119896=119873minus1
sum
119896=0
119898=119873minus1
sum
119898=0
119888119896 [119899] 119888119898 [119899] 119864 [119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(14)
which can be also written as
1205902
119908119903
= 1205902
119908119903
119896=119873minus1
sum
119896=0
1198882
119896[119899]
+
119896=119873minus1
sum
119896=0119896 =119898
119898=119873minus1
sum
119898=0119896 =119898
119888119896 [119899] 119888119898 [119899]119864[119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(15)
According to [2] expression (15) can be approximatelywritten as
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic119873 minus 1119867 (2119867 minus 1)] (16)
Figure 3 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 8 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
by using assumption (4) from the system description sectionassumptions (4) and (6)ndash(8) from this section the Holderinequality [20] and the following approximation [21]
05 [(|119898 minus 119896| minus 1)2119867minus 2(|119898 minus 119896|)
2119867+ (|119898 minus 119896| + 1)
2119867]
≃ 119867 (2119867 minus 1) |119898 minus 119896|2119867minus2
(17)
Now by substituting (12) and (16) into (11) we obtain (7)Thiscompletes our proof
4 Simulation
In this section we test our new proposed expression for theresidual ISI for the 16QAM case (a modulation using plusmn1 3levels for in-phase and quadrature components) with thealgorithm described in (5) for different SNR step-size andequalizerrsquos tap length values and for two different channeltypesThe following two channels were considered Channel1(initial ISI = 044) the channel parameters were determinedaccording to [22]
ℎ119899= (0 for 119899 lt 0 minus04 for 119899 = 0
times 084 sdot 04119899minus1 for 119899 gt 0)
(18)
Channel2 (initial ISI = 088) the channel parameters weredetermined according to
ℎ119899= (04851 minus072765 minus04851) (19)
The equalizer was initialized by setting the center tap equal toone and all others to zero
Figure 4 A comparison between the simulated and calculated residual ISI for the 16QAM source input going through channel1 for SNR = 10[dB]The averaged results were obtained in 100Monte Carlo trialsThe equalizerrsquos tap length and step-size parameter were set to 27 and 00006respectively
Figure 5 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
In the following we denote the residual ISI performanceaccording to (6) with (7) as ldquoCalculated ISIrdquo Figure 2 toFigure 8 show the ISI performance as a function of theiteration number of our proposed expression (6) with (7)for the achievable residual ISI compared with the simulatedresults for two different channels and equalizerrsquos tap length
Figure 6 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00006 respectively
and various values for 119867 SNR and step-size parameterAccording to Figures 2 3 5 6 7 and 8 a high correlation isobserved between the simulated and calculated results evenfor 119867 = 09 According to Figure 4 the calculated ISI maybe considered as an upper limit for the simulated results for119867 gt 05
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
to spend a lot of time in simulating the whole system inorder to find the best values for the step-size parameter andequalizerrsquos tap length that meet the systemrsquos requirementsfrom the residual ISI point of view In this paper we derivea closed-form approximated expression (or an upper limit)for the residual ISI obtained by nonblind adaptive equalizersthat depends on the step-size parameter equalizerrsquos taplength input signal statistics SNR channel power and Hurstexponent parameter This expression is valid for fGn inputwhere the Hurst exponent is in the region of 05 le 119867 lt 1Please note119867 = 1 is the limit case which does not havemuchpractical sense [15ndash17] It should be pointed out that a whiteGaussian process is a special case (119867 = 05) of the fractionalGaussian noise (fGn) model [18] FGn with 119867 isin (05 1)
corresponds to the case of long-range dependency (LRD)[18] Thus the new obtained expression for the achievableresidual ISI is not only valid for the special case of whiteGaussian process but also covers those cases that correspondto the case of LRD
The paper is organized as follows After having describedthe system under consideration in Section 2 the closed-formapproximated expression (or upper limit) for the residual ISIis introduced in Section 3 In Section 4 simulation results arepresented and the conclusion is given in Section 5
2 System Description
The system under consideration is illustrated in Figure 1where we make the following assumptions
(1) The input sequence 119909[119899] belongs to a two indepen-dent quadrature carriers case constellation input withvariance 1205902
119909 where 119909
119903[119899] and 119909
119894[119899] are the real and
imaginary parts of 119909[119899] respectively and 1205902119909119903
is thevariance of 119909
119903[119899]
(2) The unknown channel ℎ[119899] is a possibly nonmini-mum phase linear time-invariant filter in which thetransfer function has no ldquodeep zerosrdquo namely thezeros lie sufficiently far from the unit circle
(3) The equalizer 119888[119899] is a tap-delay line
(4) The noise 119908[119899] consists of 119908[119899] = 119908119903[119899] + 119895119908
119894[119899]
where 119908119903[119899] and 119908
119894[119899] are the real and imagi-
nary parts of 119908[119899] respectively and 119908119903[119899] and 119908
119894[119899]
are independent Both 119908119903[119899] and 119908
119894[119899] are frac-
tional Gaussian noises (fGn) with zero mean Notethat 1205902
119908119903
= 119864[1199082
119903[119899]] 1205902
119908119894
= 119864[1199082
119894[119899]] for 119898 = 119896
119864[119908119903[119899 minus 119896]119908
119903[119899 minus 119898]] = (1205902
119908119903
2)[(|119898 minus 119896| minus 1)2119867minus
2(|119898 minus 119896|)2119867+(|119898 minus 119896| + 1)
2119867] and 119864[119908
119894[119899minus119896]119908
119894[119899minus
119898]] = (1205902
119908119894
2)[(|119898 minus 119896| minus 1)2119867
minus 2(|119898 minus 119896|)2119867
+
(|119898 minus 119896| + 1)2119867] where 119864[sdot] denotes the expectation
operator on (sdot) and119867 is the Hurst exponent
(5) The variance of 119908[119899] is denoted as 119864[119908[119899]119908lowast[119899]] =1205902
119908 where 1205902
119908= 21205902
119908119894
= 21205902
119908119903
and (sdot)lowast is the conjugateoperation on (sdot)
h[n] c[n]
w[n]
x[n] y[n] z[n]+
+
Adaptive equalizer
Figure 1 Block diagram of a baseband communication system
The transmitted sequence 119909[119899] is sent through the chan-nel ℎ[119899] and is corrupted with noise 119908[119899] Therefore theequalizerrsquos input sequence 119910[119899]may be written as
where 119901[119899] is the convolutional noise namely the residualintersymbol interference (ISI) arising from the differencebetween the ideal equalizerrsquos coefficients and those chosen inthe system and 119908[119899] = 119908[119899] lowast 119888[119899] The ISI is often used asa measure of performance in equalizersrsquo applications definedby
where 120575 is the Kronecker delta function and 120577[119899] stands forthe difference (error) between the ideal and the actual valueused for 119888[119899]
Next we turn to the adaptation mechanism of theequalizer which is based on training symbols [9ndash12 19]
119888eq [119899 + 1] = 119888eq [119899] minus 120583 (119911 [119899] minus 119909 [119899]) 119910lowast[119899] (5)
where 120583 is the step-size parameter 119888eq[119899] is the equalizervectorwhere the input vector is119910[119899] = [119910[119899] sdot sdot sdot 119910[119899minus119873+1]]119879and119873 is the equalizerrsquos tap length The operator ( )119879 denotesfor transpose of the function ( ) Please note that for thenonblind adaptive case during the training period a knowndata sequence is transmitted A replica of this sequence ismade available at the receiver in proper synchronism withthe transmitter thereby making it possible for adjustments tobe made to the equalizer coefficients in accordance with theadaptive filtering algorithm employed in the equalizer design[19]
3 Residual ISI for Fractional GaussianNoise Input
In this section a closed-form approximated expression (or anupper limit) is derived for the residual ISI valid for the fGninput case
Mathematical Problems in Engineering 3
Theorem 1 Noted the following assumptions
(1) The convolutional noise 119901[119899] is a zero mean whiteGaussian process with variance 1205902
119901= 119864[119901[119899]119901
lowast[119899]]
The real part of119901[119899] is denoted as119901119903[119899] and119864[1199012
119903[119899]] =
119898119901
(2) The source signal 119909[119899] is a rectangular QuadratureAmplitude Modulation (QAM) signal (where the realpart of 119909[119899] is independent of the imaginary part of119909[119899]) with known variance and higher moments
(3) The convolutional noise 119901[119899] and the source signal areindependent
(4) The gain between the source and equalized outputsignal is equal to one
(5) The convolutional noise 119901[119899] is independent of 119908[119899](6) The added noise is fGn with zero mean(7) The channel ℎ[119899] has real coefficients(8) The Hurst exponent is in the range of 05 le 119867 lt 1
The residual ISI expressed in dB units may be defined as
ISI = 10 log10(119898119901) minus 10 log
10(1205902
119909119903
) (6)
where119898119901is defined by
119898119901= 1205902
119908119903
(120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
times (2(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
minus120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
)
minus1
)
= 1205902
119908119903
120583 (1198731205902
119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
2 minus 120583 (1198731205902119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic(119873 minus 1)119867 (2119867 minus 1)]
(7)
and119877 is the channel length1205902119908119903
is the variance of119908119903[119899] (119908
119903[119899]
is the real part of119908[119899]) and SNR is given by SNR = 1205902119909119903
1205902
119908119903
=
1205902
1199091205902
119908
Comments It should be pointed out that assumptions (1)ndash(5)from above are precisely the same assumptions made in [214]
Proof Let us first recall the expression for the adaptationmechanism of the equalizer given in (5) Then we substitute(2) into (5) and obtain
According to [13 14] when the equalizer has convergedwe may assume that 119864[Δ(1199012
119903)] cong 0 Therefore by setting
119864[Δ(1199012
119903)] = 0 into (9) we obtain
minus 2120583119898119901119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]]
+ 1205832(119898119901+ 1205902
119908119903
)
times 119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
= 0
dArr
119898119901= 1205902
119908119903
(120583119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
times (2119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] minus 120583119864
times[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
)
minus1
)
(11)
In [13] the expression 119864[(sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898])
2
]
was approximated as (119864[sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898]])
2It should be pointed out that this approximation fits theMPSK case where QPSK is a special case of it Howeversatisfying results were obtained in [13] for the 16QAM and64QAM cases in spite of the fact that the above mentionedapproximation was applied Thus it makes sense to use thesame approximation also here for our case The expression119864[sum119898=119873minus1
119898=0119910[119899 minus 119898]119910
lowast[119899 minus 119898]]may be written as
119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] = 119873120590
Figure 2 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 10 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
Next we turn to find a closed-form approximated expressionfor1205902119908119903
The real part of119908[119899] namely119908119903[119899] may be expressed
as
119908119903 [119899] =
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896] (13)
Thus the variance of 119908119903[119899] is given by
1205902
119908119903
= 119864[
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896]
119898=119873minus1
sum
119898=0
119888119898 [119899] 119908119903 [119899 minus 119898]]
=
119896=119873minus1
sum
119896=0
119898=119873minus1
sum
119898=0
119888119896 [119899] 119888119898 [119899] 119864 [119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(14)
which can be also written as
1205902
119908119903
= 1205902
119908119903
119896=119873minus1
sum
119896=0
1198882
119896[119899]
+
119896=119873minus1
sum
119896=0119896 =119898
119898=119873minus1
sum
119898=0119896 =119898
119888119896 [119899] 119888119898 [119899]119864[119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(15)
According to [2] expression (15) can be approximatelywritten as
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic119873 minus 1119867 (2119867 minus 1)] (16)
Figure 3 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 8 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
by using assumption (4) from the system description sectionassumptions (4) and (6)ndash(8) from this section the Holderinequality [20] and the following approximation [21]
05 [(|119898 minus 119896| minus 1)2119867minus 2(|119898 minus 119896|)
2119867+ (|119898 minus 119896| + 1)
2119867]
≃ 119867 (2119867 minus 1) |119898 minus 119896|2119867minus2
(17)
Now by substituting (12) and (16) into (11) we obtain (7)Thiscompletes our proof
4 Simulation
In this section we test our new proposed expression for theresidual ISI for the 16QAM case (a modulation using plusmn1 3levels for in-phase and quadrature components) with thealgorithm described in (5) for different SNR step-size andequalizerrsquos tap length values and for two different channeltypesThe following two channels were considered Channel1(initial ISI = 044) the channel parameters were determinedaccording to [22]
ℎ119899= (0 for 119899 lt 0 minus04 for 119899 = 0
times 084 sdot 04119899minus1 for 119899 gt 0)
(18)
Channel2 (initial ISI = 088) the channel parameters weredetermined according to
ℎ119899= (04851 minus072765 minus04851) (19)
The equalizer was initialized by setting the center tap equal toone and all others to zero
Figure 4 A comparison between the simulated and calculated residual ISI for the 16QAM source input going through channel1 for SNR = 10[dB]The averaged results were obtained in 100Monte Carlo trialsThe equalizerrsquos tap length and step-size parameter were set to 27 and 00006respectively
Figure 5 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
In the following we denote the residual ISI performanceaccording to (6) with (7) as ldquoCalculated ISIrdquo Figure 2 toFigure 8 show the ISI performance as a function of theiteration number of our proposed expression (6) with (7)for the achievable residual ISI compared with the simulatedresults for two different channels and equalizerrsquos tap length
Figure 6 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00006 respectively
and various values for 119867 SNR and step-size parameterAccording to Figures 2 3 5 6 7 and 8 a high correlation isobserved between the simulated and calculated results evenfor 119867 = 09 According to Figure 4 the calculated ISI maybe considered as an upper limit for the simulated results for119867 gt 05
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
(1) The convolutional noise 119901[119899] is a zero mean whiteGaussian process with variance 1205902
119901= 119864[119901[119899]119901
lowast[119899]]
The real part of119901[119899] is denoted as119901119903[119899] and119864[1199012
119903[119899]] =
119898119901
(2) The source signal 119909[119899] is a rectangular QuadratureAmplitude Modulation (QAM) signal (where the realpart of 119909[119899] is independent of the imaginary part of119909[119899]) with known variance and higher moments
(3) The convolutional noise 119901[119899] and the source signal areindependent
(4) The gain between the source and equalized outputsignal is equal to one
(5) The convolutional noise 119901[119899] is independent of 119908[119899](6) The added noise is fGn with zero mean(7) The channel ℎ[119899] has real coefficients(8) The Hurst exponent is in the range of 05 le 119867 lt 1
The residual ISI expressed in dB units may be defined as
ISI = 10 log10(119898119901) minus 10 log
10(1205902
119909119903
) (6)
where119898119901is defined by
119898119901= 1205902
119908119903
(120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
times (2(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
minus120583(1198731205902
119909
119896=119877minus1
sum
119896=0
ℎ2
119896[119899] +
1198731205902
119909
SNR)
2
)
minus1
)
= 1205902
119908119903
120583 (1198731205902
119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
2 minus 120583 (1198731205902119909sum119896=119877minus1
119896=0ℎ2
119896[119899] + (119873120590
2
119909SNR))
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic(119873 minus 1)119867 (2119867 minus 1)]
(7)
and119877 is the channel length1205902119908119903
is the variance of119908119903[119899] (119908
119903[119899]
is the real part of119908[119899]) and SNR is given by SNR = 1205902119909119903
1205902
119908119903
=
1205902
1199091205902
119908
Comments It should be pointed out that assumptions (1)ndash(5)from above are precisely the same assumptions made in [214]
Proof Let us first recall the expression for the adaptationmechanism of the equalizer given in (5) Then we substitute(2) into (5) and obtain
According to [13 14] when the equalizer has convergedwe may assume that 119864[Δ(1199012
119903)] cong 0 Therefore by setting
119864[Δ(1199012
119903)] = 0 into (9) we obtain
minus 2120583119898119901119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]]
+ 1205832(119898119901+ 1205902
119908119903
)
times 119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
= 0
dArr
119898119901= 1205902
119908119903
(120583119864[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
times (2119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] minus 120583119864
times[
[
(
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898])
2
]
]
)
minus1
)
(11)
In [13] the expression 119864[(sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898])
2
]
was approximated as (119864[sum119898=119873minus1119898=0
119910[119899 minus 119898]119910lowast[119899 minus 119898]])
2It should be pointed out that this approximation fits theMPSK case where QPSK is a special case of it Howeversatisfying results were obtained in [13] for the 16QAM and64QAM cases in spite of the fact that the above mentionedapproximation was applied Thus it makes sense to use thesame approximation also here for our case The expression119864[sum119898=119873minus1
119898=0119910[119899 minus 119898]119910
lowast[119899 minus 119898]]may be written as
119864[
119898=119873minus1
sum
119898=0
119910 [119899 minus 119898] 119910lowast[119899 minus 119898]] = 119873120590
Figure 2 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 10 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
Next we turn to find a closed-form approximated expressionfor1205902119908119903
The real part of119908[119899] namely119908119903[119899] may be expressed
as
119908119903 [119899] =
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896] (13)
Thus the variance of 119908119903[119899] is given by
1205902
119908119903
= 119864[
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896]
119898=119873minus1
sum
119898=0
119888119898 [119899] 119908119903 [119899 minus 119898]]
=
119896=119873minus1
sum
119896=0
119898=119873minus1
sum
119898=0
119888119896 [119899] 119888119898 [119899] 119864 [119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(14)
which can be also written as
1205902
119908119903
= 1205902
119908119903
119896=119873minus1
sum
119896=0
1198882
119896[119899]
+
119896=119873minus1
sum
119896=0119896 =119898
119898=119873minus1
sum
119898=0119896 =119898
119888119896 [119899] 119888119898 [119899]119864[119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(15)
According to [2] expression (15) can be approximatelywritten as
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic119873 minus 1119867 (2119867 minus 1)] (16)
Figure 3 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 8 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
by using assumption (4) from the system description sectionassumptions (4) and (6)ndash(8) from this section the Holderinequality [20] and the following approximation [21]
05 [(|119898 minus 119896| minus 1)2119867minus 2(|119898 minus 119896|)
2119867+ (|119898 minus 119896| + 1)
2119867]
≃ 119867 (2119867 minus 1) |119898 minus 119896|2119867minus2
(17)
Now by substituting (12) and (16) into (11) we obtain (7)Thiscompletes our proof
4 Simulation
In this section we test our new proposed expression for theresidual ISI for the 16QAM case (a modulation using plusmn1 3levels for in-phase and quadrature components) with thealgorithm described in (5) for different SNR step-size andequalizerrsquos tap length values and for two different channeltypesThe following two channels were considered Channel1(initial ISI = 044) the channel parameters were determinedaccording to [22]
ℎ119899= (0 for 119899 lt 0 minus04 for 119899 = 0
times 084 sdot 04119899minus1 for 119899 gt 0)
(18)
Channel2 (initial ISI = 088) the channel parameters weredetermined according to
ℎ119899= (04851 minus072765 minus04851) (19)
The equalizer was initialized by setting the center tap equal toone and all others to zero
Figure 4 A comparison between the simulated and calculated residual ISI for the 16QAM source input going through channel1 for SNR = 10[dB]The averaged results were obtained in 100Monte Carlo trialsThe equalizerrsquos tap length and step-size parameter were set to 27 and 00006respectively
Figure 5 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
In the following we denote the residual ISI performanceaccording to (6) with (7) as ldquoCalculated ISIrdquo Figure 2 toFigure 8 show the ISI performance as a function of theiteration number of our proposed expression (6) with (7)for the achievable residual ISI compared with the simulatedresults for two different channels and equalizerrsquos tap length
Figure 6 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00006 respectively
and various values for 119867 SNR and step-size parameterAccording to Figures 2 3 5 6 7 and 8 a high correlation isobserved between the simulated and calculated results evenfor 119867 = 09 According to Figure 4 the calculated ISI maybe considered as an upper limit for the simulated results for119867 gt 05
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
Figure 2 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 10 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
Next we turn to find a closed-form approximated expressionfor1205902119908119903
The real part of119908[119899] namely119908119903[119899] may be expressed
as
119908119903 [119899] =
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896] (13)
Thus the variance of 119908119903[119899] is given by
1205902
119908119903
= 119864[
119896=119873minus1
sum
119896=0
119888119896 [119899] 119908119903 [119899 minus 119896]
119898=119873minus1
sum
119898=0
119888119898 [119899] 119908119903 [119899 minus 119898]]
=
119896=119873minus1
sum
119896=0
119898=119873minus1
sum
119898=0
119888119896 [119899] 119888119898 [119899] 119864 [119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(14)
which can be also written as
1205902
119908119903
= 1205902
119908119903
119896=119873minus1
sum
119896=0
1198882
119896[119899]
+
119896=119873minus1
sum
119896=0119896 =119898
119898=119873minus1
sum
119898=0119896 =119898
119888119896 [119899] 119888119898 [119899]119864[119908119903 [119899 minus 119896]119908119903 [119899 minus 119898]]
(15)
According to [2] expression (15) can be approximatelywritten as
1205902
119908119903
cong
1205902
119909119903
SNRsum119896=119877minus1119896=0
ℎ2
119896[119899]
[1 + radic119873 minus 1119867 (2119867 minus 1)] (16)
Figure 3 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel1 forSNR = 8 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
by using assumption (4) from the system description sectionassumptions (4) and (6)ndash(8) from this section the Holderinequality [20] and the following approximation [21]
05 [(|119898 minus 119896| minus 1)2119867minus 2(|119898 minus 119896|)
2119867+ (|119898 minus 119896| + 1)
2119867]
≃ 119867 (2119867 minus 1) |119898 minus 119896|2119867minus2
(17)
Now by substituting (12) and (16) into (11) we obtain (7)Thiscompletes our proof
4 Simulation
In this section we test our new proposed expression for theresidual ISI for the 16QAM case (a modulation using plusmn1 3levels for in-phase and quadrature components) with thealgorithm described in (5) for different SNR step-size andequalizerrsquos tap length values and for two different channeltypesThe following two channels were considered Channel1(initial ISI = 044) the channel parameters were determinedaccording to [22]
ℎ119899= (0 for 119899 lt 0 minus04 for 119899 = 0
times 084 sdot 04119899minus1 for 119899 gt 0)
(18)
Channel2 (initial ISI = 088) the channel parameters weredetermined according to
ℎ119899= (04851 minus072765 minus04851) (19)
The equalizer was initialized by setting the center tap equal toone and all others to zero
Figure 4 A comparison between the simulated and calculated residual ISI for the 16QAM source input going through channel1 for SNR = 10[dB]The averaged results were obtained in 100Monte Carlo trialsThe equalizerrsquos tap length and step-size parameter were set to 27 and 00006respectively
Figure 5 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
In the following we denote the residual ISI performanceaccording to (6) with (7) as ldquoCalculated ISIrdquo Figure 2 toFigure 8 show the ISI performance as a function of theiteration number of our proposed expression (6) with (7)for the achievable residual ISI compared with the simulatedresults for two different channels and equalizerrsquos tap length
Figure 6 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00006 respectively
and various values for 119867 SNR and step-size parameterAccording to Figures 2 3 5 6 7 and 8 a high correlation isobserved between the simulated and calculated results evenfor 119867 = 09 According to Figure 4 the calculated ISI maybe considered as an upper limit for the simulated results for119867 gt 05
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
Figure 4 A comparison between the simulated and calculated residual ISI for the 16QAM source input going through channel1 for SNR = 10[dB]The averaged results were obtained in 100Monte Carlo trialsThe equalizerrsquos tap length and step-size parameter were set to 27 and 00006respectively
Figure 5 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 13 and 00006 respectively
In the following we denote the residual ISI performanceaccording to (6) with (7) as ldquoCalculated ISIrdquo Figure 2 toFigure 8 show the ISI performance as a function of theiteration number of our proposed expression (6) with (7)for the achievable residual ISI compared with the simulatedresults for two different channels and equalizerrsquos tap length
Figure 6 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00006 respectively
and various values for 119867 SNR and step-size parameterAccording to Figures 2 3 5 6 7 and 8 a high correlation isobserved between the simulated and calculated results evenfor 119867 = 09 According to Figure 4 the calculated ISI maybe considered as an upper limit for the simulated results for119867 gt 05
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
Figure 7 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00004 respectively
Figure 8 A comparison between the simulated and calculatedresidual ISI for the 16QAM source input going through channel2 forSNR = 12 [dB] The averaged results were obtained in 100 MonteCarlo trials The equalizerrsquos tap length and step-size parameter wereset to 27 and 00002 respectively
5 Conclusion
In this paper we proposed a closed-form approximatedexpression (or an upper limit) for the residual ISI obtainedby nonblind adaptive equalizers valid for the fGn input casewhere the Hurst exponent is in the region of 05 le 119867 lt
1 This new obtained expression depends on the step-sizeparameter equalizerrsquos tap length input signal statistics SNRchannel power and theHurst exponent parameter According
to simulation results a high correlation is obtained betweenthe calculated and simulated results for the residual ISI forsome cases while for others the new obtained expression is arelative tight upper limit for the averaged residual ISI results
References
[1] M Pinchas ldquoTwo blind adaptive equalizers connected in seriesfor equalization performance improvementrdquo Journal of Signaland Information Processing vol 4 no 1 pp 64ndash71 2013
[2] M Pinchas ldquoResidual ISI obtained by blind adaptive equalizersand fractional noiserdquo Mathematical Problems in Engineeringvol 2013 Article ID 972174 11 pages 2013
[3] M Pinchas andB Z Bobrovsky ldquoAmaximumentropy approachfor blind deconvolutionrdquo Signal Processing vol 86 no 10 pp2913ndash2931 2006
[4] M Pinchas and B Z Bobrovsky ldquoA novel HOS approachfor blind channel equalizationrdquo IEEE Transactions on WirelessCommunications vol 6 no 3 pp 875ndash886 2007
[5] M Pinchas Blind Equalizers by Techniques of Optimal Non-Linear FilteringTheory VDMVerlagsservice GesellschaftmbH2009
[6] M Pinchas The Whole Story Behind Blind Adaptive Equaliz-ersBlind Deconvolution Bentham Science Publishers 2012
[7] G H Im C J Park and H C Won ldquoA blind equalization withthe sign algorithm for broadband accessrdquo IEEE Communica-tions Letters vol 5 no 2 pp 70ndash72 2001
[8] D N Godard ldquoSelf recovering equalization and carrier trackingin two-dimenional data communication systemrdquo IEEE Transac-tions on Communications Systems vol 28 no 11 pp 1867ndash18751980
[9] M Reuter and J R Zeidler ldquoNonlinear effects in LMS adaptiveequalizersrdquo IEEE Transactions on Signal Processing vol 47 no6 pp 1570ndash1579 1999
[10] A H I Makki A K Dey andM A Khan ldquoComparative studyon LMS and CMA channel equalizationrdquo in Proceedings of theInternational Conference on Information Society (i-Society rsquo10)pp 487ndash489 June 2010
[11] E Tucu F Akir and A Zen ldquoNew step size control techniquefor blind and non-blind equalization algorithmsrdquo Radioengi-neering vol 22 no 1 p 44 2013
[13] M Pinchas ldquoA closed approximated formed expression for theachievable residual intersymbol interference obtained by blindequalizersrdquo Signal Processing vol 90 no 6 pp 1940ndash1962 2010
[14] M Pinchas ldquoA new closed approximated formed expression forthe achievable residual ISI obtained by adaptive blind equalizersfor the noisy caserdquo in Proceedings of the IEEE InternationalConference on Wireless Communications Networking and Infor-mation Security (WCNIS rsquo10) pp 26ndash30 Beijing China June2010
[15] J Beran Statistics for Long-Memory Processes vol 61 of Mono-graphs on Statistics and Applied Probability Chapman and HallNew York NY USA 1994
[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012
[17] M Li ldquoFractal time seriesmdasha tutorial reviewrdquo MathematicalProblems in Engineering vol 2010 Article ID 157264 26 pages2010
Mathematical Problems in Engineering 7
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
[18] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012
[19] GMalik andA S Sappal ldquoAdaptive equalization algorithms anoverviewrdquo International Journal of Advanced Computer Scienceand Applications vol 2 no 3 2011
[20] M R SpiegelMathematical Handbook of Formulas and TablesSchaumrsquos Outline Series McGraw-Hill New York NY USA1968
[21] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013
[22] O Shalvi and E Weinstein ldquoNew criteria for blind decon-volution of nonminimum phase systems (channels)rdquo IEEETransactions on Information Theory vol 36 no 2 pp 312ndash3211990
