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  • MAI and Noise Constrained LMS Algorithm forMIMO CDMA Linear Equalizer

    Khalid Mahmood1, Syed Muhammad Asad2, Muhammad Moinuddin3, Waqas Imtiaz41,4Iqra National Universty Peshawar, Pakistan

    2University of Hafr Al Batin, Kingdom of Saudi Arabia3Center of Excellence in Intelligent Engineering Systems (CEIES) King Abdul Aziz University, Jeddah

    Kingdom of Saudi Arabia

    AbstractThis paper presents a constrained least meansquared (LMS) algorithm for MIMO CDMA linear equalizer ispresented, which is constrained on spreading sequence length,number of subscribers, variances of the Gaussian noise andthe multiple access interference (MAI) plus the additive noise(introduced as a new constraint). The novelty of the proposedalgorithm is that MAI and MAI plus noise variance has neverbeen used as a constraint in MIMO CDMA systems. Convergenceanalysis is performed for the proposed algorithm in case whenstatistics of MAI and MAI plus noise are available. Simulationresults are presented to compare the performance of the proposedconstrained algorithm with other constrained algorithms and itis proved that the new algorithm has outperformed the existingconstrained algorithms.

    Index TermsLeast mean squared (LMS), multiple input,multiple output (MIMO), linear equalizer, multiple access in-terference (MAI), Variance, AWGN, adaptive algorithm

    I. INTRODUCTIONIt is shown in the literature that performance of an adaptive

    algorithm may be enhanced if partial knowledge of a particularchannel is blended in the algorithm design[1], [2]. Based onthis idea, [1] presented an algorithm (Noise-constrained LMS)for identification and tracking of finite impulse response (FIR)channels using the variance of the receiver noise. An advantageof the noise constrained least mean squared (NCLMS) is thatit outperforms the LMS algorithm while keeping the samecomputational complexity. A complementary pair LMS (CP-LMS) [3] was initiated by using constrained optimizationtechnique named augmented Lagrangian (AL) which can beutilized to solve the problem of selecting an appropriate updatestep-size in LMS algorithm. This technique was utilized in [4],in which the knowledge of the variance of MAI plus noise wasincorporated to develop what is called the constrained LMSalgorithm (MNCLMS) for CDMA systems.

    Since MAI and the additive white Gaussian noise (AWGN)effect performance of CDMA multi user, multi-antenna en-vironment, it is imperative to design a receiver architecturewhich will negate the effect of MAI and additive noise. Thisrequires a MIMO implementation of the MNCLMS algorithmpresented in [4]. The proposed constrained algorithm is de-veloped by incorporating MIMO MAI and noise variancesthus resulting in a generalized MIMO MAI plus noise con-strained LMS (MIMO-MNCLMS) adaptive algorithm. As it is

    generalized, we can deduce MAI constrained algorithm, noiseconstrained algorithm and zero constrained noise algorithm asspecial cases.

    This paper is organized as follows. After introductoryremarks, section 2 presents the motivation for the algorithmdevelopment. Algorithm development is presented in section3 whereas section 4 deals with the convergence analysis.Computational complexity of the algorithm is given in section6. In order to find the theoretical findings, simulation resultsare presented in Section 7 . Concluding remarks are shown insection 8.


    Adaptive algorithms such as LMS and RLS do not usemodels for channel coefficients and/or noise, whereas modelbased algorithms utilize various types of models such asrandom walk, auto-regressive etc. for coefficients and AWGN.Model parameters are either known or jointly estimated witha channel. Adaptive algorithms can be inferred to as modelbased algorithms with model parameters choice dependent ondata [5]. It has been reported in the literature that practicallyit is possible to enhance the performance of an adaptivealgorithm if partial knowledge of the channel is availableprovided that the computational cost of an algorithm is notincreased tremendously. According to the noise constrainedLMS algorithm [6]

    (1)wn+1 = wn + lnenXn

    (2)n+1 = 2 ln(1 + n)

    (3)n+1 = n + [(


    e2n 2 lm) n]

    Where , and are positive constants. This is a variablestep size (LMS) algorithm because step size rule is applicabledue to the constraint on the noise variance. The computationalcost of the aforementioned algorithm is the same as of LMSbut the convergence rate of the noise constrained LMS algo-rithm is much faster than the LMS due to its three independentparameters.

    (IJACSA) International Journal of Advanced Computer Science and Applications,

    Vol. 7, No. 1, 2016

    702 | P a g

  • MAI is the major limiting factor in the system performanceof a multiuser environment, it is required to design a multireceiver scheme which will negate the effect of MAI andthe additive noise. Previous research work treated MAI as apart of interfering noise. This assumption is not practicallycorrect which led this work to use MAI as an additionalconstraint by using structured information contained in it. Wealso believe that by using the combined information of MAIand the interfering noise to form into a single constraint wouldresult in an algorithm which would outperform the noise onlyconstrained algorithm. It is worth mentioning here that byusing MAI alone as a constraint is not a viable choice sincenoise is an undeniable physical constraint and may not beignored while developing such algorithms.

    As NCLMS algorithm is noise constrained only, a newconstrained algorithm is established, by incorporating MAIvariance as well as the noise variance, thus resulting in a gener-alized MAI plus noise constrained LMS (MNCLMS) adaptivealgorithm. Since this proposed algorithm is generalized, thisalgorithm is able to deduce MAI constrained, noise constrainedand zero constrained noise algorithms as special cases.

    III. ALGORITHM DEVELOPMENTIn MIMO-CDMA system with N transmitting and M receiv-

    ing antennas, output of mth matched-filter, matched to intendedsubscribers (subscriber 1) spreading sequence consists of thedesired users component, blm, MIMO-MAI, U

    lm and the white

    Gaussian noise, lm as

    (4)ylm = b

    lm +U

    lm +


    = blm + Zlm

    where ylm is the output of the matched filter and Zlm is the

    MIMO MAI and noise at the mth receiving antenna withvariance 2Zm . The statistical characterization (variance) ofMIMO CDMA MAI and noise is given in [7]. The MIMOCDMA MAI is given by

    (5)Zlm =





    AKbl,kn K,1hlmn +


    = U lm + lm

    In 5, K is the number of subscribers, whereas sl,kn (t)represents the rectangular signature waveform with a randomsignature sequence of kth subscriber defined as (l1)Tb t lTb, whereas Tb and Tc represent bit period and the chipintervals respectively related by Nc = Tb/Tc. b

    l,kn is the input

    bit stream of the kth subscriber, hlmn is the lth channel tap

    between the nth transmitter and the mth receiver, Ak is thetransmitted amplitude of the kth subscriber and m is the AWGnoise having zero mean and variance 2 at the mth receiver.The cross-correlation between the signature sequences ofsubscriber j and k for the lth symbol is given as

    (6)k, jl =


    skn (t)sjn (t)dt




    where ckl,i is the normalized spreading sequence of the sub-scriber k for the lth symbol.

    The MIMO CDMA MAI consists of two random variablesgiven as

    (7)U lm =








    C hlmn,

    where C is MAI in AWGN environment [8]. The desiredsubscribers component is written as

    blm =[(

    hlm1)T (


    . . .(


    xl1)T (


    . . .(



    In (8), hlm1 =[hlm1 h

    l1m1 . . . h


    ]Tis the time varying

    impulse response (TVIR) of the channels and is an L 1vector. xln =

    [xln x

    l1n . . .x


    ]T is an L1 vector. The filterimpulse response (FIR) of the LE which consists of an FFFis given by

    (9)wln =[(

    F ln)T]

    In (9), F ln is the nth multiple input, single output (MISO)

    FFF with dimension of ML 1, where L is the taps of FFFand M is the number of receivers.

    The mean-squared error MSE) to be minimized is



    where eln is the error between output of a matched filter andan adaptive filter and is shown to be

    (11)eln = xn wlnDlnhere, Dln is the combined input to the LE and is given by

    (12)Dln =[(


    and is of the order of (ML1).

    (13)yln =[(

    yl1)T (

    yl2)T (


    is the input to the FFF of dimension ML1 and is a collectionof vectors consisting of ylm given by

    (14)ylm =[ylm y

    l1m y




    (15)xn = wTo D


    = xln + ln


    (16)xln = wTo D

    ln ln

    (IJACSA) International Journal of Advanced Computer Science and Applications,

    Vol. 7, No. 1, 2016

    703 | P a g

  • ln is the filtered noise which passes through FFF and iscomposed of MAI and noise.

    Minimizing cost function in (10) over wln will give anappropriate value at time l. In other words, wopt = H lmn(of size1NL matrix) with J(opt) = 2 ln represents MSE. It is knownthat knowledge of 2 ln [6] is helpful in selecting the searchdirection for an adaptive algorithm in multisubscriber casewhich is quite similar to the MNCLMS algorithm in singlesubscriber environment.

    To minimize J(wln) over wln subject to the constraintJ(wln) = 2 ln ,the Lagrangian of this is given by

    (17)J1(wln,ln) = J(w

    ln) +


    [J(wln) 2


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