mai and noise constrained lms algorithm for mimo … and noise constrained lms algorithm for mimo...

MAI and Noise Constrained LMS Algorithm forMIMO CDMA Linear Equalizer

Khalid Mahmood1, Syed Muhammad Asad2, Muhammad Moinuddin3, Waqas Imtiaz4

1,4Iqra National Universty Peshawar, Pakistan2University of Hafr Al Batin, Kingdom of Saudi Arabia

3Center of Excellence in Intelligent Engineering Systems (CEIES) King Abdul Aziz University, Jeddah

Kingdom of Saudi Arabia

Abstract—This 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 Terms—Least 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.

II. MOTIVATION

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 + µlnenXn

(2)µn+1 = 2µln(1 + γλn)

(3)λn+1 = λn + β

[(

12

e2n − σ

2ν l

m)− λn

]Where λ , α and β are positive constants. This is a variable

step 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 ewww.ijacsa.thesai.org

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 intendedsubscriber’s (subscriber 1) spreading sequence consists of thedesired user’s component, bl

m, MIMO-MAI, U lm and the white

Gaussian noise, ν lm as

(4)ylm = bl

m +U lm + ν

lm

= blm + Zl

m

where ylm is the output of the matched filter and Zl

m is theMIMO MAI and noise at the mth receiving antenna withvariance σ2

Zm. The statistical characterization (variance) of

MIMO CDMA MAI and noise is given in [7]. The MIMOCDMA MAI is given by

(5)Zlm =

N

∑n=1

K

∑K=2

AKbl,kn ρ

K,1hlmn + ν

lm

= U lm + ν

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 (l−1)Tb ≤t ≤ lTb, whereas Tb and Tc represent bit period and the chipintervals respectively related by Nc = Tb/Tc. bl,k

n is the inputbit stream of the kth subscriber, hl

mn is the lth channel tapbetween the nth transmitter and the mth receiver, Ak is thetransmitted amplitude of the kth subscriber and νm is the AWGnoise having zero mean and variance σ2

ν at the mth receiver.The cross-correlation between the signature sequences ofsubscriber j and k for the lth symbol is given as

(6)ρ

k, jl =

∫ lTb

(l−1)Tb

skn (t)s j

n (t)dt

=Nc

∑i=1

ckl,ic

jl,i

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 variables

given as

(7)U l

m =N

∑n=1

K

∑K=2

AKbl,Kρ

K,1hlmn

=N

∑n=1

C hlmn,

where C is MAI in AWGN environment [8]. The desiredsubscriber’s component is written as

blm =

[(hl

m1

)T (hl

m2

)T. . .(

hlmn

)T]×[(

xl1

)T (xl

2

)T. . .(

xlN

)T]T

(8)

In (8), hlm1 =

[hl

m1 hl−1m1 . . . hl−L+1

m1

]Tis the time varying

impulse response (TVIR) of the channels and is an L× 1vector. xl

n =[xl

n xl−1n . . .xl−L+1

n]T is an L×1 vector. The filter

impulse response (FIR) of the LE which consists of an FFFis given by

(9)wln =

[(F l

n

)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

(10)J(

wln

)=[el

n

]2

where eln is the error between output of a matched filter and

an adaptive filter and is shown to be

(11)eln = xn − wl

nDln

here, Dln is the combined input to the LE and is given by

(12)Dln =

[(yl

n

)T]

and is of the order of (ML×1).

(13)yln =

[(yl

1

)T (yl

2

)T (ylM

)T]T

is the input to the FFF of dimension ML×1 and is a collectionof vectors consisting of yl

m given by

(14)ylm =

[yl

m yl−1m yl−L+1

m

]T

.and

(15)xn = wTo Dl

n

= xln + ν

ln

or

(16)xln = wT

o Dln − ν

ln


Vol. 7, No. 1, 2016


ν ln is the filtered noise which passes through FFF and is

composed of MAI and noise.Minimizing cost function in (10) over wl

n will give anappropriate value at time l. In other words, wopt = H l

mn(of size1×NL matrix) with J(opt) = σ2

ν ln

represents MSE. It is knownthat knowledge of σ2

ν 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 wl

n subject to the constraintJ(wl

n) = σ2ν l

n,the Lagrangian of this is given by

(17)J1(wln,λ

ln) = J(wl

n) + λln

[J(wl

n)− σ2ν l

n

]Since critical values of λ l

n are not unique in this case so weare using an augmented Lagrangian to get rid of this issue bydefining the under mentioned cost function

(18)J2(wln,λ

ln) = J(wl

n + γλln

[J(wl

n)− σ2ν l

n

]− γnλ

2n

Solution to (18) is given by using the method used in [6]and is shown to be

(19)wl+1n = wl

n + µlnel

nDln

where µ ln is the positive step size and is given by

µln = µn

(1+ γnλ

ln

),

m = 1,2, . . . ,M (20)

λl+1n = λn +βn

[12(el2

n −σ2ν l

n)−λ

ln

],

m = 1,2, , . . . ,M (21)

Since channel taps are independent of spreading sequenceas well as data sequence, the interferer’s components are alsoindependent of one another having zero mean.

The variance of MAI for an equal power can be written as

(22)U lm = A2

(K − 1

NC

) N

∑n=1

E[h2

mn]

whereas the variance of MAI in the unequal transmittedpower is given by

(23)U lm =

k

∑k=2

(Ak)2

NC

N

∑n=1

E[h2

mn]

In (22) and (23), E[h2

mn]

is the second moment of E [hmn].

IV. CONVERGENCE ANALYSIS

In this section, convergence behavior of our proposed algo-rithm in the presence of MAI and noise for an LE receiver isanalyzed. Following independent assumptions have been usedwhile performing the convergence analysis [9].

Assumption 1: The input process

xln

is an independentand identically distributed (i.i.d) random variable with autocorrelation matrix of Rxx = E

[xT

n xn].

Assumption 2: The noise sequence is a zero mean i.i.d.sequence, Gaussian random variable having variance σ2

νn . Thissequence is independent of the input process.

Assumption 3: MAI in AWGN environment represented byU l

m is zero mean Gaussian random variable with variance σ2Um

.It is is independent of the input process as well as the noise.

Assumption 4: For any fixed time. say l, step size µ ln and

weight vector vln (defined later) are statistically independent

of each another.

The above mentioned assumptions are justified as follows

Although assumption 1 is not true in reality but is commonlyused in literature and it has been shown in the literature thatsimulation results under this condition closely match with theanalytical results.

Assumption 2 termed as independent assumption, is alsocommonly used in the literature and could be justified in manycases.

Gaussian appropriation of MAI in AWGN is extensivelyused in the adaptive filtering literature [10]. Since MAI is in-dependent of the noise process, assumption 3 can be validated.In this way, MAI and noise is independent of the input processas well.

Generally µ ln and vl

n are dependent but if the parameters arechosen in such a way that steady-state variance of µ l

n and orvl

n is small, assumption 4 can be justified.xk and xn are also independent as both are uncorrelated for

n 6= k. Thus the input vector xn and the weight error vector(defined later)vl

n are also independent [11].Weight update equation of the proposed algorithm is written

as(24)wl+1

n = wln + µ

lnel

nDln

Where n = 1,2........,N represents the nth stream.If wn(opt) is said to be the optimum value of weight

according to the Wiener solution then weight error vector vln

can be defined as

(25)vln = wn(opt) − wl

n

Subtracting wn(opt) from both sides of (19) results in

vl+1n + wn(opt) = −wl

n + wn(opt) − µlnel

nDln

(26)vl+1n = vl

n − µlnel

nDln

The desired response of the decision device(xl

n)can be

expressed as(27)xl

n = wT(opt)D

ln

By using 27, it can be shown in case of system identificationscenario, el

n could be setup as below:

(28)eln = wT

(opt) − (wln)

T

For n = 1,2, . . .N.(28) can be compactly written as

(29)el

n =(

vln

)TDl

n

=(

Dln

)Tvl

n


Vol. 7, No. 1, 2016


Now the recursion of weight error vector can be shown as

(30)vl+1n = vl

n − µlnDl

n

(Dl

n

)Tvl

n

(31)vl+1n = I− µ

lnDl

n

(Dl

n

)Tvl

n

Taking expectation on both sides of (31) with the assump-tions made earlier, will yield

(32)vln =

[I− µ l

nE

Dln

(Dl

n

)T]

vln

µ ln = E

[µ l

n]

is the mean step size and vln = E

[vl

n]

is theweight error vector in the equation above. Whereas

E[

Dln

(Dl

n

)T]= E

[(yl

n

)T][(

yln

)T]T

or

(33)R = E[(

yln

)T]T [(

yln

)T]

or

(34)R = E[

Dln

(Dl

n

)T]

where R is the correlation matrix of input process

A. Auto-correlation Structure of MIMO-CDMA Linear Equal-izer (LE)

The correlation matrix R is given by

(35)R = RDlnDl

n

B. Eigenvalues of Linear Equalizer (LE)If ϒ1 , ϒ2 , ϒ3 . . .ϒN represent the eigenvalues of R, then

the an important condition for the convergence in the mean isgiven as ∣∣∣1− µ

lnϒN

∣∣∣ < 1,

The value of µ ln is bounded in the range

(36)0 < µ

ln

<2

ϒmax

where ϒmax is the maximum eigenvalue of R. By using astrong and a much simpler condition for convergence ofmean weight error vector [12] can be given as

(37)µmax <2

ϒmax

V. TRANSIENT ANALYSIS OF THE PROPOSED ALGORITHM

Transient analysis of an adaptive algorithm is very importantto observe the convergence behavior of an adaptive algorithmand to derive steady-state expressions for different error per-formance measures, such as EMSE and mean-square deviation(MSD). Basically, transient analysis is the observation ofthe time-evolution of the adaptive algorithms when there arevariations in the signal statistics; or in other words to studythe learning mechanism of an adaptive algorithm. Energyconservation method is used to carry out the transient analysis[9].

A. Error Measures

Transient analysis of an adaptive algorithm deals with thetime evaluation of of E

[∣∣eln∣∣2]and E

[‖vl

n‖2]

, where vln is the

weight error vector and is given by

(38)vln = wl

n(opt) − wln

For some symmetric positive definite weighting matrix Ω ,Wedefine, weighted a priori and a posteriori estimation errors as

(39)eΩan =

(vl

n

)TΩDl

n

and

(40)eΩpn =

(vl+1

n

)TΩDl

n

For the case when, Ω = I, the weighted a priori and aposteriori estimation errors defined above will be reducedto a standard a priori and a posteriori estimation errors,respectively, i.e

(41)elan =

(Dl

n

)Tvl

n

and

(42)elpn =

(Dl

n

)Tvl+1

n

It will be elaborated later in that different choices of Ω willyield different performance measures for the evaluation of anadaptive filter.

Since,

eln = xl

n −(

wln

)TDl

n

=(w(opt)

)T − ν ln −

(wl

n

)TDl

n

(43)eln =

(vl

n

)TDl

n − νln

= elan − ν l

n

Also, by using (40) and and (26) it can be shown that

(44)eΩ,lpn = eΩ,l

an − µln‖Dl

n‖2Ωel

n

B. Performance Measures

The EMSE of the proposed algorithm is given by

(45)EMSE = E[∣∣∣el

an

∣∣∣2]and steady state EMSE is

(46)E[∣∣∣el

an

∣∣∣2] = ξ∞


Vol. 7, No. 1, 2016


C. Fundamental Weighted Energy Relation

The fundamental weighted-energy conservation relation [9]is used in this section to develop the framework for thetransient analysis of the proposed MNCLMS algorithm. (44)can be expressed as

(47)eΩ,ln =

eΩ,nn − eΩ,l

pn

µ ln ‖Dl

n‖2Ω

By using (26) and (47) ,the following equation can beestablished

(48)vl+1n = vl

n −Dl

n

‖Dln‖

2Ω

[eΩ,l

an − eΩ,ln

]The fundamental weighted-energy conservation relation can

be shown as

(49)vl+1n +

1

‖Dln‖

2Ω

∣∣∣eΩan

∣∣∣2 =∥∥∥vl

n

∥∥∥2

Ω

+1

‖Dln‖

2Ω

∣∣∣el,Ωpn

∣∣∣249 describes, how the weighted energies of the error quan-

tities evolve with time. Different choices of Ω will yield dif-ferent performance measures for the evaluation of an adaptivefilter[9].

D. Time Evolution of the Weighted Variance

This section deals with derivation of time evolution of theweighted variance for the proposed MNCLMS algorithm usingthe fundamental weighted-energy conservation relation equa-tion (49). Substituting (44) into (49) and taking expectationon both sides will result in

E[∥∥∥vl+1

n

∥∥∥2

Ω

]= E

[∥∥∥vln

∥∥∥2

Ω

]−

2µ lnE[el,Ω

an en

]+(µ l

n)2E[∥∥∥Dl

n

∥∥∥2

Ω

e2n

](50)

where (µ ln)

2 is E[(µ l

n)2]. Next, the expectations in the

second and third terms on the right hand side of equation(50) is evaluated by using the following assumption

A5): For any constant matrix Ω and for all l, elan and eΩ

anare jointly Gaussian.

This assumption seems reasonable for longer filters usingthe central limit theorem [13], [14], [4]. So, E

[e

l,Ωan en

]can be

simplified as

E[el,Ω

an en

]= E

[(Dl

n

)TΩvl

n

(Dl

n

)TIvl

n

]=

E[(

vln

)T(

Ω

(Dl

n

)(Dl

n

)TI)

vln

]= E

[∥∥∥vln

∥∥∥2

ΩE[(Dl

n)(Dln)

T]]= E

[∥∥∥vln

∥∥∥2

ΩR

](51)

Now, E[∥∥Dl

n∥∥2

Ωe2

n

]is being solved by using the following

assumptionA6): The adaptive filter is long enough so that

∥∥Dln∥∥2

Ωand

e2n are uncorrelated [9].

This assumption is more realistic when the filter lengthgets longer[9]. As MAI plus noise is independent of Dl

n,expectation E

[∥∥Dln∥∥2

Ωe2

n

]can be simplified as

E[∥∥∥Dl

n

∥∥∥2

Ω

e2n

]= E

[∥∥∥Dln

∥∥∥2

Ω

]E[e2

n]=

E[∥∥∥Dl

n

∥∥∥2](

E[(

elan

)2]−σ

2ν l

n

)(52)

Now using (51) and (52) and, (50) can be written as

E[∥∥∥vl+1

n

∥∥∥2

Ω

]= E

[∥∥∥vln

∥∥∥2

Ω

]−2µ

lnE[∥∥∥vl

n

∥∥∥2

ΩR

]+

(µ ln)

2E[∥∥∥Dl

n

∥∥∥2

Ω

](ξ

ln−σ

2ν l

n

)(53)

(53) shows the time evaluation or the transient behaviorof the weighted variance E

[∥∥vln∥∥2

Ω

]for any constant weight

matrix Ω. Different performance measures can be achieved bythe proper choice of the weight matrix Ω.

E. Constructing the Learning Curves for the Excess MeanSquare Error (EMSE)

Learning curves for the EMSE can be constructed by usingEMSE = E

[(el

an)2]= E

[∥∥vln∥∥2

R

]. If Ω = I, R, . . . , RN−1, a

set of relations can be obtained. Now by using the Cayley-Hamilton theorem, following can be established

(54)Ω = −p0I − p1R− . . .− pN−1RN−1

where(55)p(x) , det

(xI − R

)= p0 + p1x + . . .+ pN−1xN−1 + xN

is the characteristic polynomial of R.Consequently,

E[∥∥∥vl+1

n

∥∥∥2

RN−1

]= E

[∥∥∥vln

∥∥∥2

RN−1

]+2µ l

n(p0E

[∥∥∥vln

∥∥∥2

I

]+ p1E

[∥∥∥vln

∥∥∥2

R

]+ . . .

+pN−1E[∥∥∥vl

n

∥∥∥2

RN−1

])+(µ l

n)2E[∥∥∥Dl

n

∥∥∥2

RN−1

](ξ

ln +σ

2ν l

n

)(56)

So,

(57)ϒln

=

1 −2µ ln 0 · · · 0 0

0 1 −2µ ln 0 · · · 0

... 0 1 −2µ ln · · ·

...0 0 0 1 −2µ l

n 00 0 · · · 0 1 −2µ l

n2µ l

n p0 2 ¯µ ln p1 2µ l

n p2 · · · 2µ ln pN−2 1 + 2µ l

n pN−1

And

(58)ϖln =

E[∥∥vl

n∥∥2

I

]E[∥∥vl

n∥∥2

R

]...

E[∥∥vl

n∥∥2

RN−1

]


Vol. 7, No. 1, 2016


Similarity

(59)aln =

E[∥∥Dl

n∥∥2

I

]E[∥∥Dl

n∥∥2

R

]...

E[∥∥Dl

n∥∥2]

By combining 57 , 58 and 59 ,

(60)ϖl+1n = ϒ

lnϖ

ln + (µ l

n)2(

ξln + σ

2ν l

n

)al

n

VI. STEADY-STATE ANALYSIS OF THE MNCLMSALGORITHMS

Steady-state analysis of an adaptive filter is performed tostudy the behavior of steady-state EMSE and MSD. Steady-state performance measures is also obtained by analyzing (50)when l→ ∞ which is presented in the next section i.e

(61)liml →∞

E[∥∥∥vl+1

n

∥∥∥2

Ω

]= lim

l→∞E[∥∥∥vl

n

∥∥∥2]

So at steady-state (53) will become

2µ∞n lim

l →∞E[∥∥∥vl

n

∥∥∥2

ΩR

]= (µ∞

n )2 lim

l→∞E[∥∥∥Dl

n

∥∥∥2

Ω

](ξ

∞n + σ

2ν l

n

)(62)

Where µ∞n , (µ∞

n )2and ξ ∞

n are steady-state values of µ ln,

(µ ln)

2and ξ ln respectively. Now using (20) and (21) it can be

shown that

(63)(

µln

)2= (µ l

n)2(

2γλln + γ

2(

λln

)2)

(λ

l+1n

)2= (1−β )2

(λ

ln

)2+β (1−β ) λ

lnξ

ln+

β 2

4

[E[∣∣∣el

n

∣∣∣4]−2σ

2ν l

n

(ξ

ln +σ

2ν l

n

)+σ

4ν l

n

](64)

(64) can be written compactly as

(65)(

λl+1n

)2= (1 + β )2 (λ l

n)2+ β (1− β ) λ

lnξ

ln +

β

2ξ

lnσ

2ν l

n

If we define mean Lagrangian multiplier as λ ln = E

[λ l

n], it

can be shown that at steady-state

(66)λln =

ξ ∞n

2

Similarly, if µ ln = E

[µ l

n], it can be shown that

(67)µ∞n = µ

(1 + γ

ξ ∞n

2

)(63) can be expressed as

(68)(µ∞n )

2 = µ2[1 + 2γλ

∞n + γ

2 (λ ∞n )2]

(65) is written as

(69)(λ ∞n )2 =

1(2− β )

[(1− β )

2(ξ ∞

n )2 + βξ∞n σ

2ν l

n

]

Using (62) with Ω = I,

2ξ∞n + γ (ξ ∞

n )2 = µTr(R)

ξ∞n +µTr

(R)

σ2ν l

n+

µγTr(R)(ξ ∞

n )2+

µγTr(R)

σ2ν l

nξ

∞n

+µγ2(

12−β

)(1−β

2

)Tr(R)(ξ ∞

n )3

+µγ2(

12−β

)(1−β

2

)Tr(R)

+µβTr(R)

σ2ν l

n(ξ ∞

n )2+

µβTr(R)

σ2ν l

nξ

∞n (70)

(70) is cubic in terms of ξ ∞n and can be expressed as

(71)A(ξ ∞n )3 + B(ξ ∞

n )2 +Cξ∞n + D = 0

In 71above,

(72)A = −µγ2Tr(R)( 1

2− β

)(1− β

2

)

(73)B = γ−µTr(R)[

γ +σ2ν l

n

(1

1− β

)(1− β

2

)+β

]

(74)C = 2− µTr(R)[

1 +(

γ + βσ2ν l

n

)σ

2ν l

n

]And

(75)D = −µTr(R)

σ2ν l

n

Now assuming that at steady-state, µTr(R) 1 [15], it

can be shown that at steady-state, the value of ξ ∞n is close

to zero meaning that higher powers of ξ ∞n can be ignored.

Consequently, (71) will become,

(76)Cξ∞n + D = 0

or(77)ξ

∞n =

−DC

Hence, steady-state EMSE of the proposed MNCLMS al-gorithm in the presence of MAI and noise can be shown tobe

(78)ξ∞

n(MNCLMS) ≈µTr

(R)

σ2ν l

n

2− µTr(R)[

1 +(

γ + βσ2ν l

n

)σ2

ν ln

]By using the same procedure developed above, steady-

state EMSE of the LMS, NCLMS, ZNCLMS, and MCLMSalgorithms in the presence of MAI and noise can be shown,respectively to be

(79)ξ∞

n(LMS) ≈µTr

(R)

σ2ν l

n

2

ξ∞NCLMS ≈

−(2 + γσ2

U

)+ 2µTr

(R)

σ2U

[1 + γσ4

U+ γ2

(1−β

2−β

)σ2

U+ σ4

U

](2 + γσ2

U

)− 2µTr

(R)

(80)


Vol. 7, No. 1, 2016


ξ∞ZNCLMS

≈−(

2 + γσ2ν l

n

)+ 2µTr

(R)

σ2ν l

n

[1 + γσ4

ν ln+ γ2

(1−β

2−β

)σ2

ν ln+ σ4

ν ln

](

2 + γσ2ν l

n

)− 2µTr

(R)

(81)

ξ∞MCLMS

≈−(

2 + γσ2ν l

n

)+ 2µTr

(R)

σ2ν l

n

[1 + γσ4

ν ln+ γ2

(1−β

2−β

)σ2

ν ln+ σ4

ν ln

](

2 + γσ2ν l

n

)− 2µTr

(R)

(82)

VII. TRACKING ANALYSIS OF THE MNCLMSALGORITHMS FOR THE RANDOM WALK CHANNEL IN THE

PRESENCE OF MAI AND NOISE

Tracking analysis of an adaptive filter is performed to studyits ability to track down the time variations in the channel.This is due to the fact that statistical properties of the weightvector and error signals are able to track the changes in theinput signal variation by relying on instantaneous data [9].In this section, tracking analysis of the proposed algorithm isperformed for a random walk model.

A. Random Walk Model

This section deals with the tracking analysis of the MN-CLMS algorithm performed for a random walk channel.A general framework for the tracking analysis of adaptivealgorithms is used in this section which can handle randomsystem nonstationarities [16] . This framework is based onan energy conservation relation and is valid for adaptivealgorithms whose recursion is of the form

(83)wl+1n = wl

n + µlnDl

n f(

eln

)Where f

(el

n)

represents a general scalar function of theoutput estimation error el

n. For an LMS algorithm f(el

n)= el

n.The random walk model for a channel is given by

(84)wl+1n = wl

o,n + qln

qln in (84) is assumed to be zero mean, i.i.d, with a positive

definite co-variance matrix E[qnqT

n]= Q and is also statis-

tically independent of the input regressor and the MAI plusnoise, whereas wo

n is the unknown system to be tracked. Fortracking analysis of an adaptive algorithm, a very importantmeasure is its steady-state tracking EMSE (ξ ∞

n ) defined as

(85)ξ

∞n = lim

l→∞E[∣∣∣el

an

∣∣∣2]= E

[∥∥∥vln

∥∥∥2

R

]Where vl

n = won −wl

n is the weight error vector for therandom walk channel.

B. Fundamental Energy Relation for the Random Walk Chan-nel

In this section, the fundamental energy conservation relation[9] is used to develop the framework for the tracking analysisof the proposed MNCLMS algorithm. By using (19) and (84),it can be shown that

(86)vl+1n = vl

n − µlnel

nDln + q

Consider the (83) , which is given by

(87)wl+1n = wl

n + µlnDl

n f(

eln

)Subtracting both sides of the (87) from wl+1

0,n

(88)(

wl+1o,n − wl+1

n

)=(

wl+1o,n − wl

n

)− µ

lnDl

n f(

eln

)In case of an LMS algorithm, f

(el

n)= el

n, so (88) becomes

(89)wl+1o,n − wl+1

n =(

wl+1o,n − wl

n

)− µ

lnDl

neln

Now we transpose both sides of (89)

(90)(

wl+1o,n − wl+1

n

)T=(

wl+1o,n − wl

n

)T− µ

lnel

n

(Dl

n

)T

By multiplying both sides of (90) with Dln from left yields(

wl+1o,n − wl+1

n

)TDl

n =(

wl+1o,n − wl

n

)TDl

n − µlnel

n

(Dl

n

)TDl

n

(91)

Equation (91) in terms of a priori error and a posteriorierrors can be expressed as

(92)elp,n = el

a,n − µlnel

n

∥∥∥Dln

∥∥∥2

For the case when Ω = I, 86 becomes

(93)vl+1n = vl

n − Dln

[el

an − elpn

‖Dln‖

2

]+ ql

n

By evaluating the energies on both sides of (93) We get

(94)∥∥∥vl+1

n − qln

∥∥∥2+

1

‖Dln‖

2

∣∣∣elan

∣∣∣2 =∥∥∥vl

nl∥∥∥2

+1

‖Dln‖

2

∣∣∣elpn

∣∣∣2Since ql

n is a zero mean stationary random vector and isindependent of the input regressor vector and the MAI plusnoise, so (94) , can be expressed as∥∥∥vl+1

n

∥∥∥2−∥∥∥ql

n

∥∥∥2+

1

‖Dln‖

2

∣∣∣elan

∣∣∣2 =∥∥∥vl

n

∥∥∥2+

1

‖Dln‖

2

∣∣∣elpn

∣∣∣2(95)

(95) is the random walk tracking model for MNCLMSalgorithm. The energy relation is used to evaluate the excessmean-square error at steady state.


Vol. 7, No. 1, 2016


TABLE ICOMPUTATIONAL COMPLEXITY PER ITERATION FOR DIFFERENTALGORITHMS FOR REAL VALUED DATA IN TERMS OF THE REAL

MULTIPLICATIONS, REAL ADDITIONS AND REAL DIVISIONS

Algorithm × +

LMS 2L+1 LRLS L2 +5L+1 L2 +3L

MNCLMS[4] 2L+1 2L+6MIMO-MNCLMS(LE) 2ML+8 2ML+4

C. Tracking Steady-State EMSE of the MNCLMS Algorithms

A7): qn is a zero-mean stationary random vector processwith a positive definite covariance matrix Q and is statisticallyindependent of the input regressor vector xl

n and MAI plusnoise Zl

m sequence .Taking expectation on both sides of equation (95) and using

equation (44) assumption A7) and the fact that at steady-state,vl+1

n = vln , the following equation can be obtained

(96)2E [µ∞n ξ

∞n ] = Tr (Q∞

n ) + E[(µ∞

n )2 ‖D∞

n ‖2 (e∞

n )2]

Where Tr (Q∞n ) =

[q∞

n (q∞n )

T]. By using equation (96) to-

gether with assumption A4 yields

(97)2µ∞n ξ

∞n = Tr (Q∞

n ) + (µ∞n )

2Tr(R)(

ξ∞n − σ

2ν l

n

)By substituting the values of µ∞

n , λ ∞n , (µ∞

n )2and (λ ∞

n )2 inequation (97) , we get the steady-state EMSE of the proposedalgorithm.

(98)ξ∞n ≈

µTr(R)

σ2ν l

n

2− µTr(R)[

1 +(

γ + βσ2ν l

n

)σ2

ν ln

] + Tr(

Q∞n

2µ

)

Since q is assumed to be an i.i.d, therefore Tr (Q∞n ) = Lσ2

q ,so equation (98) will be reduced to

ξ∞

n(MNCLMS) ≈µTr

(R)

σ2ν l

n

2− µTr(R)[

1 +(

γ + βσ2ν l

n

)σ2

ν ln

] + Lσ2q

2µ

(99)

Where L is the length of the filter.

VIII. COMPUTATIONAL COMPLEXITY OF LINEAREQUALIZER

Computational cost is an important aspect of any algorithm.Higher computational cost can render an algorithm useless.A trade off between computational cost and the performanceis possible, i.e. if the increased cost results in considerableperformance gain, then higher cost can be ignored. In thissection, we present the computational costs of few algorithms.As can be seen in table I, computational cost of the proposedalgorithm is higher when compared to [4] but MNCLMS isfor SISO- CDMA case, whereas, the proposed algorithm isfor MIMO CDMA case. This algorithm can be used for run-time applications such as channel estimation, tracking andchannel equalization. MIMO systems are being used in modernwireless standards.

0 100 200 300 400 500 600 700 800 900 1000−8

−7

−6

−5

−4

−3

−2

−1

0

iterations

MSE

(dB)

LMSMCLMSNCLMSZNCLMSMNCLMS

Fig. 1. MSE behavior for different algorithms in an AWGN environment withK = 10 at 20dB

0 100 200 300 400 500 600 700 800 900−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

iterations

MSE

(dB)


Fig. 2. MSE behavior for different algorithms in an AWGN environment withK = 20 at 20dB

IX. SIMULATIONS

Behavior of the step size of the MIMO-CDMA MNCLMSalgorithm is shown in figure 3 for 10 subscribers. As canbe seen, in the transient state, the MIMO-CDMA MNCLMSalgorithm has the largest step-size value when compared tothe other algorithms and, thus, yields the fastest convergence.Also, in the steady state, the step-size parameter of the MN-CLMS algorithm was reduced to the smallest value amongstall algorithms. Same behavior is achieved for 20 subscribersas shown in figure 4

A. Interference Cancellation in Rayleigh Fading Channel

In this section, simulation results are presented to assessthe performance of the MNCLMS algorithm for the MIMOCDMA LE case. The performance of the proposed MIMO-CDMA MNCLMS algorithm is compared with the standardLMS, MCLMS noise constrained LMS and zero noise algo-rithms. The average MSE is the performance measure throughwhich the algorithms are assessed. A 2×2 MIMO system is


Vol. 7, No. 1, 2016


0 500 1000 1500−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

iterations

Step-sizebeh

aviourof

MIM

OMNC-LMS


Fig. 3. Behavior of time-varying step size of the MNCLMS algorithm forK = 10 at 20 dB SNR

0 200 400 600 800 1000 1200 1400 1600 1800 20000

1

2

x 10−4

iterations

Step-sizebehaviourof

MIM

OMNC-LMS


Fig. 4. Behavior of time-varying step size of the MNCLMS algorithm forK = 25 at 20 dB SNR

considered here. Random signature sequences of length 31 andrectangular chip waveforms are used. The SNR is kept at 10dB and 20 dB respectively.

The results of the comparison of the convergence speed ofthese algorithms for 10 and 25 subscribers, in a Rayleighfading channel, are depicted, respectively, in Figure.5 andfigure 6. In both cases, it can be seen that the MIMO-CDMA MNCLMS algorithm converges faster than the restof the algorithms. In the case of 10 subscribers, the proposedalgorithm was able to achieve an MSE at around -4.8 dB ataround 200 iterations . In case of 25 subscribers, the proposedMNCLMS algorithm was able to achieve MSE at -5.2 dB andin around 240 iterations.

X. CONCLUSION

In this paper a new constrained MIMO CDMA LMS typealgorithm (MNCLMS) for multiuser wireless environment ispresented. The MNCLMS algorithm may be refereed to asa generalized constrained adaptive algorithm which includesthe MCLMS, the NCLMS and the ZNCLMS algorithms as

0 200 400 600 800 1000 1200 1400 1600−6

−5

−4

−3

−2

−1

0

iterations

MSE

(dB)


Fig. 5. MSE behavior for different algorithms in Rayleigh Fading environmentwith for K = 10 at 10 dB SNR

0 200 400 600 800 1000 1200 1400 1600 1800 2000−6

−5

−4

−3

−2

−1

0

iterations

MSE

(dB)


Fig. 6. MSE behavior for different algorithms in Rayleigh Fading environmentfor K = 20 at 20 dB SNR

special cases. In our analysis, we have performed a thoroughcomparison of the proposed algorithm with a number of otherconstrained algorithms and it is shown that our algorithm hasoutperformed its competitors.

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[2] RADU CIPRIAN Bilcu, Pauli Kuosmanen, and Corneliu Rusu. Anoise constrained vs-lms algorithm. In EUROCOMM 2000. InformationSystems for Enhanced Public Safety and Security. IEEE/AFCEA, pages29–33. IEEE, 2000.

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[6] Yongbin Wei, Saul B Gelfand, and James V Krogmeier. Noise-constrained least mean squares algorithm. Signal Processing, IEEETransactions on, 49(9):1961–1970, 2001.

[7] K. Mahmood, S.M. Asad, M. Moinuddin, A. Zerguine, and S. Paul.Statistical analysis of multiple access interference in rayleigh fadingenvironment for mimo cdma systems. In Statistical Signal Processing(SSP), 2014 IEEE Workshop on, pages 412–415, June 2014.

[8] M Moinuddin, AUH Sheikh, A Zerguine, and M Deriche. A unified ap-proach to ber analysis of synchronous downlink cdma systems with ran-dom signature sequences in fading channels with known channel phase.EURASIP Journal on Advances in Signal Processing, 2008(1):346465,2008.

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