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Adaptive Uncoupled Termination for Coupled

Arrays in MIMO SystemsReza Mohammadkhani, and John S. Thompson

Abstract In this paper, we present an adaptive uncoupledmatching network to compensate any performance degradationof compact Multiple-Input Multiple-Output (MIMO) systemsdue to antenna mutual coupling effects and channel variations.This method varies the antenna terminal loads per symbol-block, based on a random search algorithm, to nd an optimumtermination network that maximises the performance metric,such as received power and capacity. It uses the received signals(voltages across the resistances of the terminal loads) to estimateor calculate the performance metric, in order to include thetotal effects in the optimisation process. By applying the randomsearch algorithm, our method does not require any knowledge of the array parameters such as impedance matrix, or derivatives of the performance metric(s) unlike other optimisation techniques.We demonstrate this scheme by performing simulations tooptimise the capacity of a 3 3 MIMO considering differentpropagation scenarios. We observe signicant mean capacityimprovements (more than 2 bits/s/Hz) for all assumed propagationscenarios, when receive array antennas are spaced close as 0 . 05 and terminated non-identically.

Index Terms MIMO, mutual coupling, impedance matching,adaptive termination.

I. I NTRODUCTION

M IMO systems by using multiple antennas at both trans-mit and receive sides of the wireless link, providebetter link quality and higher data-rates [ 1], [2]. However,applying MIMO technology to small wireless devices withan array inter-element spacing less than half a wavelengthleads to antenna mutual coupling effects which degrade theMIMO performance [ 3][5]. Many studies have presentedimpedance matching techniques to compensate the electro-

magnetic coupling effects in the coupled array. Some ap-plied a complex coupled matching network [ 6] known as theMultiport-Conjugate Match (MCM) [ 4] which is shown tobe able to achieve optimal performance. Such a matchingnetwork is complicated and difcult to construct practicallyand it only offers a narrowband matching performance [ 7].These problems motivate researchers to work on simpleruncoupled termination approaches [ 8][12] which are easierto implement and can achieve near-optimal performance witha wider bandwidth [7], [12].

The major part of this work was performed at the University of Edinburghand it was nancially supported by the Ministry of Science, Research and

Technology, Iran . This paper is based in part on reference [31] which waspresented at the ISWCS 2010 conference.R. Mohammadkhani is with the Department of Electrical Engineering, Uni-

versity of Kurdistan, Sanandaj, Iran. e-mail: ([email protected]).J. S. Thompson is with the Institute for Digital Communications

(IDCoM), University of Edinburgh, Edinburgh EH9 3JL, UK. e-mail:([email protected]).

Existing uncoupled matching solutions considernumerically-optimum loads to maximise the receivedpower [8], [13 ], or the capacity [9], [10 ] of compact MIMOsystems for a given propagation scenario. These methodsrequire a priori knowledge of the propagation channel and anaccurate mutual coupling model which is difcult to measurein practice. Modelling of the mutual coupling is a challengingproblem for coupled array applications. Existing matchingnetwork approaches apply a method suggested by [ 14] toform an impedance matrix or an equivalent scattering matrixin order to model the mutual coupling effects for the coupledarray in either transmit or receive mode. However, despite itssimplicity, the accuracy of this method for the receive arrayis questioned by some other studies [ 15] [18] and alternativeapproaches [ 19][21 ] are suggested to model the mutualcoupling. Furthermore, [ 22] reveals the inuence of near-eldscatterers on the mutual coupling, which is ignored by theexisting coupling compensation methods.

Assuming known mutual coupling and channel matrices, anoptimal uncoupled matching network is obtained in [10 ] by op-timising the mean capacity with respect to the antenna terminalloads for identical termination impedances. It is extendedto non-identical antenna terminations by [23 ], [24] . Whenwe have no knowledge of the mutual coupling matrix, thesestudies are not able to nd the optimal termination network.Furthermore, these methods are not designed to track the timevariations of the propagation channel. Therefore, we proposean adaptive uncoupled termination method which compensatesthe effects of the propagation channel and mutual couplingtogether by directly dealing with the received signals. We

consider both identical and non-identical termination cases.We do not use a known mutual coupling matrix, but it will beincluded implicitly by using the voltages across the antennaterminations to estimate the channel matrix. The algorithmchanges the receive array termination for each symbol-block and calculates the MIMO performance metric (e.g. receivedpower or capacity). By comparing the current metric valueand the previously optimum one (the maximum value overprevious blocks), the algorithm selects the better impedancenetwork as the optimum termination until that iteration. It thenadds a random step to the terminal impedances to generate anew candidate impedance network for the next block. Thisprocess will be repeated until the algorithm converges to anoptimum termination network, in terms of received power orcapacity.

The rest of this paper is organised as follows. Section IIintroduces our system model. Then, the proposed adaptivetermination approach is described in section III. After that,
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section IV provides numerical results to verify the proposedmethod, assuming either a perfectly known channel matrix,or including imperfect channel state information (CSI) due to

estimation error and Doppler effects. Section V concludes thepaper.

Throughout this paper, boldface lowercase and uppercasesymbols refer to vectors and matrices, respectively and theitalic version of the symbols represent the entries of thevector or matrix. The notations ()H , E{}, det( ) denote thematrix Hermitian (conjugate-transpose), the expectation, andthe matrix determinant operators, respectively. The symbol I M also represents the identity matrix of size M M .

I I . S YSTEM M ODEL

In this section we provide a MIMO system model taking

into account the time-variation of the propagation channel,antenna mutual coupling, and the channel estimation errorissues. We also address the capacity estimation techniques atthe end of this section.

Consider a MIMO system of M transmit and M receiveantennas, communicating through a at-fading Rayleigh chan-nel. Using a discrete-time baseband model, the input-outputrelationship at time instant k is given by

y [k] = H [k]x [k] + n [k] (1)

where x [k]C M is the transmit signal vector, y [k]

C M

is the receive signal vector, H [k] C M M denotes the

channel gain matrix including the effect of transmit and receiveantennas, and n [k]

C M is the additive white complexGaussian noise vector at the receiver with zero-mean andcovariance matrix 2n I M , where I M is an M M identitymatrix.

To describe the time-variation of the channel, we use theGauss-Markov process model [ 25] as follows

H [k + 1] = 1 H [k] + W [k] (2)where W [k]

C M M includes complex Gaussian entrieswith zero-mean and unit-variance. The entries of W areindependent across rows, columns and time indices k. The

parameter R , 0 1 is introduced to control thecoherence time of the channel. In [25 ], some practical ranges

for have been calculated by tting the above Gauss-Markovmodel into real systems measurements. For instance

3 10 7 10 4 , for a slow-fading indoorenvironment with mobile speed 1-5 km/h and carrierfrequencies from 800 MHz to 5 GHz.

10 4 1.8 10 3 , for a slow-fading outdoorenvironment with mobile speeds of the order of 5 km/hand carrier frequencies from 800 MHz to 5 GHz.

= 1 .8 10 2 , for a fast-fading outdoor environmentwith mobile speed 50 km/h and carrier frequency 5 GHz.

A. Mutual Coupling

As mentioned in section I, the existing mutual couplingmodels may not be exact for the coupled receive array.However, they can reveal the total behaviour of the system

and the impact of terminations on the compact MIMO per-formance. Therefore we apply a general model described in[10] . We note that the proposed adaptive termination algorithm

is unaware of the system model and it works based on thenal received signals (voltages across the resistances of thetermination loads), or any other parameters estimated fromthem. Therefore, it does not strongly depend on the mutualcoupling model unlike [ 10] which uses an explicit matrixmodel.

Let us begin with a case that array elements of both transmitand receive sides are spaced sufciently far apart to avoid mu-tual coupling (practically over half-wavelength inter-elementspacing). Assuming a Rayleigh-fading propagation channeland using the popular Kronecker 1 structure, the channel matrixcan be expressed as [27]

H = 1/ 2R H w 1/ 2T (3)

where H w is a M M matrix with independent identicallydistributed (iid) and zero-mean unit-variance complex Gaus-sian entries, and T and R are the transmit and receivespatial channel correlation matrices, respectively.

Now consider compact receive wireless devices whoseelement spacing is less than 0.5 and which possesses animpedance matching network Z L . Assuming identical half-wavelength dipole antennas for transmit and receive arrays,the general MIMO channel matrix using Z-parameters can berepresented as [ 10]

H mc = 2 r 11 R 1/

2L (Z R + Z L ) 1 1/

2R H w 1/

2T

H ncR

1

/2T (4)

where Z R and Z T are the receive and transmit ar-ray impedance matrices, respectively with diagonal entriesZ R,ii , Z T,ii , (i = 1 , , M ) being the self impedances andthe off-diagonal entries Z R,ij , Z T,ij , (i = j , and i, j =1, 2, , M ) dening the mutual impedances. The real partsof Z L and Z T are denoted by R L and R T , respectively andr 11 = RT, 11 . For the special case of no mutual coupling(Z R,ij = Z T,ij = 0 ) and matching all the receive andtransmit antennas to their self-impedances, the channel matrixsimplies to the term represented by H nc .

Throughout this work, we assume that the transmit antennasare separated sufciently, self-impedance matched and that T = I M . Thus the channel matrix H mc from ( 4) can besimplied as [ 10]

H mc = 2 r 11 R 1/ 2L (Z R + Z L ) 1 1/ 2R H w H nc

(5)

This is the model we will use in the numerical studies in thispaper.

B. Channel Estimation

To exploit the capacity benets of the MIMO technology,accurate channel knowledge at the receiver and/or transmitter

1This model has deciencies for large number of antennas [26] , but we useit for 3 3 MIMO in our simulation results for which is still reasonably valid.
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is usually required. One of the most popular channel estima-tion approaches is the training-based estimator [28] which candirectly estimate the channel matrix including antenna mutual

coupling effects, from knowledge of the received signals and(transmitted) training signals. In this paper, we do not go intothe details of the channel estimator, rather we assume that anestimated channel matrix H mc [k] is available. Considering thechannel estimation error, we can write [ 29]

H mc [k] = H mc [k] + E [k] (6)

where E [k] is the channel estimation error matrix, whoseentries are zero-mean complex Gaussian random variableswith variance 2E . Throughout this work, we assume that eachblock includes some training-symbols and at least one channelestimate is provided per symbol-block from the knowledge of transmitted training symbols.

C. Capacity

In general, the MIMO capacity is given by [ 30]

C = maxQ : Tr( Q ) P E H log2 det I M +

12n

H mc QH H mc(7)

where Q = E{xx H }is the input covariance matrix, and P isthe total transmit power. In this paper, it is assumed that thetransmitter has no knowledge of the channel, so the transmitpower will be divided equally among all the transmit antennas,i.e., Q = ( P/M )I M [1]. Thus, the resulting ergodic capacitycan be written as

C = E H log2 det I M +

M H mc H H mc (8)

where = P/ 2n is the average signal-to-noise ratio at thereceiver. We note that the fading process {H mc [k]}is assumedto be known for the receiver. For a system with a knownchannel estimate H mc and a known estimate variance 2E ,a lower bound of ( 8) is given by [ 29]

C lower = E H log2 I M +1

1 + 2E P

M H mc H H mc (9)

If we split pilot symbols into k p groups among each block, thechannel estimate H mc can be expressed as an average of the

channel estimates H 1 , H 2 ,..., H k p from sub-blocks as follows[28]

H mc =1k p

k p

i=1

H i . (10)

D. Received Power

Assume y (t) is a vector of the received voltages across theresistance of the receive array terminal loads at time instant t .Then the received power for ith antenna can be written as

P r,i = E yi (t)yi (t)

r Li/P 0 , (i = 1 , , M ) (11)

where r Li represents the real part of the load terminal zLifor antenna i, and P 0 is the power received by a conjugatematched isolated antenna which is used to normalise theMIMO received power. An estimation of ( 11) for each symbolblock can be calculated by averaging time instances of thereceived power over a block of L symbols as described in [ 31] .

Fig. 1. Block diagram of a coupled receive/transmit array applying a matchnetwork.

III. T ERMINATION STRATEGIES

In the literature, applying a coupled or multiport matchingnetwork [4], [6], [32 ] is introduced as a decoupling network which may decouple the signals from closely spaced antennascompletely. Using network analysis, a matching network canbe placed between the coupled array and antenna loads (re-ceive mode) or excitation sources (transmit mode) such thatit can be conjugate-matched from one side to the antennaarray and from the other side to the loads/sources. Thisis shown in Fig. 1. However, constructing such a network due to the required interconnections between all ports of

the matching network is complicated and it also offers anarrowband matching performance [ 7].

Alternatively, uncoupled termination networks have beenwidely investigated, either by terminating all receive anten-nas identically [8][10] or individually applying non-identicaltermination loads [ 23], [24 ]. These studies assuming a knownmodel comprising the channel matrix and the mutual couplingmatrix. Then they either numerically or mathematically seek an optimum termination network which maximises the re-ceived power or the capacity. There are two problems with theexisting studies. First, we need to know the array impedancematrix Z R in equation (5). We would need to perform a

calibration step to do this, which is expensive and time-consuming. Secondly, even if Z R is known, it may changewith time due to temperature, environment, and any otherparameter changes. This would lead to an incorrect solutionfor the optimum termination network Z L .

Existing studies are model-based matching solutions whichassume that every parameter in the system is known except forthe matching impedance network, whereas for practical casesthe only known parameters are the termination impedancenetwork and the received signals (or any parameter estimatederived from them, such as H mc ).

Recently, two studies [31 ], [33 ] appeared roughly at thesame time which suggest applying an adaptive matchingnetwork for compact MIMO systems. The authors in [33 ]present a general idea of having an adaptive (multiport oruncoupled) impedance matching network to remedy any per-formance degradation resulting from the antenna mutual cou-pling and/or variations of the propagation channel. Although
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several possible ideas are suggested to cover the practicalissues of implementing such matching network, no particularapproach is proposed to select the matching impedances. On

the other hand, [ 31] applies a combination of a performancemetric estimation (e.g., received power or capacity) from thereceived signals, and a random search algorithm to controlan adaptive (identical) uncoupled matching network. It showsthat the proposed algorithm can optimise the performanceunder different propagation scenarios while the optimisationcomplexity is reduced by applying the random load selectionamong all possible ranges of the antenna loads.

Here, we extend the work of [ 31] to the non-identicalmatching impedance case and propose an adaptive uncoupledtermination which only relies on the knowledge of the re-ceived signals and a training-based channel estimate H mc persymbol-block. We investigate the effect of different practicalissues such as estimation error, and time-variation of thechannel matrix on the matching performance.

In what follows, we rst review the conventional termi-nations: characteristic impedance match, and self-impedanceconjugate match. Then, we describe the proposed adaptiveuncoupled termination which optimises the compact MIMOperformance (e.g. received power or capacity).

A. Characteristic Impedance Match

All receive antennas are terminated in a characteristicimpedance Z 0 . Therefore, the receive load network Z L at (5)is set to Z 0 I M . This means we have no matching network forthis case.

B. Self-Impedance Conjugate Match

In this termination case, each receive antenna is terminatedin the conjugate of its self-impedance. In other words, Z L =diag( Z R ). This termination would result in maximum powertransfer to the load network when there is no mutual couplingbetween array elements [7].

C. Adaptive Termination

The previous terminations in subsections III-A and III-B do

not consider the mutual coupling effects. Here, we proposean adaptive, uncoupled termination network to counteract theeffects of mutual coupling and propagation channel changes,and to optimise the MIMO performance. This technique uses adiagonal matrix Z L and varies it per symbol-block based on arandom search algorithm, and then calculates the performancemetric(s) to examine the effect of the current termination onthe mutual coupling and therefore system performance.

A simplied schematic diagram of the proposed method isshown in Fig. 2. This method comprises two major parts. Oneis an estimator which provides instances of the channel matrixH mc (or the statistical moments of the channel) including theactual effects of the mutual coupling and the time-varyingpropagation channel. The other part is an adaptive matchingnetwork controller which selects the loads to counteract theperformance degradation due to the mutual coupling and/orchanges of the channel, based on the performance metricestimates (capacity or received power).

Fig. 2. Schematic diagram of the proposed adaptive termination approachfor a compact MIMO system.

One may ask why we have not applied any other optimisa-tion techniques such as Gradient based or Newton-Raphsonmethods rather than a random load selection? Before weanswer this question, let us clarify our optimisation problemfor a M M MIMO system of M uncoupled transmitantennas and M coupled receive antennas. We can expressthe desired performance metric, either the received power orthe capacity, in our optimisation problem as a function of received signals, i.e., f (y ) = f (H mc x + n ) (we drop thetime index k for simplicity). According to the results fromthe previous matching solutions [3][11] , the total channelmatrix H mc which includes the mutual coupling effects, andtherefore f (y ), depend on the termination impedance matrix of the coupled array, denoted by Z L . Denoting the channel matrixwith no mutual coupling by H , we can write our optimisationproblem as below

maxZ L

f (H , Z R , Z L ). (12)

In other words, we would like to nd a terminal impedancenetwork Z L which compensates any performance degradationdue to the mutual coupling or channel matrix changes. Here,we are interested in uncoupled terminations where Z L is adiagonal matrix written as

Z L =

zL 1 0 00 zL 2 0... ... . . . ...0 0 zLM

. (13)

Suppose the channel matrix H and the array impedancematrix Z R , during the optimisation period are xed or theyvary slowly. Then, the optimisation problem (12 ) can be

simplied asmax

z L 1 , ,z LM f (zL 1 , , zLM ) (14)

where zLi = r Li + jx Li for i = 1 , , M with r Li R +and xLi R denoting the real (resistance) and imaginary
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Fig. 3. An overview of the adaptive impedance matching algorithm.

(reactance) parts of zLi , respectively. We consider two dif-ferent kinds of uncoupled terminations: (i) identical load-ing, which all diagonal entries are equal to a load zL =r L + jx L , and (ii) non-identical loading in which terminalloads zL 1 , zL 2 , , zLM are individually tuned to optimisethe compact MIMO performance.

We let z = [r L 1 , xL 1 , , r LM , xLM ]T be the vector of optimisation variables . Then, we try to nd an optimal z optin which f (z opt ) has the maximum possible value of f .This could be obtained by producing a maximising sequencez (m ) , (m = 1 , ) where f (z (m +1) ) > f (z (m ) ), from thefollowing iterative equation [34 ]

z (m +1) = z (m ) + (m ) z (m ) (15)

where m denotes the iteration number, the scalar (m ) 0is the scale factor, and the vector z (m ) determines thedirection of optimisation at iteration m . We need to specify z in accordance with each optimisation technique. Fig. 3shows an overview of the proposed idea of the adaptiveimpedance matching technique. The algorithm starts from aninitial termination load such as characteristic impedance load,and then estimates the total channel matrix H mc from theknowledge of training/pilot symbols. Then the performancemetric, the capacity or the received power, is calculated topredict the optimal load network Z L for the next symbol block.This process continues until it converges to an optimal loadnetwork which maximises the performance metric.

1) Gradient ascent method: Let us begin with the identicalloading case, and then we extend the result to non-identicaltermination. Suppose that the function f is differentiable withrespect to the variables r L and xL . Using the Gradient ascentmethod [ 34] , we can nd a solution for ( 14) by substituting z = f (z ) into ( 15) as follows

z (m +1) = z (m ) + (m )f (z (m ) ) (16)

where z (m ) = [r (m )L , x(m )L ]

T is the value of the vector of variables at iteration number m , and (m ) > 0 here is calledstep size. The term

f (z (m ) ) represents the gradient of f with

respect to the entries of z at iteration m , given by

f (r L , xL ) =f r L

,f

x L

T

. (17)

As stated previously, usually no parameter is known for prac-tical systems except the received signals, and the terminationnetwork (which can be controlled by the optimisation algo-

rithm). Therefore the gradient term at ( 16) has to be estimated.This estimate could be calculated from the following forwarddifferences [35 ]

f r L

f (r L + r, x L ) f (r L , xL ) r

(18a)

f x L

f (r L , xL + x) f (r L , xL ) x

. (18b)

It means that in order to calculate an estimate of f (r L , xL )at each iteration m , at least three channel estimates corre-

sponding to the load impedances zL 1 = r L + jx L , zL 2 =(r L + r )+ jx L , and zL 3 = r L + j (xL + x) are required. In

general, this method requires 2M +1 channel estimates at eachiteration m , for the use of adaptive non-identical impedancematching in a M M .

We numerically examine the Gradient-based method for thecapacity of a 3 3 MIMO system with different r and xvalues in the next section. This method can converge to theoptimum load impedance if we choose a proper step size (m )for each channel realisation. Furthermore, the convergence rateof the Gradient-based method depends on the quantisationlevel of z (m ) .

We also examined the Newton-Raphson method which usesthe second and rst order derivatives of f . We found that thismethod does not converge to the optimum load network underany circumstances. This might be due to complexity of thesecond order derivative approximations.

2) Random Search: In this subsection, we explain how theadaptive algorithm uses a random search (motivated by randomphase selection [36 ] and random walk [37 ] algorithms) forthe optimum load impedance. Assuming an identical loadingnetwork, and describing the receive array load network (13 )as Z L = z

(m +1)L I M at iteration (m + 1) , the impedance load

z(m +1)L is obtained from the following equation:

z(m +1)L = z(m )opt + z

(m ) , (m = 0 , 1, , N s 1) (19)where z(m )opt is the optimum load at m th iteration (initialisedby z(0)opt = Z 0 = 50 ), the non-zero scalar z(m ) de-notes a complex step size randomly selected from the set

{ r, j x, ( r j x)}with an equal probability, andN s is the number of load network variations. An overviewof the choice of step size z(m ) at iteration m is shown inFig. 4. In practice, N s can be estimated from the experimentaldata, or in a similar way the algorithm can stop after havingno change in the optimal load network for a specic numberof iterations while a small step size is used.

At each iteration (m + 1) , the capacity or received power iscalculated (or estimated from the knowledge of the received

signals) and compared to the previous value correspondsto z(m )opt . The impedance which corresponds to the highercapacity/received power is held as the optimum load z(m +1)optfor the (m + 1) th iteration. Fig. 5 illustrates the owchart of the proposed algorithm for identical loading.
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Fig. 4. Complex step size z ( m ) at iteration m is selected randomly fromthe set { r , j x ,( r j x)} with an equal probability.

Fig. 5. Flowchart of the proposed adaptive termination approach.

In this work, we have considered several values for the stepsizes r, x from the range of 1 12. Larger values of r and x result in a faster convergence but lower steady-state performance compared to the smaller step sizes. So toobtain improved performance, the algorithm can start with alarge step size and then decrease the step size after havinga completed specic number of load variations. We describethe effect of choosing the step size value in more detail insection IV-A.

The algorithm can also be extended to the non-identicalloading case. For any antenna # i, (i = 1 , , M ), theload impedance zLi can be tuned individually according to(19) . This can be represented as (20) where z(m +1)Li , (i =1, , M ), are the selected terminal impedances at (m + 1) thiteration, z(m )opt,i , (i = 1 , , M ) are the optimum loads for them th iteration, and z

(m )i , (i = 1 , , M ) are independentrandom complex step sizes at m th iteration. Let us summarise

the algorithm to optimise the capacity C as follows.1) Initialise the array termination load z(0)L = Z 0 (or Z L =

Z 0I M );2) Estimate the corresponding C from ( 8) for perfect CSI

or ( 9) for imperfect CSI;3) Set C opt = C , and zopt = z

(0)L (or Z opt = Z

(0)L for

non-identical loading);4) Calculate the next termination for the following symbol-

block from ( 19) or (20 );5) Estimate the corresponding C from ( 8) or ( 9);

6) If ( C > C opt ) then ( zopt = zL , and C opt = C );otherwise zL = zopt and go back to step 4.Some matching solutions [ 8], [13] have found the optimummatch by maximising the received power. This algorithmcan also be applied to maximise the received power P r bysubstituting P r from Section II-D for C , and P r,opt for C opt .However, most commonly it is of interest to increase the datarate and the capacity of MIMO wireless systems rather thanthe received power.

IV. N UMERICAL R ESULTS

In this section we provide a numerical study to evaluate the

matching performance of the proposed adaptive terminationapproach for a 3 3 MIMO system with a coupled receivearray. Here, we consider the MIMO capacity optimisation,but the result can similarly be extended to the receivedpower as well. We assume linear arrays of identical half-wavelength dipoles are applied for both transmit and receivesides. The receive array antenna element spacing is assumedto be d = 0 .05 , where is the wavelength, and the transmitantennas are considered to be placed far enough such that weneglect the mutual coupling effect at the transmit side. Mutualcoupling impedances for the receive array are calculated usingthe electromotive force (EMF) method [38]. Following ourassumptions in section II-A, we apply the channel model of (5)including a time-variant term H w [k], under different scatteringdistributions: 2D uniform, and a truncated 2D Laplaciandened by the mean angle of incidence and an angularspread of 40 for a signal-to-noise-ratio (SNR) 20dB at thereceiver.
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z(m +1)L 1 0 00 z(m +1)L 2 0...

..

.

. . ....0 0 z

(m +1)LM

=

z(m )opt 1 0 00 z(m )opt 2 0...

..

.

. . ....0 0 z

(m )optM

+

z(m )1 0 00 z(m )2 0...

..

.

. . ....0 0 z(m )M

(20)

0 20 40 60 80 100 120 140100

50

0

50

6 7

8 9

1 0

1 0

rL

( )

x L ( )

10.27

9.90

Mean Capacity (bits/s/Hz)

10.48

Fig. 6. Contour plot of the mean capacity versus real and imaginary partsof the antenna load impedance zL for uniform scattering distribution at thereceiver. Three points are marked for: self-impedance match z11 (square),Z 0 = 50 (triangle), and optimum load zopt (circle) for the identical loading.

0 50 100 150 200 250 300 350 400 450 5000.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

Normalised capacity: C/Cz0

N o r m a

l i s e

d c a p a c i

t y

C instCz11Cz0

= 0.3 = 0.003

= 3x10 5

Fig. 7. Normalised capacity of one channel realisation using the Gradient-based adaptive identical impedance matching technique for different step sizevalues.

A. Perfect CSI at the receiver

Let us begin with a simple case that the channel matrixis perfectly known at the receiver. We generate 200 randomtime-invariant channel realisations assuming a uniform scat-tering distribution such that the entries of the receive spatialcorrelation matrix R are given by

R,ii = 1 ( i = 1 , 2, , M ) (21a)R,ij = J 0(

2d

) (i = j ), (i, j = 1 , 2, , M ). (21b)Fig. 6 illustrates the average capacity as a function of real

Fig. 8. Normalised capacity values for 10 channel realisations using theadaptive impedance matching by applying the Gradient algorithm with thestep size = 0 .01.

0 500 1000 15001

1.05

1.1

1.15

1.2

1.25

1.3a

M e a n

( C Z

L / C

Z 0

)

0 50 100 150 200 250 300 350 400 450 5001

1.01

1.02

1.03

1.04

1.05

1.06

1.07

Iteration, m

M e a n

( C Z

L / C

Z 0

)

(b)

r = x = 1

r = x = 2

r = x = 4

r = x = 8

r = x = 12

Fig. 9. Normalised mean capacity versus iteration number for the adaptive (a)non-identical, and (b) identical uncoupled terminations considering r = xvalues from the set {1, 2, 4, 8, 12}() .

and imaginary parts of the antenna load for a non-adaptiveidentical termination case. It shows three different terminat-ing cases of self-impedance match (marked by a square),Z 0 = 50 match (marked by a triangle), and the numericallyoptimum match (marked by a circle) and the correspondingaverage capacity values. We observe that the average capacityhas a maxima for a specic identical termination called theoptimum load [9], [10]. The optimal load can be extendedto the non-identical case as well [ 23] . Although these papersrevealed the relevance of compact MIMO capacity/received
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8

0 500 1000 1500 2000 2500 30009.5

10

10.5

11

11.5

12

12.5

iteration

M e a n

C a p a c

i t y

( b i t s

/ s / H z

)

Adaptive nonidenticalAdaptive identical

Selfconjugate match

50 match

10.27

9.901

12.39

10.5

Fig. 10. Mean capacity of a 3 3 MIMO system for the following matchingconditions: adaptive non-identical, adaptive identical, self-impedance conju-gate, and Z 0 = 50 matched termination networks.

1.15 1.2 1.25 1.3 1.35 1.40

5

10

15

20

25

30

C/CZ0

P e r c e n

t a g e

1 1.05 1.1 1.15 1.20

10

20

30

40

50

C/CZ0

IdenticalLoading

NonidenticalLoading

(a) C/C Z 0 instances

5 6 70

25

50

75

rL1

( )

P e r c e n

t a g e

37 36 35 340

25

50

75

100

xL1

( )

P e r c e n

t a g e

0 1 20

25

50

75

100

rL2

( )

0 1 2 30

25

50

75

100

xL2

( )

5 6 70

25

50

75

rL3

( )

36 35 340

25

50

75

100

xL3

( )

(b) Non-identical Loads: zLi = r Li + jx Li , (i = 1 , 2, 3)

10 20 30 40 500

10

20

30

40

50

60

P e r c e n

t a g e

rL

( )

36 35 34 33 32 310

10

20

30

40

50

60

xL

( )

(c) Identical Loads: zLi = r L + jx L , (i = 1 , 2, 3)

Fig. 11. Histogram plots of (a) C/C Z 0 instances for both adaptivetermination cases, and the real and imaginary parts of the termination loadsfor (b) non-identical, and (c) identical terminations at iteration number 3000.

5 dB

3 dB

1 dB

1 dB

3 dB

45

225

90

270

135

315

180

0

123

(a) z0 = 50 match

5 dB

3 dB

1 dB

1 dB

3 dB

45

225

90

270

135

315

180

0

123

(b) Self-conjugate match

5 dB

3 dB

1 dB

1 dB

3 dB

45

225

90

270

135

315

180

0

123

(c) Adaptive identical match

5 dB

3 dB

1 dB

1 dB

3 dB

45

225

90

270

135

315

180

0

123

(d) Adaptive non-identical match

Fig. 12. Array element patterns in dB for (a) z0 = 50 match, (b) self-impedance conjugate match, (c) adaptive identical, and (d) adaptive non-identical impedance matching networks. Zero degrees corresponds to the arraybroadside.

2

4

6

8

10

E i g e n v a

l u e

# 1

0

0.5

1

1.5

E i g e n v a

l u e

# 2

0 500 1000 1500 2000 2500 3000

2

4

6

8

10

12

x 104

E i g e n v a

l u e

# 3

iteration

Nonidentical LoadingIdentical LoadingZ

0= 50

Selfconjugate matchno mutual coupling

Fig. 13. Eigenvalues of ( H mc H H mc ) versus iteration number for differentantenna terminations.

power to the termination, they require prior knowledge of the propagation channel and mutual coupling model for anumerical search over all possible termination loads for theoptimal load. Furthermore, these studies perform the optimisa-

tion process over the mean values of the performance metric(s)with respect to the termination load(s), whereas our proposed algorithm seeks the optimum load network for each channelrealisation.

Now, we apply the adaptive uncoupled terminations for the




9

assumed propagation channel scenario. We rst investigateusing the Gradient algorithm for the adaptive impedancematching technique. Fig. 7 shows the convergence behaviour

of this method for one channel realisation applying differentvalues of the step size . It can be seen in Fig. 7 that too largea value of (e.g. 0.3) leads to instability in the algorithm per-formance while making too low (e.g. 310 5 ) leads to veryslow convergence performance. The value = 0 .03 seems togive the best compromise of stability and convergence

Since the mutual coupling model in our problem is un-known, we can not separate the channel matrix H from thetotal channel matrix H mc which includes the mutual couplingeffects and is measured/estimated from the received signals.However, assuming H is known, we applied the followingnormalisation

H n =

NM H

||H ||F , (22)to further investigate the Gradient-based adaptive impedancematch. As we see from Fig. 8, even for the normalised H n ,there is a range of convergence characteristics, so the step size could even be adjusted further according to the entries of thechannel matrix H - though such a solution is beyond the scopeof this paper. For practical implementation, the available set of impedance values may be quantized to a nite set of distinctimpedances, which will limit the accuracy of the convergedsolution and possibly reduce performance. We recall that foridentical impedance loads, at each iteration we need at least3 channel estimates.

Alternatively, we use the random search algorithm for theadaptive (identical) impedance matching technique. We inves-tigate the effect of having different values of step size z bychoosing xed r = x values from the set {1, 2, 4, 8, 12}.At each iteration, the algorithm applies an estimate of thechannel matrix including the realistic mutual coupling effectsand selected antenna loads. This is provided from the receivedsignal and knowledge of the training signals. Convergenceresults of the normalised mean capacity for adaptive identicaland non-identical terminations versus the number of changesof the termination network ( m th iteration) are illustrated inFig. 9. The most signicant feature for both termination cases

is that applying a smaller step size leads to a higher steady-state performance but requires a longer convergence time.Assuming the same step size for both termination cases,the identical adaptive termination rises sharply and reachesits steady-state about ve to ten times faster than the non-identical termination. However, as depicted in Fig. 9, theadaptive non-identical termination can achieve much highersteady performance albeit with a slower convergence time, forall step sizes except for the case with r = x = 12 .

In Fig. 10, the mean capacity of the system is shown for fourdifferent matching networks: adaptive non-identical, adaptiveidentical, self-impedance conjugate, and Z 0 = 50 matchedtermination networks. Steady state values of the mean capacityfor all termination strategies are shown on the right hand sidey-axis. In order to achieve a higher steady-state performancein a shorter convergence time, the adaptive algorithm appliesa variable step size for both termination cases according tothe results reported in Fig. 9. Therefore, initially r = x is

assumed to be 8 for both termination cases and then reducedto 4 , 2, and 1 at iterations 100, 400, and 1000 for the non-identical case, and at iterations 30, 80, and 200 for the identical

case respectively. Any change in the propagation channel ormutual coupling would result in a new optimal Z L which maychange with time. In order to track these possible changes,the algorithm can also decide to update the selected optimumvalues of C (m )opt , Z

(m )opt and to increase the step size, whenever

(C (m ) C (m )opt ) is not positive or C

(m )opt is not changed for a

large number of steps. These occurrences may indicate thatthe optimal Z L has changed and the receiver should nd thenew best solution. As we observe from Fig. 10, both adaptiveterminations nd an optimum load network which results in ahigher mean capacity than the conventional terminations. Theadaptive identical case reaches to its steady state performance

just after 85 iterations while for the non-identical case it takesabout 1000 iterations to achieve to a point above 99% of the steady-state. However, the latter case achieves about 2(bits/s/Hz) higher capacity gain in the expense of a longerconvergence time.

Unlike earlier studies, the adaptive termination algorithmperforms the optimisation process over the capacity instancesrather than the mean capacity. For further investigation of theadaptive algorithm behaviour, percentage histogram plots of the normalised capacity instances C/C (Z 0), and the real andimaginary parts of the termination loads for both adaptivetermination cases at iteration number 3000 are shown in

Fig. 11. The y-axes are relative frequencies for the total of 200channel realisations. Fig. 11a shows that the capacity instancesare improved for both adaptive terminations compared to theconventional Z 0 = 50 match by at least 20% for the non-identical case and by 5% for the identical loading case. Ascan be seen from the results in Fig. 11b and 11c, more than75% of the channel realisations are optimised by selecting anon-identical terminal network as Z L = diag(6 j 35, 1 + j 1, 6 j 35), and about 50% of them by having an identicalnetwork equals to Z L = (32 j 35)I 3 3. The antenna patternof the array elements for the above optimal load networksare plotted in Fig. 12 with = 0 as the array broadside.The results conrm the beamforming behaviour [ 12], [ 23] anddecoupling effect of the antennas by having optimal uncoupledterminations, which is more effective for the non-identicalcase.

Furthermore, we examine the eigenvalues of H mc H H mcusing the above terminations in addition to the case with nomutual coupling as shown in Fig. 13. The gure clearly showsthat the non-identical termination tends to improve strongereigenvalues to achieve higher capacity, while the identical casehas a similar trend for all eigenvalues.

Since the 3rd eigenvalue of the channel in Fig. 13 issmall in value, and the impedance load of the middle receiveantenna for the adaptive non-identical match is small, onemay argue that the middle receive antenna does not affectthe performance. In other words, the middle antenna can beremoved from the system with no change to the performance.This is similar to antenna selection for the receive array inthe presence of mutual coupling in [ 39] when there is no
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10

0 200 400 600 800 1000 1200 1400 1600 1800 200010

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

10.9

iteration, m

C a p a c i

t y ( b i t s / s /

H z

)

Adaptive nonidentical

Adaptive identical

Selfconjugate matchz

0match

Fig. 14. Mean capacity of a 3 2 MIMO with the element spacing of d = 0 .1 at the receiver, for the following matching conditions: adaptivenon-identical, adaptive identical, self-impedance conjugate, and z0 = 50matched termination networks.

5 dB

3 dB

1 dB

1 dB

3 dB

45

225

90

270

135

315

180

0

1

2

Fig. 15. Element pattern of the receive antennas with d = 0 .1 for theassumed 3 2 MIMO using the adaptive impedance matching techniques.Results for the non-identical and identical matching scenarios are similar .

impedance matching solution. To investigate this issue, we

simulate the performance of a 3 2 MIMO considering thesame propagation scenario as the above, except the elementspacing between the receive antennas which is doubled here,i.e. 0.1 . Fig. 14 shows the mean capacity of this systemover 200 channel realisations using the following matchingnetworks: adaptive non-identical match, adaptive identicalmatch, self-conjugate match and z0 = 50 match. We seethat removing the middle receive antenna does not changethe performance improvement of the adaptive identical match(about 6% above the capacity for z0 = 50 ), while it reducesthe performance of the adaptive non-identical match. To fur-ther the investigation, the element patterns for both adaptivematching networks are shown in Fig. 15. Both identical andnon-identical match result in the same optimum loads andconsequently similar antenna pattern for such a receiver withtwo antennas. We note that the 2nd antenna in Fig. 15 is in theplace of the 3rd antenna used in Fig. 12. Comparing Figures 12and 15 reveals the impact of the middle antenna (which

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

12

14

16

18

C a p a c i

t y ( b i t s / s / H z )

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.58

10

12

14

16

18

C a p a c i

t y ( b i t s / s /

H z

)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.58

9

10

11

12

13

14

Element Spacing (d/ )

C a p a c

i t y ( b i t s / s

/ H z )

Adaptive NonidenticalAdaptive IdenticalSelfimpedance Match50 Match

(a)

(b)

(c)

Fig. 16. Mean capacity as a function of element spacing for different

matching methods and three different propagation scenarios: (a) Uniform,(b) Laplacian ( = 0 , = 40 ), and (c) Laplacian( = 90 , = 40 ).

is terminated in a small load impedance, i.e. roughly shortcircuited) on antennas 1 and 3 such that the performance isimproved signicantly. It can be interpreted as a beamformingprocess where antenna currents are weighted by employingproper impedance loads.

To complete this part, we plot the mean capacity at itera-tion number 4000 for the above terminations versus antennaelement spacing in Fig. 16, under the following scattering

distributions: (a) uniform, (b) Laplacian centred at = 0

(broadside), and (c) Laplacian with = 90 (endre). Weobserve that for an element spacing d < 0.2 , the non-identical adaptive termination gives a much higher perfor-mance improvement compared to other terminations but atthe cost of a longer convergence time, while the identicaladaptive termination has a similar performance to the self-impedance conjugate match. For d > 0.2 , both adaptiveimpedance matching techniques achieve roughly the samecapacity performance as the self impedance conjugate match,and higher than the characteristic impedance ( Z 0 = 50 ).However, the self conjugate match requires knowledge of theself-impedances of antennas, whereas the adaptive algorithmsdo not. As stated in [4], [40], [41] , the self-impedances of coupled antennas are different than the case when antennasare isolated, and that measuring these impedances is difcultin practice. Furthermore, antenna elements of practical arraysmay not be identical. For such cases, the adaptive non-identical
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11

impedance matching technique can be applied to improve theperformance.

B. Imperfect CSI at the receiver: channel estimation error

We extend our investigation to the case that the channelmatrix is unknown for the receiver and it has to be estimated.We assume a training-based estimation approach is applied andthe channel estimate H mc is provided with an estimation errorvariance 2E . The algorithm decides to select the terminationload(s) based on the capacity estimates calculated from theknowledge of H mc , either by applying ( 9) for block-at fadingchannels or directly substituting H mc into ( 8). In order toassess the performance of the algorithm, we calculate and

plot the actual capacity corresponds to the selected terminalnetwork at each iteration. We include Doppler effects and time-variations of the channel by applying Gauss-Markov channelmodel described in section II.

We consider the block data transmission with a block length of L = 100 symbols, including L p = 18 trainingsymbols per block which are split into 6 groups among theblock. For each block, a channel estimate H mc is obtained byaveraging the channel estimates from sub-blocks according to(10) , where each sub-block channel estimate has an estimationerror 2E . We also assume a uniform scattering distributionat the receiver. To investigate the effect of time-variations of

the channel, we use the Gauss-Markov channel model with= 1 .8 10 3 , and 1.8 10 2 for the slow- and fast-fadingscenarios, respectively.

The mean value of the normalised capacity instances forthe adaptive and non-adaptive uncoupled terminations aredepicted in Fig. 17 for slow-fading with solid lines and fast-fading scenario with dashed lines. Capacity instances for alltermination cases have been normalised to the correspondinginstances for the 50 match. Mean capacity values for theadaptive terminations are plotted for three different iterationsm = 100 , 500, and 2000, in order to evaluate the convergencebehaviour of these termination cases. We observe that for bothslow and fast fading scenarios, better channel estimates (i.e.,lower 2E ) result in larger capacity improvements.

Furthermore, we can see that the adaptive identical termi-nation achieves a slightly better performance than the self-conjugate match for both fading scenarios after sufcientiterations. It is noted that the self-conjugate match requiresthe knowledge of the diagonal elements of the receive arrayimpedance matrix Z R , whereas the adaptive identical does not.In comparison, the non-identical termination provides largercapacity improvements even for less accurate channel estima-tion scenarios (i.e. larger 2E ). We recall that self-impedancesof coupled antennas are different than the case when theyare isolated [ 4], [10] , and that measuring these impedancesis difcult in practice. We also note that practical antennaarrays and consequently their impedance matrices may not beToeplitz in structure. The adaptive non-identical terminationwould be a reliable solution to optimise the performance forthese corresponding scenarios.

0.95

1

1.05

1.1

1.15

1.2

1.25

M e a n

( C / C

Z 0

)

0 0.002 0.004 0.006 0.008 0.010.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

Channel Estimation Error, 2E

M e a n

( C / C

Z 0

)

Adaptive NonidenticalSelfconjugate

50

match

Adaptive IdenticalSelfconjugate50 match

(a)

(b)

m=2000

m=500

m=100

m=100

m=2000

m=500

Fig. 17. Mean value of the normalised capacity instances C/C (Z 0 ) as afunction of estimation error 2E , applying adaptive (a) identical , and (b) non-identical terminations for slow-fading and fast-fading scenarios at iterationsm = 100 , 500, 2000 .

V. C ONCLUSION

Antenna mutual coupling and channel variations may de-grade the performance of compact MIMO systems. This paperpresented single-port, adaptive, uncoupled matching networksallowing both identical and non-identical component values

to optimise the performance of MIMO systems with coupledarrays. The proposed method uses a random search algorithmto change the termination network, in order to compensate theperformance degradation. It requires neither knowledge of thearray parameters nor derivatives of the performance metric, butoptimises performance by dealing with the received signalsat antenna terminals (voltages across the load resistances).Simulation results for the capacity of a 3 3 MIMO underdifferent propagation scenarios indicated that these adaptivenetworks are capable of optimising the performance in thepresence of strong mutual coupling and time-variations of the channel. The adaptive non-identical termination gives asignicant improvement in the mean capacity (about 2bits/s/Hzfor d = 0 .05 ) at the expense of a longer convergencetime (more than ten times longer) compared to the identicalcase. The latter approach can also be suggested as a reliablecandidate to the practical arrays whose antenna elements orarray parameters may vary along the array.
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[34] Stephen Boyd and Lieven Vandenberghe, Convex Optimization , Cam-bridge University Press, 2009.

[35] C. T. Kelley, Iterative Methods for Optimization , Society for Industrialand Applied Mathematics (SIAM), 1999.

[36] R. Mudumbai, D.R. Brown, U. Madhow, and H.V. Poor, Distributedtransmit beamforming: challenges and recent progress, IEEE Commu-nications Magazine , vol. 47, no. 2, pp. 102 110, Feb. 2009.

[37] D. Aldous and J. Fill, Reversible Markov Chains and Random Walkson Graphs , (Monograph in preparation.) [Online]. Available: http://stat-www.berkeley.edu/users/aldous/RWG/book.html.

[38] C. A. Balanis, Antenna Theory: Analysis and Design , John Wiley &Sonc, Inc., 2nd edition edition, 1997.

[39] Di Lu, D.K.C. So, and A.K. Brown, Receive antenna selection schemefor v-blast with mutual coupling in correlated channels, in IEEE 19th International Symposium on Personal, Indoor and Mobile RadioCommunications, 2008. PIMRC 2008. , Sep. 2008, pp. 1 5.

[40] Y. Fei, Compact MIMO Terminals with Matching Networks , Ph.D.thesis, The University of Edinburgh, 2008.

[41] R. Mohammadkhani, Adaptive Impedance Matching to Compensate Mutual Coupling Effects on Compact MIMO Systems , Ph.D. thesis,The University of Edinburgh, 2012.

Reza Mohammadkhani received the BSc degreein Electronics Engineering from Isfahan Universityof Technology (IUT) in 2001, the MSc degreein Communication Systems Engineering from IranUniversity of Science and Technology (IUST) in2004, and the PhD degree in Signal Processing fromThe University of Edinburgh in 2012. From 2003to 2005, he worked on several research projectsat Antenna and Microwave Research Center, IUST.Currently, he is an assistant professor of SignalProcessing at the Electrical Engineering Department,

University of Kurdistan (UoK), Iran.

His research interests are in various elds of signal processing include:wireless communication systems, Multiple-Input Multiple-Output (MIMO)wireless systems, adaptive array processing, phased array radars, and passiveFM radars.


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John Thompson was appointed as a lecturer at whatis now the School of Engineering at the Universityof Edinburgh in 1999. He was recently promotedto a personal chair in Signal Processing and Com-

munications. His research interests currently includesignal processing, energy efcient communicationssystems, and multi-hop wireless communications.He has published over 200 papers to date includinga number of invited papers, book chapters and tuto-rial talks, as well as co-authoring an undergraduatetextbook on digital signal processing.

During 2012-2014, Professor Thompson will serve as member-at-largefor the Board of Governors of the IEEE Communications Society. He wastechnical programme co-chair for the IEEE Globecom Conference in Miamiin 2010 and will serve in the same role for the IEEE Vehicular TechnologyConference Spring in Dresden in 2013.

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