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Interference Avoidance and Multiaccess Vector Channels Dimitrie C. Popescu Otilia Popescu and Christopher Rose Dept. of Electrical Engineering Wireless Information Network Laboratory University of Texas at San Antonio Rutgers University [email protected] otilia, crose @winlab.rutgers.edu October 12, 2003 Abstract Distributed codeword adaptation through greedy interference avoidance methods has been shown to yield optimum codeword ensembles for simple CDMA channel models. In this paper, we show how greedy interference avoidance can be applied to general multiaccess vector chan- nels in a distributed fashion. We present two algorithms for codeword optimization based on greedy interference avoidance for which convergence to maximum sum capacity is guaranteed, and in so doing contrast interference avoidance with a recently introduced iterative water filling procedure. In addition we discuss structural properties (identical signal to interference/noise ra- tios for each codeword and identical receiver structures for each codeword) which might prove useful for simple integrated adaptive receivers. 1 Introduction Vector channels provide a theoretical framework for the analysis of a wide variety of communica- tion channels – multiple access channels, channels with memory, or channels with multiple antennas in the transmitter and/or receiver – and have thus of late received increasing attention from the re- search community. Capacity results for multiaccess vector channels have been derived in [18,22], and an asymptotically optimal water filling algorithm for multiaccess vector channels can be found in [21]. A characterization of the capacity region for multiaccess vector channels and an iterative water filling algorithm to evaluate optimal transmit spectra that maximize the sum capacity of the channel can be found in [24]. Code Division Multiple Access (CDMA) schemes have a natural vector channel representation implied by the signature sequences (codewords) corresponding to distinct users in the system. The selection of optimal signature sequences (or codeword ensembles) that maximize the sum capacity This work was presented in part at the 2002 IEEE International Symposium on Information Theory – ISIT’02 and is submitted to the EURASIP Journal on Wireless Communications and Networking, special issue on Innovative Signal Transmission and Detection Techniques for Next Generation Cellular CDMA Systems.
Transcript

Interference Avoidance and Multiaccess Vector Channels�

Dimitrie C. Popescu Otilia Popescu and Christopher RoseDept. of Electrical Engineering Wireless Information Network Laboratory

University of Texas at San Antonio Rutgers [email protected]

�otilia, crose � @winlab.rutgers.edu

October 12, 2003

Abstract

Distributed codeword adaptation through greedy interference avoidance methods has beenshown to yield optimum codeword ensembles for simple CDMA channel models. In this paper,we show how greedy interference avoidance can be applied to general multiaccess vector chan-nels in a distributed fashion. We present two algorithms for codeword optimization based ongreedy interference avoidance for which convergence to maximum sum capacity is guaranteed,and in so doing contrast interference avoidance with a recently introduced iterative water fillingprocedure. In addition we discuss structural properties (identical signal to interference/noise ra-tios for each codeword and identical receiver structures for each codeword) which might proveuseful for simple integrated adaptive receivers.

1 Introduction

Vector channels provide a theoretical framework for the analysis of a wide variety of communica-tion channels – multiple access channels, channels with memory, or channels with multiple antennasin the transmitter and/or receiver – and have thus of late received increasing attention from the re-search community. Capacity results for multiaccess vector channels have been derived in [18, 22],and an asymptotically optimal water filling algorithm for multiaccess vector channels can be foundin [21]. A characterization of the capacity region for multiaccess vector channels and an iterativewater filling algorithm to evaluate optimal transmit spectra that maximize the sum capacity of thechannel can be found in [24].

Code Division Multiple Access (CDMA) schemes have a natural vector channel representationimplied by the signature sequences (codewords) corresponding to distinct users in the system. Theselection of optimal signature sequences (or codeword ensembles) that maximize the sum capacity

�This work was presented in part at the 2002 IEEE International Symposium on Information Theory – ISIT’02 and

is submitted to the EURASIP Journal on Wireless Communications and Networking, special issue on Innovative SignalTransmission and Detection Techniques for Next Generation Cellular CDMA Systems.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 2

in a CDMA system has been addressed by several researchers and algorithms that yield such optimalcodeword ensembles can be found in [2, 14, 18–20]. Optimal signature sequences provide all usersin the system with a uniform signal-to-interference plus noise ratio. In addition, the optimal linearreceiver for such ensembles is a matched filter for each codeword [20].

Recently, optimal codeword ensembles for CDMA systems have also been obtained by applica-tion of interference avoidance methods [12, 13, 17]. We note that interference avoidance providesdistributed algorithms for codeword optimization in which users independently adjust codewordsin response to changing patterns of interference. As opposed to centralized optimization methodsperformed at the receiver and which are based on complete knowledge of the system, distributed op-timization through interference avoidance requires only that each user know its associated channeland have access to the system covariance information (which may be obtained through a feedbackchannel broadcast from the base station). We also note that, while the mathematics of interferenceavoidance allows for centralized processing as well, it is the distributed version of the algorithmwhich may prove most useful in unlicensed/uncoordinated environments.

In this paper we discuss application of greedy interference avoidance methods based on a min-imum eigenvector approach [12, 13] and extend its application to general multiaccess vector chan-nels. We note that application of these greedy interference avoidance methods has already beenshown to yield codeword ensembles that maximize sum capacity for particular vector channel mod-els corresponding to multiaccess dispersive channels or multiple-input multiple-output (MIMO)channels [7, 8, 10].

We use the same general multiaccess vector channel model as in [24] along with a multicodeCDMA scheme for transmission of information and show how greedy interference avoidance ap-plies for a given user through projection of the received signal onto its corresponding signal space.We then show that application of greedy interference avoidance for any codeword/user monoton-ically increases sum capacity and present codeword adaptation algorithms based on greedy inter-ference avoidance. We note that these algorithms yield codeword ensembles which maximize sumcapacity, and for which user transmit covariance matrices satisfy a simultaneous water filling solu-tion [24]. However, we emphasize that application of greedy interference avoidance to codewordoptimization in CDMA systems is in general not a water filling procedure although particular code-word update sequences can be designed to correspond to the iterative water filling scheme presentedin [24], and convergence to the simultaneous water filling solution is an emergent property of algo-rithms based on greedy interference avoidance.

This work firmly establishes how greedy interference avoidance can be applied to all commu-nication problems in which the underlying model is a multiaccess Gaussian vector channel. Inaddition, the potential for distributed implementation of greedy interference avoidance in multiusersystems along with the identical receiver structures implied by a common SINR for all codewordsowed to (optimal) matched filter detection, could make algorithms based on greedy interferenceavoidance good candidates for integrated receiver structures.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 3

2 The Vector Multiple Access Channel

A single user vector channel is defined by [24]

��������� (1)

where � and � are the input and output vectors of dimension � � , respectively ��� , is the additivenoise vector, and � is the ��������� channel matrix. We assume that the channel matrix � isknown and stable for a large number of transmission frames, which is a reasonable assumption forhigh data rates systems operating in environments with reduced degree of mobility [3]. From anabstract mathematical perspective, the vector channel defined in equation (1) represents a lineartransformation from an input signal space of dimension � � to an output signal space of dimension��� defined by the channel matrix � . We note that for memoryless channels the channel matrix �merely relates the bases of the input and output signal spaces [16, p. 116], and that for channels withmemory the channel matrix � incorporates also channel attenuation [7,10] and multipath [1,11,24].

Extending the definition in equation (1) to multiple users, a multiaccess vector channel is ob-tained in which a set of � users communicate with a common receiver. The multiaccess vectorchannel is defined by the equation

� � �� ���� �

� � � � (2)

where � � of dimension ��

is the input vector corresponding to user � , � ��� �"!#!#!$� � , � of dimen-sion � is the received vector at the common receiver corrupted by additive noise vector of thesame dimension, and � �

is the �%�&��

channel matrix corresponding to user � . It is assumed that�('��� �*) � �+� �"!#!#!"� � . This is a general approach to a multiuser communication system in which

different users reside in different signal spaces, with different dimensions and potential overlap be-tween them, and all being subspaces of the receiver signal space. We note that each user’s signalspace as well as the receiver signal space are of finite dimension as implied by a finite time interval,

and finite bandwidths -�

for each user � , respectively and - (which includes all -�’s corre-

sponding to all users) for the receiver [4]. Figure 1 provides a graphical illustration of such a signalspace configuration for 2 users residing in 2-dimensional subspaces with a 3-dimensional receiversignal space.

In this signal space setting we assume that in an interval of duration,

users transmit “frames”of data using a multicode CDMA approach as described schematically in Figure 2. That is, user� , � � � �"!#!#!$� � transmits the sequence of information symbols .

� � /10#2�43� !"!#!50"2

�43687*9;: as a linear

superposition of distinct, unit-energy waveforms < 2�43=?>A@CB

D � >A@CB � 6 7�= ��� 0 2

�43= < 2

�43= >A@CB (3)

as if each symbol in the frame corresponded to a distinct virtual user.In the �

�-dimensional signal space corresponding to user � , each waveform can be represented

as an ��-dimensional vector, and the transmitted vector � � corresponding to user � is equivalent

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 4

to a linear superposition of unit norm codeword column vectors � 2 �43= scaled by the correspondingsymbol 0#2

�43= . That is, each user uses an �

�� � �

codeword matrix � �� � � �� � � �� 2 � 3� !#!#! � 2 �43= ����� � 2 �436 7� � � �

(4)

and the transmitted signal vector by user � can be written as

� � � � � . � (5)

Therefore, the received signal at the common receiver can be rewritten as

� � �� ���� �

� � � . � � (6)

We note that if� �

'+��

then the ��� �

�transmit covariance matrix of user � , defined as � �� / � � � :

�9 � � � � :� , can have full rank and may span user � signal space.

The capacity region for the multiaccess vector channel defined in equation (2) has been estab-lished in [24]. In the same paper [24], maximization of sum capacity for the multiaccess vectorchannel described by equation (2)�� � ������������������ �� �

��� �� � � :

� �! #"�$&% ��'���(� > ���)� B (7)

is formulated as a convex optimization problem*,+.-/ 7 �� subject to Trace / �9 �10 � � �

'32 � � �+� �#!#!"! � � (8)

and it is shown that optimal transmit covariance matrices � � � � � �#!"!#!"� � satisfy a simultaneouswater filling condition and can be found through an iterative water filling procedure.

3 Greedy Interference Avoidance: A Brief Review

Interference avoidance methods allow users in a multiaccess system to adapt their signatures toachieve better performance by maximizing signal-to-interference plus noise-ratio. Introduced orig-inally in the context of DS-CDMA systems [17] interference avoidance methods have been subse-quently developed [13] more fully, but not in complete generality. More precisely, the frameworkfor interference avoidance in [13] assumes an arbitrary � -dimensional signal space for the receiverand all users in the multiuser system as opposed to the multiuser system in equation (2) where dif-ferent users might reside in different signal subspaces and have non-identical, non-uniform gainsacross signal space dimensions at the receiver.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 5

For background we review the basic greedy interference avoidance procedure presented in [13],which we will subsequently adapt to our general problem. Each user � is assigned a finite-durationunit-energy signature waveform �

�>A@CB , or equivalently a unit norm � -dimensional codeword � � , to

convey one information symbol 0�

which is a random variable with zero mean and unit variance.The received signal vector at the common receiver is

� � �� ���� 0

� � � � (9)

where is the additive noise vector that corrupts the received signal. We define the � � � codewordmatrix containing user codewords as columns

� ��� � � �� � !#!"! ��� !"!#! � �� � � � � (10)

the vector containing the information sent by users . � /10 � !#!#!50 � 9 : , and rewrite the received signalin vector-matrix form as � � � . � (11)

Assuming simple matched filters at the receiver for all users, the signal-to-interference plusnoise-ratio (SINR) for user � is

�� � �� :� � � � � � �� :� > � � : �! � ��� �

% � � � :� B � � (12)

with��

being the autocorrelation matrix of the interference-plus-noise seen by user � , � the auto-correlation matrix of the received signal, and � � /1 : 9 the noise covariance matrix.

In this framework, greedy interference avoidance is defined by replacement of user � code-word � � with the minimum eigenvector of

��. Sequential application by all users of this greedy

SINR maximization procedure defines the eigen-algorithm for interference avoidance [13], for-mally stated below:

1. Start with a randomly chosen codeword ensemble specified by the codeword matrix �2. For each user � � �8!#!#! �

replace user � codeword � � with the minimum eigenvector � � of the autocorrelationmatrix of the corresponding interference-plus-noise process

��

3. Repeat step 2 until a fixed point is reached.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 6

It has been shown [12] that in a colored noise background the eigen-algorithm1 converges to theoptimal fixed point where the resulting codeword ensemble performs an aggregate water filling ofthe signal space and maximizes sum capacity�� � �� � ��� > � ��� � B % �� � ��� > ����� B (13)

We note that this implies maximization of the determinant of the received signal autocorrelationmatrix � ��� � since the second term in the sum capacity expression is constant. In fact, codewordreplacement based on greedy interference avoidance monotonically increases sum capacity, as canbe easily seen when looking at ����� � before and after codeword replacement. Before codewordreplacement we have

� � �� � � � � :� , which after codeword replacement becomes

� � �� ��� � � :

�,

where � � is the minimum eigenvector of��, and we need to show that� ����� � � � � � � :���

' ������� � � � � � � :��� (14)

With��

is assumed invertible, one can factor out ����� � �from both sides of equation (14) and

obtain � ������� � �� �� �� � � � :��� �� �� �

' ��������� � �� �� �� � � � :� �� �� �� �(15)

Using the identity ���)� > � = �� �� B � ���)� > ��� ���� B where and � are � ��� , respectively � ��� ,matrices, and � = , respectively ��� , is the identity matrix of order � , respectively � , equation (15)can be re-written as > � �?� :

�� � �� � � B ' > � � � :� � � �� � � B (16)

which further reduces to � :�� � �� � � ' � :� � � �� � � (17)

When � � is the minimum eigenvector of��, then it is also the maximum eigenvector of

� � ��, and

properties of the Rayleigh quotient [16, p. 348-349] imply that equation (17) is true. Thus, equa-tion (14) is true, and as a consequence we have that greedy interference avoidance monotonicallyincreases sum capacity. This result, which is not mentioned in previous work on greedy interfer-ence avoidance [12, 13], is useful in extending greedy interference avoidance to multiaccess vectorchannels. An alternate proof of this result based on majorization theory [6] can be found in [7].

Maximization of sum capacity along with the implied water filling solution is an emergentproperty of greedy interference avoidance, as individual users do not directly attempt to maximizesum capacity through an individual or ensemble water filling scheme, but rather, they greedilymaximize the SINR of their own codeword. In fact, individual water filling schemes over thewhole signal space are impossible in this framework since each user’s transmit covariance matrix � � � � � :� is of rank one and cannot possibly span the � -dimensional signal space.

It has been pointed out to us [23] that the problem of maximizing sum capacity in equation (13)can also be formulated as a spectral optimization problem. In this case one finds user transmitcovariance matrices �

as solution of the following constrained optimization problem:

maximize�� � ����� � ����� �� �

��� � �! #"�$&% ��'� ��� > � ��� B (18)

1In which step 3 is augmented with a procedure to escape eventual suboptimal fixed points.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 7

subject to ���� Trace / �9 �+� �

' 2rank > �

B �+� � � � �"!#!#!"� � (19)

We note that, while the optimization problem defined in equation (18) subject to the constraintsin equation (19) does not enjoy the usual global convergence properties of convex optimizationproblems due to the non-convex rank constraint, application of the eigen-algorithm for interferenceavoidance is guaranteed to reach the optimal (water filling) solution [12].

Finally, we mention that the optimal linear receiver which minimizes the mean squared error(MSE) for each user is a matched filter which provides sufficient statistics for the received signal� . This result was derived in [20] and we include it for completeness here. Let ��� be the unit normreceiver filter for user � so that �0 � � � :� � (20)

The MSE is given by [5]

MSE � � � / > � :� � % 0 � B � 9 � � :� � ��� � > � :� �� % � B � (21)

and we first note that when a matched filter ��� � �� is used then the second term in equation(21) is zero. Furthermore, for the optimal codeword ensemble which maximizes sum capacity,each codeword ��� is a minimum eigenvector of its corresponding interference-plus-noise covariancematrix

� � . Therefore, the first term in equation (21), which corresponds to the Rayleigh quotientof matrix

� � , will achieve its minimum value thus implying that the MSE is minimized. Hence,the matched filter and MMSE receiver filter are identical in this case.

4 Greedy Interference Avoidance for Vector Channels

Returning to the multiaccess vector channel in equation (6) we note that, in our approach transmitcovariance matrices are expressed in terms of user codeword matrices as � � � � � :� , � � � �#!"!#!"� �and sum capacity is written in terms of user codeword matrices as�� � �� ����� � � ��� � �� �

��� �� � � � :� � :

� �! " $ % �� ���(� > ���)� B (22)

We also note that in the context of interference avoidance we are interested in sum capacity max-imization through codeword adaptation. Thus, the problem of maximizing sum capacity throughcodeword adaptation has a slightly different formulation than that in [24] which was stated here inequation (8). More precisely, in our case we are interested in finding codeword ensembles whichmaximize sum capacity, that is we need to solve the following constrained optimization problem:* +.- 7 ��

subject to Trace � � � � :��� � � � � � � � �"!#!#!"� � (23)

An additional constraint in this case is given by the fact that all matrices � � have unit norm columns.We note that this is a convex optimization problem only when each user has at least as many

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 8

codewords� �

as signal space dimensions ��, so that its corresponding transmit covariance matrix �

is allowed to span the whole signal space. We also note that sum capacity maximization subjectto additional constraints on the rank of �

matrices is still an open research problem [23].The fact that algorithms based on greedy interference avoidance have been shown to yield code-

word ensembles that maximize sum capacity for dispersive channels [7, 10] – a particular case ofthe general multiaccess vector channel in equation (6) – suggests that greedy interference avoid-ance may also be used in the general context of multiaccess vector channels to obtain such optimalcodeword ensembles. We note that, since the second term in the sum capacity expression in equa-tion (22) is fixed, one needs to investigate only maximization of the determinant of the receivedsignal covariance matrix

� � � / � � : 9 � �� ���� �

� � � � :� � :� �! (24)

To show how greedy interference avoidance can be applied to this determinant maximizationproblem, we start by rewriting the received signal in equation (6) from the perspective of user �

� � � ��� � . � � ������ �

��� � �

� � � . � �&� � �.� �".�� ��� � (25)

where � � represents the interference-plus-noise seen by user �� � � ���

��� ���� � �

� � � . � � (26)

with covariance matrix

� � � � /�� � � :� 9 � ������ �

��� � �

� � � � :� � :� � (27)

Since� � is symmetric it can be diagonalized

� � ��� � � � :� (28)

Furthermore, because� � is a positive definite covariance matrix we can define the whitening trans-

formation � � � � �� �� � :� (29)

such that, in transformed coordinates equation (25) is equivalent to

�� � � � � � � � � �.� �".�� � � � � � � �� � � �".�� �� � (30)

where�� � � � � � � is the channel matrix seen by user � in the new coordinates and � � � � � �

is the equivalent “white noise” with covariance matrix� /� � :� 9 � � � � � � :� � � equal to the

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 9

identity matrix. We note that the received signal covariance matrix in the transformed coordinatesis related to the original signal covariance matrix by the equation

�� � � / �� �� : 9 � � � � � :� (31)

and any procedure that attempts to increase ���)� �� through adaptation of user � codeword matrix � �will also increase � ��� � since they are related by����� �� � ���)� > � � � � :� B � ���)� � > � ��� � � B � (32)

We now apply the singular value decomposition (SVD) [16, p. 442] to the transformed channelmatrix corresponding to user � �� � ��� ��� ��� :� (33)

where matrix � � of dimension � � � has as columns the eigenvectors of�� � �� :� , matrix � � of

dimension � � � � � has as columns the eigenvectors of�� :� �� � , and matrix � � of dimension � � � �

contains the singular values of�� � on the main diagonal and zero elsewhere. We note that because� � is invertable the rank of

�� � will be equal to that of � � . Without loss of generality we assumethat � � has full rank2 � � . Thus, the singular value matrix � � can be partitioned as

� � � � �� �� � (34)

with�� � an � � � � � diagonal matrix containing the non-zero singular values along the diagonal

and zeros in rest. The left inverse of � � is defined as

�� � � �� � �� � �(35)

and it is obvious that ��� � � � � ��� (36)

Returning to equation (30) in which the SVD for transformed channel matrix�� � has been

applied we have �� ��� ��� � � :� � �".�� �� � (37)

We pre-multiply by � :� � � ��� :� � � � ��� :� � �".�� ��� :� � (38)

and define�� � � � :� � � and

� � ��� :� � . Note that because both � � and � � are orthogonalmatrices they preserve norms of vectors. Thus, columns of

�� � are also unit norm as were thecolumns of � � . Also, because the equivalent noise term � is white, then

� � will remain whitewith the same covariance matrix equal to the identity matrix.

� � � � � �� � . � � � � (39)

2This is not a restriction since if ��� is not full rank then some dimensions the user � signal space will have zeroprojection on the output space. Therefore we can redefine a reduced codeword matrix ��� which uses only dimensionswith nonzero projections on the output space.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 10

with covariance matrix given by

� 2 � 3 � � / � � � :� 9 � � :� �� � � (40)

for which ���)� � 2 � 3 � � ��� �� since � � is an orthogonal transformation. Following a similar lineof reasoning as above we note that any procedure which will attempt to increase ����� � 2 � 3 throughadaptation of

�� � will also increase � ��� � . We also note that the partitioning of the singular valuematrix in equation (34) implies the following partition on

� 2 � 3� 2 � 3 � � �� � �� � �� :� �� � � � ��� �� � � � � � � (41)

in which � � denotes the identity matrix of order � and�

denotes a matrix with all elements equal tozero.

At this point we define an equivalent problem for user � by pre-multiplying with the left inverseof � � and obtain �� � � ��� � � � �� �".�� � �� � (42)

which is identical in form with equation (11) and allows straightforward application of greedyinterference avoidance to optimizing the codeword matrix

�� � . The “noise” term�� � in equation (42)

represents the interference-plus-noise from the rest of the system that is present in user � signalspace and has covariance matrix

�� � � � / �� � �� :� 9 � � � � / � � � :� 9 � :� � �� � �� (43)

The transformed codeword matrix�� � in equation (42) is completely equivalent to the original code-

word matrix � � since they are related through an orthogonal transformation � :� . The covariancematrix of

�� � is given by �� 2 � 3 � � / �� � �� :� 9 � �� � �� :� � �� � �� (44)

and using the partition in equation (41) we have���)� � 2 � 3 � � ��� > �� � �� � �� :� �� � � � � � B � ���)� � � �� � ��� �� 2 � 3 (45)

which implies again that increasing � ��� �� 2 � 3 will also increase ���)� � 2 � 3 which in turn implies in-creasing ���)� � – the determinant of the original received signal covariance matrix.

Greedy interference avoidance for multiaccess vector channels consists of replacing codewordcorresponding to symbol � of user � by the minimum eigenvector of the autocorrelation matrix ofthe corresponding interference-plus-noise process in the transformed problem, that is

� 2 � 3= � �� � �� :� % �� 2 � 3= �� 2 � 3 := � �� � �� (46)

Applying the result in equation (14) we have that � ��� �� 2 � 3 � ����� > � 2 � 3= � �� 2 � 3= �� 2 � 3 := B is monotonicallyincreased by application of greedy interference avoidance, and as a consequence sum capacity willbe monotonically increased.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 11

5 Interference Avoidance Algorithms for MultiaccessVector Channels

Numerous interference avoidance algorithms can be formulated based on repeated application of thegreedy interference avoidance procedure presented in the previous section. These are defined by thevarious ways in which user codewords are selected for replacement. For example, one algorithmcould be defined by replacement at a given step of one codeword of a given user, followed byreplacement of a randomly selected codeword of a randomly selected user. Alternatively, at a givenstep of the algorithm, one could replace the codeword with the lowest SINR over all codewordsand users. Or one could replace the codeword which will yield the maximum increase in sumcapacity. One can even posit “lagged” versions of the algorithm that instead of using the minimumeigenvector, a linear superposition of the minimum eigenvector and the original codeword is used.

Nonetheless, we note that empirically we have observed that repeated application of greedyinterference avoidance with various codeword replacement reaches an optimal fixed point – unlessdeliberately placed in a suboptimal fixed point at initialization. Unfortunately, we have been unableto prove this result in general. However, simulations have shown that when users have at least asmany codewords as signal space dimensions, this fixed point reached by interference avoidanceinvariably corresponds to a simultaneous water filling solution for all users and must therefore be amaximum sum capacity ensemble [24].

We now formally state two algorithms for which convergence to maximum sum capacity can beproven.

The Maximum Capacity Increase Algorithm For Interference Avoidance

1. Start with a randomly chosen codeword ensemble specified by user codeword matrices ��� ��� �� ���2. Define the equivalent problem in equation (42) for all users � � � !"!#! �3. Identify that codeword � 2 � 3= whose replacement will maximally increase sum capacity. If no

codeword will increase sum capacity, and escape methods [12] are ineffective, then STOP.Else

(a) apply greedy interference avoidance: replace � 2 � 3= by the minimum eigenvector of thecorresponding

� 2 � 3= in equation (46)

(b) return to step 2

We note that, in addition to monotonically increasing sum capacity as a consequence of applyinggreedy interference avoidance, the algorithm stops only if sum capacity can no longer be increased.Therefore, sum capacity must be strictly increasing. Using a similar line of reasoning as in [7]for the similar algorithm defined for dispersive channels, it can be shown that in the limit, as thenumber of iterations goes to infinity, the algorithm yields codeword matrices which maximize sumcapacity and satisfy a simultaneous water filling solution [24].

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 12

The algorithm may not seem very attractive from a practical implementation point of viewsince it involves finding the poorest performer. However, one could imagine a stochastic updateprocedure where update probability increased with decreasing codeword SINR. Though not exactlythe algorithm stated above, it is close enough that one could imagine similar performance. But thepractical utility of this specific algorithm aside, its theoretical importance is that it is not a waterfilling procedure but still which converges to a simultaneously water-filled solution.

An alternative algorithm consists of a codeword update sequence which simulates iterative waterfilling:

The Eigen-Algorithm for Multiaccess Vector Channels

1. Start with a randomly chosen codeword ensemble specified by user codeword matrices ��� ��� �� ���2. For each user � � �8!#!#! �

(a) Define the equivalent problem in equation (42)

(b) adjust user � codewords sequentially by applying greedy interference avoidance: thecodeword corresponding to symbol � of user � is replaced by the minimum eigenvectorof the corresponding

� 2 � 3= in equation (46)

(c) Iterate previous step until convergence (making use of escape methods [12] if necessary)

3. Repeat step 2 iteratively for each user until a fixed point is reached for which further modifi-cation of codewords will bring no additional improvement.

Application of the eigen-algorithm for user � in steps 2(b)–(c) above corresponds to water fillingof user � ’s signal space while regarding the other users as noise. The water filling process isdone for all users sequentially and repeated (step 3) until convergence to a fixed point. From thisperspective, iterative application of the eigen-algorithm by all users in the system is an instance ofthe iterative water filling procedure in [24] in which each user adapts its corresponding codewordmatrix, regarding all other users’ signals as noise while maximizing � ��� � . Such a procedure isguaranteed to converge to maximum sum capacity.

6 Discussion and Conclusions

We showed how interference avoidance can be applied to general multiaccess vector channels wherethe channel was regarded as a linear transformation between the input signal space (transmitter) andthe output signal space (receiver). The approach is general and allows different, possibly overlap-ping, signal spaces for distinct users in a multiuser system, all being subspaces of the signal spaceat the common receiver(s).

Information is transmitted using multicode CDMA in which symbols of a given user are con-veyed by a distinct signature waveforms. Using an arbitrary signal space representation, the set ofwaveforms corresponding to a given user is equivalent to a codeword matrix, and optimal codeword

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 13

matrices that maximize the sum capacity of the multiaccess channel can be obtained by applicationof greedy interference avoidance.

By whitening the interference-plus-noise and using the SVD, greedy interference avoidance isapplied in a given user’s signal space. We show that application of greedy interference avoidance forany codeword/user monotonically increases sum capacity. Next, we note that various interferenceavoidance algorithms can be defined based on the codeword update sequence and present two suchalgorithms for which convergence to maximum sum capacity is assured: one which is equivalent toan iterative water filling procedure [24] while the other is not, though it also yields a sum capacitymaximizing codeword ensemble satisfying a simultaneous water filling solution. We also note thatempirically, we have observed convergence to maximum sum capacity under a variety of codewordreplacement sequences, although we have been unable to prove convergence in general.

We note that, from a practical standpoint, the assumption of known and stable channels may berelaxed and replaced with knowledge of the average channel behavior, in which case interferenceavoidance will be applied to the average channel [9]. We also note that the distributed natureof algorithms based on greedy interference avoidance requires that each user know only its ownchannel and the received signal covariance, and allows independent codeword adjustments by usersin the system. This, in addition to the identical SINRs attained by all codewords of a given useralong with the identical optimal linear receiver structure for each codeword could prove useful forinexpensive integrated realizations.

One natural question which arises when considering interference avoidance is whether it mightbe simpler to do iterative water filling using orthogonal codewords based on the eigendecompositionof the received covariance matrix. Certainly this is possible, but in general such codewords will havedifferent (perhaps greatly different) SINRs. Though from a theoretical signal processing standpoint,this non-uniformity is irrelevant to sum capacity, from a practical standpoint (receiver front endsdynamic range, peak to average power ratios, quantization, etc.) having uniform SINRs may beespecially useful.

However, if uniform SINRs are desirable, one could also imagine calculating the optimal covari-ance and then using known codeword construction algorithms at the transmitter (e.g., [18]) to obtainappropriate equal power (and thereby equal SINR) codewords. In comparison, greedy interferenceavoidance adjusts one codeword at a time and may therefore seem both slower and computationallymore costly. This issue needs to be carefully evaluated.

As a beginning here are some initial thoughts based on our experience with interference avoid-ance. The distributed nature of the algorithm posits codeword tracking at the receiver which im-plies incremental codeword adaptation [15]. Otherwise an explicit (and potentially costly) feedbackchannel to convey the proper codewords to the receiver must be used. Per-codeword interferenceavoidance is intuitively more incremental in its total effect on the received covariance when manyusers are involved. And even in cases where the new codeword differs significantly from the old,lagged versions of the algorithm where linear superpositions of the old and new are used for updatealso increase sum capacity at each step (and have empirically always converged to the optimum).In contrast, such gradual adjustment renders precalculated codewords useless in the face of similargradual adjustments by other users. So, whether one pursues direct water filling and precalcu-lates codewords, or incrementally adjusts codewords may depend on practicalities of the implicit

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 14

necessity for feedback to keep the receiver and transmitter in “sync”.In any case, practicalities aside, our results allow direct application of greedy interference avoid-

ance methods to a variety of problems including arbitrary dispersive channel models, multiusermultiple antenna systems, and asynchronous multiuser systems.

References

[1] J. I. Concha and S. Ulukus. Optimization of CDMA Signature Sequences in Multipath Channels.In Proceedings ��� rd IEEE Vehicular Technology Conference – VTC’01, volume 3, pages 1227–1239,Rhodes, Greece, May 2001.

[2] T. Guess. Optimal Sequences for CDMA with Decision-Feedback Receivers. IEEE Transactions onInformation Theory, 49(4):886–900, April 2003.

[3] H. Hashemi. The Indoor Radio Propagation Channel. Proceedings of the IEEE, 81(7):943 – 968, July1993.

[4] H. J. Landau and H. O. Pollack. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty– III: The Dimension of the Space of Essentially Time- and Band-Limited Signals. The Bell SystemTechnical Journal, 41(4):1295–1335, July 1962.

[5] U. Madhow and M. L. Honig. MMSE Interference Suppression for Direct-Sequence Spread-SpectrumCDMA. IEEE Transactions on Communications, 42(12):3178–3188, December 1994.

[6] A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and its Applications. AcademicPress, Orlando, FL, 1979.

[7] D. C. Popescu. Interference Avoidance for Wireless Systems. PhD thesis, Rutgers University, Depart-ment of Electrical and Computer Engineering, 2002. Thesis Director: Prof. C. Rose.

[8] D. C. Popescu and C. Rose. A New Approach to Multiple Antenna Systems. In Proceedings ��� th

Conference on Information Sciences and Systems – CISS’01, volume II, pages 868–871, Baltimore,MD, March 2001.

[9] D. C. Popescu and C. Rose. Fading Channels and Interference Avoidance. In Proceedings ��� th AllertonConference on Communication, Control, and Computing, pages 1073–1074, Monticello, IL, October2001.

[10] D. C. Popescu and C. Rose. Interference Avoidance Applied to Multiaccess Dispersive Channels. InProceedings ��� th Annual Asilomar Conference on Signals, Systems, and Computers, volume II, pages1200–1204, Pacific Grove, CA, November 2001.

[11] G. S. Rajappan and M. L. Honig. Signature Sequence Adaptation for DS-CDMA with Multipath. IEEEJournal on Selected Areas in Communications, 20(2):384–395, February 2002.

[12] C. Rose. CDMA Codeword Optimization: Interference Avoidance and Convergence Via Class Warfare.IEEE Transactions on Information Theory, 47(6):2368–2382, September 2001.

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[13] C. Rose, S. Ulukus, and R. Yates. Wireless Systems and Interference Avoidance. IEEE Transactionson Wireless Communications, 1(3):415–428, July 2002.

[14] M. Rupf and J.L. Massey. Optimum Sequence Multisets for Synchronous Code-Division Multiple-Access Channels. IEEE Transactions on Information Theory, 40(4):1226–1266, July 1994.

[15] J. Singh and C. Rose. Distributed Incremental Interference Avoidance. In Proceedings 2003 IEEEGlobal Telecommunications Conference - GLOBECOM ’03, San Francisco, CA, December 2003.

[16] G. Strang. Linear Algebra and Its Applications. Harcourt Brace Jovanovich College Publishers, SanDiego, CA, third edition, 1988.

[17] S. Ulukus. Power Control, Multiuser Detection and Interference Avoidance in CDMA Systems. PhDthesis, Rutgers University, Department of Electrical and Computer Engineering, 1998. Thesis Director:Prof. R. D. Yates.

[18] P. Viswanath and V. Anantharam. Optimal Sequences and Sum Capacity of Synchronous CDMA Sys-tems. IEEE Transactions on Information Theory, 45(6):1984–1991, September 1999.

[19] P. Viswanath and V. Anantharam. Optimal Sequences for CDMA Under Colored Noise: A Schur-Saddle Function Property. IEEE Transactions on Information Theory, 48(6):1295–1318, June 2002.

[20] P. Viswanath, V. Anantharam, and D. Tse. Optimal Sequences, Power Control and Capacity ofSpread Spectrum Systems with Multiuser Linear Receivers. IEEE Transactions on Information Theory,45(6):1968–1983, September 1999.

[21] P. Viswanath, D. Tse, and V. Anantharam. Asymptotically Optimal Waterfilling in Vector MultipleAccess Channels. IEEE Transactions on Information Theory, 47(1):241 – 267, January 2001.

[22] P. Viswanath and Anantharam V. Total Capacity of Multiaccess Vector Channels. Technical Memoran-dum M99/47, Electronics Research Laboratory, University of California, Berkeley, 1999.

[23] W. Yu. Interference Avoidance and Iterative Water Filling: A Connection. Private communication, May2001.

[24] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi. Iterative Water-Filling for Gaussian Vector Multiple AccessChannels. In Proceedings 2001 IEEE International Symposium on Information Theory - ISIT’01, page322, Washington, DC, June 2001. Submitted for journal publication.

Popescu, Popescu, and Rose: Interference Avoidance and Multiaccess Vector Channels 16

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