Interference Avoidance Via Waveform Agility Christopher Rose, Sennur Ulukus and Roy Yates Correspondence to: Christopher Rose Telephone: Fax: E-mail: Subject Areas:
WINLAB { Rutgers University 73 Brett Road Piscataway NJ 08854-8060 (732) 445-5250 (732) 445-2820
[email protected] Globecom'99 Symposia Communication Theory Advanced Signal Processing for Communications Future Wireless Communication Systems
Abstract Motivated by the emergence of programmable radios, we seek to understand a class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a metric, we present a class of iterative distributed algorithms for synchronous systems which results in an ensemble of optimal waveforms for multiple users connected to a common receiver (or co-located independent receivers). That is, the waveform ensemble meets the Welch Bound with equality and therefore achieves minimum average interference over the ensemble of signature waveforms. We provide convergence results for a number of scenarios and propose further research to help develop a theory and practice of interference avoidance for future licensed and unlicensed band systems.
Interference Avoidance Via Waveform Agility Christopher Rose, Sennur Ulukus, Roy Yates WINLAB { Rutgers University
[email protected] [email protected] [email protected]
Abstract Motivated by the emergence of programmable radios, we seek to understand a class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a metric, we present a class of iterative distributed algorithms for synchronous systems which results in an ensemble of optimal waveforms for multiple users connected to a common receiver (or co-located independent receivers). That is, the waveform ensemble meets the Welch Bound with equality and therefore achieves minimum average interference over the ensemble of signature waveforms. We provide convergence results for a number of scenarios and propose further research to help develop a theory and practice of interference avoidance for future licensed and unlicensed band systems.
1 Introduction Wireless system designers have always had to contend with interference from both natural sources and other users of the medium. Thus, the classical wireless communications design cycle has consisted of measuring or predicting channel impairments, choosing a modulation method, signal pre-conditioning at the transmitter and processing at the receiver to reliably reconstruct the transmitted information. These methods have evolved from simple (like FM and pre-emphasis) to relatively complex (like CDMA and adaptive equalization). However, all share a common attribute { once the modulation method is chosen, it is dicult to change. For example, a frequency hopped system cannot be simply modi ed to obtain a TDM system owing to the complexities of the transmission and reception hardware. Universal radios1 [1, 2] change this paradigm by providing the communications engineer with a radio which can be programmed to produce almost arbitrary output waveforms and act as an almost arbitrary receiver type. Thus, it is no longer unthinkable to instruct the transmitting and receiving radios to use a more e ective modulation in a given situation. Of course, practical radios of this sort are probably many years away. Nonetheless, if Moore's law holds true, they are certainly on the not-too-distant horizon. 1
Also often called software radio.
1
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It is therefore probable that wireless systems of the near future will have elements which adapt dynamically to changing patterns of interference by adjusting modulation and processing methods in much the same way that power control [3{7], is used today, albeit on a possibly slower time-scale. Furthermore, if the recent release of 300MHz of unlicensed spectrum in the 5GHz range [8] is any indication, one might expect there to be an abundance of mutually interfering independent systems and no central control for ecient coordination. This provides added impetus to understand mutual interference of systems at some general level and implicit coordination in a multi-domain environment. In this paper we consider radios which can vary their output waveforms as well as their demodulation method. We start with optimal waveform selection to maximize the signal to interference ratio (SIR) for a power-constrained user in the presence of interference. We then consider ensembles of users and describe a class of distributed greedy algorithms which can optimize their shared use of the medium. That is, through local self-interested action, a social optimum can always be reached. Moreover, the algorithms we will describe are simple and amenable to adaptive implementation.
2 Background Consider the classical continuous time digital communications model, in which during an interval [0; T ], a signal bppS (t) is transmitted where b = 1 equiprobably, p is the received power, and S (t) is the (unit energy/power) signal waveform. A receiver recovers r(t) = bppS (t) + Z (t), where Z (t) is an independent interference stochastic waveform that may be composed of both thermal noise and interfering signals of other transmitters. For a single bit, the fundamental problem is to build a receiver which guesses b with minimum probability of error. Alternatively, when b is one bit in a stream of coded bits, we would like to produce a soft estimate of b with high signal to interference ratio (SIR). When Z (t) is P composed of known waveforms in addition to independent Gaussian noise, that is p Z (t) = i bi piSi (t)+N (t), multiuser receivers have been designed for a variety of objectives, e.g. minimum probability of error, maximum SIR, or zero interference from other users [9]. These multiuser systems share the property that the receiver does as best it can given the set of transmitter signals Si(t). From the perspective of the receiver for S (t), the interference Z (t) is simply a stochastic process. Without loss of generality, we will assume Z (t) is a zero mean process. Ideally, we would like to obtain a set of uncorrelated (and preferably independent) sucient statistics and then optimally combine these either to detect the bit b or to derive an estimate of b. When Z (t) is Gaussian, these projections would indeed be independent Gaussian random variables and the optimal detection problem would be easily solved. A complete and rigorous development of the ideas can be found in [10]. Here we provide a brief recapitulation. In general, given a stochastic process, Z (t) we seek an orthonormal representation
Z (t) = Nlim !1 with ai =
RT 0
N X i=1
ai i(t)
(1)
Z (t)i (t)dt. Note that in Equation (1), our convergence requirement is not
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the usual pointwise limit, but the limit in the mean criterion. The interested reader is referred to [10, 11] for further details. For our purposes, we assume that such an expansion for Z (t) exists and converges. Now we seek a special set of orthonormal i which produce uncorrelated projections.
E [hi (t); Z (t)ihj (t); Z (t)i] = E
"Z
TZT 0
0
#
i ( )Z ( )j (t)Z (t) dt d = j ij
(2)
The solution to this integral equation requires
ii (t) =
ZT 0
RZ (t; )i( )d
(3)
For the interested reader, the properties of Equation (3) are discussed at some length in [10]. Since integral equations are in general dicult to solve, it is useful to derive an equivalent discrete representation of Equation (3). This will allow us to use simple methods from linear algebra. Now let us assume that Z (t), and therefore the function set fi(t)g, can be wellapproximated by a nite set of orthonormal basis functions f n(t)g on the interval [0; T ]. That is, we assume that the process Z (t) has no signi cant energy outside some nite signal space. As an example, a process \almost" limited to bandwidth W has a basis function set with about 2WT orthonormal functions [10,12,13]. Likewise, for a synchronous CDMA system with N chips per bit, the appropriate orthonormal set consists of the N time-shifted chip pulses. One could also use a space-time orthogonalization for reception/transmission antenna diversity and/or a frequency-time orthogonalization for a frequency-hopped system. Regardless of the speci cs, once we assume a nite basis function set over the interval, the i (t) can then be represented by the nite sum i (t) = Combining equations (3) and (4) yields
i
N X n=1
in n(t) =
N X n=1
ZT 0
in n(t)
RZ (t; )
N X n=1
(4)
in n( )d
(5)
Further projecting the right and left hand sides onto k (t) yields
i ik =
N X n=1
in
ZTZT |0
0
RZ (t; ) k (t) n( ) dt d = {z rkn
}
N X n=1
rkn in
We may also rewrite the rnk directly in terms of the projections Zn =
rnk = E [
ZTZT 0
0
RT 0
Z (t)Z ( ) k (t) n( )dtd ] = E [Zk Zn]
(6)
Z (t)n(t)dt as (7)
Thus, equation (6) is a reduction of the continuous time integral equation (3) to a standard matrix eigenvalue/eigenvector equation of the form i h (8) E ZZ> i = R i = i i
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where i = [ i1 iN ]> and Z = [Z1 ZN ]>. Each eigenvector corresponds to an eigenfunction of equation (3) and it is easily veri ed that each eigenvalue is the amount of interference signal energy carried by that eigenfunction. It is also easy to verify that since RZ (t; ) is an autocorrelation function, R is symmetric and positive semi-de nite. This implies that R has non-negative eigenvalues and an associated full set of orthonormal eigenvectors which span < 0 say 0
(11)
where the frng are the projections of r(t) onto the decorrelating basis functions n(t). This detection method is called a whitening lter since it can be written as say 1 r s n n p p >< 0 n n n=1 say 0 N X
(12)
which is an initial rescaling of the input p to make interference components (fzng), already uncorrelated, have equal energy (fzn= ng) { just as would be the case for a white noise p process without rescaling. A matched lter on the rescaled signal vector components sn= n is then performed to complete the detection process. It is worthwhile to note that in a CDMA system where Z (t) consists of the other users' known signature waveforms and additive white Gaussian noise, the vector c with components
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cn = sn=n is a scaled version of the well known minimum mean squared error (MMSE) linear lter [16] and the decision rule (11) is the MMSE multiuser detector. We see that the lter output (and decision statistic) X
X = c>r = c
n rn
n
N s2 ! N X n b + X sn zn n=1 n n=1 n
=
(13)
contains both signal and interference terms and that the output SIR is
SIRX =
E b PNn=1 s2nn
2
PN sn zn 2 E n=1 n
=
P
N s2n 2 n=1 n PN PN sn sm E [zn zm ] n=1 m=1 n m
N s2 X = n n=1 n
(14)
It is well known that among all linear lters c, the MMSE lter maximizes the output SIR [16]. However, equation (14) demonstrates that it remains possible to obtain a higher output SIR by altering the components sn of the desired signal S (t). That is, when S (t) is subject to the unit energy constraint Pn s2n = 1, we can maximize SIRX by choosing sn = 1 for any n = = mink k . In this case, we have S (t) = n (t). Equivalently, we could distribute the signal energy in some arbitrary way over all such n(t). Regardless, this result has a simple intuitively pleasing physical interpretation: To obtain maximum SIR, place all the signal energy where there is least interference We call this procedure interference avoidance and for a single user with a given interference process, the method is straightforward. We now examine the implications of this simple Karhunen-Loeve inspired rule for an ensemble of users. We will nd that the greedy objective in which a user adapts its signature to improve its SIR has desirable consequences for multiuser systems.
3 Interference Avoidance for Multiple Users We now consider a multiuser system in which the received signal r(t) explicitly includes M users and white Gaussian noise. Given the existence of a nite set of N orthonormal basis functions i(t) for the signal space, we can express the received signal as the vector
r=
M p X i=1
pibi si + N
(15)
where N is the projection of the additive white Gaussian noise onto the basis. The classical communications scenario presumes that each user signature Si(t) is xed. Assuming universal radio receivers and transmitters, we now allow the use of tailored signature waveforms Si(t). Without loss of generality, we assume each Si(t) has unit energy. The relationship between signature selection and multiuser system capacity has been studied in several papers [17{19]. In [17], it is shown that for a set of users' rates R1; : : : ; RM belonging to the information theoretic achievable rate region C , the sum capacity is M X 1 log hdet I + ?2 SPS>i (16) Cs = (R ;max R = N i 2 1 ;RM )2C i=1
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In equation (16), IN is the N N identity matrix, P is the diagonal matrix of users powers pk , and S = [s1 ; ; sM ] is the N M matrix with columns si . Note that these sk are the projections of the Si(t) onto the arbitrary spanning orthogonal basis f k (t)g so that the si may be called generalized codeword vectors and represent arbitrary signals inside an arbitrary (but nite) signal space. When the powers of the users are the same, pk = p for all k, (16) reduces to 1 1 p p > > Cs = 2 log det IN + 2 SS = 2 log det IM + 2 S S (17) where the last equality follows from the fact that for any two matrices AK M and BM K , det (IK + AB) = det (IM + BA). It was shown in [20] that the sum capacity for equal received powers is maximized if the signature sequences are chosen such that if M N , S>S = IM (18) and if M N , SS> = (M=N )IN (19) In [19], the user capacity of a CDMA system is de ned in terms of the maximum number of admissible users. Given the signal space dimensionality N and a common SIR target , M users are said to be admissible if there exist positive powers pi and signature sequences si such that each user has an SIR at least as large as . The user capacity was found for two kinds of linear receiver structures in [19]: matched lters and MMSE lters [16, 21]. It was shown in [19] that the user capacity with MMSE receivers is maximized if the signature sequence set is chosen to satisfy (18) if M N and (19) if M N , and if the received powers of the users are chosen to be the same. When M N and (18) is satis ed or when M N and (19) is satis ed, the MMSE and matched lters are the same. Thus, reference [19] concludes that the user capacity of a system with matched lter receivers is the same as that using MMSE lters. In [20], the unit energy sequence sets satisfying (19) are called Welch Bound Equality (WBE) sequence sets. Welch [22] derived the following lower bound on the sum of the squared cross correlations, which we will call total squared correlation (TSC). TSC = Trace[(SS>)2] =
M X M X (s>i sj )2 i=1 j =1
M 2 =N
(20)
For a simple derivation of the bound (20), see [23, 24]. Note that sequence sets satisfying (19) satisfy the bound (20) with equality. We observe that the set of sequences satisfying either equation (18) for M N or equation (19) for M N have the property that the sequence set has minimum TSC. That is, to maximize both sum capacity and user capacity, we should choose sequence sets with minimum TSC.
4 Iterative Methods of TSC Reduction There are a number of methods which might be used to determine codeword sets which minimize TSC and thereby maximize sum capacity [18, 19, 25]. Here we explore simple
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iterative methods which can be applied by each transmitter/receiver pair asynchronously and independently. For a single user k, we observe that SS> = Rk + sk s>k where Rk = Pi6=k si s>i , the correlation matrix of the interference faced by user k, is analogous to the matrix R introduced in Section 2. When user k replaces its signature vector sk with a vector x, the resulting di erence in TSC is = Trace[(Rk + sk s>k )2 ] ? Trace[(Rk + xx>)2]
(21)
After some linear algebraic manipulations we nd that 0 i 2s>k Rk sk + jsk j2 2x>Rk x + jxj2 which reduces to
s>k Rk sk x>Rk x
(22)
(23) if jxj = jsk j as we will hereafter assume. When the interference faced by user k includes additive white Gaussian noise with power spectral density 2, we may replace Rk by Zk = Rk + 2I if desired. As previously, we note that minimizing Trace[(SS>)2] isPequivalent to minimizing Trace[(SS> + 2I)2] because the trace of SS> is xed at E = Mk=1 jsk j2, the total energy in the signal constellation. Thus, in terms of TSC minimization, operations on Rk or Zk are equivalent. Note that equation (23) de nes a class of replacement algorithms whereby a given user can reduce (or at least not increase) the total squared correlation assuming other users' codewords remain xed during the replacement. Such an algorithm may be used by each user sequentially until all users have updated their codewords. At that point the cycle may begin anew. Cycles (iterations) would then be repeated until there were no further change in the TSC by individual codeword updates. Consideration of this process begs at least two questions. First, what is an example of such an algorithm? Second, do such algorithms eventually minimize TSC? In answer to the rst question we present two algorithms. The most obvious method we will call the eigen-algorithm { let x = k where k is a minimum eigenvalue eigenvector of Rk . From equation (14), we see that one step of the eigen-algorithm maximizes the SIR of user k by allowing nonzero signal energy only along those basis functions with absolute minimum n. From equation (23) we see that such a choice guarantees 0 (and thereby no increase in the TSC) since both the right and left hand sides of the condition are under-bounded by ( k )>Rk k . The second less obvious algorithm we will call the MMSE algorithm since we replace sk by the unit energy MMSE receiver lter ck = s>k [Zk ]?2sk ?1=2 [Zk ]?1 sk . We note that in the presence of additive white Gaussian noise, Zk is always invertible. The lter ck is equivalent to forming cn = ?1s in the decorrelated space and then renormalizing where is the eigenvalue matrix of Zk . The MMSE algorithm has been proposed in [6,26{28] and it is proven in [6,26] that the MMSE algorithm is indeed an interference avoidance algorithm. The question of convergence is addressed next where we nd that the eigen-algorithm and the MMSE algorithm share a number of common properties:
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Both algorithms decrease the TSC monotonically. Since TSC is bounded below by the
Welch bound, they must converge. Furthermore both sides of equation (23) are lower bounded by ( k )>Rk k where k is an eigenvector of Rk with smallest eigenvalue k . Thus, for the eigen-algorithm, = 0 i each sk is a minimum eigenvalue eigenvector of its Rk . By not permitting gratuitous changes in signatures without reductions in TSC, the eigen-algorithm converges. For the MMSE algorithm, if TSC converges, then the signatures must converge [28]. At xed points of both algorithms, each sk is an eigenvector of Zk . For neither algorithm is the resulting codeword set unique. Any rotation of the codeword set will have the same cross-correlation properties. When M N , the signatures converge to an orthonormal set. For the MMSE algorithm, this may take several cycles. For the eigen-algorithm, this occurs after one cycle since each user chooses an eigenvector orthogonal to the previously chosen signatures. When M N , the algorithms may converge to a WBE signature set S satisfying SS> = (M=N )IN . Alternatively, the algorithms may converge to a local minimum for TSC. In [6, 28] mild conditions are derived under which the MMSE algorithm converges. The eigen-algorithm has always converged to minimum TSC in experiments starting from random codeword vectors, but general conditions for convergence remain unknown. Further detail for eigen-algorithm xed points is provided in section 5 and ad hoc methods which seem to force social optima are discussed in section 6.3. The intuition behind all interference avoidance algorithms which obey equation (23) is embodied by the simple requirement x>Rk x s>k Rk sk { the replacement vector x attempts to reduce the interference from the ensemble of other user vectors and noise. From the standpoint of implementation, in the MMSE algorithm, user k must identify (Rk + 2I)?1sk . In the eigen-algorithm, user k seeks the minimum eigenvalue eigenvector k of Rk . These points, taken together suggest that the class of algorithms governed by equation (23) could be implemented by blind techniques at the receiver along with a feedback channel to the transmitter. Speci cally, in the MMSE algorithm, the receiver for user k could be a blind adaptive MMSE lter [29] based on the observable Zk = Rk +2I. Likewise, for the eigen-algorithm, of equation (23) is maximized when x>Rk x, the sum interference experienced by user k with new codeword x, is minimized { equivalent to minimizing x>Zk x. The vector x can also be found using blind techniques. Thus, interference avoidance algorithms are based on a measurable quantity { the interference/noise signal correlation Zk . In the MMSE algorithm, a codeword replacement by user k requires rst that the receiver lter for user k converges. Further, the MMSE lter coecients ck must be communicated to the transmitter via a feedback channel. Consequently, at each iterative step, the speed of the algorithm is limited since (1) the convergence to the MMSE lter may require several hundred bits and (2) several hundred bits may be needed for the feedback transmission of the new signature. These same conclusions will also hold for the eigen-algorithm. Therefore, these signature adaptation algorithms operate on a slower time scale than the algorithms for multiuser interference suppression. However, there are applications such as wireless local
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loop and indoor wireless LANs where slow convergence is less likely to be a problem. In these systems, signature adaptation may o er potentially large capacity increases.
5 Fixed Point Properties: greedy interference avoidance Unfortunately, for M > N there is no guarantee that interference avoidance always leads to an optimal xed point. For example, from equation (23) can be zero over a full cycle of an interference avoidance algorithm even though the minimum eigenvalues k might not all be equal as required for SS> = (M=N )I. We now examine the nature of such suboptimal xed points in more detail.2 The convergence properties of the MMSE algorithm are described in [6,28]. In this paper, our focus will be on the eigen-algorithm, or more generally, on any greedy algorithm which always chooses codeword replacements which increase its SIR when at all possible. Or equivalently, for all interference avoidance algorithms whose only xed points are those such that the equilibrium codewords for agile users are minimum eigenvalue eigenvectors of their respective Rk . For brevity be tting a conference paper, we omit proofs which can be found in [30,31].
5.1 Fixed Points for Equal-Power Agile Users
Lemma 1 If fsk g is a xed point of a greedy interference avoidance algorithm with minimum eigenvalue ensemble fk g, the eigenvalue set of each Rk contains k and fj + 1g where j = 6 k . Furthermore, all corresponding sj are also eigenvectors of Rk . Theorem 1 Let fsk g be a xed point with SS> having eigenvalues f(k + 1)g. The distinct eigenvalues, 1 ; : : : p satisfy ji ? j j 1 for i; j = 1; : : : ; p. Furthermore, the signal vectors associated with each distinct eigenvalue i form mutually orthonormal subspaces which collectively span having eigenvalues (k + 1) = , k = 1; 2; ; N , then = MN and SS> = MN IN . In summary, there are the optimal xed points with k = M=N ? 1 for k = 1; : : : ; M , and the suboptimal xed points with vectors fsk g in mutually orthogonal subspaces corresponding to each di erent k { which cannot di er from each other by more than 1. If all the k are identical, then the ensemble correlation SS> is an identity matrix. That is, the ensemble is essentially whitened. Our worry, of course, is that the algorithm will converge to a suboptimal point. Fortunately, in all our experience with numerical studies suboptimal minima have never been For simplicity, we assume no uniform background noise, noting that the minimization of TSC is the same whether we consider Rk or Zk . 2
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obtained when the starting ensemble is chosen randomly. Only when the initial set is not full rank or when the component vectors can be partitioned into mutually orthogonal subspaces does the algorithm not always converge to an optimal codeword ensemble. We note, however, that unlike the MMSE algorithm [6, 28], non-full rank and/or mutually orthogonal partitioning of the starting set does not necessarily guarantee a suboptimal solution for the eigen-algorithm. Furthermore, we have identi ed a procedure for use with the eigen-algorithm which allows escape from local optima through greedy and aggressive user action (see section 5.4).
5.2 Fixed Points for Mixtures of Fixed and Agile Users
It is likely that users with both xed and agile waveform radios may need to occupy the same signal space. We now consider this scenario and nd the intuitively pleasing result that the agile users perform a sort of aggregate water lling over the portion of the signal space with least xed user energy to achieve an optimum SIR. Put another way, the xed users appear as colored noise to the agile users which causes an appropriate Shannon-esque distribution of agile user signal energy over the signal space. It is readily understood that this feature is of bene t, on average, to the xed users as well since it implies that the agile users avoid the xed users where possible. Formally, we let the set fak j1 k Lg be the set of signal vectors associated with waveform-agile users. Let ffi j1 i M ? Lg be the signal vectors associated with xedwaveform users. Greedy interference avoidance will only be applied to the fak g. We de ne
SS> =
L X k=1
ak a>k +
MX ?L i=1
fifi> = AA> + FF>
(24)
We also de ne the mutually orthogonal eigenvectors of FF> as i with associated eigenvalues i2 0. We will assume throughout that i2 i2+1 and notice that Trace[SS>] = M . First consider the case where there are fewer agile users L than dimensions N . The agile users cannot span the entire signal space. Thus, we must ask in what portion of the signal space should the agile users reside to achieve minimum interference. The answer is stated as a theorem.
Theorem 2 If there are L < N agile users, then the interference experienced by at least
one agile user can be reduced (while not increasing the interference seen by other agile users) unless the set fak g is contained in the space spanned by the L eigenvectors of FF> with the smallest eigenvalues.
We can therefore assume, with no loss of generality, that in what follows L N { since if not, we simply recast the problem in a space of dimension N 0 = L following Theorem 2. We have previously shown that an interference avoidance algorithm will attempt to reduce TSC = Trace[(SS>)2]. It is therefore only necessary to (1) derive a lower bound on TSC for this scenario and (2) show that there always exists at least one codeword set which achieves the lower bound which we state as a theorem. A proof and consideration of item (2) are provided in [30,31].
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Theorem 3 When SS> = AA> + FF> with F xed, Ph 2 !2 h X 4 M ? i=1 i > 2 Trace[(SS ) ] (N ? h ) i (25) + N ? h i=1 P where h = arg minh (M ? hi=1 i2 )=(N ? h). This bound is met P with equality when the eigenvalues of SS> are f12; : : : ; h2 ; c; : : : ; cg where c = (M ? hi=1 i2 )=(N ? h ) has multiplicity N ? h Experimentally, for randomly chosen initial codeword vectors fak g and ffk g, invariably
the lower bound of Theorem 3 was attained. Therefore, through Theorem 3, equation (23) and experimentation, we have compelling evidence that greedy interference avoidance algorithms seek a minimum mutual interference set of agile vectors by \water lling" the energy levels provided by the xed users, and avoiding completely energetic interference above a certain threshold. As a byproduct, we have also shown that the algorithm seeks a minimum mutual interference set of vectors in a background of colored noise. That is, all the contentions of this section rely only on the sum of agile and user energies being equal to M . If we relax the constraint that M be an integer, then FF> is an autocorrelation matrix over some orthogonal projection of an arbitrary colored noise process. Finally, we note that as in the agile-only case, there are optimal xed points where the agile users obtain uniformly minimum SIR, and suboptimal xed points where groups of di erent agile users obtain di ering SIRs. In the suboptimal case, it is easily shown that once again, the agile users are partitioned into mutually orthogonal subspaces according to the SIR obtained. Fortunately, also as with the agile-only case, random choice of initial vectors fak g seems to preclude convergence to a suboptimum minimum.
5.3 Fixed Points for Interference Avoidance with Unequal Power
In this section we consider the case where each user has arbitrary but xed received power pk . We will nd that interference avoidance achieves eigenvalues for SS> identical to those shown in [25] to maximize the sum capacity. We explicitly add white Gaussian background noise so that capacity is well-de ned although this is not a necessary feature for interference avoidance to be e ective. We remind the reader that we assume at least as many users M as signal dimensions N since otherwise the trivial orthogonal channel codeword set is obtained. Note also once again that we have incorporated the signal power pk into the code vector sk ; i.e., jsk j2 = pk . In an additive white Gaussian noise background, at equilibrium we require (SS> ? sk s>k + 2 I)sk = (k + 2 )sk (26) With pk = jsk j2 we have (SS> + 2I)sk = ([k + 2] + pk )sk = pk ( k + 1)sk (27) 2 The quantity k = kp+k is the inverse of the SIR achieved by the kth user. We choose k as small as possible which in turn implies we choose the minimum eigenvalue eigenvector of
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SS> as previously discussed. We also have Trace[SS>] = E where E = PMj=1 pk . Likewise Trace[(SS>)2 ] is not increased by an interference avoidance algorithm which replaces sk with an x of equal power (see equation (22)). Since the interference avoidance algorithm cannot increasePTrace[(SS>)2], we now seek a lower bound for Trace[(SS>)2]. We have Trace[(SS>)2] = Nn=1 2n where the fng are the eigenvalues of SS>. Consider then that for any signal with power pk , the corresponding eigenvalue of SS> is at least pk . That is, SS>sk = Rk sk + sk s>k sk = (k + pk )sk and we have in general
Trace[(SS>)2] =
N X i=1
(ci + pi)2
(28)
with ci 0 and where we assume ordered energies, pi pi+1. Since the eigenvalues must PN sum to E , we have i=1(ci + pi) = E . The minimization of Trace[(SS>)2 ] requires ci = 0 if pi > c and ci = c ? pi when pi c. Once again, the \water level" c which satis es the power constraint equation is Ph E ? c = N ?ih=1 pi (29)
Thus c is an eigenvalue of SS> with multiplicity N ? h and the complete set of eigenvalues of SS> is (30) fp1 ; : : : ; ph ; c; : : : ; cg From equation (27), we see that ( 2 (31) k = (2=p+k c)=p kh E=N we then have the usual uniform solution of c = E=N (h = 0) and consequently 2 k = p + pEN ? 1 (32) k k Overall, these results have a facetious but memorable interpretation Might makes right That is, those users with greater received power pk obtain better performance. In fact, the excessively energetic h users command private channels. It is worth noting again that minimizing Trace[(SS>)2] also minimizes Trace[(SS> + 2IN )2 ]. Since minimizing Trace[(SS> +2IN )2 ] is equivalent to maximizing the sum capacity (see [30,31]), the eigenvalues of SS> given in equation (30) imply a codeword set which can achieve the sum capacity. Furthermore, when no pk > E=N , we have h = 0 and k = 8k. This implies an absolute minimum TSC which in turn implies an absolute maximum sum capacity. This result is in agreement with that provided in [25] where an existence proof for such codeword sets can also be found. Finally, we note once again, that suboptimal minima may be obtained in which users are partitioned into mutually orthogonal subspaces with di ering SIR characteristics. Also as before, these subpoptimal xed points seem to be avoided by starting from initially random codewords.
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5.4 Escape from Local Minima: a formal method
We have now seen that if each codeword vector sk is an eigenvector of SS>, then a xed point of an interference avoidance algorithm has been achieved. We have also seen that some such xed points will be suboptimal if di erent codewords correpond to di erent eigenvalues. We will now show that if a user with greater interference (larger eigenvalue) adjusts its codeword so it interferes with codewords in a space with lower interference, the local minimum can be escaped. Essentially, the user receiving poorer performance (say in subspace I) intentionally interferes with users in a better performing subspace (say subspace II). This action, though it may increase TSC intially, actually forces the reduction of TSC BELOW the previously established local minimum when an a ected user in subspace II modi es its codeword using the eigen-algorithm. Let us start with some speci cs. Assume with no loss of generality that the equilibrium codeword ensemble consists of three mutually orthogonal sets fak ; b`; cj g such that Ka
Kb
X X SS> = a a> + b b> +
k=1
k k
`=1
` `
M ?X Ka?Kb j =1
cj c>j
(33)
and that SS>ak = I ak , SS>b` = II b` and SS>cj = j cj with I > II and j 6= I ; II . Codewords from the set fak g su er more interference than codewords from the set fb`g. By Theorem 1 we have cj , b` and ak mutually orthogonal as well as I ? II 1. We assume a basis set f ig, i = 1; 2; ; n1 , which spans fak g and is orthogonal to fb`g and fcj g. Likewise we assume a basis set f j g, j = 1; 2; ; n2, (n1 + n2 N ) which spans fb`g and is orthogonal to fak g and fcj g, and a basis set fmg, m = 1; 2; ; N ? n1 ? n2 which spans fcj g and is orthogonal to fak g and fb`g. Collectively, f j g, f ig and fm g span ak = I ak and p sets > SS b` = II b` where I > II . Replacement of a1 by 1 ? 2 a1 + b1 (0 < jj < 1) and then subsequent application of the eigen-algorithm to replace b1 strictly decreases total square correlation (TSC). Theorem 4 has an interesting interpretation. In the speci ed two-step replacement, an \aggrieved user" with poorer performance essentially \attacks" a user with better performance. The attacked user then seeks relief through interference avoidance which strictly reduces TSC BELOW that of the starting suboptimal xed point. Since subsequent application of an interference avoidance algorithm cannot increase TSC, this \warfare" technique causes local minima to be escaped.
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6 Discussion
6.1 Eigen-Algorithm: numerical examples
Here we provide example applications of interference avoidance to the scenarios analyzed in this paper. FIGURE 1 shows CDMA chip sequences for ve agile users with 3 chips after 5 interference avoidance cycles. The resultant associated SS> is approximately diagonal 2 SS> = 64
1:667 ?0:00035 ?0:00002 ?0:00035 1:666 0:00017 ?0:00002 0:00017 1:666
3 7 5
and each unit energy user achieves a signal to interference ratio of approximately MN?N = 1:5. In FIGURE 2 we allow only the rst three signals to be agile and x the remaining two. After 5 cycles the three agile users reside in a space of dimension 2 (coplanar) achieving approximately = 1:6 and a concomitant signal to interference ratio of approximately ( ? 1)?1 = 1:66. They avoid a strong xed user interference component (with energy 1:8) in the remaining dimension. In FIGURE 3 we assume four users, one of whom has much larger power than the others (p1 = 4). This energetic user commands a private channel and the remaining three users are forced to share two dimensions of the signal space { again the weaker users are coplanar { and have = 3=2 for a shared SIR of 2.
6.2 Addition and Deletion of Users: convergence speed and codeword stability
We have not conducted rigorous convergence-speed experiments for greedy interference avoidance algorithms. For the MMSE algorithm however, the settling time seems rapid [6,28] with near-convergence to minimum TSC within a three or four iteration cycles. In 100 trials with the eigen-algorithm using M = 24 and N = 16, the evolution of Trace[(SS>)2] also converged within a few iteration cycles. However, much more interesting from a system standpoint is the issue of what occurs when a given set of users are disturbed by the addition or deletion of a user. How rapidly does the system settle after the perturbation? Perhaps even more important is the question of how much the signatures change since large signature adjustments may imply large signaling load on the codeword feedback channel. After 100 trials where a single user was either added to or deleted from an already settled system with 24 users in 16 dimensions, we found that convergence was once again rapid and occurred within three ensemble waveform update cycles. The magnitude squared of the di erence between the pre-addition/deletion codewords and the convergent codewords was used as a measure of codeword volatility. The results are plotted as histograms in FIGURE 4 and FIGURE 5. As can be seen, the post deletion codewords are more volatile than the post addition codewords. However, neither convergent set often di ers much much from their pre-addition/deletion counterparts. Where they do di er signi cantly (square di erence approximately 4), the original codeword has simply been
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inverted. We believe this to be an artifact of the manner in which eigenvectors are chosen for the eigen-algorithm although we have not pursued the issue carefully. Regardless, these limited results seem to indicate that codeword adaptation under arrivals and departures could be more orderly and rapid than might initially be imagined. We close this section with a caveat. Studies of convergence rate based on perfect information are chimeric. In practical systems where information is imperfect, stochastic convergence studies [5] are necessary, and as a general rule, the convergence rate is much slower and is often messier. Nonetheless, perfect information can often be used to suggest an upper bound on convergence properties. That is, were convergence slow and codewords volatile, there would be little hope of rapid and stable behavior under imperfect information.
6.3 Suboptimal Fixed Point Resolution Via Class Warfare
The simplest example of a suboptimal waveform ensemble for an interference avoidance algorithm has three users who reside in a signal space of dimension 2. One of the users is orthogonal to the other two which are colinear. The eigenvalues of the SS> matrix are 1 and 2 corresponding to the orthogonal subspaces in which the signals reside. The optimal SS> would be a diagonal matrix with MN = 1:5 along the diagonal. Given that such suboptimal xed points are possible in any number of dimensions, it is natural to ask how one might perturb such solutions, preferably in an autonomous and distributed fashion, to allow convergence to an optimal waveform ensemble. An ad hoc method is suggested by the result of section 5.4 and we dub it \class warfare". Those users having inferior SIR use some portion of their signal energy to \attack" those users with higher SIR. In the case of our simple example, the colinear users would reduce their principal signal power slightly so that a small signal component in the space of the o ending user (user with better SIR) could be added without increasing their power budget. We have experimented with this procedure for a variety of systems. For higher dimensions, each aggrieved user \attacks" all the eigenvectors in all the spaces where a single user's SIR might be larger than their own. Please note that such an attack is based on a measureable quantity, SS>, as opposed to explicit knowledge of all codewords. Invariably, (systems with up to 20 dimensions) the ensemble settled into the optimal SS> = MN IN constellation. This result, though anecdotal and supported only indirectly by the formal result of section 5.4, may suggest that self-regulation of waveform-agile users in an unlicensed environment o ers an alternative to a formal etiquette and its associated costs of enforcement.
7 Summary and Conclusion Starting from a general signal space foundation, we have derived a class of interference avoidance algorithms whereby individual users asynchronously adjust their transmitted waveforms and corresponding matched lter receivers to minimize interference from other sources, including other users. This method presupposes that transmitters and receivers are waveform agile as would be the case assuming universal radios [1,2] are inexpensive and ubiquitous. The interference avoidance procedure, based on SIR as opposed to individual capacity maximization, minimizes total square correlation (TSC). The minimization/maximization is
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achieved through a satisfyingly information theoretic \water lling" of the signal space by the ensemble of users. Note that this water lling is essentially an \emergent" property of the ensemble since individual users seek not to maximize their capacity (via water lling) but rather, only seek to maximize their signal to interference ratio. Combinations of xed and agile users (or agile users in colored background noise, see section 5.2) as well as users with unequal received powers (section 5.3) were analyzed as examples. It was also shown that for equal energy codewords, a suboptimal xed point can always be escaped through application of a simple intuitive procedure (section 6.3). Numerical experiments were conducted which corroborated the basic analytic results. Further experiments, aimed at determining whether interference avoidance might be used in practical systems were also conducted. These included convergence speed, codeword set volatility under the addition of a single user, elimination of socially suboptimal codeword sets through greedy malicious action (class warfare). In all cases the results were encouraging. It is also worth noting that experimentally, suboptimal sets were never obtained when starting from randomly chosen codewords. We close with a (non-exhaustive) sampling of problem areas, both theoretical and practical, which must be addressed if interference avoidance is to become a useful tool for wireless system design. First, the issue of the proper venue for interference avoidance must be established. By virtue of the necessity for interference measurement over a large number of bit intervals for each waveform update and the asynchronicity of the algorithm over multiple users, it is not obvious whether interference avoidance can be e ective in situations where the wireless channel is changing rapidly { as with moving vehicles where fading is a primary impairment. Perhaps a xed wireless environment is most appropriate for application of interference avoidance methods. Next, despite the assumption of perfect channels used in this paper. Wireless channels are notoriously dispersive, and the when signal energy for each bit is not contained within a single received bit interval, the problem formulation changes. Related work [32,33] suggests that the application of interference avoidance to dispersive channels where intersymbol interference is an issue, will require some care. Similar considerations may apply for the asynchronous case where users may have noncoincident bit intervals and di ering bit rates. We are hopeful that the methodology developed for asynchronous multi-user detection [9] may provide a useful starting point. We have also found the speed and stability with which codeword ensembles converge surprising. Especially intriguing is the seemingly certain convergence to an optimal set when starting from a randomly chosen starting set and the ease with which suboptimal xed points are escaped through purturbation. Theorem 4 suggests that suboptimal xed points might be unstable with respect to interference avoidance, and may be useful { in an expanded form { as a route to better understanding the convergence properties of interference avoidance. The issue of codeword representation and delity in a real software/universal radio has not yet been considered. Speci cally, instead of uniform-amplitude codeword \chips", interference avoidance presumes real-valued \chips" or more generally, real-valued coecients for a set of orthonormal signal basis functions used by the transmitter and receiver. To be e ective, these values must be communicated to the transmitter, and the amount of information which may be fed back to the transmitter will determine quantization methods for
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codewords. Rough preliminary studies indicate that three or four bit quantization of each chip is sucient to preserve the quality of the waveform ensemble. Finally, it must be emphasized that only a single receiver (or co-located multiple receivers) were assumed for this study. Limited experiments with multiple receivers and power control showed unstable behavior when interference avoidance was directly applied. An understanding of the multiple receiver problem is therefore paramount in determining the utility of interference avoidance in real systems. Nevertheless, owing to the simplicity of the concept and the ever increasing sophistication of radio hardware, we expect interference avoidance to be an interesting new addition to the study of wireless systems. Furthermore, with any luck, it may also evolve into a practical method for wireless system design. This may be especially useful in unlicensed bands such as the U-NII, where users can mutually interfere with ocially sanctioned inpunity [8].
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8 Figures
0.8 0.6 0.4 1
0.2 0.8
0.6
0.4
0.2 –0.2
0.4
0.6
0.8
–0.6
–0.4 –0.6
Figure 1: Vector plot of 5 signal vectors in 3-space after 5 cycles of interference avoidance. Signal vectors are represented as hollow inverted pyramids for greater clarity of the 3-dimensional representation.
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0.8 0.6 0.4 0.2 0.8
0.6
0.2 0.4
0.4
0.6
0.8
0.2 –0.2
–0.4
–0.6
–0.4 –0.6
Figure 2: Vector plot of 5 signal vectors (3 agile and 2 xed) after 5 cycles of interference avoidance. Notice the co-planarity of the three agile signal vectors in avoidance of a strong xed interferer component along the remaining dimension.
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1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 –0.4
–0.2
–1 0.2
–0.2
0.4
0.6
0.8
1
–0.4
Figure 3: Vector plot of 4 signal vectors, three with power 1 and the remaining with power 4. Notice the co-planarity of the three power 1 vectors in avoidance of the strong interferer with power 4.
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Square Difference Histogram for Codeword Addition 1500
Occurences
1000
500
0
0
0.1 Square Difference
0.2 3.7
3.8 3.9 Square Difference
4
Figure 4: Histogram of norm squared di erence between initial and convergent waveforms after addition of a single new user to an optimal signal set of 24 users in 16 signal dimensions. Square Difference Histogram for Codeword Deletion 1500
Occurences
1000
500
0
0
0.1 0.2 0.3 Square Difference
0.4 3.5
3.6 3.7 3.8 3.9 Square Difference
4
Figure 5: Histogram of norm squared di erence between initial and convergent waveforms after deletion of a single user from an optimal signal set of 24 users in 16 signal dimensions.
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