Interference Avoidance, Sum Capacity and ... - Semantic Scholar

Interference Avoidance, Sum Capacity and Convergence Via Class Warfare Christopher Rose June 15, 2000

WINLAB, Dept. of Electrical and Computer Engineering Rutgers University 94 Brett Road Piscataway NJ 08854-8060 email: [email protected]

Abstract Interference avoidance has been shown to reduce total average interference (TSC) for given ensembles of user signature waveforms (codewords) in a synchronous CDMA system. In all experiments we have conducted, sequential application of interference avoidance produces an optimal codeword set when starting from randomly chosen initial codewords. Here we provide the first formal proof of convergence to optimal codeword ensembles for greedy interference avoidance algorithms augmented by a technique called “class warfare” whereby users which reside in more heavily loaded areas of the signal space purposely interfere with the reception of users in less crowded areas. Coordination of deliberate interference by a complete class of aggrieved user is also sometimes necessary. Such “attacks” and subsequent codeword adjustment by attacked users are shown to strictly decrease TSC. Along the way we also show, using linear algebra and a variant of stochastic ordering, equivalence between minimization of total square correlation (TSC) and maximization of sum capacity. The existence proof for capacity maximizing (TSC minimizing) codeword sets is the provable convergence to optimal of interference avoidance augmented by class warfare.

i

Rose: Convergence Via Class Warfare

ii

Contents 1 Introduction

1

2 Greedy Interference Avoidance: a brief review

1

3 Convergence Via Class Warfare 3.1 Pioneers: the devil you know . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pioneers: the devil you don’t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Manifest Destiny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 7 11

4 Sum Capacity and TSC 4.1 Sum Capacity and TSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Properties of λ-Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Lower Bound Is the Optimal λ-Constellation . . . . . . . . . . . . . . . . .

13 13 15 18

5 Warfare Maximizes/Minimizes Sum Capacity/TSC 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 5.2 The Bounds of Equation (76) . . . . . . . . . . . . . 5.3 The λ-Constellation Bound Is the Stopping Condition 5.4 Manifest Destiny Attained, Eventually . . . . . . . .

19 20 20 22 23

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6 Summary and Conclusion

25

A When Single User Warfare Fails

25

B Proof of Theorem 5

26

C Derivation of Equation (19)

27


1

1 Introduction Interference avoidance has been identified as a method to iteratively obtain optimal signature waveforms (codewords) in multiple access systems [1–5]. The notion behind interference avoidance is that each user, with feedback from the receiver, is allowed to adjust its waveform to minimize interference. Empirically speaking, sequential iterative updates by all users always results in a set of codewords which maximize various measures of system capacity [6–8] and minimize a measure of mutual interference called the total square correlation (TSC). There are even analytic hints that interference avoidance should always converge to an optimal ensemble [3]. Unfortunately, even copious empirical evidence where never has a suboptimal set been obtained starting from random codewords [1–5] and strong analytic hints are unsettling and hamper work which depends on existence proofs for optimal codeword sets. Therefore, in this paper we modify the basic interference avoidance procedure to include an aggressive component whereby users who reside in a more crowded portion of the signal space deliberately interfere with other users in more sparsely populated dimensions. The procedure, dubbed class warfare, allows escape from suboptimal minima, and coupled to the finite number of possible fixed point TSC values for greedy interference avoidance, forces convergence to codeword sets which minimize the average mutual interference between codewords. Along the way it will also be shown directly via elementary linear algebra and a variant of stochastic ordering methods that TSC minimization is equivalent to sum capacity maximization, and thereby, that greedy interference avoidance with class warfare achieves signal sets which meet sum capacity bounds. We note that a slightly more compact statement of this equivalence can be obtained by an appeal to matrix majorization theory [8–10]. However, the direct approach presented here is useful in that it can be wholly understood from first principles without collateral references.

2 Greedy Interference Avoidance: a brief review We assume that user signals can be represented by N-vectors sk in some arbitrary signal space. The power of each signal vector is defined as pk = |sk |2 . Information is carried by corresponding independent zero mean, unit average power bk and the superposition bathed in some noise process, represented by a noise vector w. The result is that the received signal vector y is given by y = Sb + w

(1)

where b is the vector of modulations corresponding to each codeword and S has M columns composed of the sk .


2

The total interference power experienced by user k assuming a simple matched filter receiver is 

!2  s⊤(SS⊤ + W − sk s⊤ k )sk E [Sb − sk bk + w]  = k 2 |sk | |sk | s⊤ k

where SS⊤ = SS⊤ =

(2)

M

∑ sk s⊤k

(3)

k=1

and W is the covariance matrix of the noise vector w. Equation (2) suggests that user k could minimize the interference seen at the receiver by choosing sk proportional to the eigenvector associated with the minimum eigenvalue of SS⊤ + W − sk s⊤ k. This simple intuitively pleasing greedy procedure when applied sequentially by all users, results at each iteration in the reduction of a quantity called the total square correlation (TSC) – a measure of the average interference seen by all users [1–5]: TSC = Trace[(SS⊤ + W)2 ]

(4)

Furthermore, empirically speaking, the fixed point reached by this iterative procedure invariably has the absolute minimum TSC attainable [5]. Other procedures have also been shown to reduce TSC [1,3], but here we consider only greedy procedures which reduce interference for each user if at all possible. At any fixed point of the algorithm, each sk must obviously be an eigenvector of SS⊤ + W. Equally obvious, the sk associated with different eigenvalues λk must be orthogonal [11]. This orthogonality leads to the observation that if λI is the eigenvalue for sk and λII the eigenvalue for sℓ , k 6= ℓ, then we must also have λI − pk ≤ λII since otherwise user k could reduce its interference by √ setting sk = pk sℓ /|sℓ |.

3 Convergence Via Class Warfare It is easily seen that there may exist many such fixed points with differing TSC values. Thus, if we seek to show that greedy interference avoidance can absolutely minimize TSC, we must first provide some means of escape from local minima. Then, since interference avoidance monotonically reduces TSC toward fixed points, if we show that the possible number of fixed point TSC values is finite, eventual convergence to the absolute minimum TSC is unavoidable. To these ends we provide formal escape methods in which users “attack” other users by deliberate interference, or artful encroachment on less crowded portions of the signal space. It should be noted that since we assume all user signals are collected at common antennas and it is the reception point which feeds optimal codewords back to users, the notion of unilateral “attack”


3

is more a useful analogy than an operational principle. That is, class warfare is more an analytic method to escape suboptimal minima than a distributed algorithmic method of guaranteeing minimum TSC “in the wild”. Of course, as software radios become more powerful and sophisticated, a useful systems-level paradigm might have semi-autonomous software agents responsible for decoding each user’s signal as opposed to the centralized multiuser architectures of today [12–17]. In such a case, where agents may or may not collaborate directly, a warfare analogy might be more accurate. However, since all previous numerical experiments indicate that interference avoidance algorithms seem to converge without the help of escape methods, even here the point is somewhat moot, and we reiterate that the term class warfare is a conceptual crutch for an analytic method of TSC minimization.

3.1 Pioneers: the devil you know Assume with no loss of generality that the equilibrium codeword ensemble consists of three known sets {ak , bℓ , c j } such that SS⊤ =

Ka

Kb

M−Ka −Kb

k=1

ℓ=1

j=1

∑ ak a⊤k + ∑ bℓ b⊤ℓ +

∑

c j c⊤j

(5)

and (W + SS⊤ )ak = λI ak , (W + SS⊤ )bℓ = λII bℓ and (W + SS⊤ )c j = λ j c j with λI > λII and λ j 6= λI , λII . By virtue of their different eigenvalues, codewords of different classes, {ak , bℓ , c j }, are mutually orthogonal. Finally, since it is possible that the codewords might not span ℜN , we also admit the possibility of a set {dm } of eigenvectors for (W + SS⊤ ) with cardinality Kd and associated eigenvalues λm which are themselves mutually orthogonal as well as orthogonal to the codeword sets {ak , bℓ , c j }. For equal power codewords, each codeword from the set {ak } suffers greater interference than each codeword from the set {bℓ }. For unequal power, we can only say that set {ak } is in a more energetic (including the set energy) portion of the signal space than set {bℓ }. The assumption that the codeword constellation be a fixed point of a greedy interference avoidance algorithm requires λI − λi ≤ mink |ak |2 where λi is any other eigenvalue of W + SS⊤ . We assume a basis set {φi }, i = 1, 2, · · ·, n1 , which spans {ak } and is orthogonal to {bℓ } and {c j }. Likewise we assume a basis set {ψ j }, j = 1, 2, · · ·, n2 , (n1 + n2 ≤ N) which spans {bℓ } and is orthogonal to {ak } and {c j }, and a basis set {θm}, m = 1, 2, · · ·, N − n1 − n2 which spans both the {c j } and {dm } and is therefore orthogonal to {ak } and {bℓ }. We will later show (Theorem 2) that we can always choose these eigenvector sets as appropriate partitions of the eigenvector set of W. But for this case where we assume all codewords are known, we leave {φi }, {ψ j } and {θm } as convenient bases for their respective codeword sets. As a specific example, since all the elements of {ak } have the same eigenvalue, λI (and similarly for {bℓ }), we can assign with no loss of generality, φ1 = a1 /|a1 | and ψ1 = b1 /|b1 |.


4

Now consider a scenario where a single aggrieved user a1 with eigenvalue λI attacks a known user b1 with eigenvalue λII < λI . We must have λI − λ j ≤ |a1| = p1 where λ j is any eigenvalue of W + SS⊤ and therefore λI − λII ≤ p1 . We assume that p1 = 1 with no loss of generality1 so that φ1 = a1 . Likewise we assume b1 = βψ1 where |b21 | = β2 . The aggrieved user a1 “attacks” user b1 by adjusting its codeword to a′1 = cos ωφ1 + sinωψ1

(6)

2 ⊤ ⊤ ⊤ a′1 (a′1 )⊤ = (cos2 ω)φ1 φ⊤ 1 + (sin ω)ψ1 ψ1 + sinω cos ω(ψ1 φ1 + φ1 ψ1 ) 2 1 ⊤ ⊤ ⊤ = (cos2 ω)φ1 φ⊤ 1 + (sin ω)ψ1 ψ1 + 2 sin(2ω)(ψ1 φ1 + φ1 ψ1 )

(7)

so that

for some non-zero ω. User b1 will react to this challenge by replacing b1 with b′1 , the new minimum eigenvalue eigenvector. That is, ′ ′ ⊤ ⊤ ′ ∗ ′ (W + SS⊤ − a1 a⊤ (8) 1 + a1 (a1 ) − b1 b1 )b1 = λ b1 where λ∗ is the minimum eigenvalue of ′ ′ ⊤ ⊤ G = W + SS⊤ − a1 a⊤ 1 + a1 (a1 ) − b1 b1

(9)

We then note that

for i = 2, 3, · · ·, n1 for j = 2, 3, · · ·, n2 , and

Gφi = λI φi

(10)

Gψ j = λII ψ j

(11)

Gθm = λm θm

(12)

for m = 1, 2, · · ·, N − n1 − n2 . We therefore have N − 2 eigenvectors for G. Since φ1 and ψ1 are orthogonal to these, the remaining two (new) eigenvectors must be linear superpositions of φ1 and ψ1 , x = x1 φ1 + x2 ψ1 . Depending upon ω, the best new codeword b′1 might be one of these two new eigenvectors, or could be another one entirely chosen from equations (10), (11) and (12). However, we will restrict our attention to b′1 of the form b′1 = β [(sinχ)φ1 + (cos χ)ψ1 ]

(13)

and note that a potentially better choice might exist. That is, by showing suitable selection of χ in that we can always effectively set |a1 | = 1 by normalizing SS⊤ + W with |a1 |2 = p1 . Thus, any codeword can be used as a1 with |a1 | = 1. 1 Note


5

equation (13) can strictly reduce TSC, we will have also shown that a better response to attack can only decrease TSC even more. With b′1 as in equation (13) we then have b′1 (b′1 )⊤ = β2 (sin2 χ)φ1 φ⊤ + (cos2 χ)ψ1 ψ⊤ + sin χ cos χ(ψ1 φ⊤ + φ1 ψ⊤ ) 1 1 1 1 1 2 ⊤ ⊤ ⊤ = β2 (sin2 χ)φ1 φ⊤ 1 + (cos χ)ψ1 ψ1 + 2 sin(2χ)(ψ1 φ1 + φ1 ψ1 )

(14)

Now we calculate the difference in TSC after the attack and response,

∆ = Trace[(SS⊤ + W)2 ] − Trace[(G + b′1 (b′1 )⊤ )2 ]

(15)

Making the required substitutions and defining Q(ω, χ) = = + = +

′ ′ ⊤ ⊤ a′1 (ω)(a′1 (ω))⊤ − a1 a⊤ 1 + b1 (χ)(b1 (χ)) − b1 b1 1 2 ⊤ ⊤ ⊤ −(sin2 ω)φ1 φ⊤ 1 + (sin ω)ψ1 ψ1 + 2 sin(2ω)(ψ1 φ1 + φ1 ψ1 ) 2 1 ⊤ ⊤ ⊤ β2 (sin2 χ)φ1 φ⊤ 1 − (sin χ)ψ1 ψ1 + 2 sin(2χ)(ψ1 φ1 + φ1 ψ1 ) 2 2 2 ⊤ −[sin2 ω − β2 sin2 χ]φ1 φ⊤ 1 + [sin ω − β sin χ]ψ1 ψ1 1 2 ⊤ ⊤ 2 [sin(2ω) + β sin(2χ)](ψ1 φ1 + φ1 ψ1 )

(16)

we obtain ∆(ω, χ) = Trace[(SS⊤ + W)2 ] − Trace[((SS⊤ + W) + Q(ω, χ))2] = −2Trace[(SS⊤ + W)Q(ω, χ)] − Trace[Q(ω, χ)Q(ω, χ)]

(17)

Performing the indicated substitution and remembering that (SS⊤ + W)φ = λI φ and (SS⊤ + W)ψ = λII ψ yields 1 2 ∆(ω, χ)

= δ(sin2 ω − β2 sin2 χ) − [sin2 ω − β2 sin2 χ]2 − 14 [sin(2ω) + β2 sin(2χ)]2

(18)

where δ = λI − λII . We note that 0 < δ ≤ 1. We need to determine whether ∆ > 0 for some choices of ω. However, we first note that for any given ω, the response b′1 (χ) to the attack by a′1 (ω) will maximize ∆ [5]. Therefore, we first seek the maximizing χ for equation (18). Algebraic manipulations and trigonometric identities applied to equation (18) yield,2 1 2 ∆(ω, χ)

= (δ − (1 − β2 )) sin2 ω − 12 β2 (δ + β2 ) + 12 β2 (δ − (1 − β2 )) cos(2χ) + cos(2χ + 2ω)

(19)

q We then note that A cos 2x + B cos(2x + 2ω) has extremal values ± (A + B cos 2ω)2 + B2 sin2 2ω. 2A

derivation is provided in Appendix C, although the interested reader might simply compare equation (18) and c equation (19) with a symbolic mathematics program such as Maple .


6

Therefore letting Γ = δ + β2 we have for A = Γ − 1 and B = 1, max 12 ∆(ω, χ) = (Γ − 1) sin2 ω χ q h i Γ−1 1 2 − 2 (Γ − δ)Γ 1 − 1 − 4 Γ2 sin ω

(20)

We then note that ∆(ω, χ) > 0 relies on having Γ − 1 > 0 which in turn implies Γ − δ > 0 since δ ≤ 1. Thus, the positivity of equation (20) rests on whether ∃ω such that "

1 (Γ − 1) sin2 ω > (Γ − δ)Γ 1 − 2

r

1−4

Γ−1 2 sin ω Γ2

#

(21)

and we note that the term inside the radical is non-negative so that the right hand side of equation (21) is non-negative. Rearranging we have Γ−1 1 2 sin ω < 1−2 Γ−δ Γ

r

1−4

Γ−1 2 sin ω Γ2

(22)

and we again note that since the term inside the radical is non-negative (4(Γ−1) sin2 ω/Γ2 ≤ 1) that the left hand side of equation (22) is non-negative as well. So, squaring both sides and simplifying leads to Γ−1 2 sin ω < δ (23) Γ−δ

which is true for some small enough ω so long as δ > 0. Thus, δ>0

(24)

and Γ − 1 = δ − (1 − β2 ) > 0

(25)

suffice for the existence of an ω such that ∆(ω, χ) > 0. We summarize and generalize the result as a theorem: Theorem 1 Suppose a single user with codeword power α2 and eigenvalue λI attacks another user with codeword power β2 and eigenvalue λII < λI by adjusting its codeword according to equation (6). Further suppose that the attacked user responds by adjusting its codeword to obtain the maximum possible SINR. Then there exists an attack parameter ω that will be successful in strictly reducing TSC if δ = λI − λII > 0 and δ > α2 − β2 . This result is intuitively pleasing since it makes little sense to engage in warfare if at best what you gain (λII − β2 ) is no better than what you already have (λI − α2 ). Reworking these two expressions leads directly to β2 > α2 − δ. Another looser but more memorable condition can be had by noting that since δ > 0, any β2 ≥ α2 allows ∆(ω, χ) > 0. Since β2 = α2 implies equal power


7

aggrieved and offending users, β2 > α2 implies a weaker user attacking a stronger user. Thus, we may say that TSC can always be strictly reduced whenever a less powerful user with associated eigenvalue λI attacks an equal or more powerful user with associated eigenvalue λII < λI .

3.2 Pioneers: the devil you don’t Now suppose that user codewords in one class are unknown to those in another so that a directed attack against a particular user is infeasible. However, if the background noise covariance W is known to all users, then an aggrieved user could measure the signal energy along dimensions (eigenvectors) of the noise covariance and attack specific offending dimensions. We will find the following two theorems useful. The first theorem pertains to the relationship between equilibrium codeword sets and the eigenspace of the noise covariance matrix W. Theorem 2 At equilibrium, codewords with different eigenvalues reside in mutually orthogonal spaces. These spaces are spanned by mutually orthogonal partitions of the eigenvector set for W. Furthermore, the eigenvectors of W are also eigenvectors of SS⊤ + W. Proof: Theorem 2 Consider a set of codewords {ai } with cardinality m1 , dimensionality n1 and associated unique eigenvalue λa . Let us then define SS⊤ = AA⊤ + ZZ⊤ where

m1

AA = ∑ ai a⊤ i ⊤

(26)

(27)

i=1

and Z is a matrix containing the codewords with eigenvalues other than λa . We then have (SS⊤ + W)ai = (AA⊤ + ZZ⊤ + W)ai = (AA⊤ + W)ai = λa ai

(28)

Now consider vectors φm in the N − n1 -dimensional orthogonal complement of {ai }. Construct a basis for this set such that (AA⊤ + W)φm = Wφm = λm φm (29) Since there are exactly N −n1 of these eigenvectors, the remaining n1 eigenvectors φi , i = 1, 2, · · ·, n1 of W must form the basis for the set {ai }. Extending this result, we see that for all sets of codeword with the same eigenvalue, there must be an associated set of eigenvectors of W which exactly spans the space of these codewords. Therefore, equilibrium codewords which share different eigenvalues reside in different partitions of the noise covariance signal space.


8

Finally, each of the eigenvectors of W which span the codeword space can be expressed as a suitable linear combination of codewords m1

∑ µi j ai = φ j

(30)

i=1

j = 1, 2, · · ·, n1 . Therefore, since (SS⊤ + W)sk = λa sk we also have (SS⊤ + W)φ j = λa φ j and all the φ j are eigenvectors of SS⊤ + W with eigenvalue λa which completes the proof. • It is worth mentioning that from this point onward we will express codewords as linear superpositions of the noise covariance eigenvector partition in which they reside as opposed to the arbitrary basis set used in section 3.1. The next theorem establishes a relationship between the mutual correlations of codewords in a given class and their projections onto a noise covariance eigenvector contained in that class. Theorem 3 Let φi , i = 1, 2, · · ·, N be the eigenvectors of W with eigenvalues σi . Assume {φi }, i = 1, 2, · · ·, L exactly spans the set of codewords sk , k = 1, 2, · · ·, K which share the same eigenvalue λ, (SS⊤ + W)sk = λsk . ⊤ If we define αkℓ = s⊤ k φℓ , then for ρi j = si s j we must have K

α α ρ − α α =0 i j iℓ jℓ iℓ jℓ ∑

(31)

j=1

Proof: Theorem 3 By Theorem 2 and Mercer’s theorem [18] we have L

SS + W = λ ∑ ⊤

N

φi φ⊤ i +

i=1

∑

λi φi φ⊤ i

(32)

i=L+1

where the {φi } is the complete eigenvector set for W. Since the {sk }, k = 1, 2, · · ·, K are exactly spanned by the φi , i = 1, 2, · · ·, L we then have L

λ ∑ φi φ⊤ i = i=1

K

L

k=1

i=1

∑ sk s⊤k + ∑ σiφiφ⊤i

(33)

which implies K

L

k=1

i=1

∑ sk s⊤k = ∑ (λ − σi)φiφ⊤i

(34)

We define Ei = λ − σi , the codeword energy along eigenvector φi and note that for αki = φ⊤ i sk we K 2 have Ei = ∑k=1 αki . We then have K K (35) ∑ αiℓα jℓ ρi j − αiℓα jℓ = ∑ αiℓρi j α jℓ − α2iℓEℓ j=1

j=1


9

and note that K

K

L

j=1

j=1

i=1

∑ αiℓρi j α jℓ = ∑ φ⊤ℓ (sis⊤i )(s j s⊤j )φℓ = φ⊤ℓ (sis⊤i ) ∑ Eiφiφ⊤i

!

2 φ⊤ ℓ = αiℓ Eℓ

(36)

which completes the proof. • Now, let A be the aggrieved set of codewords with elements ai such that (SS⊤ + W)ai = λI ai . We can write ai = αi φ1 + xi (37) where φ1 is one of the eigenvectors of W which spans the aggrieved codeword set. We note that (SS⊤ + W)φ1 = λI φ1 , that Wφ1 = σ1 φ1 and that (SS⊤ + W)xi = λI xi . Let S be the (non-empty) set of offending codewords with eigenvalue λII < λI . And by Theorem 2 let the noise covariance dimension φN be contained in the span of S with (SS⊤ + W)φN = λII φN . For the codewords {si }, i ∈ S in this set we write si = βi φN + yi

(38)

⊤ ⊤ with βi = φ⊤ N si . As for a we note that (SS + W)φN = λII φN , that WφN = σN φN and that (SS + W)yi = λII yi . To attack, we rotate each ai into φN as

a′i = αi (cos ωφ1 + sin ωφN ) + xi

(39)

and then script the evasion response by the offending set as s′i = βi (cos χφN + sinχφ1 ) + yi

(40)

To calculate the change in TSC we first form 2 ⊤ ⊤ ⊤ ⊤ ai a⊤ i = αi φ1 φ1 + xi xi + αi (φ1 xi + xi φ1 )

(41)

2 ⊤ ⊤ ⊤ ⊤ si s⊤ i = βi φN φN + yi yi + βi (φN yi + yi φN )

(42)

2 ⊤ + cos ω sinω(φ φ⊤ + φ φ⊤ ) a′i (a′i )⊤ = α2i cos2 ωφ1 φ⊤ + sin ωφ φ N N 1 N N 1 1 ⊤ + x φ⊤ ) + α sinω(φ x⊤ + x φ⊤ ) + α cos ω(φ x + xi x⊤ i N i 1 i N i i 1 i i

(43)

and

and then

Rose: Convergence Via Class Warfare and

10

2 ⊤ ⊤ ⊤ s′i (s′i )⊤ = β2i cos2 χφN φ⊤ N + sin χφ1 φ1 + cos χ sinχ(φ1 φN + φN φ1 ) ⊤ ⊤ ⊤ ⊤ + yi y⊤ i + βi cos χ(φN yi + yi φN ) + βi sinχ(φ1 yi + yi φ1 )

(44)

which allows us to write

and

2 sin2 ωφ φ⊤ − sin2 ωφ φ⊤ + cos ω sin ω(φ φ⊤ + φ φ⊤ ) a′i (a′i )⊤ = ai a⊤ + α N N 1 1 i i N N 1 1 ⊤ ) + α sin ω(φ x⊤ + x φ⊤ ) + αi (cos ω − 1)(φ1 x⊤ + x φ i 1 i N i i N i

(45)

2 2 2 ⊤ ⊤ ⊤ ⊤ s′i (s′i )⊤ = si s⊤ i + βi sin χφ1 φ1 − sin χφN φN + cos χ sinχ(φ1 φN + φN φ1 ) ⊤ ⊤ ⊤ + βi (cos χ − 1)(φN y⊤ i + yi φN ) + βi sin χ(φ1 yi + yi φ1 )

(46)

which in turn allows us to write 2 ⊤ ⊤ ⊤ S′ (S′ )⊤ = SS⊤ + ∑ α2i sin2 ωφN φ⊤ N − sin ωφ1 φ1 + cos ω sin ω(φ1 φN + φN φ1 ) i∈A

+

∑ αi(cos ω − 1)(φ1x⊤i + xiφ⊤1 ) + ∑ αi sinω(φN x⊤i + xiφ⊤N )

i∈A 2 sin2 χφ φ⊤ − sin2 χφ φ⊤ + cos χ sinχ(φ φ⊤ + φ φ⊤ ) β N 1 N N 1 N 1 1 ∑ i

i∈A

+

(47)

i∈S

+

∑ βi(cos χ − 1)(φN y⊤i + yiφ⊤N ) + ∑ βi sinχ(φ1y⊤i + yiφ⊤1 ) i∈S

i∈S

We then define the total aggrieved codeword energy in φ1 as α2 =

∑ α2i , the total offending

i∈A

codeword energy in dimension φN as β2 = ∑ β2i and form Q(ω, χ) as in section 3.1; i∈S

⊤ − φ φ Q(ω, χ) = (α2 sin2 ω − β2 sin2 χ) φN φ⊤ 1 N 1 ⊤ + (α2 cos ω sin ω + β2 cos χ sinχ)(φ1 φ⊤ N + φN φ1 ) ⊤ ⊤ ⊤ + ∑ αi (cos ω − 1)(φ1 x⊤ i + xi φ1 ) + ∑ αi sinω(φN xi + xi φN ) +

i∈A

i∈A

i∈S

i∈S

(48)

∑ βi(cos χ − 1)(φN y⊤i + yiφ⊤N ) + ∑ βi sin χ(φ1y⊤i + yiφ⊤1 )

We then define ∆(ω, χ) as in equation (17) and reduce it to 1 2 ∆(ω, χ)

= δ(α2 sin2 ω − β2 sin2 χ) − (α2 sin2 ω − β2 sin2 χ)2 − 14 (β2 sin 2χ + α2 sin2ω)2 − ((cos ω − 1)2 + sin2 ω) ∑ αi α j (κi j − αi α j ) i, j∈A

−

((cos χ − 1)2 + sin2 χ)

∑ βiβ j (ρi j − βiβ j )

i, j∈S

(49) κi j = a⊤ i a j,

ρi j = s⊤ i s j,

where i, j ∈ A and i, j ∈ S . 2 By Theorem 3 the terms in (cos ω − 1) + sin2 ω and (cos χ − 1)2 + sin2 χ must be identically


11

zero so we have 1 2 ∆(ω, χ)

2 = δ(αh2 sin2 ω − β2 sin2 χ) − (α2 sin2 ω − β2 sin2 χ)2 − 14 (β2 sin 2χ + α2 sin2ω) i 2 2 2 = α4 αδ2 (sin2 ω − αβ2 sin2 χ) − (sin2 ω − αβ2 sin2 χ)2 − 14 (sin2ω + αβ2 sin2χ)2 (50) 4 2 ˜ ˜ Equation (49) is identical to equation (18) within a factor of α if we define δ = δ/α and β = β/α2 . Therefore the result is identical to that in section 3.1: TSC will be strictly reduced by dimensional attack so long as δ˜ > 0 and δ˜ > 1 − β˜ 2 which reduces to δ > 0 and δ > α2 − β2 . The only difference is that we have launched a coordinated attack and have “scripted” an ensemble response (specifically, a rotation by all offending users from the attacked dimension φN toward the attacking signal dimension φ1 ) as opposed to assuming individual greedy responses by each codeword.

3.3 Manifest Destiny Now suppose that application of greedy interference avoidance has resulted in a fixed point such that there exists an eigenvector ψ of SS⊤ + W with eigenvalue λII < λI where λI is the eigenvalue of one of the user codewords. Further, suppose that no sufficiently powerful codeword (or none at all) resides in the offending dimension ψ so that attack and response warfare will be ineffective. 3 We will show that in such a case a coordinated migration into the offending dimension will strictly reduce TSC unless a simple set of conditions is violated. As with dimensional attack assume there exist codewords {si }, i = 1, 2, · · ·, K such that (SS⊤ + W)si = λI si and we assume the codewords occupy n < N dimensions. Each codeword “attacks” ψ by forming √ s′i = cos ωi si + sin ωi pi ψ (51) so that

2 2 ⊤ ⊤ s′i (s′i )⊤ = si s⊤ i − sin ωi si si + pi sin ωi ψψ

+

1√

2

(52)

pi sin2ωi (si ψ⊤ + ψs⊤ i )

Now we form the new S′ (S′ )⊤ + W as S′ (S′ )⊤ + W = SS⊤ + W + Q 3 See

Appendix A for an example.

(53)


12

K

where Q = ∑ [s′i (s′i )⊤ − si s⊤ i ]. Using equation (52) we have i=1

K

⊤ Q = − ∑ sin2 ωi (si s⊤ i − pi ψψ ) i=1 K

1√ + ∑ pi sin 2ωi (si ψ⊤ + ψs⊤ i ) 2 i=1

(54)

As found previously (see equation (17)), the difference in TSC before and after migration is ∆ = −2Trace[(SS⊤ + W)Q] − Trace[Q2 ]. So defining δ = λI − λII as before and ρi j = s⊤ i s j we have 1 K √ √ pi p j ρi j sin2ωi sin 2ω j ∑ 4 i,∑ j=1 i=1 i, j=1 (55) Now, for small enough ωi , terms on the order of sin4 (·) become insignificant and we have

1 2 ∆(Ω)

K

= δ ∑ pi sin2 ωi −

K

1 2 ∆(Ω)

sin2 ωi sin2 ω j (ρ2i j + pi p j ) −

K

≈ δ ∑ pi ω2i − i=1

If we define

where yi =

√



pi ωi we see that

  y=  

K

∑

√

√ pi ωi ρi j p j ω j

(56)

i, j=1

y1 y2 .. . yK

     

(57)

K 1 ∆(Ω) ≈ δ ∑ pi ω2i − y⊤ Zy 2 i=1

where Z is a matrix whose entries are zi j = ρi j . We then note that the K × K matrix    –s1 –   | | |  |  –s2 –     Z=  ..   s1 s2 · · · sK   –.–  | | | | –sK –

(58)

(59)

will be singular if the si are linear combinations of only n < K eigenvectors φ j . In such a case there K

exists a non-zero vector y for which y⊤ Zy = 0. And since δ ∑ pi ω2i > 0 for any of the ωi nonzero, i=1

we can always find a set of suitably small {ωi } for which ∆(Ω) > 0.


13

We summarize the result as a theorem: Theorem 4 Let {si } be the set of all codewords which share a common eigenvalue λI . Let ψ be an eigenvector of W orthogonal to all the {si } and for which (SS⊤ +W)ψ = λII ψ where λI −λII = δ > 0. TSC can be strictly reduced by coordinated attack on dimension ψ if the number of codewords is at least one more than the number of dimensions they span. We call this “migration” strategy manifest destiny after geographic expansion policies of the 19th century United States.

4 Sum Capacity and TSC In [5] equivalence between TSC minimization and sum capacity maximization was obtained simply through application of majorization theory [7–9]. However, an approach based directly on elementary linear algebra [11] and a variant of stochastic ordering is also useful and possibly accessible to a wider audience. Specifically, sum capacity maximization and TSC minimization both depend on the eigenvalues of the received signal covariance matrix. The sum capacity function is concave and the TSC convex in these eigenvalues. We establish that both TSC and sum capacity are optimized when identical bounds on the eigenvalues are met with equality thereby establishing an equivalence between TSC minimization and sum capacity maximization.

4.1 Sum Capacity and TSC We modify the approach in [6] to include the nonwhite, possibly correlated Gaussian channel described by equation (1). The mutual information between Y and b is [19] I(Y; b) = h(Y) − h(Y|b) = h(Y) − h(w)

(60)

This quantity is upper bounded by assuming that Y is a Gaussian random vector. Since b and w are assumed zero mean and independent and the components of b are also assumed independent, we have cov(Y) = SS⊤ + W where W is the covariance of the noise vector. This leads directly to [19, 20] i 1 h 1 Cs = log (2πe)N SS⊤ + W − log (2πe)N |W| (61) 2 2 which reduces to 1 1 ⊤ Cs = log SS + W − log|W| (62) 2 2 We then define the eigenvalues of the noise covariance matrix W as {σi }, i = 1, · · · , N. Likewise we define the eigenvalues of the matrix SS⊤ + W as {λi }, i = 1, · · ·, N and obtain the sum capacity


14

in terms of eigenvalues 1 N 1 N Cs = ∑ logλi − ∑ logσi 2 i=1 2 i=1

(63)

Since the eigenvalues {σi } are fixed, capacity maximization depends only on the choice of the {λi }. We note that the codeword energies {pk } are fixed which leads to N

∑ λi = Trace[W + SS⊤] =

i=1

M

∑

k=1

N

pk + ∑ σi

(64)

i=1

since the trace of a square matrix is identical to the sum of its eigenvalues [11] and |sk |2 = pk .4 The received vector Y has covariance matrix SS⊤ + W. Thus, the total square correlation [1,3– 5] is defined as N

TSC = Trace[(SS + W) ] = ∑ λ2i ⊤

2

(65)

i=1

where again, the {λi } are the eigenvalues of SS⊤ + W and the sum constraint of equation (64) still holds. We then note that were we to pursue the constrained optimization [21, 22] of equation (63) and equation (65), that the stationary point equations would be virtually identical. Specifically, for the optimization of sum capacity, define the Lagrange multiplier as Ω and we obtain 1 N 1 N GC (λ) = ∑ logλi − ∑ log σi + Ω 2 i=1 2 i=1

"

N

M

N

i=1

k=1

i=1

∑ λi − ∑ pk − ∑ σi

#

(66)

which results after partial differentiation with respect to each λi in 1 ∂GC (λ) = +Ω ∂λi 2λi

(67)

The second partials are negative and the cross partials are zero. Therefore the equality-constrained sum capacity is concave in the λi . Setting equation (67) to zero yields λi = −

1 2Ω

(68)

Likewise for TSC we have N

GTSC (λ) = ∑ λ2i + Ξ i=1

4 Please

"

N

M

N

i=1

k=1

i=1

∑ λi − ∑ pk − ∑ σi

#

(69)

note that unlike [8] and much of the multiuser detection literature, we incorporate the fixed signal power into the signal vector power (i.e., pk = |sk |2 ) for notational simplicity.


15

Taking partials as before results in ∂GTSC (λ) = 2λi + Ξ ∂λi

(70)

The second partials with respect to the λi are positive and the cross partials are zero which implies GTSC (λ) is convex in the {λi }. Setting equation (70) to zero yields 2λi + Ξ = 0

(71)

which is identical in form to equation (68). The λ-constellation which maximizes and minimizes equation (68) and equation (71) respectively will be identical, depending on the convexity of the λ-constellation solution space [21, 23] – the classic Kuhn-Tucker conditions. That is, unless the uniform λi solution implied by the two extremal equations is contained in the solution space, the solution will lie in a “corner” of a convex solution space. In the next sections we derive bounds on possible λ-constellations with an implicitly convex λ-constellation solution space. We will then show that when the bound is met with equality, TSC and sum capacity are minimized and maximized respectively.

4.2 Properties of λ-Constellations As in previous sections we define the eigenvalue and eigenvector set of the noise covariance matrix W as {σi } and {φi } respectively, i = 1, 2, · · ·, N. With no loss of generality, we assume that the {σi } and the signal energies {pk }, k = 1, 2, · · ·, M are ordered as σi ≥ σi+1 and pk ≥ pk+1 . For N convenience, we also define P = ∑M k=1 pk and U = ∑i=1 σi . We will assume at least as many codewords as dimensions (M ≥ N) since if not, optimality dictates that the codewords be contained in the space spanned by the M least noisy dimensions – those with energies σN , σN−1 , · · · , σN−M+1 . Our basic approach is to find upper and lower bounds to sums of ordered eigenvalues of SS⊤ + W, λi ≥ λi+1 . It is clear from the definition that if λ-constellations λ1 and λ2 satisfy the bounds, then for 0 ≤ α ≤ 1, so does λ3 = αλ1 + (1 − α)λ2 . The solution space is therefore convex. Now, for any matrix Q with eigenvalues {µi } ordered from largest to smallest we have [11], k

k

max

∑ x⊤i Qxi = ∑ µi

x⊤ i x j =δi j i=1

Consider then

M

SS + W = ∑ ⊤

i=1

(72)

i=1

si s⊤ i +

N

∑ σiφiφ⊤i

j=1

(73)


16

and that max x⊤ (SS⊤ + W)x = λ1

(74)

|x|=1

It follows immediately that λ1 ≥ σ1 and λ1 ≥ |s1 |2 + σN = p1 + σN . Equally obvious is that λ1 ≥ (P +U)/N [5, 6, 8]. Slightly less obvious is that λ1 ≥

1 k ∑ (pi + σN−i+1) k i=1

(75)

for k = 1, 2, · · ·, N − 1. This may be shown by noting that ∑ki=1 λi ≥ ∑ki=1 (pi + σN−i+1 ) and that the minimum maximum λ1 must then be at least this quantity divided by k. Therefore, we must have in total " # 1 k P +U λ1 ≥ max σ1 , max ∑ (pi + σN−i+1 ), (76) N 0