AD-A236 847 EIII Hill IliillIEllIllllllINFORMATION CAPACITY OF THE MATCHED GAUSSIAN CHANNEL WITH JAMMING

II. INFINITE-DIMENSIONAL CHANNELS

C.R. Baker* I.F. ChaoDepartment of Statistics Department of MathematicsUniversity of North Carolina National Central UniversityChapel Hill, NC 27599, U.S.A. Chungli, Taiwan. ROC

Present Address: Department of Mathematics, EPFLCH-1015 Lausanne, Switzerland

LISS-47

October 1990

Abstract

The additive infinite-dimensional Gaussian channel subject to jamming is

modeled as a two-person zero-sum game with mutual information as the payoff

function. The jammer's noise is added to the ambient Gaussian noise. The

coder's signal energy is subject to a constraint given in terms of the RKHS of

the covariance of the ambient noise; such a constraint is necessary in order

that the capacity without feedback be finite. It is shown that use of this same

RKHS constraint on the jammer's process is too strong; the jammer would then

not be able to reduce capacity, regardless of the amount of jamming energy

available. The constraint on the jammer is thus on the total jamming energy,

without regard to its distribution relative to that of the ambient noise

energy. The existence of a saddle value for the problem does not follow from

the von Neuman minimax theorem in the original problem formulation. However, a

solution is shown to exist. A saddle point, saddle value, and the jammer's

minimax strategy are determined. The solution is a function of the problem

parameters: the constraint on the coder, the constraint on the jammer, and the

covariance of the ambient Gaussian noise. The essential effect of jamming is

_ to convert the infinite-dimensional channel into a finite-dimensional channel

, with the same constraints, with the dimensionality depending upon the problem

parameters.

J Research supported by ONR Grant N00014-89-J-I175 and NSF Grant NCR-8713726.

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11. TITLE (Include Seuity Cw,ftcatin) *Information capacity of the matched Gaussian

12. PERSONAL AUTHOR(S)C.R. Baker and I.F. Chao

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TECHNICAL FROM TO 1990 October 17

16. SUPPLEMENTARY NOTATION

17. COSATI CODES I. SUBJECT TERMS (Continue on reuere if necesary and identify by block number)

FIELD GROUP SUB. GR.Gaussian channels, jamming channels, Shannon theory,channel capacity.

19. ABSTRACT (Continue on rev.e if nceaaary and identify by block number)

The additive infinite-dimensional Gaussian channel subject to jamming is modeled as atwo-person zero-sum game with mutual information as the payoff function. The jammer'snoise is added to the ambient Gaussian noise. The coder's signal energy is subject to aconstraint given in terms of the RKHS of the covariance of the ambient noise; such aconstraint is necessary in order that the capacity without feedback be finite. It isshown that use of this same RKHS constraint on the jammer's process is too strong; thejammer would then not be able to reduce capacity, regardless of the amount of jammingenergy available. The constraint on the jammer is thus on the total jamming energy,without regard to its distribution relative to that of the ambient noise energy. Theexistence of a saddle value for the problem does not follow from the von Neuman minimaxtheorem in the original problem formulation. However, a solution is shown to exist. Asaddle point, saddle value, and the jammer's minimax strategy are determined. The

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11. TITLE CONT.: Channel with Jamming II. Infinite-dimensional Channels.

19. ABSTRACT CONT.: solution is a function of the problem parameters: the constrainton the coder, the constraint on the jammer, and the covariance of the ambientGaussian noise. The essential effect of jamming is to convert the infinite-dimensional channel into a finite-dimensional channel with the same constraints,with the dimensionality depending upon the problem parameters.

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6-1

Introduction

Information capacity of the additive Gaussian channel subject to jamming

is determined in [3] under the assumption that the channel is finite-

dimensional. By "information capacity" we mean here a saddle point solution

to a zero-sum two-person game in which mutual information is the payoff

function, and the admissible strategies for coder and jammer are determined by

average-energy constraints on the stochastic signals of the coder and jammer.

In this paper this problem is solved for the infinite-dimensional

channel: a channel in which all sample paths belong to a real separable

Hilbert space, H. with inner product (.-> and norm 11-11. Very substantial

differences exist between this problem and that for the finite-dimensional

channel. In the former, a saddle point solution is guaranteed by the von

Neunmnn minimax theorem. Moreover, the solution can be obtained, after a

preliminary development, by application of constrained optimization and the

Kuhn-Tucker conditions. For the infinite-dimensional channel, it will be seen

that the von Neumann theorem does not apply. Thus one does not know in advance

if a saddle value exists, much less a saddle point. Moreover, constrained

optimization is not available. Nevertheless, we shall prove the existence of

a saddle point, obtain the saddle value, and in the process give the jammer's

optimum strategy.

This problem is of interest from several viewpoints. One, of course, is

that information capacity is an intrinsic measure of a channel's ability to

convey information. The problem has a long history of interest for

researchers in information theory, based on comments in the literature.

Second, as a potentially very practical application, the information capacity

is typically used to obtain coding capacity. The finite-dimensional results

of [3] are sufficient for this purpose in the case of the discrete-time

Info. Cap.II-LISS 47 - 12/17/90 - 1

channel. However, for continuous-time channels, one obtains the upper bound

on coding capacity by computing the information capacity for each value of T.

thus obtaining a quantity CT, and then taking limit CT/T. Cr is computed forT-vo

the channel having sample functions in L2 [O.T] . Thus, in order to obtain

coding capacity for continuous-time channels subject to jamming, the

information capacity of the infinite-dimensional channel subject to jamming is

required. Finally, in addition to applications for channels with jamming. the

results obtained here can be used to obtain worst-case capacity in non-jamming

applications when the channel noise is only partially known. An example that

often arises is when the total noise consists of known receiver noise and

additive medium noise having unknown statistical properties.

The results given here are for the general channel; there is no

limitation such as stationarity, memory, or univariate nature. It is

anticipated that limit CT./T can be given in terms of spectral densities forT- O

the stationary Gaussian channel subject to jamming. This will require

limiting arguments based on the results given here. in the same way that the

results of [2] have been used to obtain liraT Cr/T of stationary Caussian

channels without jamming [4].

Mathematical Model

The channel sample paths are described by Y = X + W + J. where X is the

coder's signal, W is the additive Gaussian noise, and J is the jammer's

signal. X. W. and J are mutually independent. These quantities are described

by the probabilities pX" 4. and p j on the Borel sets of H; all are assumed

countably additive, and second order (i.e., fHllxll2 dp(x) < -). The mutual

information of interest is

Info. Cap.II-LISS 47 - 12/17/90 - 2

f1

I(X.Y) = I(Pxy)= !jlog - -X. y) j (x.Y)

where PXOPy denotes product measure and pjy(A) = % pjW{(xwv):

(x. x+w+v) C A}.

Under these assumptions. X. PW. and Pj have covariance operators RX. RW.

and R.: e.g., <Rwu.v> = fH<x.u>X,v>dpw(x). We assume WLOG that RW is

strictly positive on H and that all probabilities are zero-mean.

The assumptions imply the existence of a self-adJoint operator S.

possibly unbounded, satisfying RW + Rj = (I+S)Rw

or, equivalently. Rj = R:4,

where range(R) C T(S). the domain of S. The operator (I+S)- exists, since S

is non-negative; and necessarily bounded, since range(R) C range(Rw+RJ)K.

RW = InOl Xnen 0 en ,where X n > 0 and Xn Xn+1 for all n 1 1 nlXn < .

e n , n 1 1} is a CONS for H, and (en Oe)V = <env>en

The constraints on the coder are given by pX[range(R.)] = 1 and

2 % 2 -JA 2E x1IV W P1 . where for x in range(R). I1xIIW = IRw x11 . Such constraints are

necessary in order that the capacity without jamming be finite [1). This

constraint amounts to a RKHS (reproducing kernel Hilbert space) constraint on

the coder's energy, in terms of the RKHS of the ambient channel noise

covariance. As such, it limits the amount of energy that the coder can place

into regions where the ambient noise energy is small.

The jammer's constraint is given by E Pixil2 P The stronger

constraints .j[range(RN)] = 1 and E Pix11 K P 2 might be thought reasonable,

since they are consistent with those imposed on the coder. However, they are

too strong; the jammer would not be able to have any effect on the channel

capacity under these constraints. This can be seen from the results of [2];

Info. Cap.II-LISS 47 - 3/26/91 - 3

from Theorem 3 of that paper, if S is non-negative and has zero as the only

limit point of its spectrum, then the capacity is equal to P1/2. This

situation would hold if one used the constraints pj [range(R)] = 1 and

E IIXIIW 2 P2" since the inequality (using the fact that Rj = RX) is the

same as Trace S P2. Thus, the Jammer's constraint must be weaker than that

of the coder.

The constraint E 11xl 2 P2 places a constraint on the total jammer

energy, but not on the relative jammer/noise energy. In terms of frequency

ranges, this means that the jammer can use signals that have relatively large

energy compared to the noise energy in appropriate frequency ranges. Such a

constraint has an immediate intuitive interpretation. The jammer's optimum

policy should be that of adding his available energy to the ambient noise

energy in such a way that the sum provides maximum interference to the coder.

The jammer will thus want to place his energy in regions available to the

coder in which the noise energy is small, and this is provided by the above

constraint. The use of the RIMS constraint E 11x112 P2 " while it limits the

jammer's energy in regions that can be used by the coder in the same way that

the coder's energy is limited, also limits the jammer's energy relative to the

ambient noise energy, and is evidently too strong for the jammer to have an

effect. This may be regarded as somewhat surprising and is one of the first

differences that one sees in going from the finite-dimensional to the infinite-

dimensional channel. It means that, with optimum coding-decoding by the

channel users, the jammer cannot reduce the rate of data transmission if his

energy limitation is subject to the same type of RKHS constraint as that

applied to the coder, regardless of the value of P2. i.e.. regardless of the

total amount of energy available to the jammer. As already mentioned, the

Info. Cap.II-LISS 47 - 12/17/90 - 4

constraint applied to the coder is implied by any constraint that gives finite

capacity in the absence of jamming.

As a practical consideration, it may be noted that the total jamming

energy will typically be much easier to estimate than its frequency

distribution. Thus, the constraint used here is not only the appropriate

constraint from the viewpoint of the jammer's capability to reduce the

capacity, but is also one that can be viewed as feasible for the coder to

estimate in designing his coding-decoding strategy.

In the form given above, there is no constraint on the probability

distribution of the Jammer's signal other than E AxiI2 P2. However, it

follows from [7] that for a given Rj, the channel capacity (coder's viewpoint)

will be minimized by taking tij to be Gaussian. Thus, the Jammer should always

choose pj to be Gaussian, and this will be assumed henceforth. With this

assumption, the jamming channel is a special case of the mismatched Gaussian

channel: a Gaussian channel such that the constraint covariance RW is not the

same as the noise covariance [2].

The jammer's strategy is uniquely determined by the choice of the

operator S. For a given strategy, the mutual information is

l(pAy) = F(z,a) =I log[1 + zn(l+an) - ]

where (z n) and (a n) are defined as follows. The coder's covariance operator

Rx is given by

RX = I T n(Rw+Rj)%un (Rw+Rj) uR,n

where {un. n 1) is a c.o.n.s. in H. Tn > 0 for n 1, nTn ( o. Then:

a = <S n Un>

Info. Cap.II-LISS 47 - 12/17/90 - 5

Zn = T i (I+S)%U n 112

where U" is the unitary operator satisfying (RW+R) = (I+S)%U.

The problem to be considered is to determine if there exists a saddle

value for the zero-sum two-person game with I(pXy) as the payoff function;

further, if a saddle value exists, then determine whether or not a saddle

point exists; if such a point exists, then give its definition. That is, we

seek to determine if

sup inf F(za) = inf sup F(z,a)z a a z

where the sup and inf are taken over the admissible signals for the coder and

jammer. respectively. If this equality holds, then it defines the saddle

value. In that case, one seeks to determine if the saddle value is actually

attained by an admissible pair (za); i.e., if a saddle point exists.

The constraint on the coder is equivalent to InZn K PI. The constraint

on the Jammer is Trace Rj = Trace RR K P2. Since Trace R =

In(R un , un>. the constraint in this form cannot be effectively used, so

will subsequently be given a different formulation.

6 will denote the smallest limit point of the spectrum of S. The limit

points of the spectrum (= the essential spectrum, a ess(S)) of S consists of

all eigenvalues of infinite multiplicity, limit points of distinct

eigenvalues. and points of the continuous spectrum. (-n) will denote the

sequence of eigenvalues of S that are strictly less than 0. repeated according

to their multiplicity. {v n . n 1} will denote the corresponding eigenvectors.

The following result is essential to our development.

Info. Cap.II-LISS 47 - 3/26/91 - 6

Lemma 1 [2: Suppose that the jammer's strategy is fixed and that e ( c. The

capacity CW(PI) is then given as follows.

(a) If {'r n , n~l} is not empty, and I (-i) Pl" then

[1+9 1 1+ a

PI

(b) If f n 1} is empty, then CW(Pj) -2(l#)

(c) In (a), the capacity can be attained if and only if P1 = I (0--n). Inn l

that case, it is uniquely attained by a Gaussian Ax with covartance

.1. . -

RX nI Tn R;u OR;U , where u = Uv andT 7 for alln 1

n 1 if (7rn) is an infinite sequence; for 1 n K and Tn = 0 for n ) K

when (n) is a finite sequence with K elements. If P1 ) 1 - nn~l

then capacity can be approached as closely as desired by using a

covartance Rx of the above form.

As previously noted, the above constraint on the Jammer is not in a form

suitable for a well-defined game theoretic problem. That is, F involves (a )

determined by both the coder and the Jammer, defined by an = <SUn U n un> .

where the c.o.n.s. {u n , n 1} is chosen by the coder.

Lemma 2 (Energy-sautng principle): When H is infinite dimensional and dim

[supp pX] = =, the jammer's minimax strategy can be achieved by taking

S = 1 tet e, where I t=i Xt i i P2.

Proof: From Lemma 1. only the smallest limit point e of a(S) and those

eigenvalues 7i strictly less than 6 will affect the information capacity. One

can thus assume without loss of generality that S has pure point spectrum with

its eigenvalues consisting of non-negative real numbers less than 0 and/or

Info. Cap.II-LISS 47 - 12/17/90 - 7

equal to 6. We can order the eigenvalues ('i) of S such that -Y 0 i .6,

all 1 1. By the definition of S, Domain(S) D range(R). All the eigenvectors

of RW lie in the domain of S; therefore <Sen en> is well-defined for all n~l.

Let (v,, il} be the CXNS eigenvectors of S corresponding to t,., eigenvalues

(7i). Then consider

TrRXSR = TrRWS I= l<RWY"'i,vi> I M .i M IT Xj<vi ej > 2

Slvi,.e >2 = I

;=l<vi. >ej>2 =1.

We use the facts that (X) is positive and decreasing, and that -i is positive

and increasing to 0. Let p = 6 - I Consider

TrR;C(I-S)R; = i j=l =lX <vi• ej>2

By Lemma 1B of [6]. one has

-i=l -J=l~i jiej i i=l i i

Hence, 1 0i 1 'i iXj vi e> 2 I.i_0lXi- i. Thus, given any sequence ( n) of

eigenvalues of the operator S' satisfying = R '%. ' = n 1=rnvnvn (v

n~l} a CONS in H, TrRj P2. the jammer can use instead the operator R

R , S = Inlrnenen, and then TrRj TrR' g P2.

Hereafter, we define RJ = Inn Xne nen (the set {en , n 1) being fixed by

RW).

Lemma 3 I"2. Suppose that 0 ( -. {'rn, n 1) ts an tnftntte set, and P1 ) 0.

Then P 1 1n - K K for all K 1 tf and only if P1 .1 (O' n).nI4

Info. Cap. II-LISS 47 -12/17/90 -8

Lemma 4. When dtm(H) = m, the Jammer's mintmax strategy requires a (-Y)

sequence such that P 1 I n(0'n). Moreover, WLOG it can be assumed that the

coder chooses his covartance so that the vectors {un , n 1) are defined by

u = Ue for all n 1.

Proof: Suppose that the jammer chooses (in) such that M+ _ P + n=lYn )n ~ M+1 1 n=n

M M for some integer M. Then, Lemma 7 of [3] shows that such a (in) sequence

is not a minimax policy for the jammer.

Thus, for a minimax strategy the jammer must choose (7n) such that

P 1 + IXrn MiY for all M 1. From Lemma 3. this implies that (in) and 0

satisfy P1 Inl(O-7n)" If equality holds, then by part (c) of Lemma 1, the

coder must choose the vectors {un , n 1) by un = Uen for n > 1; if strict

inequality holds, then the coder can choose these vectors in order to approach

capacity as closely as desired, again by part (c) of Lemma 1. 0

We shall always assume hereafter that both coder and jamer use the {e n '

n~l} vectors. That is, they are the o.n. eigenvectors of S. S = ni nen en'

and the coder has covariance operator RX = nlT n(l+n)Xne ne n

The objective function and the constraints now have the following form:

I(pXy) = F(z.,) = % I. log[1 + zn(l+1n)-] ,nYl

where z and 7 are admissible if and only if their components are non-negative,

I _)IZn PI. and Ini Xnin P2. F is thus strictly concave/strictly convex.

To apply the von Neumann minimax theorem, guaranteeing the existence of a

(unique) saddle point, it is necessary ([5]) that the set of admissible z

constitutes a bounded and closed subset of a reflexive space H and that the

set of admissible i constitutes a bounded and closed subset of a reflexive

space H. As defined, the set of admissible z constitutes a closed and

Info. Cap.II-LISS 47 - 12/17/90 - 9

bounded subset of the non-reflexive space 8 Of course, the admissible z

constitute a bounded set in the reflexive space e2. However. the set of

admissible z is not closed in e2 , since the unit ball in e1 is not closed in

the e2 norm. This can be seen by considering the sequence (fn) in e1 defined

by fn(k) = 1/k 2 cn/k for k n, fn(k) = 0 for k > n, with c n -0 chosen so

that enIfn(k)I = 1 for n 1. Moreover, if one considers F(z',r) by setting

zI = V then F is no longer concave-convex. In summary, the von Neumann

minimax theorem cannot be applied to guarantee the existence of a saddle

point. We shall prove, however, that a saddle point exists, define the saddle

value, and give the jammer's optimum strategy.

The above assumption on the representations of S and Rx fixes the problem

in a coordinate system determined by the ambient channel noise. In view of

Lemma 2. a minimax strategy for the jammer in the original problem can be

obtained within this coordinate system: choosing S to have pure point

spectrum, and with its eigenvectors the same as those of the channel noise

covariance operator, RW. By Lemma 4. one can then assume that Rx is defined

by taking un = Uen for n 1, which gives {e n . n 1} as CON eigenvectors for

Rx. Thus. by showing that a saddle value and saddle point exist within this

coordinate system, one will also show their existence for the original

problem. However. this approach will not show that the saddle point is

unique.

Lemma 5 F31. Define (It), (zt), 6K, ga, and .. as foltows, for K 1 0,K L i K gK K

in K+I:

gK(6) = = P2 + h--K+X + (Pl-K)Xt

Info. Cap.II-LISS 47 - 3/26/91 - 10

T4 = 1"K+1

P2 * t=K#+1t + K+I

1 K,t P2 t K+I

-ra =0 t KK, i

where is the solution to I + 0 = gK(), and

ea gK) andI

z t = " K,t i 1.

A unique solution exists for 0K. If P (r-I)P 2 A and it is required

that the coder's covartance satisfy z n = 0 for all n 2 m+l, then the jamming

problem has a saddle point (z X, ), with -rt = KL for i a and 1 0 for

a ) ; zt = z a for i K a and zt = 0 for t ) m; and K the smallest integer h

such that Okm . This K satisfies K K a-2, and P > X

The condition P1 K (m-1)P2/X\ assumed in Lemma 5 will of course be

satisfied for all sufficiently large m; in fact, for all m 2 k. where k is

such that P1 kP2/('Ik+lXj)"

Lemma 6. If K is any integer 2 0 such that P1 ( (K+I)P2 "t1iK+ 2Xt' then

e.. ) T' for all a 2 K+I.K K

Proof. We will show that gK(TM.) > I + Te. for all m 2 K+l. This will prove

the statements, since otherwise one would have mK for some m; since gm(0)

is a strictly decreasing function of 0 for fixed m and K, this would give

+!". 2 1 I m0. I + ,I,

+K 1 K = gK('K) 2 gK(TK) > 1 K

To see that gK(Tm.) > 1 + T. this is equivalent to

Info. Cap.II-LISS 47 - 3/26/91 - 11

XiEP 2+1'j'K+lXJ+KXK+l 2 +J=K+l +K"1

i=K+l (P 2+lmkK+Xk)Ep 2 1=K+n+QK+l+pl?i) P2 + l=K+1"k + (Pl+K)XIK+l

or

x'=~ & + Pl("K+1-Xi) ( 2

This inequality holds if

P1XK+1ml'K+2Xi < P2[P2 + 1K2Xj + K1N+]

which is satisfied, since LHS P2"K+I(K+1). c

Now define TK and gK by

PI K+I

TK =

P 2 + J=K+l j + ""K+I

gK(G) =11=Kl 0 0gK( O) =K+I P +XK+IXJ + (PI-KG)XiJ (6 O)

and let OK be the solution to 1 + 0 = gK(O). It is obvious that TK = lir m K;

we now show that the equation 1 + 6 = gK(O) has a unique solution. OK; and

that OK = lim Om6K m

Lemma 7. (a) gK(8) ts a strictly decreasing function of a for fixed K and 0;

(b) a is a strictly decreasing function of m;

(c) OK isuiqe defined and KK=Li '

(d) 0 TK if and only if IX p 2t=K+l P2 + =1 2

2 J=K#I j + K+1 I i'

(e) OK TK if and only if "K ) T. for all a K I.K K

Info. Cap.II-LISS 47 - 3/26/91 - 12

Proof. (a) It is immediate that I l (0)1-1 > Eg(O)]- 1 if e P1/K.

(b) Since gK(6) is a strictly decreasing function of m. the solution to

1 + 6 = gK (0) must be strictly less than the solution to 1 + 6 = gm(,).

(c) Let f(6) = 1 + 6. f is continuous and stictly monotone increasing.

gK is continuous and strictly monotone decreasing, and f(O) = 1 < gK(O). If

P1/K 2 P2/Ej K+lXJ, then f(P1/K) = I + P1/K >1 + P2/EJ K+lXj = gK(P1/K).

Thus a unique solution exists to f = gK if P1/K > P2/'j K+lj. if P1 /K-

P2/1j)K+1X"J then the same result holds, since then f(P2/:j K+lXj) >

g (P2/:Ej K+1Xj ) "

Since (6 ) is monotone decreasing as m increases, and bounded below by

zero, lim m m 0 exists. Since gK(O) = lim gm(O) for 0 > 0. gK(O0 ) =l m m m

lim gK(O0 ) lim g(m) = 1 + 00. Conversely. gK(O0) = limm gK(eK)

m gK() I + . Thus, 1 + 00 = gK(G0); since this solution is unique,

e0 =-

(d) gK(TK) 1 + TK , since otherwise (proceeding as in the proof of part

(c)) the solution to gK(O) = 1 + 6 will occur for 6 < TK . The inequality of

(d) then follows from gKI(TK ) K (TnTK)-li

(e) As in the proof of (d) m > T!K if and only if a(nd) > 1 +K K onl if K

this occurs if and only if

m P1("K+1-Xi)Xi2 - < P2"

i=K-1 P2 +" X + 2P2 j =K+ 1 KX+ K+l 1 1

The LHS of this inequality is a strictly increasing function of m. Moreover.

K+1 OK~ K~l K+l1= 1 + 1K K+1

P+ [P I-KOKi +1]XK+l

so that

K+1 P 2 + P >"~ P~~~ - TK+l +6K = (K+I)XK+I > P2 + (K+I)XK+I - K

Info. Cap.II-LISS 47 - 3/26/91 - 13

Thus.I ii 0 m rn". <=> for all m K+l. E

The following theorem is the main result of this paper. It will be seen

(see Remark 2 concluding the paper) that the effect of the Jammer is

essentially to convert the infinite-dimensional channel into a channel of

dimension K K (K ( ). This integer K is defined in the following theorem.

and depends on P1. P2, and the eigenvalues ( n) of RW.

Theorem.

The Joxaming problem has a saddle value and a saddle point. The saddle value

is given by

I(AzXy ) = I log[1 + (PI-KOK)Xn + log(I+0K)n K+l P 2 + ItQK+1 tj 2K

where 0K is the unique solution of

I + 0 = .2 n + K8)1 P2 + J K+l i (P1 -K)n]

and K is the smallest integer k 0 such that

i ,IPP2"t=k+l P2 + Ij=k+lXJ x +

1+ + Pixt

*xA saddle point is given by (z , i ), where

x xzL = K t for all i 1

xIT 1 = 0 for i g K

1 + -r x P2+j K,1Xj(I+K) for i K+1.P2 + ItK+Xt + (P 1 -KOK)Xi

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Proof: Define Z = ((zi): z, 0 for i 1 and li Z, 9 PI);

Zm = {(zi) C Z: z, = 0 for i > m);

r' = ( ,i): -, 0 for i 1 and i P2).

It is sufficient [5] to show that sUpZ infr F(z.r) infrf SUpZ F(z.-).Supz infr Fz, ) SUpzm infr F(z.-) = (Lemma 5) F(zm-,) where (zm-r) is

m m

defined in Lenmm 5. Thus. sUpZ infr F(z,v) lim inf F(z. k) =

M m m m Thlim inf % 2 log[l + zi/(l+ k i)] Define k(m) by m = M. Then

M # W ilg[ + i ki 7k(m) k

rnm inf k(m) = lim inffk: 0m > TT.) = (Lemma 7) K. with K defined as in

in m m m n

the Theorem. From the definitions of iand z, lim eK = 0K implies YK,i i

and zM -+Z for all i 1 as m-- o. By Fatou's Lemma

zm z

lrm inf % I log 1 + I log I + .m- 1=1 1+ K.i i=1 1 +

Thus. supZ infr F(z,T) I(p y) as given in the Theorem.

Conversely, one notes that i * e and that I Thus. byi K i=l(Ok-'i) = P1.Thsb

Lemma 1. F(z*. Y) = supZ F(z.r) infr supZ F(z.-). This shows

infr sUpZ F(z.i) 9 F(z*, ) supz infr F(z,r). and completes the proof.

Remark 1. The integer K defined in the Theorem can be no larger than the

smallest integer k satisfying P1 9 (k+l)P2 /lj=k+2 .J" This follows from Lemma

6, and'yields the following Corollary.

Corollary. The saddle value of F on A has the upper bound

F(z,) 9 2. log[1 + RK1T]

where K is the smaltest integer k sattsfying P1 g ( +l)P21 J--*+2 J"

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Proof. From Lemma 1(a). this is the value of the channel capacity that vould

be obtained if the Jammer used (n) such that -n = 0 for n K+l. and

-rn = P1/(K+I) for n > K+I. In that case. nrlnXn = I ,nn=

(P 1 /K+I)nK+2 X n P2" This is thus an admissible strategy for the jammer.

and the result follows by Lemma l(a).

0

Remark 2. The capacity of the channel in the absence of jamming is P 1/2; see

[2]. This equals llm i- log1I + One thus sees that the minimum effect

K 2 One-l

of jamming can be immediately gauged by determining the value of K. the

largest integer such that P1 > KP2/ n=K+IXn . Since the capacity of the2[ P]

(K+l)-dimensional channel in the absence of jamming is 2 log 11 + . one

can view the effect of jamming as converting the infinite-dimensional channel

into a finite-dimensional channel. The value of this K depends on all the

channel parameters: the coder's constraint PI. the jammer's constraint P2- and

the covariance operator of the ambient Gaussian noise.

The statement of the Theorem can be interpreted by considering the sum of

the jamming and the ambient noise. That is, if k is the largest integer such

thatr k = 0 (note that k K as defined in the Theorem and k K as defined in

the Corollary). then corresponding to the eigenvectors (en . n 1} of RW + RJ.

the eigenvalues are given by X for j k. while for j > k. the eigenvalues

are equal to X (l1-y). Thus, as j -. . the eigenvalues are approximately

equal to X (1+0). Since the coder will typically wish to place his energy, so

far as possible, according to small eigenvalues of the total channel noise.

the effect of the janming is to increase those small elgenvalues by a factor

that converges upward to (1+0).

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REFERENCES

[1] Baker. C.R. Capacity of the Gaussian channel without feedback. Inform.and Control, 37, 70-79. 1978

[2] . Capacity of the mismatched Gaussian channel, IEEE Trans.Information Theory, IT-33, 802-812, 1987.

[3] and I.F. Chao. Information capacity of the matched Gaussianchannel with jamming. I. Finite-dimensional channels, to appear.

[4] and S. Ihara. Information capacity of the stationaryGaussian channel. 1990, to appear in IEEE Trans. Information Theory.

[5] Barbu and Precupanu. Convexity and Optimization in Banach Spaces.Reidel, Boston. 1986.

[6] Fan, K. Maximum properties and inequalities for the eigenvalues of com-pletely continuous operators. Proc. Nat. Acad. Sci., vol. 37, 760-766,1951.

[7] Ihara, S. On the capacity of channels with additive non-Gaussiannoise, Inform. and Control, 37, 34-39, 1978.

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