Channel Capacity Subject to Frequency Domain
Normed Uncertainties - SISO and MIMO Cases
Stojan Z. Denic ∗ , Charalambos D. Charalambous † , Seddik M. Djouadi‡
∗ School
of Info. Tech. and Eng., University of Ottawa, Ottawa, Canada
E-mail: sdenic@site.uottawa.ca
† Depart. of Elect. and Comp. Engineering, University of Cyprus, Nicosia, Cyprus
Also with School of Info. Tech. and Eng. University of Ottawa, Ottawa, Canada
This work was supported by the European Commission under the project
ICCCSYSTEMS and the NSERC under an operating grant (T-810-289-01).
Email: chadcha@ucy.ac.cy
‡ Depart. of Elect. and Comp. Engineering, University of Tennessee, Knoxville, USA
Email: djouadi@ece.utk.edu
Abstract— This paper gives a precise definition of robustness
of channel capacity of additive Gaussian channels, with respect to
channel transfer function uncertainty and noise power spectral
density uncertainty. Three robust channel capacity problems and
their solutions are presented in the frequency domain.
I. I NTRODUCTION
There are many reasons why uncertainties arise in communication and some of them are due to errors in the channel
estimation, the network operating conditions, and the jamming [18], [16]. Any uncertainty regarding the channel state
information on the transmitter and/or receiver side leads to
degradation of system’s performance. The computation of a
channel capacity subject to uncertainty can help to determine
the level of performance degradation due to uncertainty.
In this paper, we constrain ourselves to uncertain Gaussian
channels with memory. Both SISO and MIMO channels are
discussed, which are the examples of infinite dimensional
channels. We use a well known fact that the channel capacity
of a Gaussian channel depends on the frequency response
of the channel, the power spectral density (PSD) of the
transmitted signal, and the PSD of the noise [10]. This enables
the application of uncertainty models in frequency domain that
are common in the robust control theory [9]. The modeling
in frequency domain is advocated for the three reasons: 1.
The parameters of uncertainty models in frequency domain
can be extracted from the practical measurements; 2. The
frequency domain models are closer in the physical sense to
real channels, (see [20]), as compared to other models found
in literature [16]; 3. Clearly, it often very difficult to convert
H ∞ normed uncertainty into uncertainty description in the
time domain using another norm [25].
Here, we study three capacity problems, of which two are
related to SISO channels, and one is related to MIMO channel.
The channel capacities are defined as max-min optimization
problems with mutual information as a pay-off function. We
explicitly compute the channel capacities and optimal transmitted powers for the following cases: 1. The uncertainty of a
channel frequency response is described in H ∞ normed linear
space, while the PSD of the noise is known; 2. The uncertainty
of the PSD of the noise is described through a subset of L1
space, while the channel frequency response uncertainty is described as in problem 1.; 3. The noise uncertainty of a MIMO
channel is described through the uncertainty of PSD matrix of
the noise that belongs to the subset of L1 space. Further, for the
problems when the noise uncertainty is present, which can be
treated as the examples of communication subject to jamming,
the optimal jammers’ strategies are computed as well. Our
results can be understood as a generalization of the results of
Blachman [4] for colored noise case.
There is a vast body of literature concerning the computation
of the channel capacity under uncertainties. The standard
model of Gaussian uncertain channels is called Gaussian
arbitrary varying channel (GAVC). The capacities of finite
dimensional GAVC’s are considered in [12], [13], [1]. These
problems are related to the channel capacities of communication systems subject to jamming, which are studied in [2],
[8], [17], [23], [15], [5]. As opposed to previous references, [3]
discusses jamming problem of infinite dimensional channels
subject to energy constraint. The consideration of the problems
related to the uncertainty of MIMO channels can be found
in, for instance, [24], [14], [21]. For an overview of MIMO
channels subject to uncertainties see [11]. For comprehensive
survey of communication under uncertainties, including universal encoding/decoding see [16].
Unlike most treatments found in the literature, the problems
that we consider are defined as optimization problems over
subsets of infinite dimensional function spaces, which may not
be compact. This implies that we cannot rely on the saddle
point property to convert the max-min optimization problem
into min-max problem to compute the channel capacities as
often done [5]. Further, we prefer to solve the optimization
problems by applying Lagrange optimization technique (as
opposed to numerical technique found in [23]) because that
gives us better insight into the relations between optimal
transmitters’ and jammers’ strategies. The proof of coding and
converse to coding theorems for considered capacity formulas
comes from generalization of [22] (see [7]).
The organization of the paper is as follows. The capacities of
SISO channels are considered in Sections II,III,IV. The MIMO
case is discussed in Section V. At the end, an example is given
to illustrate theoretical results.
II. P ROBLEMS D EFINITION - SISO C ASE
The continuous time communication system under consideration is described by
Z +∞
y(t) =
h̃(t − τ )x(τ )dτ + n(t).
(1)
Second problem. (Channel unknown, disturbance unknown.)
Here, the uncertainty of the channel frequency response H̃(f )
is modeled as in the first problem. The PSD of the noise
Sn (f ) is unknown, and it belongs to the set defined by
R +∞
4
A3 = {Sn ; (f ) −∞ Sn (f )df ≤ Pn }. The channel capacity
is defined by
Z ∞
Sn |H̃|2
log(1 +
C2 = sup inf inf
)df
(3)
Sn
Sx ∈A1 Sn ∈A3 H̃∈A2 −∞
If ∆H(f ) = 0, the channel capacity in the presence of just
noise uncertainty is obtained.
−∞
4
The transmitted signal x = {x(t) : −∞ < t < +∞} and
4
additive noise n = {n(t) : −∞ < t < +∞} are real
valued independent wide sense stationary Gaussian random
processes, while the impulse response h̃(t) belongs to L2 ;
Sx (f ) denotes the power spectral density of a transmitted
signal, and Sn (f ) denotes the power spectral density of the
noise. A power constraint on the input signal is assumed to
R +∞
4
be A1 = {Sx (f ) : −∞ Sx (f )df ≤ Px }.
First problem. (Channel unknown, disturbance known.) The
channel is modeled by using an additive uncertainty description given by H̃(f ) = Hnom (f ) + ∆H(f ), where Hnom (f )
is a known part representing the nominal channel, that corresponds to one’s limited knowledge about a channel, and
∆H(f ) is the perturbation part modeling channel uncertainty
satisfying ∆H(f ) ∈ H ∞ , where H ∞ is the space of proper,
rational, and analytical transfer functions in the closed right
half-plane (<{s} ≥ 0). The perturbation part of the transfer function H̃(f ) is represented as a product ∆H(f ) =
∆(f )W (f ), where ∆(f ) is a variable stable transfer function
with k∆(f )k∞ ≤ 1, where k.k∞ is an infinity system norm
4
defined by kG(f )k∞ = supf |G(f )|. W (f ) is a fixed stable
transfer function representing the weight. Now, the uncertainty
4
set of possible frequency responses is defined as A2 =
n
H̃(f ) ∈ H ∞ : H̃ = Hnom + ∆W ; Hnom ∈ H ∞ , W ∈
H ∞ , ∆ ∈ H ∞ }. Also, other types of uncertainty descriptions may be used, as for instance, multiplicative description
H̃(f ) = Hnom (f )(1 + ∆(f )W (f )) [9]. The technique for the
computation of a capacity remains similar. The choice of the
type of an uncertainty description depends on the situation
at hand. The power spectral density of disturbance Sn (f )
is known. The channel capacity for the above description is
defined by
Z ∞
Sx |H̃|2
C1 = sup inf
log(1 +
)df
(2)
Sn
Sx ∈A1 H̃∈A2 −∞
When the infimum is removed, (2) is a well known formula
for the channel capacity of additive Gaussian noise channel,
with average power constraint on the transmitter side, when
the transfer function of the channel H̃(f ) is known, i.e., when
∆H(f ) = 0 implying H̃(f ) = Hnom (f ). The infimum gives
the worst case channel rate.
III. F IRST P ROBLEM : C HANNEL U NKNOWN ,
D ISTURBANCE K NOWN
The solution for this problem is given by the following
theorem.
Theorem 3.1: Consider an additive uncertainty description
)|+|W (f )|)2
of H̃(f ), and suppose that (|Hnom (f
is bounded,
Sn (f )
integrable, and that |Hnom (f )| 6= |W (f )|. Then the following
hold.
1) The capacity of a continuous time Gaussian channel with
additive channel uncertainty is given parametrically by
µ ∗
¶
Z
1
ν (|Hnom (f )| − |W (f )|)2
C1 =
log
df, (4)
2 S
Sn (f )
where ν ∗ is a Lagrange multiplier found via
¶
Z µ
Sn (f )
ν∗ −
df = Px
(|Hnom (f )| − |W (f )|)2
S
(5)
where S is defined by
{f : ν ∗ (|Hnom (f )| − |W (f )|)2 − Sn (f ) > 0, ν ∗ > 0}. (6)
2) The infimum over the channel uncertainty in (2) is
achieved at
∆∗ (f ) = exp[−j arg(W (f )) + j arg(Hnom (f )) + jπ]
k∆∗ (f )k∞ = 1,
and the resulting mutual information rate after minimization is given by
µ
¶
Z +∞
Sx (f )|Hnom (f ) + ∆(f )W (f )|2
inf
log 1 +
df
Sn (f )
−∞
µ
¶
Z +∞
Sx (f )(|Hnom (f )| − |W (f )|)2
log 1 +
=
df,
Sn (f )
−∞
where the infimum is over k∆k∞ ≤ 1. The supremum
of the previous equation over A1 yields the water-filling
equation
Sx∗ (f ) +
Sn (f )
= ν∗.
(|Hnom (f )| − |W (f )|)2
(7)
Proof. The proof is given in Appendix A.
Thus, (4) and (7) show that the channel capacity and the optimal transmitted power are affected by the uncertainty through
fixed transfer function W (f ). Since |W (f )| determines the
size of uncertainty set A2 [9], it can be seen that the channel
capacity decreases as the size of uncertainty set increases.
The effect of the uncertainty on the channel capacity and the
optimal transmitted power is illustrated by an example at the
end of the paper.
IV. S ECOND P ROBLEM : C HANNEL U NKNOWN ,
D ISTURBANCE U NKNOWN - JAMMING P ROBLEM
To solve the optimization problem given by (3), we first
apply the solution of the first problem to resolve the min with
respect to channel uncertainty. Then, the max-min optimization problem remains to be solved, where the min is with
respect to noise uncertainty, and the maximum is with respect
to transmitted signal PSD. This problem may be seen in a
game theoretical framework, where transmitter x, wishes to
maximize mutual information, while the jammer n, wishes
to minimize it. The existence of a saddle point cannot be
proved by using standard results of a game theory [19] because
this optimization problem is over infinite dimensional function
spaces (see sets A1 and A3 , which may not be compact). Thus,
the channel capacity in a presence of jamming is solved by
directly solving max-min problem. The solution is given by
the following theorem.
Theorem 4.1: Consider an additive uncertainty description
of H̃(f ), A1 , and noise description given by A2 . Suppose
(f )|+|W (f )|)2
that Sx (f )(|Hnom
is bounded, integrable, and that
Sn (f )
|Hnom (f )| 6= |W (f )|. Then the following hold
¶
µ
Z
1 +∞
λo1
2
C2 =
log 1 + o (|Hnom (f )| − |W (f )|) df, (8)
2 −∞
λ2
cancel it and make the white noise like situation. The similar
explanation holds for the transmitter. For the transmitter, the
most favorable situation is also to cancel the effect of the
jammer and to impose the white noise like situation. This is
actually the consequence of a saddle point existence. Namely,
the existence of a saddle point for this case is proved by the
authors, by solving min-max problem, and then showing that
the optimal strategies in that case are identical with the optimal
strategies for the max-min problem. Further, as in the classical
case the water-filling type formula holds
Sxo (f ) + Sno (f )(|Hnom (f )| − |W (f )|)2 =
1
.
2λo1
(12)
although Sno (f ) has to be computed via (9). It should be
emphasized that by using our approach, it is possible to distinguish and measure the impact of two types of uncertainties,
one that comes from the frequency response of the channel,
and the other that comes from the uncertainty of the PSD of
the noise. Hence, this approach represents the generalization
of the work found in [4]. In Section V, it will be demonstrated
that this result can be extended to MIMO case, which shows
the usefulness of SISO result.
V. U NCERTAIN N OISE - MIMO C ASE
Consider a discrete time MIMO system defined by the equation
y(t) =
+∞
X
h(t − j)x(j) + n(t),
(13)
j=−∞
4
Proof. The proof is given in Appendix B.
where x = {x(t) : t ∈ Z} is a C m -valued stationary stochastic
4
process representing transmitted signal, n = {n(t) : t ∈ Z} is
p
a C -valued Gaussian stochastic process representing noise,
4
y = {y(t) : t ∈ Z} is a C p -valued stationary stochastic
4
process representing received signal, and h = {h(t) : t ∈
Z} is a sequence of C p×m -valued matrices representing the
impulse response of the MIMO communication channel. It
is assumed that x generates a Hilbert space. Here, H(ejθ )
represents the channel frequency response matrix, which is
the discrete Fourier transform of h given by H(ejθ ) =
P
+∞
−jθt
, where frequency θ ∈ [0, 2π]. It is ast=−∞ h(t)e
P+∞
sumed that t=−∞ h(t)e−jθt converges to H(ejθ ) in L2 (Fx ),
which represents a Hilbert space of complex-valued LebesgueStieltjes measurable functions H(ejθ ) of a finite norm
Z 2π
4
jθ
kH(e )kL2 (Fx ) = Trace
H(ejθ )dFx (θ)H ∗ (ejθ ).
The solution for the channel capacity provides the optimal
strategies for the transmitter and the jammer as the optimal
PSD’s of two signals. In this case, the optimal transmitter’s
and jammer’s strategies are affected by the channel uncertainty.
It should be noted that the optimal PSD’s of the transmitter
and jammer are proportional. An explanation for this result
is obtained from the formula for the channel capacity (8).
From the jammer’s point of view, it can be concluded that
the jammer tries to mimic the transmitter’s signal in order to
Fx denotes the matrix spectral distribution of x [6], which
is assumed to be absolutely continuous with respect to the
Lebesgue measure on [0, 2π]. Hence, dFx (θ) = Wx (θ)dθ,
where Wx (θ) represents the PSD matrix of x. Wn (θ) represents the PSD matrix of n.
In this section, the problem of computing the channel
capacity of a Gaussian MIMO channel when the PSD matrix
of the noise Wn (θ) is unknown, is considered. It is assumed
that although unknown, the matrix Wn (θ) belongs to the set
where
Sno (f ) =
Sxo (f )
(|Hnom (f )| − |W (f )|)2
,
2(λo1 (|Hnom (f )| − |W (f )|)2 + λo2 )
=
=
Z
+∞
−∞
λo1 (|Hnom (f )| − |W (f )|)2
o
2λ2 (λo1 (|Hnom (f )| − |W (f )|)2 +
λo1 o
S (f ).
λo2 n
Z
Sxo (f )df
+∞
= Px ,
−∞
Sno (f )df = Pn ,
(9)
λo2 )
(10)
(11)
in which λo1 , and λo2 are positive Lagrange multipliers of the
two constraint sets A3 , A1 , respectively.
0
4
A2 = {Wn (θ) :
R 2π
0
Trace(Wn (θ))dθ ≤ Pn }. The same
VI. E XAMPLE
4
constraint isR introduced for the transmitter such that A1 =
2π
{Wx (θ) : 0 Trace(Wx (θ))dθ ≤ Px }. These are natural
constraints that limit the sum power of the transmitted and
noise/jamming signals. The capacity of an uncertain MIMO
channel is defined similarly to its SISO counterpart as
This example is an application of Theorem 4.1, in the presence
of the channel uncertainty and jamming. The channel is
represented by the second order frequency response H(s) =
2
ωn
2 , where a damping ration ξ is uncertain, and
s2 +2ξωn s+ωn
ωn = 2π104 rad/s. It is assumed that the damping ratio ξ,
C = sup
inf I(Wx (θ), Wn (θ))
(14) while unknown, belongs to a certain interval, 0 < ξlow ≤
Wx ∈A1 Wn ∈A2
ξ ≤ ξup < 1. This set is approximated by using the following
procedure. We choose the nominal damping ratio ξnom = 0.3,
where
and 0.2 ≤ ξ ≤ 0.5. Further, the size of the uncertainty set
I(Wx (θ), Wn (θ)) =
is defined by |W1 | = |Hnom | − |Hlow |, where Hlow (s) =
Z 2π
2
2
1
ωn
ωn
jθ
∗ jθ
−1
2 , Hnom (s) = s2 +2ξ
2 . The values of
ln det(I + H(e )Wx (θ)H (e )Wn (θ))dθ,
s2 +2ξup ωn s+ωn
nom ωn s+ωn
4π 0
ξlow = 0.2 and ξup = 0.5 are deliberately chosen such that
and the notation (.)∗ means complex conjugate transpose [6]. |Hub | = |Hnom | + |W1 | is a good approximation of Hup (s) =
2
ωn
Again, as in the SISO case, the problem may be treated as
2 . Thus, the frequency response uncertainty set
s2 +2ξlow ωn s+ωn
a game between the transmitter x and the noise/jammer n. is roughly described by |H
nom | ± |W1 |. However, |W1 | =
The channel capacity of MIMO Gaussian channel with noise |H
|−|H
|
implies
|H
nom
low
low | = |Hnom |−|W1 |. This means
uncertainty is given by the following theorem.
that the robust capacity is determined by the transfer function
Theorem 5.1: The capacity of a discrete time MIMO H . The uncertainty set is identified by the range of the
low
Gaussian channel modeled by (13), subject to a noise uncer- damping ratio, ∆ξ = ξ − ξ , ∆ξ = 0.30 for this particular
up
low
tainty described by L1 norm constraint on the PSD matrix case. The value ∆ξ = 0 corresponds to the nominal channel
Wn (θ), A2 , is given by
model. The transmitted power is limited by Px = 0.1W. Fig.
Z 2π
1 shows the change in the capacity vs. the SNR ration, defined
1
C=
ln det(I + H(ejθ )Wxo (θ)H ∗ (ejθ )Wno−1 (θ))dθ
as SN R = 10 log Px /Pn , for different values of ∆ξ. For
4π 0
instance, the channel capacity decreases by 3 knat/s due to
where the optimal PSD matrices satisfy
the noise uncertainty, when the SN R decreases from 2dB
H ∗ Wno−1 (I + HWxo H ∗ Wno−1 )H = λo2 I,
(15) to 1dB for ∆ξ = 0, and decreases for additional 3 knat/s,
due to channel uncertainty ∆ξ = 0.15. Fig. 2 shows that the
1
1 o
1
o ∗
o2
o ∗
o
o ∗
optimal
transmitter’s strategy to combat the noise uncertainty
Wn + HWx H Wn + Wn HWx H − o HWx H = 0 (16)
2
2
λ1
is to shrink the bandwidth and to regroup the power towards
and λo1 and λo2 are Lagrange multipliers of constraint sets A2 the lower frequencies.
and A1 , which are computed from
55
Z 2π
o
Trace(Wx (θ))dθ = Px ,
(17)
50
0
Z 2π
'[ 0.15
45
Trace(Wno (θ))dθ = Pn .
(18)
0
Thus, the optimal PSD matrix Wxo (θ) of the transmitted signal
that achieves the channel capacity is a scaled version of the
optimal PSD matrix Wno (θ) of a jammer. Thus, the equation
(19) shows the advantage of direct solution of optimization
problem (14) since it directly relates the optimal transmitted
and jamming strategies, which cannot be seen if the problem
is solved through numerical techniques.
40
C (knats/s)
Proof. The max-min problem (14) is resolved by introducing
positive Lagrange multipliers λ1 and λ2 , and finding necessary
conditions (15) and (16) by applying calculus of variations and
Kuhn-Tucker conditions.
Remark. First, note that (15) and (16) corresponds to SISO
equations (9) and (10), respectively. Further, if it is assumed
that H(ejθ ) is square and invertible, after some manipulation
of (15) and (16), it is obtained
λo
Wxo = 1o H ∗ Wno H(H ∗ H)−1 .
(19)
λ2
35
'[
0
30
25
'[
20
0 .3
15
10
0
1
2
3
Fig. 1.
4
5
SNR (dB)
6
7
8
9
10
Capacity vs. uncertainty
VII. C ONCLUSION
This paper considers a problem of defining and computing
the channel capacity of additive Gaussian channels subject to
-6
3.5
x 10
Thus, from (20), the inequality in (21) holds with equality
giving the formula for infimum. The problem of finding
supremum over A1 is exactly the same as in the classical
case [10], and will be omitted. After optimization, the optimal
transmitted power is given by (7).
Pn=0.1 W
3
PSD [W/Hz]
Pn=0.01 W
Pn=0.055 W
2.5
IX. A PPENDIX B
2
Proof of Theorem 4.1. For the notation simplicity, assume
that ∆H(f ) = 0. Due to the existence of a saddle point,
and taking into account power constraints, the Lagrangian is
defined as
µ
¶
Z
Sx |Hnom |2
1 +∞
log 1 +
df
L(Sx , Sn , λ1 , λ2 ) =
2 −∞
Sn
µ Z +∞
¶
µ Z +∞
¶
+λ1
Sn df − Pn − λ2
Sx df − Px .
1.5
1
0.5
0
-60
-40
Fig. 2.
-20
0
f [kHz]
20
40
60
−∞
−∞
After applying the calculus of variation, and Kuhn-Tucker
conditions, the necessary conditions are obtained, which gives
the main expressions of the theorem.
Optimal PSD of the communicator
R EFERENCES
uncertainties, which are described as the subsets of normed
spaces L1 , and H ∞ in frequency domain. Two SISO, and
one MIMO problems are presented. The explicit formulas for
the channel capacities are derived, accompanied with optimal
transmitted power formulas.
VIII. A PPENDIX A
Proof of Theorem 3.1. The condition
of boundedness, and
)|+|W (f )|)2
integrability of (|Hnom (f
comes
from Lemma 8.5.7,
Sn (f )
[10] (page 423). Further, the infimum of mutual information
rate over the set A2 is derived. First, note that
¶
µ
Z +∞
Sx |H̃|2
inf H̃∈A2
df
log 1 +
Sn
−∞
¶
µ
Z +∞
Sx |H̃|2
≥
df
inf log 1 +
Sn
−∞ H̃∈A2
From k∆(f )k∞ ≤ 1, and since log is a monotonically
increasing function, it follows that
µ
¶
Z +∞
Sx |Hnom + ∆W |2
inf
log 1 +
df
Sn |
H̃∈A2 −∞
µ
¶
Z +∞
Sx (|Hnom | − |W |)2
≥
log 1 +
df,
(20)
Sn
−∞
with equality when ∆(f ) = ∆∗ (f ) = exp[−j arg(W (f )) +
j arg(Hnom (f )) + jπ]. The opposite inequality of (20) holds
as well. Observe that
µ
¶
Z +∞
Sx (|Hnom | − |W |)2
log 1 +
df
Sn
−∞
Z
+∞
=
−∞
Z
µ
¶
Sx |Hnom + ∆∗ W |2
log 1 +
df
Sn
+∞
≥ inf
H̃∈A2
−∞
µ
¶
Sx |Hnom + ∆W |2
log 1 +
df
Sn
(21)
[1] Ahlswede, R., ”The capacity of a channel with arbitrary varying Gaussian
channel probability functions,” Trans. 6th Prague Conf. Information
Theory, Statistical Decision Functions, and Random Processes, pp. 13-31,
Sept. 1971.
[2] Baker, C. R., Chao, I.-F., ”Information capacity of channels with partially
unknown noise. II. Infinite-dimensional channels,” SIAM J. Appl. Math.,
vol. 56, no. 3, pp. 946-963, June 1996.
[3] Baker, C. R., Chao, I.-F., ”Information capacity of channels with partially
unknown noise. I. Finite dimensional channels,” SIAM J. Control and
Optimization, vol. 34, no. 4, pp. 1461-1472, July 1996.
[4] N. M. Blachman, “The effect of statistically dependent interference upon
channel capacity,” IRE Trans. Inform. Theory, vol. IT-8, pp. 553557, Sep.
1962.
[5] Boche, H., Jorsweick, E. A., ”Multiuser MIMO systems, worst case noise,
and transmitter cooperation,” Proc. of the 3rd IEEE Inter. Sym. on Signal
Processing and Information Technology, 2003. ISSPIT 2003. , 2003.
[6] Caines, P., E. Linear Stochastic Systems. New York: John Wiley & Sons,
1988.
[7] Denic, S.Z., Charalambous, C.D., Djouadi, S.M., ”Robust capacity for additive colored Gaussian uncertain channels,” preprint,
http://www.uottawa.ca/∼ sdenic/colored
[8] Diggavi, S. N., Cover, T. M., ”The worst additive noise under a covariance
constraint,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp.
3072-3081, November, 2001.
[9] Doyle, J. C., Francis, B. A., Tannenbaum, A. R., Feedback Control
Theory. New York: Mcmillan Publishing Company, 1992.
[10] Gallager, R., Information theory and reliable communication. New York:
Wiley, 1968.
[11] A. J. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, ”Capacity
limits of MIMO channels,” IEEE J. Select. Areas Commun., vol. 21, no.
5, pp. 684-702, Jun. 2003.
[12] Hughes, B., Narayan P., ”Gaussian arbitrary varying channels,” IEEE
Transactions on Information Theory, vol. 33, no. 2, pp. 267-284, Mar.,
1987.
[13] Hughes, B., Narayan P., ”The capacity of vector Gaussian arbitrary
varying channel,” IEEE Transactions on Information Theory, vol. 34, no.
5, pp. 995-1003, Sep., 1988.
[14] D. Hoesli, A. Lapidoth, ”The Capacity of a MIMO Ricean Channel
is Monotonic in the Singular Values of the Mean,” Proc. of the 5th
International ITG Conference on Source and Channel Coding (SCC),
Erlangen, Nuremberg, January 14-16, 2004
[15] Kashyap, A., Basar, T., Srikant, R., ”Correlated jamming on MIMO
Gaussian fading channels,” IEEE Transactions on Information Theory,
vol. 50, no. 9, pp. 2119-2123, Sep., 2004.
[16] Lapidoth, A., Narayan, P., ”Reliable communication under channel
uncertainty,” IEEE Transactions on Information Theory, vol. 44, no. 6,
pp. 2148-2177, October, 1998.
[17] McEliece, R. J., ”Communication in the Presence of Jamming-An
Information Theory Approach,” in Secure digital communications, CSIM
courses and lectures, no. 279. New York: Springer, 1983.
[18] Medard, M., ”Channel Uncertainty in Communications,” IEEE Information Society Newsletter, vol. 53, no. 2, p. 1, pp. 10-12, June 2003.
[19] Osborne, M. J., Rubinstein, A., A course in game theory, MIT Press,
1994
[20] L. H. Ozarow, S. Shamai (Shitz), and A. D. Wyner, ”Information
Theoretic considerations for cellular mobile radio,” IEEE Tran. on Veh.
Technol., vol. 43, pp. 359-378, May, 1994.
[21] D. P. Palomar, J. M. Cioffi, M. A. Lagunas, ”Uinform power allocation
in MIMO channels: a game theoretic approach,” IEEE Transactions on
Information Theory, vol. 49, no. 7, pp. 1707-1727, July, 2003.
[22] Root, W. L., Varaiya, P. P., ”Capacity of Classes of Gaussian Channels,”
SIAM J. Appl. Math., vol. 16, no. 6, pp. 1350-1393, November 1968.
[23] Vishwanath, S., Boyd, S., Goldsmith, A., ”Worst-case capacity of
Gaussian vector channels,” Proceedings of 2003 Canadian Workshop on
Information Theory, 2003.
[24] T. Yoo, E. Yoon, and A. J. Goldsmith, ”MIMO Capacity with Channel
Uncertainty: Does Feedback Help?”, IEEE GlobeCom 2004, Dallas,
Texas, Dec. 2004.
[25] Zames, G., ”Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE
Transactions on Automatic Control, vol. 26, pp. 301-320, 1981.