IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 1715
Min-Capacity of a Multiple-Antenna Wireless
Channel in a Static Ricean Fading Environment
Mahesh Godavarti, Member, IEEE, Alfred O. Hero, III, Fellow, IEEE, and Thomas L. Marzetta, Fellow, IEEE
Abstract—This paper presents the optimal guaranteed per-
formance for a multiple-antenna wireless compound channel with
M antennas at the transmitter and N antennas at the receiver
on a Ricean fading channel with a static specular component.
The channel is modeled as a compound channel with a Rayleigh
component and an unknown rank-one deterministic specular
component. The Rayleigh component remains constant over a
block of T symbol periods, with independent realizations over
each block. The rank-one deterministic component is modeled as
an outer product of two unknown deterministic vectors of unit
magnitude. Under this scenario, to guarantee service, it is required
to maximize the worst case capacity (min-capacity). It is shown
that for computing min-capacity, instead of optimizing over the
joint density of T · M complex transmitted signals, it is sufficient
to maximize over a joint density of min { T,M} real transmitted
signal magnitudes. The optimal signal matrix is shown to be
equal to the product of three independent matrices—a T × T
unitary matrix, a T × M real nonnegative diagonal matrix, and
an M × M unitary matrix. A tractable lower bound on capacity
is derived for this model, which is useful for computing achievable
rate regions. Finally, it is shown that the average capacity
(avg-capacity) computed under the assumption that the specular
component is constant but random with isotropic distribution is
equal to min-capacity. This means that avg-capacity, which, in
general, has no practical meaning for nonergodic scenarios, has a
coding theorem associated with it in this particular case.
Index Terms—Capacity, compound channel, information
theory, multiple antennas, Ricean fading.
I. INTRODUCTION
T
HE need for higher rates in wireless communications has
never been greater than in the present. Due to this need
and the derth of extra bandwidth available for communica-
tion, multiple antennas have attracted considerable attention
[6], [7], [16], [20], [21]. Multiple antennas at the transmitter
and the receiver provide spatial diversity that can be exploited to
improve spectral efficiency of wireless communication systems
and to improve performance.
Manuscript received May 31, 2003; revised February 20, 2004; ac-
cepted May 31, 2004. The editor coordinating the review of this paper
and approving it for publication is A. Saglione. This work was performed
in part while M. Godavarti was a summer intern at the Mathematical
Sciences Research Center, Bell Laboratories, and in part while M. Godavarti
was a Ph.D. candidate at the University of Michigan. Parts of this work were
presented at ISIT 2001 held in Washington, D.C., USA.
M. Godavarti is with the Ditech Communications Inc., Mountain View, CA
94043 USA.
A. O. Hero III is with the Department of Electrical Engineering, University
of Michigan, Ann Arbor, MI 48109 USA.
T. L. Marzetta is with Bell Laboratories, Lucent Technologies, Murray Hill,
NJ 07974 USA.
Digital Object Identifier 10.1109/TWC.2005.850261
Two kinds of models widely used for describing fading in
wireless channels are the Rayleigh and Ricean models. For
wireless links in Rayleigh fading environment, it has been
shown by Foschini and Gans [6], [7] and Telatar [20] that with
perfect channel knowledge at the receiver, for high signal-to-
noise ratio (SNR), a capacity gain of min(M, N) bits/s/Hz,
where M is the number of antennas at the transmitter and
N is the number of antennas at the receiver, can be achieved
with every 3-dB increase in SNR. The assumption of complete
knowledge about the channel might not be true in the case of
fast mobile receivers and large number of transmit antennas be-
cause of insufficient training. Marzetta and Hochwald [16] con-
sidered the case when neither the receiver nor the transmitter
has any knowledge of the fading coefficients. They considered
a model where the fading coefficients remain constant for T
symbol periods and instantaneously change to new independent
realizations after that. They derived the structure of capacity
achieving signals and also showed that under this model, the
complexity for capacity calculations is considerably reduced.
In contrast, the attention paid to Ricean fading models has
been fairly limited. Ricean fading components traditionally
have been modeled as independent Gaussian components with
a deterministic nonzero mean [1], [4], [5], [12], [17], [19].
Farrokhi et al. [5] used this model to analyze the capacity of
a MIMO channel with a specular component. They assumed
that the specular component is deterministic and unchanging
and unknown to the transmitter but known to the receiver. They
also assumed that the receiver has complete knowledge about
the fading coefficients (i.e., has knowledge about the Rayleigh
component as well). They worked with the premise that since
the transmitter has no knowledge about the specular compo-
nent, the signaling scheme has to be designed to guarantee a
given rate irrespective of the value of the deterministic specular
component. They concluded that the signal matrix has to be
composed of independent circular Gaussian random variables
of mean 0 and equal variance to maximize the rate.
Godavarti et al. [13] considered a nonconventional ergodic
model for the case of Ricean fading where the fading channel
consists of a Rayleigh component, modeled as in [16] and an
independent rank-one isotropically distributed specular compo-
nent. The fading channel is assumed to remain constant over a
block of T consecutive symbol periods but take a completely
independent realization over each block. They derived similar
results on optimal capacity achieving signal structures as in
[16]. They also established a lower bound to capacity that
can be easily extended to the model considered in this paper.
The model described in [13] is applicable to a mobile-wireless
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1716 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
link where both the direct line of sight component (specular
component) and the diffuse component (Rayleigh component)
change with time.
In [12], Godavarti and Hero considered the standard Ricean
fading model. The capacity calculated for the standard Ricean
fading model is a function of the specular component since
the specular component is deterministic and known to both the
transmitter and the receiver. The authors established asymptotic
results for capacity and conclude that beamforming i s the op-
timum signaling strategy for low SNR, whereas for high SNR,
the optimum signaling strategy is the same as that for purely
Rayleigh fading channels.
In this paper, we consider a quasi-static Ricean model where
the specular component is nonchanging while the Rayleigh
component is varying over time. The only difference between
this model and the standard Ricean fading model is that in this
model, the specular component is of single rank and is not
known to the transmitter. We can also contrast the formulation
here to that in [13] where the specular component is also mod-
eled as stochastic and the ergodic channel capacity is clearly
defined. In spite of a completely different formulation, we
obtain surprisingly similar results as in [13].
Modeling the specular component to be of rank one is fairly
common in the literature [14], [15], [18]. The rank of the
specular component is determined by the number of direct line
of sight paths between the transmitter and the receiver, which is
typically much lower than the number of transmit and receive
antennas leading to ill-conditioned specular components [10],
[11]. Furthermore, if the distance between the transmit and
receive antennas is much greater than the distance between
individual antenna elements, then the rank of the specular
component can only be one [3], [4].
The Ricean channel models considered in [12], [13] and in
this paper are all extensions of the Rayleigh model considered
in [16] in the sense that all models reduce to the Rayleigh model
of [16] when the specular component goes to zero.
The channel model considered here is applicable to the case
where the transmitter and receiver are either fixed in space
or are in motion but sufficiently far apart with a single direct
path so that the specular component has single r ank and is
practically constant while the diffuse multipath component
changes rapidly. This can be contrasted with the channel model
in [13] where the specular component is changing as rapidly
as the diffuse multipath component. This allows modeling the
channel in [13] as an ergodic channel. On the other hand, the
channel model in [12] is almost exactly the same as the model
proposed in this paper except that in [12], it is assumed that
there is a feedback path from the receiver to the transmitter
and as a result the transmitter can be modeled to have complete
knowledge of the specular component.
In this paper, since the transmitter has no knowledge about
the specular component, the transmitter can either maximize the
worst case rate over the ensemble of values that the specular
component can take or maximize the average rate by estab-
lishing a prior distribution on the ensemble. We address both
approaches in this paper. Note that when the transmitter has
no knowledge about the specular component, knowledge of it
at the receiver makes no difference on the worst case capacity
[2]. We however assume the knowledge as it makes it easier to
analyze the fading channel.
Similar to [5], the specular component is an outer product
of two vectors of unit magnitude that are nonchanging and
unknown to the transmitter but known to the receiver. The
difference between our approach and that of [5] is that in [5],
the authors consider the channel to be known completely to
the receiver. We assume that the receiver’s extent of knowledge
about the channel is limited t o the specular component. That is,
the receiver has no knowledge about the Rayleigh component
of the model. Considering the absence of knowledge at the
transmitter, it is important to design a signal scheme that guar-
antees the largest overall rate for communication irrespective of
the value of the specular component. This is formulated as the
problem of determining the worst case capacity in Section II.
This is followed by derivation of upper and lower bounds on the
worst case capacity in Section III and optimum signal properties
in Section IV. In Section V, the average capacity is considered
instead of the worst case capacity, and it is shown that both
formulations imply the same optimal signal structure and the
same maximum possible rate. In Section VI, we use the results
derived in this paper to compute capacity regions for some
Ricean fading channels. For interested readers, we show the
existence, in the Appendix, of a coding theorem corresponding
to the worst case capacity for the fading model considered here.
II. S
IGNAL MODEL AND PROBLEM FORMULATION
Let there be M transmit antennas and N receive antennas.
It is assumed that the fading coefficients remain constant over
a block of T consecutive symbol periods but are independent
from block to block. Keeping that in mind, the channel is
modeled as carrying a T × M signal matrix S over an M × N
MIMO channel H, producing X at the receiver according
to the model
X =
ρ
M
SH + W (1)
where the elements, w
tn
of W are independent circular com-
plex Gaussian random variables with mean 0 and variance 1
[CN(0, 1)].
The MIMO Ricean model for the matrix H is H =
(1 − r)
1/2
G +(rNM)
1/2
αβ
†
where G consists of indepen-
dent CN(0, 1) random variables and α and β are deterministic
vectors of length M and N, respectively, such that α
†
α =1
and β
†
β =1. The parameter r, 0 ≤ r ≤ 1 denotes the frac-
tion of the energy propagated via the specular component.
r =0 and r =1 correspond to purely Rayleigh and purely
specular fading, respectively. Irrespective of the value of r,the
average variance of the elements of H is 1, that is, H satisfies
E[tr{HH
†
}]=M · N .
It is assumed that α and β are known to the receiver. Since
the receiver is free to apply a coordinate transformation by post-
multiplying X by a unitary matrix, without loss of generality,
β can be taken to be identically equal to [1 0 ... 0]
T
.We
will sometimes write H as H
α
to highlight the dependence of
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GODAVARTI et al.: MIN-CAPACITY OF A WIRELESS CHANNEL IN A STATIC RICEAN ENVIRONMENT 1717
H on α. G remains constant for T symbol periods and takes on
a completely independent realization every T th symbol period.
The problem in this section is to find the distribution p
∗
(S)
that attains the maximum in the following maximization defin-
ing the worst case channel capacity
C
∗
= max
p(S)
I
∗
(X; S) = max
p(S)
inf
α∈A
I
α
(X; S)
and also to find the maximum value C
∗
.
I
α
(X; S)=
p(S)p(X|S, αβ
†
)
×log
p(X|S, αβ
†
)
p(S)p(X|S, αβ
†
)dS
dS dX
is the mutual information between X and S when the specular
component is given by αβ
†
and A
def
= {α : α ∈ C
M
and α
†
α =
1}. Since A is compact the “inf” in the problem can be replaced
by “min.” For convenience, we will refer to I
∗
(X; S) as the
min-mutual information and C
∗
as min-capacity.
The above formulation is justified for the Ricean fading
channel considered here because there exists a corresponding
coding theorem that we prove in the Appendix. However,
the existence of a coding theorem can also be obtained from
[2, chap. 5, pp. 172–178]. The min-capacity defined above
is just the capacity of a compound channel. We will use
the notation in this paper to briefly describe the concept of
compound channels given in [2]. Let α ∈ A denote a candidate
channel. Let C
∗
= max
p(S)
min
α
I
α
(X; S) and P
∗
(e, n)=
max
α
P
α
(e, n), where P
α
(e, n) is the maximum probability of
decoding error for channel α when a code of length n is used.
Then for every R<C
∗
, there exists a sequence of (2
nR
,n)
codes such that
lim
n→∞
P
∗
(e, n)=0.
It is also shown in [2, Prob. 13, p. 183] that min-capacity
does not depend on the receiver’s knowledge of the channel.
Hence, it is not necessary for us to assume that the specular
component is known to the receiver. However, we do so be-
cause it facilitates easier computation of min-capacity and avg-
capacity in terms of t he conditional probability distribution
p(X|S).
Note that since A is unitarily invariant, it means that no
preference is attached to the direction of the line of sight
component and, therefore, it is intuitive to expect the optimum
signal to attach no significance to the direction of the line of
sight component as well. Moreover, since all α ∈ A have the
same strength, it is intuitive to expect the optimum signal to be
such that it generates the same mutual information irrespective
of the choice of the s pecular component. This intuition is made
concrete in the following sections.
III. C
APACITY UPPER AND LOW E R BOUNDS
Theorem 1: Min-capacity C
∗
H
when the channel matrix H is
known to the receiver but not to the transmitter is given by
C
∗
H
= TElog det
I
N
+
ρ
M
H
†
e
1
H
e
1
(2)
where e
1
=[1 0 ... 0]
T
is a unit vector in C
M
. Note that
e
1
in (2) can be replaced by any α ∈ A without changing the
answer.
Proof: The idea for this proof has been taken from the
proof of Theorem 1 in [20]. First note that for T>1,givenH,
the channel is memoryless, and hence, the rows of the input
signal matrix S are independent of each other. That means
the mutual information I
α
(X; S)=
T
i=1
I
α
(X
i
; S
i
) where
X
i
and S
i
denote the ith row of X and S, respectively. The
maximization over each term can be done separately, and it
is easily seen that each term will be maximized individually
for the same density on S
i
. That is, p(S
i
)=p(S
j
) for i = j
and max
p(S)
I
α
(X; S)=T max
p(S
1
)
I
α
(X
1
; S
1
). Therefore,
WLOG assume T =1.
Given H, the channel is an AWGN channel, therefore, capac-
ity is attained by Gaussian signal vectors. Let Λ
S
be the input
signal covariance. Since the transmitter does not know α, Λ
S
cannot depend on α and the min-capacity is given by
max
Λ
S
:tr{Λ
S
}≤M
F(Λ
S
) = max
Λ
S
:tr{Λ
S
}≤M
min
α∈A
E
×log det
I
N
+
ρ
M
H
†
α
Λ
S
H
α
(3)
where F(Λ
S
) is implicitly defined in an obvious manner. First
note that F(Λ
S
) in (3) is a concave function of Λ
S
.(This
follows from the fact that log det K is a concave function of
K.) Also, F(Ψ
†
Λ
S
Ψ) = F(Λ
S
) for any M × MΨ:Ψ
†
Ψ=
I
M
since Ψ
†
α ∈ A for any α ∈ A and G has independent
identically distributed (i.i.d.) zero mean complex Gaussian
entries. Let Q
†
DQ be the singular value decomposition (SVD)
of Λ
S
, then we have F(D)=F(Q
†
DQ)=F(Λ
S
). Therefore,
we can choose Λ
S
to be diagonal. Moreover, F(P
†
k
Λ
S
P
k
)=
F(Λ
S
) for any permutation matrix P
k
, k =1,...,M!. There-
fore, if we choose Λ
S
=(1/M !)
M!
k=1
P
†
k
Λ
S
P
k
, then by con-
cavity and Jensen’s inequality, we have
F (Λ
S
) ≥
1
M !
M!
k=1
F
P
†
k
Λ
S
P
k
= F(Λ
S
).
Therefore, it can be concluded that the maximizing input
signal covariance Λ
S
is a multiple of the identity matrix.
It is quite obvious to see that to maximize the expres-
sion in (3), we need to choose tr{Λ
S
} = M or Λ
S
= I
M
,
and since E log det[I
N
+(ρ/M)H
†
α
1
H
α
1
]=E log det[I
N
+
(ρ/M )H
†
α
2
H
α
2
] for any α
1
, α
2
∈ A, (2) easily follows.
By the data processing theorem, additional information at
the receiver does not decrease capacity, hence, the following
propositions.
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1718 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
Proposition 1: An upper bound on the channel min-capacity
when neither the transmitter nor the receiver has any knowledge
about the channel is given by
C
∗
≤ T · E log det
I
N
+
ρ
M
H
†
e
1
H
e
1
. (4)
Now, we establish a lower bound.
Proposition 2: A lower bound on min-capacity when the
transmitter has no knowledge about H and the receiver has no
knowledge about G is given by
C
∗
≥C
∗
H
− NE
log
2
det
I
T
+(1− r)
ρ
M
SS
†
(5)
≥C
∗
H
− NM log
2
1+(1− r)
ρ
M
T
. (6)
Proof: Proof is a slight modification of the proof of
Theorem 3 in [13], therefore, only the essential steps will be
shown here.
First note that
I
α
(X; S)=I(X; S|α)
= I(X; S, H|α) − I(X; H|S, α)
= I(X; H|α)+I(X; S|H, α) − I(X; H|S, α)
≥I(X; S|H, α) − I(X; H|S, α)
where the last inequality follows from the fact that I(X;
H|α) ≥ 0. Therefore
C
∗
(X; S) ≥ max
p(S)
min
α
[I(X; S|H, α) − I(X; H|S, α)].
The lower bound is obtained by observing that the second
term is the mutual information between the “input” H =
(1 − r)
1/2
G +(rNM)
1/2
αe
†
1
, and the “output” X through the
“channel” X =
(ρ/M )SH + W . Since α is fixed and α and
S are known at the “receiver,” the mutual information between
H and X is the same as the mutual information between G and
X
, where X
= X − (ρ/M)
1/2
(rNM)
1/2
Sαe
†
1
. Therefore,
the second term can be evaluated, irrespective of the value
of α,as
NE
log
2
det
I
T
+(1− r)
ρ
M
SS
†
and the first term can be maximized by choosing p(S) such
that the elements of S are independent CN (0, 1) random
variables.
Notice that the second term at right-hand side of the lower
bound is
NE
log
2
det
I
T
+(1− r)
ρ
M
SS
†
instead of NE[log
2
det(I
T
+(ρ/M)SS
†
)], which occurs in
the lower bound derived for the model in [13]. The second term
I(X; H|S) is the mutual information between the output and
the channel given the transmitted s ignal. In other words, this
is the information carried in the transmitted signal about the
channel. Therefore, the second term in the lower bound can
be viewed as a penalty term for using part of the available
rate to learn the channel. When r =1 or when the channel
is purely specular, it can be seen that the penalty term for
training goes to zero. This makes perfect sense because the
specular component is known to the receiver and the penalty for
learning the specular component is zero in the current model as
contrasted to the model in [13].
Combining (4) and (6) gives us the following.
Corollary 1: The normalized min-capacity, C
∗
n
= C
∗
/T in
bits per channel use as T →∞is given by
C
∗
n
= E log det
I
N
+
ρ
M
H
†
e
1
H
e
1
.
Note that this is the same as the capacity when the receiver
knows H, so that as T →∞, perfect channel estimation can
be performed.
IV. P
RO PE RT IES O F CAPACITY ACHIEVING SIGNALS
In this section, the optimum signal structure for achieving
min-capacity is derived. The optimization is being done under
the power constraint E[tr{SS
†
}] ≤ TM.
The results in this section theoretically establish what can
be gauged intuitively. It has already been established in [16]
that when the specular component is zero, the optimum signal
density is invariant to unitary transformations. This is no longer
true if the specular component is nonzero. However, if the set
of nonzero specular components (called A in this paper), from
which the worst case specular component is selected and the
corresponding performance maximized, is invariant to unitary
transformations, then it is natural to expect the optimum signal
to be invariant to unitary transformations as well.
The basic ideas for showing invariance of optimum signals
to unitary transformations in this section, and also in the next,
have been taken from [16].
Lemma 1: I
∗
(X; S) as a functional of p(S) is concave
in p(S).
Proof: First, note that I
α
(X; S) is a concave functional
of p(S) for every α ∈ A.LetI
∗
(X; S)
p(S)
denote I
∗
(X; S)
evaluated using p(S) as the signal density. Then
I
∗
(X; S)
δp
1
(S)+(1−δ)p
2
(S)
= min
α∈A
I
α
(X; S)
δp
1
(S)+(1−δ)p
2
(S)
≥ min
α∈A
δI
α
(X; S)
p
1
(S)
+(1− δ)I
α
(X; S)
p
2
(S)
≥ δ min
α∈A
I
α
(X; S)
p
1
(S)
+(1− δ) min
α∈A
I
α
(X; S)
p
2
(S)
= δI
∗
(X; S)
p
1
(S)
+(1− δ)I
∗
(X; S)
p
2
(S)
.
Lemma 2: For any T × T unitary matrix Φ and any M ×
M unitary matrix Ψ,ifp(S) generates I
∗
(X; S), then so does
p(ΦSΨ
†
).
Proof:
1) Note that p(ΦX|ΦS)=p(X|S), therefore, I
α
(X;
ΦS)=I
α
(X; S) for any T × T unitary matrix Φ and
all α ∈ A.
2) Also, Ψα ∈ A for any α ∈ A and any M × M unitary
matrix Ψ. Therefore, if I
∗
(X; S) achieves its minimum
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GODAVARTI et al.: MIN-CAPACITY OF A WIRELESS CHANNEL IN A STATIC RICEAN ENVIRONMENT 1719
value at α
0
∈ A, then I
∗
(X; SΨ
†
) achieves its minimum
value at Ψα
0
because I
α
(X; S)=I
Ψα
(X; SΨ
†
) for α ∈
A and Ψ an M ×M unitary matrix.
Combining 1) and 2), we get the lemma.
Lemma 3: The min-capacity achieving signal distribution,
p(S) is unchanged by any pre- and postmultiplication of S by
unitary matrices of appropriate dimensions.
Proof: It will be shown that for any signal density p
0
(S)
generating min-mutual information I
∗
0
, there exists a density
p
1
(S) generating I
∗
1
≥ I
∗
0
such that p
1
(S) is invariant to pre-
and postmultiplication of S by unitary matrices of appropriate
dimensions. By Lemma 2, for any pair of permutation ma-
trices, Φ(T × T ) and Ψ(M × M ) p
0
(ΦSΨ
†
) generates the
same min-mutual information as p(S). Define u
T
(Φ) to be
the isotropically random unitary density function of a T × T
unitary matrix Φ. Similarly, define u
M
(Ψ).Letp
1
(S) be a
mixture density given as follows
p
1
(S)=
p
0
(ΦSΨ
†
)u(Φ)u(Ψ)dΦdΨ.
It is easy to see that p
1
(S) is invariant to any pre- and postmul-
tiplication of S by unitary matrices and if I
∗
1
is the min-mutual
information generated by p
1
(S), then from Jensen’s inequality
and concavity of I
∗
(X; S),wehaveI
∗
1
≥ I
∗
0
.
Corollary 2: p
∗
(S), the optimal min-capacity achieving sig-
nal density lies in P = ∪
I>0
P
I
where
P
I
= {p(S): I
α
(X; S)=I ∀α ∈ A}. (7)
Proof: This follows immediately from Lemma 3 because
any signal density that is invariant to pre- and postmultiplication
of S by unitary matrices generates the same mutual information
I
α
(X; S) irrespective of the value of α.
The above result is intuitively obvious because all α ∈ A
are identical to each other except for unitary transformations.
Therefore, any density function that is invariant to unitary
transformations is expected to behave the same way for all α.
Theorem 2: The signal matrix that achieves min-capacity
can be written as S =ΦV Ψ
†
, where Φ and Ψ are T × T and
M × M isotropically distributed matrices independent of each
other, and V is a T × M real nonnegative diagonal matrix,
independent of both Φ and Ψ.
Proof: From the SVD, we can write S =ΦV Ψ
†
, where
Φ is a T × T unitary matrix, V is a T × M nonnegative real
diagonal matrix, and Ψ is an M × M unitary matrix. In general,
Φ, V , and Ψ are jointly distributed. Suppose S has probability
density p
0
(S) that generates min-mutual information I
∗
0
.Let
Θ
1
and Θ
2
be isotropically distributed unitary matrices of size
T × T and M × M independent of S and of each other. Define
a new signal S
1
=Θ
1
SΘ
†
2
, generating min-mutual information
I
∗
1
. Now conditioned on Θ
1
and Θ
2
, the min-mutual infor-
mation generated by S
1
equals I
∗
0
. From the concavity of the
min-mutual information as a functional of p(S) and Jensen’s
inequality, we conclude that I
∗
1
≥ I
∗
0
.
Since Θ
1
and Θ
2
are isotropically distributed, Θ
1
Φ and
Θ
2
Ψ are also isotropically distributed when conditioned on Φ
and Ψ, respectively. This means that both Θ
1
Φ and Θ
2
Ψ are
isotropically distributed making them independent of Φ, V , and
Ψ. Therefore, S
1
is equal to the product of three independent
matrices, a T × T unitary matrix Φ,aT × M real nonnegative
matrix V , and an M × M unitary matrix Ψ.
Now, it will be shown that the density p(V ) on V is un-
changed by rearrangements of diagonal entries of V . There
are min{M!,T!} ways of arranging the diagonal entries of
V . This can be accomplished by pre- and postmultiplying
V by appropriate permutation matrices P
Tk
and P
Mk
, k =
1,...,min{M!,T!}. The permutation does not change the
min-mutual information because ΦP
Tk
and ΨP
Mk
have the
same density functions as Φ and Ψ. By choosing an equally
weighted mixture density for V , involving all min{M!,T!}
arrangements, a higher value of min-mutual information can
be obtained because of concavity and Jensen’s inequality. This
new density is invariant to the rearrangements of the diagonal
elements of V .
V. A
VERAGE CAPACITY CRITERION
In this section, we will investigate how much worse the worst
case performance is compared to the average performance.
To find the average performance, we maximize I
E
(X; S)=
E
α
[I
α
(X; S)], where I
α
is defined earlier and E
α
denotes
expectation over α ∈ A under the assumption that all α are
equally likely. That is, under the assumption that α is unchang-
ing over time, isotropically random and known to the receiver.
Note that this differs from the model considered in [13] where
the authors consider the case of a piecewise constant time
varying i.i.d. specular component.
Therefore, the problem can be stated as finding p
E
(S) the
probability density function on the input signal S that achieves
the following maximization
C
E
= max
p(S)
E
α
[I
α
(X; S)] (8)
and also to find the value C
E
. We will refer to I
E
(X; S) as
avg-mutual information and C
E
as avg-capacity.
Like in the previous section, we would expect the optimum
signal to be such that it generates the same mutual information
irrespective of the choice of the specular component because
the density function attaches no significance to any particular
α ∈ A. In other words, since the set of nonspecular components
A and the density function on A are such that the density
function is invariant under unitary transformations, we would
expect the optimum signal density to be invariant to unitary
transformations as well. Moreover, since all α ∈ A are identical
to each other except for unitary transformations, intuition tells
us that Corollary 2 should hold here also. Therefore, the average
mutual information over all α should be equal to the mutual
information for a single α. That is, the average performance
should be equal to the worst case performance.
Formally, it will be shown that the signal density p
∗
(S) that
attains C
∗
also attains C
E
. For that, we need to establish the
following lemmas. We omit some of the proofs because the
proofs are very similar to the proofs in Section IV.
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