1

Fundamental Limits of Covert Communication over
MIMO AWGN Channel
Amr Abdelaziz and C. Emre Koksal
Department of Electrical and Computer Engineering
The Ohio State University
Columbus, Ohio 43201

arXiv:1705.02303v4 [] 13 Mar 2018

Abstract
Fundamental limits of covert communication
have been studied for different models of scalar channels. It was shown that,
√
over n independent channel uses, O( n) bits can be transmitted reliably over a public channel while achieving an arbitrarily
low probability of detection (LPD) by other stations. This result is well known as the square-root law and even to achieve this
diminishing rate of covert communication, all existing studies utilized some form of secret shared between the transmitter and the
receiver. In this paper, we establish the limits of LPD communication over the MIMO AWGN channel. In particular, using relative
entropy as our LPD metric, we study the maximum codebook size for which the transmitter can guarantee reliability and LPD
conditions are met. We first show that, the optimal codebook generating input distribution under δ-PD constraint is the zero-mean
Gaussian distribution. Then, assuming channel state information (CSI) on only the main channel at the transmitter, we derive the
optimal input covariance matrix, hence, establishing scaling laws of the codebook size. We evaluate the codebook scaling rates
in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO).
We show that, in the asymptotic block-length regime, square-root
law still holds for the MIMO AWGN. Meanwhile, in massive
√
MIMO limit, the codebook size, while it scales linearly with n, it scales exponentially with the number of transmitting antennas.
Further, we derive equivalent results when no shared secret is present. For that scenario, in the massive MIMO limit, higher covert
rate up to the non-LPD constrained capacity still can be achieved, yet, with much slower scaling compared to the scenario with
shared secret. The practical implication of our result is that, MIMO has the potential to provide a substantial increase in the file
sizes that can be covertly communicated subject to a reasonably low delay.
Index Terms
LPD communication, Covert MIMO Communication, MIMO physical layer security, LPD Capacity.

I. I NTRODUCTION
Conditions for secure communication under a passive eavesdropping attack fall in two broad categories: 1) low probability
of intercept (LPI). 2) low probability of detection (LPD). Communication with LPI requires the message exchanged by two
legitimate parties to be kept secret from an illegitimate adversary. Meanwhile, LPD constrained communication is more
restrictive as it requires the adversary to be unable to decide whether communication between legitimate parties has taken
place. Fundamental limits of LPD constrained communication over scalar AWGN has been established in [1] where the
square-root law for LPD communication was established. Assuming a shared secret of sufficient length between transmitter
√
and receiver, square-root law states that, over n independent channel uses of an AWGN channel, transmitter can send O( n)
bits reliably to the receiver while keeping arbitrary low probability of detection at the adversary. In this paper, we study the
fundamental limits of communication with LPD over MIMO AWGN channels.
Consider the scenario in which a transmitter (Alice) wishes to communicate to a receiver (Bob) while being undetected by a
passive adversary (Willie) when all nodes are equipped with multiple antennas. To that end, Alice wish to generate a codebook
that satisfies both reliability, in terms of low error probability , over her channel to Bob and, in the same time, ensures, a
certain maximum PD, namely δ, at Willie. Denote the maximum possible size of such codebook by Kn (δ, ). In this paper,
we are interested in establishing the fundamental limits of Kn (δ, ) in the asymptotic length length regime and in the limit
of large number of transmitting antenna. First we show that, the maximum codebook size is attained when the codebook is
generated according to zero mean circular symmetric complex Gaussian distribution. We establish this result building upon the
the Principle Minimum Relative Entropy [2] and Information Projection [3].
Some
p of our findings can be summarized as follows. For an isotropic Willie channel, we show that Alice can transmit
O(N n/M ) bits reliably in n independent channel uses, where N and M are the number of active eigenmodes of Bob and
Willie channels, respectively. Further, we evaluate δ-PD rates in the limiting regimes for the number of channel uses (asymptotic
block length) and the number of antennas (massive MIMO). We show that, while the square-root law still holds for the MIMO
AWGN, the number of bits that can be transmitted covertly scales exponentially with the number of transmitting antennas.
This work was submitted in part to IEEE CNS-2017.
This work was in part supported by the National Science Foundation under Grants NSF NeTs 1618566 and 1514260 and Office of Naval Research under
Grant N00014-16-1-2253.

2

TABLE I
S UMMARY OF R ESULTS
Result

Main Channel

Adversary Channel

Shared
Secret

Kn (δ, ) Scales Like

Theorems 2&3

Deterministic

Bounded Spectral norm

Yes

N

Theorem 4

Deterministic and of Unit
Rank

Deterministic and of Unit
Rank

Yes

Theorems 5&6

Deterministic

Bounded Spectral norm

No

N/M

Theorem 7

Deterministic and of Unit
Rank

Deterministic and of Unit
Rank

No

1/cos2 (θ)

Theorem 8

Deterministic and of Unit
Rank

Unit Rank chosen uniformly at random

Yes

Theorem 9

Deterministic and of Unit
Rank

Unit Rank chosen uniformly at random

No

√

p
n/M
n/ cos2 (θ)

n
c
(1 + √ )(Na −2)/2
K 2 Na
n
where c is constant independent
on n and Na .
r

1
c
√
(1 + )(Na −2)/2 where
n
K Na
c is constant independent on n and
Na .

r

n
c
(1 + √ )(Na −2)/2 where Na is the
K 2 Na
n
number of transmitting antennas, K is a universal constant and c is constant independent on n and Na . Further, we derive the
scaling of Kn (δ, ) with no shared secret between Alice and Bob. In particular, we show that achieving better covert rate is
a resource arm race between Alice, Bob and Willie. Alice can transmit O(N/M ) bits reliably in n independent channel uses,
i.e., the covert rate is in the order of the ratio between active eigenmodes of both channels. The practical implication of our
findings is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated
subject to a reasonably low delay. The results obtained in this paper are summarized in Table I1 .
The contributions of this work can be summarized as follows:
• Using the Principle Minimum Relative Entropy [2] and Information Projection [3], we show that the Kn (δ, ) is achievable
when the codebook is generated according to zero mean complex Gaussian distribution in MIMO AWGN channels.
• With the availability of only the main CSI to Alice, we evaluate the optimal input covariance matrix under the assumption
that Willie channel satisfies a bounded spectral norm constraint [4], [5]. Singular value decomposition (SVD) precoding
is shown to be the optimal signaling strategy and the optimal water-filling strategy is also provided.
• We evaluate the block-length and massive MIMO asymptotics for Kn (δ, ). We show that, while the square-root law
cannot be avoided, Kn (δ, ) scales exponentially with the number of antennas. Thus, MIMO has the potential to provide
a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay.
• We evaluate scaling laws of Kn (δ, ) when there is no shared secret between Alice and Bob in both limits of large block
length and massive MIMO.
Related Work. Fundamental limits of covert communication have been studied in literature for different models of scalar
channels. In [6], LPD communication over the binary symmetric channel was considered. It was shown that, square-root law
holds for the binary symmetric channel, yet, without requiring a shared secret between Alice and Bob when Willie channel is
significantly noisier. Further, it was shown that Alice achieves a non-diminishing LPD rate, exploiting Willie’s uncertainty about
his own channel transition probabilities. Recently in [7], LPD communication was studied from a resolvability prespective for
the discrete memoryless channel (DMC). Therein, a trade-off between the secret length and asymmetries between Bob and
Willie channels has been studied. Later in [8], the exact capacity (using relative entropy instead of total variation distance
as LPD measure) of DMC and AWGN have been characterized. For a detailed summary of the recent results for different
channel models on the relationship between secret key length, LPD security metric and achievable LPD rate, readers may
refer to Table II in [6]. LPD communication over MIMO fading channel was first studied in [9]. Under different assumption
of CSI availability, the author derived the average power that satisfies the LPD requirement. However, the authors did not
obtain the square-root law, since the focus was not on the achievable rates of reliable LPD communication. Recently in [10],
LPD communication with multiple antennas at both Alice and Bob is considered when Willie has only a single antenna over
Rayleigh fading channel. An approximation to the LPD constrained rate when Willie employs a radiometer detector and has
uncertainty about his noise variance was presented. However, a full characterization of the capacity of MIMO channel with
LPD constraint was not established.
More precisely, for a unit rank MIMO channel, we show that Kn (δ, ) scales as

1θ

is the angle between right singular vectors of main and adversary channels in the unit rank channel model.

3

Despite not explicitly stated, the assumption of keeping the codebook generated by Alice secret from Willie (or at least a
secret of sufficient length [1], [7]) is common in all aforementioned studies of covert communication. Without this assumption,
LPD condition cannot be met along with arbitrarily low probability of error at Bob. This is because, when Willie is informed
about the codebook, he can decode the message using the same decoding strategy as that of Bob [1]. Only in [6], square-root
law was obtained over binary symmetric channel without this assumption when Willie channel is significantly noisier than that
of Bob, i.e., when there is a positive secrecy rate over the underlying wiretap channel. Despite the availability of the codebook
at Willie, [6] uses the total variation distance as the LPD metric.
In short, the square root law is shown to be a fundamental upper limitation that cannot be overcome unless the attack model
is relaxed to cases such as the lack of CSI or the lack of the knowledge of when the session starts at Willie. Here, we do not
make such assumptions on Willie and solely take advantage of increasing spatial dimension via the use of MIMO.
II. S YSTEM M ODEL AND P ROBLEM S TATEMENT
In the rest of this paper we use boldface uppercase letters for vectors/matrices. Meanwhile, (.)∗ denotes conjugate of complex
number, (.)† denotes conjugate transpose, IN denotes identity matrix of size N , tr(.) denotes matrix trace operator, |A| denotes
the determinant of matrix A and 1m×n denotes a m × n matrix of all 1’s. We say A  B when the difference A − B is
positive semi-definite. The mutual information between two random variables x and y denoted by I(x; y) while lim denotes
the limit inferior. We use the standard order notation f (n) = O(g(n)) to denote an upper bound on f (n) that is asymptotically
tight, i.e., there exist a constant m and n0 > 0 such that 0 ≤ f (n) ≤ mg(n) for all n > n0 .
A. Communication Model
We consider the MIMO channel scenario in which a transmitter, Alice, with Na ≥ 1 antennas aims to communicate with a
receiver, Bob, having Nb ≥ 1 antennas without being detected by a passive adversary, Willie, equipped with Nw ≥ 1 antennas.
The discrete baseband equivalent channel for the signal y and z, received by Bob and Willie, respectively, can be written as:
y = Hb x + eb ,
z = Hw x + ew ,

(1)

where x ∈ CNa ×1 is the transmitted signal vector constrained by an average power constraint E[tr(xx† )] ≤ P . Also,
Hb ∈ CNb ×Na and Hw ∈ CNw ×Na are the channel coefficient matrices for Alice-Bob and Alice-Willie channels respectively.
Throughout this paper, unless otherwise noted, Hb and Hw are assumed deterministic, also, we assume that Hb is known to all
parties, meanwhile, Hw is known only to Willie. We define N , min{Na , Nb } and M , min{Na , Nw }. Finally, eb ∈ CNb ×1
and ew ∈ CNw ×1 are an independent zero mean circular symmetric complex Gaussian random vectors for both destination
and adversary channels respectively, where, eb ∼ CN (0, σb2 INb ) and ew ∼ CN (0, σe2 INw ).
We further assume Hw lies in the set of matrices with bounded spectral norms:
n
√ o
Sw = Hw : kHw kop ≤ γw
n
o
= Ww , H†w Hw : kWw kop ≤ γw ,
(2)
where kAkop is the operator (spectral) norm of A, i.e., the maximum eigenvalue of A. The set Sw incorporates all possible
Ww that is less than or equal to γw Î (in positive semi-definite sense) with no restriction on its eigenvectors, where Î is diagonal
matrix with the first M diagonal elements equal to 1 while the rest Na − M elements of the diagonal are zeros. Observe that,
kWw kop represents the largest possible power gain of Willie channel. Unless otherwise noted, throughout this paper we will
assume that Hw ∈ Sw .
B. Problem Statement
Our objective is to establish the fundamental limits of reliable transmission over Alice to Bob MIMO channel, constrained
by the low detection probability at Willie. Scalar AWGN channel channel have been studied in [8], we use a formulation that
follows closely the one used therein while taking into consideration the vector nature of the MIMO channel. Alice employs a
stochastic encoder with blocklength2 nNa , where n is the number of channel uses, for message set M consists of:
1) An encoder M 7→ CnNa , m 7→ xn where x ∈ CNa .
2) A decoder CnNb 7→ M, yn 7→ m̂ where y ∈ CNb .
Alice chooses a message M from M uniformly at random to transmit to Bob. Let us denote by H0 the null hypothesis under
which Alice is not transmitting and denote by P0 the probability distribution of Willie’s observation under the null hypothesis.
2 Note that, when Alice has nN bits to transmit, two alternative options are available for her. Either she splits the incoming stream into N streams of
a
a
n bits each and use each stream to select one from 2n messages for each single antenna, or, use the entire nNa bits to choose from 2nNa message. The
latter of these alternatives provides a gain factor of Na in the error exponent, of course, in the expense of much greater complexity [11], [12]. However, in
the restrictive LPD scenario, Alice would choose the latter alternative as to achieve the best decoding performance at Bob.

4

Conversely, let H1 be the true hypothesis under which Alice is transmitting her chosen message M and let P1 be the probability
distribution of Willie’s observation under the true hypothesis. Further, define type I error α to be the probability of mistakenly
accepting H1 and type II error β to be the probability of mistakenly accepting H0 . For the optimal hypothesis test generated
by Willie we have [13]
α + β = 1 − V(P0 , P1 ),

(3)

where V(P0 , P1 ) the total variation distance between P0 and P1 and is given by
1
kp0 (x) − p1 (x)k1 ,
(4)
2
where p0 (x) and p1 (x) are, respectively, the densities of P0 and P1 and k.k1 is the L1 norm. The variation distance between
P0 and P1 is related to the Kullback–Leibler Divergence (relative entropy) by the well known Pinsker’s inequality [14]:
r
1
V(P0 , P1 ) ≤
D(P0 k P1 )
(5)
2
where
V(P0 , P1 ) =

D(P0 k P1 ) = EP0 [log P0 − log P1 ] .

(6)

Note that, since the channel is memoryless, across n independent channel uses, we have
D (Pn0 k Pn1 ) = nD (P0 k P1 )

(7)

by the chain rule of relative entropy. Accordingly, for Alice to guarantee a low detection probability at Willie’s optimal detector,
she needs to bound V(Pn0 , Pn1 ) above by some δ chosen according to the desired probability of detection. Consequently, she
ensure that the sum of error probabilities at Willie is bounded as α + β ≥ 1 − δ. Using (5), Alice can achieve her goal by
designing her signaling strategy (based on the amount of information available) subject to
2δ 2
.
(8)
n
Throughout this paper, we adopt (8) as our LPD metric. Thus, the input distribution used by Alice to generate the codebook
has to satisfy (8). As in [8], our goal is to find the maximum value of log |M| for which a random codebook of length nNa
exists and satisfies (8) and whose average probability of error is at most . We denote this maximum by Kn (δ, ) and we
define
Kn (δ, )
.
(9)
L , lim lim √
↓0 n→∞
2nδ 2
√
Note that L has unit nats. We are interested in the characterization of L under different conditions of Bob and Willie
channels in order to derive scaling laws for the number of covert bits over MIMO AWGN channel. We first give the following
Proposition which provides a general expression for L by extending Theorem 1 in [8] to the MIMO AWGN channel with
infinite input and output alphabet.
D(P0 k P1 ) ≤

Proposition 1. For the considered MIMO AWGN channel,
r
L=

max

lim

{fn (x)}
n→∞
tr(En [xx† ])≤P

n
2δ 2

I(fn (x), fn (y))

Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0

(10)

where {fn (x)} is a sequence of input distributions over CNa and En [·] denotes the expectation with respect to fn (x).
Before we give the proof of Proposition 1, we would like to highlight why the second moment constraint on the input signal
is meaningful in our formulation. It was explicitly stated in [8] that, an average power constraint on the input signal is to be
superseded by the LPD constraint. The reason is that, the LPD constraint requires the average power to tend to zero as the
block length tends to ∞. However, unlike the single antenna setting, over a MIMO channel, there exist scenarios in which the
LPD constraint can be met without requiring the Alice to reduce her power. In the sequel, we will discuss such scenarios in
which the power constraint remains active.
Proof. First, using the encoder/decoder structure described above, we see that the converse part of Theorem 1 in [8] can be
directly applied here. Meanwhile, the achievability part there was derived based on the finiteness of input and output alphabet.
It was not generalized to the continuous alphabet input over scalar AWGN channel. Rather, the achievability over AWGN
channel was shown for Gaussian distributed input in Theorem 5. Here, we argue that, showing achievability for Gaussian
distributed input is sufficient and, hence, we give achievability proof in Appendix B that follow closely the proof of Theorem

5

5 in [8]. Unlike the non LPD constrained capacity which attains its maximum when the underlying input distribution is zero
mean complex Gaussian, it is not straightforward to infer what input distribution is optimal. However, using the Principle
Minimum Relative Entropy [2] and Information Projection [3], we verify that, the distribution P1 that minimizes D(P0 k P1 )
is the zero mean circularly symmetric complex Gaussian distribution.

Further, we provide a more convenient expression for L in the following Theorem which provides an extension of Corollary
1 in [8] to the MIMO AWGN channel.
Theorem 1. For the considered MIMO AWGN channel,
r
n
L = lim
n→∞
2δ 2

max
fn (x)
tr(En [xx† ])≤P

I(fn (x), fn (y))

Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0

(11)

where fn (x) is the input distribution over CNa and En [·] denotes the expectation with respect to fn (x).


Proof. The proof is given in Appendix C.

Now, since we now know that zero mean circular symmetric
Gaussian input distribution is optimal, the only
 complex

remaining task is to characterize the covariance matrix, Q = E xx† , of the optimal input distribution. Accordingly, (11) can
be rewritten as:
r
Wb Q
n
(12)
max log INa +
L = lim
n→∞
2δ 2 Q0
σb2
tr(Q)≤P

Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0,
where Wb , H†b Hb . Further, we can evaluate the relative entropy at Willie as follows (see Appendix A for detailed derivation):
1
Hw QH†w + INw
2
σw
(
−1 )
1
†
Hw QHw + INw
− Nw .
+ tr
2
σw

D (P0 k P1 ) = log

(13)

In this paper, we are mainly concerned with characterizing L when Alice knows only Hb . To that end, let us define:
Cpd (δ) , max log INa +
Q0
tr(Q)≤P

Wb Q
σb2

(14)

Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0.
r
Clearly, L = limn→∞

n
Cpd (δ). In what follows, we characterize Cpd (δ) and, hence, L under different models of Hb and
2δ 2

Hw .
Remark 1. Observe that, since Bob and Willie channels are different, Willies does not observe the same channel output as
Bob. Hence, there exist situations in which D(Pn0 k Pn1 ) does not increase without bound as n tends to infinity. In the next
Section, we provide some examples.
III. M OTIVATING E XAMPLES
Consider the scenario in which both of Willie’s and Bob’s channels are of unit rank. Accordingly, we can write H◦ = λ◦ v◦ u†◦ ,
where v◦ ∈ CN◦ and u◦ ∈ CNa are the left and right singular vectors of H◦ where the subscript ◦ ∈ {e, b} used to denote
Bob and Willie channels respectively.
Under the above settings, consider the scenario in which Alice has a prior (non-causal) knowledge about both channels.
Alice task is to find Q∗ that solve (14). Note that, since both channels are of unit rank, so is Q∗ and it can be written as
Q∗ = Pth q∗ q∗† where Pth ≤ P is the power threshold above which she will be detected by Willie. Now suppose that Alice
choose q∗ to be the solution of the following optimization problem:
max
< q † , ub >
q
kqk=1

Subject to < q † , uw >= 0,

(15)

6

whose solution is given by


I − uw u†w ub
i
.
q∗ = h
I − uw u†w ub

(16)

The beamforming direction q∗ is known as null steering (NS) beamforming [15], that is, transmission in the direction orthogonal
to Willie’s direction while maintaining the maximum possible gain in the direction of Bob. Recall that Willies channel is of unit
rank and is in the direction uw , thus, the choice of Q = Q∗ implies that Hw Q∗ H†w = 0. Accordingly, Σ1 = Σ0 , i.e, Willie
is kept completely ignorant by observing absolutely no power from Alice’s transmission. More precisely, D (Pn0 k Pn1 ) = 0.
However, this doesn’t mean that Alice can communicate at the full rate to Bob as if Willie was not observing, rather, the
LPD constraint forced Alice to sacrifice some of its power to keep Willie oblivious of their transmission. More precisely, the
effective power seen by Bob scales down with cosine the angle between ub and uw . In the special case when < ub , uw >= 0,
Alice communicate at the full rate to Bob without being detected by Willie. In addition, the codebook between Alice and Bob
need not to be kept secret from Willie. That is because the power observed at Willie from Alice transmission is, in fact, zero.

Fig. 1. Radiation pattern as a function of the number of transmitting antennas. When number of antennas gets large, < ub , uw >→ 0.

IV. Cpd (δ) WITH S ECRET C ODEBOOK
With uncertainty about Willie’s channel, Hw ∈ Sw , it is intuitive to think that Alice should design her signaling strategy
against the worst (stronger) possible Willie channel. We first derive the worst case Willie channel, then, we establish the saddle
point property of the considered class of channels in the form of min max = max min, where the maximum is taken over all
admissible input covariance matrices and the minimum is over all Hw ∈ Sw . Thus, we show that Cpd (δ) equals to the Cpd (δ)
evaluated at the worst possible Hw .
A. Worst Willie Channel and Saddle Point Property
w
To characterize Cpd (δ) when Hw ∈ Sw , we need first to establish the worst case Cpd (δ) denoted by Cpd
(δ). Suppose we
w
have obtained Cpd (δ) for every possible state of Hw , then, Cpd (δ) is the minimum Cpd (δ) over all possible state of Hw . First,
let us define
Wb Q
.
(17)
R(Ww , Q, δ) = log INa +
σb2
w
We give Cpd
(δ) in the following proposition.

Proposition 2. Consider the class of channels in (2), for any Q < 0 satisfies tr{Q} ≤ P and Ww ∈ Sw we have:
w
Cpd
(δ) = min

Ww ∈Sw

Subject

max R(Ww , Q, δ)

Q0
tr(Q)≤P
to: D(Pn0

k Pn1 ) − 2δ 2 ≤ 0

= max R(γw Î, Q, δ)
Q0
tr(Q)≤P

Subject to: D(Pn0 k Pn1 ) − 2δ 2 ≤ 0

(18)

7

i.e., the worst Willie channel is isotropic.


Proof. See Appendix D.

w
w
Proposition 2 establishes Cpd
(δ). The following proposition proves that Cpd (δ) = Cpd
(δ) by establishing the saddle point
property of the considered class of channels.

Proposition 3. (Saddle Point Property.) Consider the class of channels in (2), for any Q < 0 satisfies tr{Q} ≤ P and
Ww ∈ Sw we have:
Cpd (δ) = min

Ww ∈Sw

Subject
= max

max R(Ww , Q, δ)

Q0
tr(Q)≤P
to: D(Pn0

k Pn1 ) − 2δ 2 ≤ 0

min R(Ww , Q, δ)

Q0 Ww ∈Sw
tr(Q)≤P
Subject to: D(Pn0
w
= Cpd
(δ).

k Pn1 ) − 2δ 2 ≤ 0
(19)

Proof. By realizing that, for any feasible Q, the function D (P0 k P1 (Ww )) is monotonically increasing in Ww , we have that
min R(Ww , Q, δ) = R(γw Î, Q, δ)

Ww ∈Sw

Subject to: D(Pn0 k Pn1 ) − 2δ 2 .

(20)


Hence, the required result follows by using proposition 2.
B. Evaluation of Cpd (δ)

In light of the saddle point property established in the previous Section, in this Section we characterize Cpd (δ) by solving
(19) for the optimal signaling strategy, Q∗ . We give the main result of this Section in the following theorem.
Theorem 2. The eigenvalue decomposition of the capacity achieving input covariance matrix that solves (14) is given by
Q∗ = Ub ΛU†b where Ub ∈ CNa ×Na is the matrix whose columns are the right singular vectors of Hb and Λ is a diagonal
matrix whose diagonal entries, Λii , are given by the solution of
−1
λ =(σb2 λ−1
i (Wb ) + Λii )
 2
−2  2
−1 !
σw
σw
+η
+ Λii
+ Λii
−
γw
γw

where λ and η are constants determined from the constraints tr {Q} ≤ P and (8), respectively. Moreover,


N
X
Λii λi (Wb )
log 1 +
Cpd (δ) =
σb2
i=1

(21)

(22)

where λi is the ith non zero eigenvalue of Wb .


Proof. See Appendix E.

The result of Theorem 2 provides the full characterization of Cpd (δ) of the considered class of channels. It can be seen that,
the singular value decomposition (SVD) precoding [12] is the optimal signaling strategy except for the water filling strategy
in (21) which is chosen to satisfy both power and LPD constraints. Unlike both MIMO channel without security constraint
and MIMO wiretap channel, transmission with full power is, indeed, not optimal. Let
X
Pth ,
Λii ,
(23)
i

be the maximum total power that is transmitted by Alice. An equivalent visualization of our problem is that Alice need to
choose a certain power threshold, Pth , to satisfy the LPD constraint. However, again, Pth is distributed along the eigenmodes
using conventional water filling solution. Although it is not straightforward to obtain a closed form expression3 for Cpd (δ)
and, hence, L, we could obtain both upper and lower bounds on Cpd (δ) which leads to upper and lower bounds on L. Based
on the obtained bounds, we give the square-root law for MIMO AWGN channel in the following Theorem.
3 Using

Mathematica, Λii was found to be an expression of almost 30 lines which does not provide the required insights here.

8

Theorem 3 (Square-root Law of MIMO AWGN channel). For the considered class of channels, the following bounds on
Cpd (δ) holds
!
√ 2
N
X
2σw δλi (Wb )
√
log 1 +
≤ Cpd (δ)
σb2 γw nM
i=1
!
√ 2
N
X
2σw ξδλi (Wb )
√
≤
(24)
log 1 +
σb2 γw nM
i=1
where ξ ≥ 1 is a function of δ that approaches 1 as δ goes to 0. Moreover,
N
N
2
2
X
X
σw
λi (Wb )
σw
ξλi (Wb )
√
√
≤
L
≤
.
2γ
2γ
σ
M
σ
b w
b w M
i=1
i=1

Proof. We give both achievability and converse results in Appendix F.

(25)


Theorem 3 extends the square-root lawpfor scalar AWGN channel to the MIMO AWGN channel. In particular, it states that
Alice can transmit a maximum of O(N n/M ) bits reliably to Bob in n independent channel uses while keeping Willie’s
sum of error probabilities lower bounded by 1 − δ. The interesting result here is that, the gain in covert rate scales linearly
with number of active eigenmodes of Bob channel. Meanwhile, it scales down with the square-root of the number of active
eigenmodes over Willie channel. This fact will be of great importance when we study the case of massive MIMO limit. Further,
the bounds on L in (25) can, with small effort, generate the result of Theorem 5 in [8] by setting N = M = 1, λ(Wb ) = γw
and σb = σw .
It worth mentioning that, in some practical situations, compound MIMO channel can be too conservative for resource
allocation. In particular, the bounded spectral norm condition in (2) not only leads us to the worst case Willie channel, but it
also does not restrict its eigenvectors leaving the beamforming strategy used by Alice (SVD precoding) to be of insignificant
gain in protecting against Willie. Although we believe that the eigenvectors of Willie channel plays an important role in the
determination of the achievable covert rate, the ignorance of Alice about Willie channel leaves the compound framework as
our best option.
V. U NIT R ANK MIMO C HANNEL
As pointed out in the previous Section, the distinction between the eigenvectors of Bob and Willie channels would have a
considerable effect on the achievable covert rate. However, unavailability of Willie’s CSI left the compound framework as the
best model for Hw . In this Section, we consider the case when either Hw or Both Hw and Hb are of unit rank. This scenario
not only models the case when both Bob and Willie have a single antenna, but it also covers the case when they have a strong
line of sight with Alice. Moreover, this scenario allow us to evaluate the effect of the eigenvectors of Hw and Hb on the
achievable covert rate.
A. Unit Rank Willie Channel
In this Section we analyze the scenario in which only Willie channel is of unit rank. In this case, we can write Hw =
1/2
λw vw u†w , where vw ∈ CNw and uw ∈ CNa are the left and right singular vectors of Hw . Accordingly, Ww = λw uw u†w
and the product Ww Q has only one non zero eigenvalue. The nonzero eigenvalue λ(Ww Q) is loosely upper bounded by
λw λmax (Q). Accordingly, following the same steps of the proof of Theorem 3 we can get (assuming well conditioned Bob
channel, i.e. λi (Wb ) = λb for all i)
!
√ 2
2σw δλb
N log 1 + 2 √
≤ Cpd (δ)
σb λw n
!
√ 2
2σw ξδλb
≤ N log 1 + 2 √
(26)
σb λw n
consequently,
N

2
σw
λb
σ 2 ξλb
≤ L ≤ N w2
.
2
σb λw
σb λw

(27)

√
Again, Bob gets O(N n) bits in n independent channel uses. We note also that, the achievable covert rate increases linearly
with N .

9

B. Both Channels are of Unit Rank
Consider the case when both Hb and Hw are of unit rank. In this case, we have N = 1. However, setting N = 1 in the
results established so far will yield a loose bounds on the achievable LPD constrained rate. The reason is that, the bound
λ(We Q) ≤ λw λmax (Q) is, in fact, too loose especially for large values of Na . Although it is hard to establish a tighter upper
bound on λ(We Q) when Q is of high rank, it is straightforward to obtain the exact expression for λ(We Q) when Q is of
1/2
unit rank (which is the case when rank{Hb } = 1). Given that Hb = λb vb u†b , Alice will set Q = Pth ub u†b . Accordingly, we
have
2

λ(We Q) = λw Pth |< ub , uw >|
= λw Pth cos2 (θ),

(28)

where θ is the angle between ub and uw . We give the main result of this Section in the following theorem.
Theorem 4. If rank{Hb } = rank{He } = 1, then,
(

√

! )
2
2σw
δλb
√
min log 1 + 2
, C ≤ Cpd (δ)
σb λw cos2 (θ) n
(
! )
√ 2
2σw ξδλb
√
≤ min log 1 + 2
,C
σb λw cos2 (θ) n

where C is the non LPD constrained capacity of Alice to Bob channel. Accordingly,

if θ = π/2
L = ∞,
2
2
.
σw
λb
σw
ξλb
 2
≤L≤ 2
, otherwise
2
2
σb λw cos (θ)
σb λw cos (θ)

(29)

(30)

Proof. Follows directly by substituting (28) into (13) and following the same steps as in the proof of Theorem 3 while realizing
that Pth ≤ P .

√
2
Theorem 4 proves that Alice can transmit a maximum of O( n/ cos (θ)) bits reliably to Bob in n independent channel
uses while keeping Willie’s sum of error probabilities lower bounded by 1 − δ. In the statement of Theorem 4, the minimum
is taken since the first term diverges as θ → π/2, i.e., when ub and uw are orthogonal. In such case, we will have L = ∞.
This fact proves that, over MIMO channel Alice can communicate at full rate to Bob without being detected by Willie. An
interesting question is, how rare is the case of having ub and uw to be orthogonal? When the angle of the vectors (i.e., the
antenna orientation) are chosen uniformly at random, as the number of antennas at Alice gets large, we will see in Section
VII that cos2 (θ) approaches 0 exponentially fast with the number of antennas at Alice.
Remark 2. It should not be inferred from the results in this Section that the unit rank Bob channel can offer covert rate better
than that of higher rank. In fact, we used a loose upper bound on the eigenvalue of Willie’s channel for higher rank case. That
is due to the technical difficulty in setting tight bounds to the power received by Willie when Bob channel has higher rank.
Also, we see that unit rank channel offers better covert rate than that shown under the compound settings for Willie channel.
That is because, unlike the scenario of this Section, compound settings does not restrict the eigenvectors of Willie’s channel.
VI. Cpd (δ) W ITHOUT S HARED S ECRET
So far, we have established fundamental limits of covert communication over MIMO AWGN channel under the assumption
that the codebook generated by Alice is kept secret from Willie (or at least a secret of sufficiet length). In this Section, we
study the LPD communication problem without this assumption. The assumption of keeping the codebook generated by Alice
secret from Willie (or at least a secret of sufficient length [1], [7]) is common in all studies of covert communication. Without
this assumption, LPD condition cannot be met along with arbitrarily low probability of error at Bob, since, when Willie is
informed about the codebook, he can decode the message using the same decoding strategy as that of Bob [1]. Also we note
that, acheiving covertness does not require positive secrecy rate over the underlying wiretap channel. Indeed, recalling the
expression for relative entropy at Willie
1
D (P0 k P1 ) = log 2 Hw QH†w + INw
σw
|
{z
}
Willie’s Channel Capacity
(
−1 )
1
†
+ tr
Hw QHw + INw
− Nw ,
2
σw
|
{z
}
≤0, Willie’s penalty due to codebook ignorance

(31)

10

we observe that, the first term in (31) is the channel capacity of Willie channel with the implicit assumption of the knowledge
of the codebook generated by Alice. In particular, the first term in (31) equals to I(x; (z, Hw )). Meanwhile, it can be easily
verified that the remaining difference term in (31) is always non positive. This term represents Willie’s penalty from his
ignorance of the codebook. Analogous result for the scalar AWGN channel can be found in [1] for which the same arguments
can be made. This in fact provides an interpretation to the scenario on which the secrecy capacity of the main channel may
be zero, meanwhile, Alice still can covertly communicate to Bob. Let us define
Kn (δ, )
√
.
n 2δ 2

(32)

n
Cpd (δ).
L̂ = lim √
n→∞
2δ 2

(33)

L̂ , lim lim

↓0 n→∞

Following Proposition 1 and theorem 1, we can show that

√
Observe that, in the definition of L̂, we used the normalization over n instead of n. Now suppose that Alice chooses Q
such that the first term in (31), which is the capacity of Willie’ channel, is upper bounded by 2δ 2 /n. Indeed this signaling
strategy satisfies the LPD metric (8) and, thus, achieves covertness. Moreover, we have limn→∞ I(x; (z, Hw )) = 0, thus,
strong secrecy condition is also met. In particular, if I(x; (z, Hw )) ≤ 2δ 2 /n, Willie can reliably decode at most 2δ 2 nats
of Alice’s message in n independent channel uses. It worth mentioning that, requiring limn→∞ I(x; (z, Hw )) = 0 is more
restrictive than the strong secrecy condition. In principle, if Alice has a message m to transmit, strong secrecy condition
requires limn→∞ I(m; (z, Hw )) = 0. Meanwhile, since m = f −1 (x) for some encoding function f : m 7→ x, we have
I(m; (z, Hw )) ≤ I(x; (z, Hw )). Now, Cpd (δ) without any shared secret can be reformulated as follows:
Cpd (δ) = max log INa +
Q0
tr(Q)≤P

Subject to: log INa +

Wb Q
σb2

(34)

Ww Q
− 2δ 2 /n ≤ 0.
2
σw

(35)

In light of the saddle point property established in Section IV, we characterize Cpd (δ) without shared secret by solving (34)
for the optimal signaling strategy, Q∗ . We give the main result of this Section in the following theorem.
Theorem 5. The eigenvalue decomposition of the capacity achieving input covariance matrix that solves (34) is given by
Q∗ = ub Λu†b where ub ∈ CNa ×Na is the matrix whose columns are the right singular vectors of Hb and Λ is a diagonal
matrix whose diagonal entries, Λii , are given by the solution of
 2
−1
σw
2 −1
−1
λ =(σb λi (Wb ) + Λii ) − η
+ Λii
(36)
γw
where λ and η are constants determined from the constraints tr {Q} ≤ P and (8), respectively. Moreover,


N
X
Λii λi (Wb )
log 1 +
Cpd (δ) =
σb2
i=1

(37)

where λi is the ith non zero eigenvalue of Wb .
Proof. See Appendix G.



Theorem 5 provides the full characterization of the Cpd (δ) of the considered class of channels without requiring any shared
secret between Alice and Bob. Again, it is not straightforward to obtain a closed form expression for Cpd (δ). Thus, we obtain
both upper and lower bounds on Cpd (δ) as we did in Section IV. Based on the obtained bounds, we give the square-root law
for MIMO AWGN channel without shared secret in the following theorem.
Theorem 6. For the considered class of channels without any shared secret between Alice and Bob, the following bounds on
Cpd (δ) holds


N
2 2
X
2σw
δ λi (Wb )
log 1 +
≤ Cpd (δ)
σb2 γw nM
i=1


N
X
2σ 2 ξδ 2 λi (Wb )
≤
log 1 + w 2
(38)
σb γw nM
i=1

11

where ξ =

nM
. Accordingly,
nM − 2δ 2
N
X
i=1

√

N

2
X
2σw
δλi (Wb )
≤ L̂ ≤
2
σb γw M
i=1

√

2
2σw
ξδλi (Wb )
2
σb γw M

Proof. We give both achievability and converse results in Appendix H.

(39)


Theorem 6 extends the result of Theorem 3 to the scenario when Alice and Bob do not share any form of secret. It proves
that Alice can transmit a maximum of O(N/M ) bits reliably to Bob in n independent channel uses while keeping Willie’s
sum of error probabilities lower bounded by 1 − δ.
Now let us consider the case when both Hb and Hw are of unit rank. Under this assumption, we give converse and
achievability results of Cpd (δ) over Alice to Bob channel without a shared secret in the following theorem.
Theorem 7. If rank{Hb } = rank{He } = 1, then, δ-PD constrained capacity over Alice to Bob channel without a shared
secret between Alice and Bob is bounded as
 
 
2σ 2 δ 2 λb
, C ≤ Cpd (δ)
min log 1 + 2 w 2
σ λw cos (θ)n
 
 b 
2
2σw
ξδ 2 λb
,C
(40)
≤ min log 1 + 2
σb λw cos2 (θ)n
where ξ is as defined in Theorem 6 and C is the non LPD constrained capacity of Alice to Bob channel. Accordingly,


= ∞,
if θ = π/2
L̂ √
√ 2
2
2σw
δλb
2σw ξδλb

≤ L̂ ≤ 2
, otherwise
 2
σb λw cos2 (θ)
σb λw cos2 (θ)

(41)

Proof. Follows directly by substituting (28) into (35) and following the same steps as in the proof of Theorem 6 while realizing
that Pth ≤ P .

Again, in (40), the minimum is taken since the first term diverges as θ → π/2, i.e., when ub and uw are orthogonal. The
theorem proves that Alice can transmit a maximum of O(1/ cos2 (θ)) bits reliably to Bob in n independent channel uses while
keeping Willie’s sum of error probabilities lower bounded by 1 − δ. This fact proves that, over MIMO channel Alice can
communicate at full rate to Bob without being detected by Willie without requiring Alice and Bob to have any form of shared
secret.
VII. C OVERT C OMMUNICATION WITH M ASSIVE MIMO
In Theorems 4 and 7, it was shown that Alice can communicate at full rate with Bob without being detected by Willie
whenever cos(θ) = 0 regardless of the presence of a shared secret. In this Section, we study the behavior of covert rate as
the number of antennas scale, which we call the massive MIMO limit, with and without codebook availability at Willie. In
particular, the high beamforming capability of the massive MIMO system can provide substantial gain in the achievable LPD
rate. However, a quantitative relation between the achievable LPD rate and the number of transmitting antennas seems to be
unavailable. To that end, we address the question: how does the achievable LPD rate scale with the number of transmitting
antennas? We also study how does the presence of a shared secret between Alice and Bob affects the scaling of the covert rate
in the massive MIMO limit. Before we answer these questions, we state some necessary basic results on the inner product of
unit vectors in higher dimensions [16].
A. Basic Foundation
In this Section, we reproduce some established results on the inner product of unit vectors in higher dimensions.
Lemma 1. [Proposition 1 in [16]] Let a and b any two vectors in the unit sphere in Cp chosen uniformly at random. Let
θ = cos−1 (< a, b >) be the angle between them. Then


π
√
Pr θ −
≤ ζ ≥ 1 − K p(cos ζ)p−2
(42)
2
 π
for all p ≥ 2 and ζ ∈ 0,
where K is a universal constant.
2
The statement of Lemma 1 states that, the probability that any two vectors chosen uniformly at random being orthogonal
increases exponentially fast with the dimension p. Indeed, note that, for any 0 < a < 1, ap has the same decay rate as (2−a)−p .
√
Thus, the probability that θ is within ζ from π/2 scales like (2 − cos(ζ))p−2 /K p.

12

Corollary 1. Let a and b any two vectors in the unit sphere in Cp chosen uniformly at random and let θ = cos−1 (< a, b >)
be the angle between them. Let A, B ∈ Cp×p be two matrices of unit rank generated as A = λa aa† and B = λb bb† . Then,
the probability that the eigenvalue of of the product λ(AB) approaches 0 grows to 1 exponentially fast with the dimension p.
Proof. It can be easily verified that λ(AB) = λa λb cos2 (θ). Using Lemma 1, we see that, the probability that θ approaches
π/2 increases exponentially with p. Hence, the probability that cos(θ) approaches 0 increases in the same order. Then so is
cos2 (θ).

B. Massive MIMO Limit With Shared Secret
In the previous Section it was demonstrated that, in higher dimensions every two independent vectors chosen uniformly
at random are orthogonal with very high probability. More generally, using spherical invariance [16], given ub , for any uw
chosen uniformly at random in CNa , the result of Lemma 1 still holds. This scenario typically models the scenario when Alice
knows her channel to Bob, meanwhile, she models uw as a uniform random unit vector.
Recall that, when Alice has nNa bits to transmit, two alternative options are available for her. Either she splits the incoming
stream into Na streams of n bits each and use each stream to select one from 2n messages for each single antenna, or, use
the entire nNa bits to choose from 2nNa message. The latter of these alternatives provides a gain factor of Na in the error
exponent, of course, in the expense of much greater complexity [11], [12]. However, in the restrictive LPD scenario, Alice
would choose the latter alternative as to achieve the best decoding performance at Bob. Therefore, we need to analyze the
LPD rate in the limiting case of the product nNa when both n and Na grow. Of course, the number of antennas at Alice is
a physical resource which can not be compared to n that can approach ∞ very fast. The more interesting question is, how
fast cos2 (θ) approaches 0 as Na increase. As illustrated in corollary 1, we know that cos(θ) approaches 0 exponentially fast
with Na . Consequently, we conclude that cos2 (θ) , also, approaches 0 exponentially fast with Na . For proper handling of the
scaling of Kn (δ, ) in massive MIMO limit, let us define
S , lim

lim

↓0 nNa →∞

Kn (δ, )
√
.
Na 2nδ 2

(43)

Note that the in the definition of S, both n and Na are allowed to grow without bound compared to L in which only n was
allowed to grow while Na was treated as constant. Now observe that, following Proposition 1 and Theorem 1, we can show
that
r
n
Cpd (δ).
(44)
S = lim Na
nNa →∞
2δ 2
We give the result of the massive MIMO limit with a pre-shared secret between Alice and Bob in the following Theorem.
Theorem 8. Assume that rank{Hb } = rank{He } = 1. Given ub , for any uw chosen uniformly at random, Cpd (δ) is as
given in Theorem 4 and
S = ∞.

(45)
√

!

2
δ
n
c (Na −2)/2
2σw
√
)
where
K
is
a
universal
constant
and
c
=
.
(1
+
K 2 Na
λw P
n
r
n
Proof. Combining the result of Theorem 4 and Corollary 1, multiplying (29) by Na
and taking the limit as both of n
2δ 2
and Na tend to infinity we obtain
r
n
lim lim Na
Cpd (δ)
Na →∞ n→∞
2δ 2
σ 2 λb
= lim Na 2 w 2
Na →∞
σb λw cos (θ)
= ∞,
(46)

r

Moreover, Kn grows like

where the last equality follow since cos2 (θ) → 0 as Na → ∞. On the other hand, we also can verify that
r
n
lim lim Na
Cpd (δ) = ∞.
(47)
n→∞ Na →∞
2δ 2
√
To show how Kn scales in this massive MIMO limit, we first note that, for fixed Na , Kn scales like n. Also note that,
S = ∞ implies that LPD constraint becomes inactive and full non-LPD capacity is achieved. This happens when the quantity

13

Cpd (δ) = C. The question we adress now is, how does Cpd (δ) behave in between these two extreme regimes. Following the
same steps of the proof of Theorem 3, we can obtain the following bound on Pth :
(
)
√ 2
2σw δ
Pth ≤ min √
,P .
(48)
nλw cos2 (θ)
Thus, we have Pth = P , and hence Cpd (δ) = C, when
√

P ≤√

2
2σw
δ
nλw cos2 (θ)

(49)

equivalently,
√ 2
2σ δ
cos (θ) ≤ √ w
nλw P
2


√ 2
2σ δ
√ w .
nλw P

(50)

√ 2 !(Na −2)/2
2σ δ
1− √ w
nλw P

(51)

s
⇒ |θ − π/2| ≤ π/2 − cos−1 
This happens with probability no less than:
p
P r(Cpd (δ) = C) ≥ 1 − K Na

where (51) follows by setting ζ in Lemma 1 equal
√ to the RHS of (50) and using the following basic trigonometry facts:
2 . It can be seen that, the probability that C (δ) = C scales as
cos(π/2 − x) = sin(x) and sin(cos−1 (x)) = 1 − x!
pd
√ 2
√
2σw δ
(Na −2)/2
.

(1 + g)
/K Na up to 1, where g = √
nλw P
Theorem 8 states that Alice can communicate at full rate to Bob while satisfying the LPD constraint (8). Note that, the limit
in both orders yields S = ∞.
As Na → ∞, the radiation pattern of a wireless MIMO transmitter becomes so extremely directive (pencil beam). We call this
limit the wired limit of wireless MIMO communication. In the wired limit, Willie cannot detect Alice’s transmission unless he
wiretaps this virtual wire. Theorem 8 provides a rigorous characterization of the wired limit of wireless MIMO communication.
In principle, it answers the fundamental question: How fast does the LPD constrained rate increase with the√number of antennas
at Alice? It can be seen that the probability that Alice fully utilizes the channel scales like 2(Na −2)/2 /K Na n up to 1 using
the same justification given after Lemma 1.
C. Massive MIMO Limit Without Shared Secret
In Section VI it was shown that, only diminishing covert rate, O(N/M ), can be achieved without requiring a shared secret
between Alice and Bob. Again, we note that this diminishing rate was shown to be achievable when Willie’s channel is isotropic.
Also, we have shown shown that, in the massive MIMO limit, the achievable covert rate grows exponentially with the number
of transmitting antennas when there is a shared secret between Alice and Bob. Thus, it is also instructive to consider LPD
communication problem without a shared secret in the massive MIMO limit. As illustrated in Section III, if Alice has CSI of
both channels, not only can she communicate covertly and reliably at full rate whenever the eigen directions of both channels
are orthogonal, but also she does not need a shared secret to achieve this rate. Building on our analysis in Section VII, we
give the massive MIMO limit of the δ-PD capacity when there is no shared secret between Alice and Bob.
Now, let us consider the scenario in which uw is chosen uniformly at random and fixed once chosen. For proper handling
of the scaling of Kn (δ, ) in massive MIMO limit without a shred secret, let us define
Kn (δ, )
√
.
(52)
↓0 nNa →∞ nNa 2δ 2
√
Observe that, unlike S, Kn (δ, ) is normalized to n instead of n in the expression of Ŝ. Now, following Proposition 1 and
Theorem 1, we can show that
nNa
Ŝ = lim √
Cpd (δ).
(53)
nNa →∞
2δ 2
We give the result of this scenario in the following Theorem.
Ŝ , lim

lim

14

Theorem 9. Assume that rank{Hb } = rank{He } = 1 and suppose that there is no shared secret between Alice and Bob.
Given ub , for any uw chosen uniformly at random, Cpd (δ) is as given in Theorem 4 and
Ŝ = ∞.
r
Moreover, Kn grows like

1
c
(1 + )(Na −2)/2 where K is a universal constant and c =
K 2 Na
n

Proof. The proof follows exactly the same steps as in the proof of Theorem 8.

(54)
√

!

2
2σw
δ
.
λw P



Again, it can be seen that Alice achieve the maximum achievable non-LPD rate even under the LPD constraint.However,
the rate at which Cpd (δ) converges to C is much slower, compared to the case with shared secret codebook. Hence, it can be
deduced from Theorems 7 and 9 that, in the limit of large Na , Alice can transmit O(n) bits in n independent channel uses
while satisfying the LPD constraint without the need for any form of shared secret. Even though, it has to be considered that
the number of antennas required at Alice under this scenario is much larger than that when she shares a secret of sufficient
length with Bob. The following numerical example demonstrates the covert rates in massive MIMO limit with and without a
shared secret between Alice and Bob.
Example
1. Assume that Alice intend to use the channel for n = 109 times over a channel of bandwidth of 10M Hz, hence,
√
2
n = 3.1623 × 104 . Suppose that Alice is targeting δ = 10−2 . Let σw
= σb2 = 10−2 and λw = λb = 10−3 . Assume that Alice
is targeting SN R = 15dB
at
Bob,
hence,
P
=
316.228.
Then,
for
N
a = 100 it can be verified that, Alice can transmit O(n)
√
covert bits instead of O( n). Observe that, Alice needed only Na = 100 to communicate covertly at near full rate to Bob.
Also note that, at 6GHz, two dimensional array of 100 elements can fit within an area of a single sheet of paper. See Fig.
(2) for the relation between the δ-PD capacity and number of transmitting antenna for different values of number of antennas
at Willie with δ = 10−2 .

Fig. 2. The relation between achievable covert rate with and without a shared secret (plotted in log scale), in bits per second, and Na for different values of
Nw with target δ = 10−2 . It shows that Alice can communicate near full rate with Na around 100 when she share a secret with Bob. A large gap can be
observed when there is no shared secret.

As can be seen from Fig. (2), Alice could achieve a covert rate very close to the non-LPD constrained capacity of her
channel to Bob with Na ≥ 100 with a block of length n = 109 . Also we see that there is a significant gap (nearly 4 orders
of magnitude) between the achievable covert rate with and without a preshared secret. Although both rates converges to C as
Na → ∞, we see that without a shared secret, the number of antennas required to achieve near full rate is significantly greater
than that required when Alice and Bob are sharing a secret of sufficient length. For practical consideration, this result leaves
the massive MIMO limit of the δ-PD capacity without a shared secret of theoretical interest only.
VIII. D ISCUSSION
Impact of CSI level. Throughout this paper we have considered perfect CSI at Alice about the her channel to Bob. In
practical scenarios, this assumption might not hold true as CSI always suffer from imperfection due to e.g. channel estimation
error or non error free CSI feedback link. Despite these potential impairments, we argue that the case of imperfect CSI at Alice
does not affect the obtained results. That is because we have made the assumption that Willie has perfect CSI as well about
his channel. Further, in case when Alice has absolutely no CSI is of special interest as communication under these critical
conditions may not allow Bob to share his √
CSI to Alice, specially when dealing with passive Bob. In this scenario, we can
verify that the Alice can transmit O(N/M n) bits in n independent channel uses (details are omitted here). We see that,

15

√
the covert rate scales with N/M compared to N/ M when Alice has CSI. More interestingly, in massive MIMO limit with
absolutely no CSI at Alice, we can verify that the covert rate → 0 even with Na → ∞. While it was recognized as the most
favorable scenario for Alice when she has CSI, massive MIMO may make matters worse when she has absolutely no CSI.
Contrary, when Alice has CSI about both channels, covert rates up to the non LPD constrained rate of the channel may be
achieved (under certain conditions, see Section III) without the need for a shared secret. As it is the case for MIMO channel
with no secrecy constraint, CSI availability play an important role in achieving higher covert rates.
Impact of Willie’s ignorance. All results obtained in this paper assumes that Willie has perfect CSI about his channel
and he is aware of his channel noise statistics. Ignorance of Willie about one of these parameters is expected to have positive
impact on the achievable covert rate. For example, in [6], it was shown that O(n) bits can be transmitted reliably with low
probability of detection over BSC whose error probability is unknown to Willie except that it is drawn from a known interval.
This scenario is subject for future research.
Length of the Shared Secret. In our analysis, we have considered scenarios in which the entire codebook is either available
or unavailable at Willie. Meanwhile, secrets of shorter length were reported to be enough for fulfilling
√ LPD requirements. For
scalar AWGN channel, it was shown that the required secret secret length is in order of O(log n n) [1]. Similar result was
established for DMC in [8]. In this work, while we showed that there is a significant gap between the achievable covert rate
with and without a shared secret, the minimum length of the required shared secret has not been addressed.
IX. S UMMARY AND C ONCLUSIONS
We have established the limits of LPD communication over the MIMO AWGN channel. In particular, using relative entropy
as our LPD metric, we studied the maximum codebook size, Kn (δ, ), for which Alice can guarantee reliability and LPD
conditions are met. We first showed that, the optimal codebook generating input distribution under δ-PD constraint is the
zero-mean Gaussian distribution. We based our argumentspon the the principle of minimum relative entropy. For an isotropic
Willie channel, we showed that Alice can transmit O(N n/M ) bits reliably in n independent channel uses, where N and
M are the number of active eigenmodes of Bob and Willie channels, respectively. Further, we evaluated the scaling rates of
Kn (δ, ) in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive
MIMO). We showed that, while the square-root law still holds for the MIMO-AWGN channel, the number of bits that can
be transmitted covertly scales exponentiallyrwith the number of transmitting antennas. More precisely, for a unit rank MIMO
n
c
channel, we show that Kn (δ, ) scales as
(1 + √ )(Na −2)/2 where Na is the number of transmitting antennas, K
K 2 Na
n
is a universal constant and c is constant independent on n and Na . Also, we derived the scaling of Kn (δ, ) with no shared
secret between Alice and Bob. In particular, we showed that achieving better covert rate is a resource arm race between Alice,
Bob and Willie: Alice can transmit O(N/M ) bits reliably in n independent channel uses, i.e., the covert rate is in the order
of the ratio between active eigenmodes of both channels. Despite this diminishing rate, in the massive MIMO limit, Alice
can still achieve higher covert rate up to the non LPD constrained capacity of her channel to Bob, yet, with a significantly
greater number of antennas. Although the covert rates both with and without a shared secret are shown to converge the non
LPD constrained capacity as Na → ∞, numerical evaluations showed that without a shared secret, the number of antennas
required to achieve near full rate can be orders of magnitude greater. The practical implication of our result is that, MIMO
has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably
low delay.
R EFERENCES
[1] B. A. Bash, D. Goeckel, and D. Towsley, “Limits of reliable communication with low probability of detection on awgn channels,” IEEE Journal on
Selected Areas in Communications, vol. 31, no. 9, pp. 1921–1930, 2013.
[2] A. D. Woodbury, “Minimum relative entropy, bayes and kapur,” Geophysical Journal International, vol. 185, no. 1, pp. 181–189, 2011.
[3] I. Csiszár and F. Matus, “Information projections revisited,” IEEE Transactions on Information Theory, vol. 49, no. 6, pp. 1474–1490, 2003.
[4] R. F. Schaefer and S. Loyka, “The secrecy capacity of compound gaussian mimo wiretap channels,” IEEE Transactions on Information Theory, vol. 61,
no. 10, pp. 5535–5552, 2015.
[5] A. Abdelaziz, A. Elbayoumy, C. E. Koksal, and H. El Gamal, “On the compound MIMO wiretap channel with mean feedback,” in 2017 IEEE International
Symposium on Information Theory (ISIT) (ISIT’2017), Aachen, Germany, Jun. 2017.
[6] P. H. Che, M. Bakshi, and S. Jaggi, “Reliable deniable communication: Hiding messages in noise,” in Information Theory Proceedings (ISIT), 2013
IEEE International Symposium on. IEEE, 2013, pp. 2945–2949.
[7] M. R. Bloch, “Covert communication over noisy channels: A resolvability perspective,” IEEE Transactions on Information Theory, vol. 62, no. 5, pp.
2334–2354, 2016.
[8] L. Wang, G. W. Wornell, and L. Zheng, “Fundamental limits of communication with low probability of detection,” IEEE Transactions on Information
Theory, vol. 62, no. 6, pp. 3493–3503, 2016.
[9] A. O. HERo, “Secure space-time communication,” IEEE Transactions on Information Theory, vol. 49, no. 12, pp. 3235–3249, 2003.
[10] S. Lee, R. J. Baxley, J. B. McMahon, and R. S. Frazier, “Achieving positive rate with undetectable communication over mimo rayleigh channels,” in
Sensor Array and Multichannel Signal Processing Workshop (SAM), 2014 IEEE 8th. IEEE, 2014, pp. 257–260.
[11] R. G. Gallager, Information theory and reliable communication. Springer, 1968, vol. 2.
[12] E. Telatar, “Capacity of multi-antenna gaussian channels,” European transactions on telecommunications, vol. 10, no. 6, pp. 585–595, 1999.
[13] E. L. Lehmann and J. P. Romano, Testing statistical hypotheses. Springer Science & Business Media, 2006.
[14] T. M. Cover and J. A. Thomas, Elements of information theory. John Wiley & Sons, 2012.

16

[15] B. Friedlander and B. Porat, “Performance analysis of a null-steering algorithm based on direction-of-arrival estimation,” IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 37, no. 4, pp. 461–466, 1989.
[16] T. T. Cai, J. Fan, and T. Jiang, “Distributions of angles in random packing on spheres.” Journal of Machine Learning Research, vol. 14, no. 1, pp.
1837–1864, 2013.
[17] D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge university press, 2005.

A PPENDIX A
K ULLBACK –L EIBLER D IVERGENCE AT W ILLIE
Assuming that Willie is informed about its own channel to Alice, Willie’s observation when Alice is silent is distributed as
P0 , meanwhile, it takes the distribution P1 whenever Alice is active where

−1
P0 = |πΣ0 | exp −z† Σ−1
0 z ,

−1
P1 = |πΣ1 | exp −z† Σ−1
(55)
1 z ,
 †
2
†
2
where Σ0 = σw INw and Σ1 = Hw QHw + σw INw . Here, Q = E xx is the covariance matrix of signal transmitted by Alice.
Note that, the choice of Q is highly dependent on the amount of CSI available at Alice. Thus, we evaluate the KL divergence
at Willie in general, then, in the following Sections we will discuss the effect of CSI availability at Alice on the performance
of Willie’s optimal detector. Assuming that Willie channel is known and fixed, the KL divergence between P0 and P1 is given
as follows:
D

=
=

=
=
=
=
=

EP0 [log P0 − log P1 ]

EP0 − log |πΣ0 | − z† Σ−1
0 z

† −1
+ log |πΣ1 | + z Σ1 z


|Σ1 |
† −1
+ EP0 z† Σ−1
log
1 z − z Σ0 z
|Σ0 |

 
−1
log Σ1 Σ−1
+ EP0 z† Σ−1
z
0
1 − Σ0
  −1
 † 
−1
−1
log Σ1 Σ0 + EP0 tr Σ1 − Σ0 zz

 −1
log Σ1 Σ−1
+ tr Σ−1
0
1 Σ0 − tr Σ0 Σ0

log Σ1 Σ−1
+ tr Σ−1
0
1 Σ0 − Nw

(56)

1
1
1
Hw QH†w + INw and that 2 Hw QH†w + INw = 2 Ww Q + INa . Define Ww , H†w Hw
2
σw
σw
σw
we note that, the non-zero eigenvalues of Hw QH†w and Ww Q are identical, hence, we ca write:
Now observe that, Σ1 Σ−1
0 =

D (P0 k P1 )

=

1
log 2 Hw QH†w + INw
σw
|
{z
}
Willie’s Channel Capacity
(
−1 )
1
†
Hw QHw + INw
+ tr
− Nw
2
σw
|
{z
}
≤0, Willie’s penalty due to codebook ignorance

Nw 
Y


λi (Ww Q)
= log
1+
2
σw
i=1
−1
Nw 
X
λi (Ww Q)
+
− Nw
1+
2
σw
i=1


Nw
X

λi (Ww Q)
=
log 1 +
2
σw
i=1

−1

λi (Ww Q)
+ 1+
−1
2
σw


Nw
X

λi (Ww Q)
=
log 1 +
2
σw
i=1

−1 !−1

λi (Ww Q)
− 1+
,
2
σw
where λi (Ww Q) is the ith eigenvalue of Ww Q.

(57)

17

A PPENDIX B
ACHIEVABILITY P ROOF OF P ROPOSITION 1
We show the achievability for Gaussian input. As discussed in [8], the sequence {Kn } is achievable provided that:
Kn
1
f ×n (zn |xn )
lim √ ≤ P − lim inf √ log
,
n→∞
n→∞
f ×n (zn )
n
n

(58)

where P − lim inf denotes the limit inferior in probability, namely, the largest number such that the probability that the random
variable in consideration is greater than this number tends to one as n tends to infinity. Meanwhile, f ×n (xn ) denotes the nth
extension of the probability density function of a random vector x. Thus, we need to show
1
f ×n (zn |xn ) √
√ log
− nI(fn (x), fn (z|x)) → 0
f ×n (zn )
n

(59)

in probability as n tends to infinity. First observe that,
f ×n (zn |xn ) =
f ×n (zn ) =

n
Y
i=1
n
Y

−1


exp −(zi − xi )† Σ−1
0 (zi − xi ) ,

−1



exp −z†i Σ−1
z
,
i
1

|πΣ0 |

|πΣ1 |

(60)

i=1



2
2
INw and let Q = E xx† be chosen such that (8) is satisfied. Note that, Q has to
INw , Σ1 = Hw QH†w + σw
where Σ0 = σw
be a decreasing function of n. Then,
f ×n (zn |xn )
n
= Σ1 Σ−
01 ×
×n
n
f (z )
!
n
n
X
X
†
†
−1
−1
exp
tr(Σ1 zi zi ) −
tr(Σ0 ei ei ) .
i=1

(61)

i=1

Accordingly,
1
f ×n (zn |xn ) √
√ log
= n log Σ1 Σ−
01 +
f ×n (zn )
n
!
n
n
X
X
1
†
†
−1
−1
√
tr(Σ1 zi zi ) −
tr(Σ0 ei ei ) ,
n i=1
i=1
whose expectation can be found as


√
1
f ×n (zn |xn )
E √ log
= n log Σ1 Σ−
01 +
f ×n (zn )
n
!
n
n
X
X
1
−1
−1
√
tr(Σ1 Σ1 ) −
tr(Σ0 Σ0 )
n i=1
i=1
√
= n log Σ1 Σ−
01
√
1
= n log INw + 2 Hw QH†w
σw
√
= nI(fn (x), fn (z|x)).
It then follows by Chebyshev’s inequality that, for any constant a > 0,


1
f ×n (zn |xn ) √
P r √ log
nI(f
(x),
f
(z|x))
−
≥
a
n
n
f ×n (zn )
n


f ×n (zn |xn )
1
1
≤ 2 var √ log
a
f ×n (zn )
n

(62)

(63)

(64)

and, it remains to show that

lim var

n→∞

1
f ×n (zn |xn )
√ log
f ×n (zn )
n


= 0.

(65)

18

Note that,

1
f ×n (zn |xn )
√ log
f ×n (zn )
n
!!
n
n
X
X
1
† −1
† −1
z Σ zi −
ei Σ0 ei
= var √
n i=1 i 1
i=1
n


1X
† −1
=
var z†i Σ−1
1 zi − ei Σ0 ei
n i=1


† −1
= var z†i Σ−1
1 zi − ei Σ0 ei

h
† −1
Σ
e
z
−
e
= E z†i Σ−1
i
i
1
i 0

† 
† −1
† −1
zi Σ1 zi − ei Σ0 ei
h
† −1
= E z†i Σ−1
1 z i z i Σ1 z i


var

† −1
− z†i Σ−1
1 zi ei Σ0 ei
† −1
− e†i Σ−1
0 ei zi Σ1 zi
i
† −1
−e†i Σ−1
e
e
Σ
e
i i 0
i
0

(66)
Now observe that, since Q → 0 as n tends to infinity, we can verify that each term in (66) tends to zero as n tends to infinity.

A PPENDIX C
P ROOF OF T HEOREM 1


We show that the limit in (11) always exists. Note that, for every n, fn (x) is zero mean Gaussian. Let Q∗n = En xx† be
Q∗n = arg max I(fn (x), fn (y))

(67)

Q0
tr(Q)≤P

where the maximum is subject to (8). Hence, we have
max I(fn (x), fn (y)) = log I +

Q0
tr(Q)≤P

Hb Q∗n H†b
.
σb2

(68)

Now, we have two cases to consider:
1) D(Pn0 k Pn1 ) = 0. In this case, δ can be made 0 causing the limit to be infinity.
2) D(Pn0 k Pn1 ) > 0. In this case, Q∗n has to be a decreasing function of n, otherwise, the constraint (8) can not be met. In
this case, the limit is ≥ 0 and < ∞.
In either cases, the limit exist and, hence, limit can be used in place of limit inferior and, also, the order of limit and maximum
can be interchanged.
A PPENDIX D
P ROOF OF P ROPOSITION 2
It is enough to show that
∗
Ww
= arg max D (P0 k P1 (Ww ))
Ww ∈Sw

= γw Î,

(69)

hence, we need to show that the function D (P0 k P1 (Ww )) is monotonically increasing in Ww , i.e., D1 , D (P0 k P1 (Ww1 )) ≥
D (P0 k P1 (Ww2 )) , D2 whenever Ww1 < Ww2 . Recalling the expression of D (P0 k P1 (Ww )) in (13), we note that the

19

function log |I + WQ| is monotonically increasing in W for any Q. Meanwhile, the second term in (13) is negative and
decreases monotonically in W. Even though, we have that
D1 − D 2

=

EP0 [log P0 − log P1 (Ww1 )

=

EP0 [log P1 (Ww2 ) − log P1 (Ww1 )]


P1 (Ww1 )
−EP0 log
P1 (Ww2 )

− log P0 + log P1 (Ww2 )]
(a)

≥
(b)

≥

0,

(70)

where (a) follows from Jensen’s inequality using convexity of log and (b)
with some standard matrix algebra
 follows because,

P1 (Ww1 )
as in Appendix A, we can show that, for Ww1 < Ww2 , we have EP0 log
≤ 1. 
P1 (Ww2 )
A PPENDIX E
P ROOF OF T HEOREM 2
∗
∗
We solve (14) for Ww = Ww
. Without loss of generality, assume that Nw ≥ Na , hence, Ww
= γw INw . For Nw < Na ,
∗
we can set the γw = 0 for the Na − Nw minimum eigenvalues of Q∗ . Plugging Ww
into (13), we get 4
(
!
−1 )
γ
γ
w
w
− Nw
(71)
INa + 2 Q
Dn =
n log INa + 2 Q + tr
σw
σw
Now observe that,
|I + Wb Q| ≤

Na
Y

(1 + λi (Wb )λi (Wb ))

(72)

i=1
∗
is
with equality if and only if Wb and Q have the same eigenvectors and note that this choice does not affect (71) since Ww
isotropic. Hence, the eigenvectors of Q∗ is the same as the eigenvectors of Wb which is the same as the right singular vectors
of Hb .
Now, we form the following Lagrange dual problem
1
L = log INa + 2 Wb Q + λ(tr(Q) − P ) − tr(MQ)
σb


2δ 2
+η D−
,
(73)
n
where λ, η ≥ 0 are the Lagrange multipliers that penalize violating the power and LPD constraints, respectively, and M  0
penalizes the violation of the constraint Q  0. Where the associated KKT conditions can be expressed as:

λ, η ≥ 0,

λ(tr(Q) − P ) = 0, MQ = 0,

Q  0, M  0, tr(Q) ≤ P,

(74)

where the equality constraints in (74) ar the complementary slackness conditions. Note that, (14) is not a concave problem in
general. Thus, KKT conditions are not sufficient for optimality. Yet, since the constraint set is compact and convex and the
objective function is continuous, KKT conditions are necessary for optimality. Hence, we proceed by finding the stationary
points of the gradient of the dual Lagrange problem in the direction of Q and obtain the stationary points that solve the KKT
conditions. By inspecting the objective function at these points, the global optimum can be identified. To identify the stationary
points of the Lagrangian (73), we get its gradient with respect to Q as follows:

−1
Wb
1
5Q L = − INa + 2 Wb Q
+ λINa − M
σb
σb2

−1 
−2 !
ηγw
γw
γw
+ 2
INa + 2 Q
− INa + 2 Q
σb
σw
σw

−1
1
Wb
+ λINa − M
= − INa + 2 Wb Q
σb
σb2
 2
−1  2
−2 !
σw
σw
+η
IN + Q
−
IN + Q
γw a
γw a
4 For

n
simplicity, we write Dn instead of D Pn
0 k P1



(75)

20

Assume, without loss of generality, that Q  0, then, from MQ = 0 it follows that M = 0. Now, from 5Q L = 0 we obtain

−1
λINa = σb2 Wb−1 + Q
−2  2
−1 !
 2
σw
σw
−
+η
IN + Q
IN + Q
(76)
γw a
γw a
Since we know that Q∗ and Wb have the same eigenvectors, hence, the eigenvalues of Q∗ , Λii , can be found from
−1
λ =(σb2 λ−1
i (Wb ) + Λii )
 2
−2  2
−1 !
σw
σw
+η
−
+ Λii
+ Λii
,
γw
γw

(77)

as required. 
A PPENDIX F
P ROOF OF T HEOREM 3
Achievability. Starting from (71), we obtain
(a)

D

≤

(b)

=

=

(c)

≤

=

(

−1 )
γw
γw
− Nw
Q + tr
tr
INa + 2 Q
2
σw
σw




Nw
X
 γw Λii

 σ2 +


1
− 1

γw Λii
w
i=1
1+
2
σw


γw Λii
N
w
X  γw Λii
2

σw


 σ2 −
γw Λii 
w
i=1
1+
2
σw


γw Pth
2

 γw Pth
N σw

Nw 
 N σ2 −
γw Pth 
w
1+
2
N σw


2 2
γw Pth
4
 N 2 σw

,
Nw 

γw Pth 
1+
2
N σw
(78)

where (a) follows from the inequality log |A| ≤ tr {A − I}, (b) is straightforward matrix algebra, (c) follows since the RHS
is maximized for Λii = Pth /N for all i. For Alice to ensure (8), we need the RHS of (78(e)) to be less than or equal 2δ 2 .
After some manipulation, if Alice sets
√
2
2N σw
δ
√
Pth ≤
,
(79)
γw nNw
we can verify that (8) is satisfied. Now let Alice set Pth for (79) to be met with equality. Given that choice of Pth , the LPD
constraint is met and the rest of the problem is that of choosing the input distribution to maximize the achievable rate. The
solution of the problem is then the SVD precoding with conventional water filling [12], [17] as follows:

+
(µ − σb2 λ−1
for 1 ≤ i ≤ N
i (Wb ))
Λii =
(80)
0
for N < i ≤ Na ,
where N = min{Na , Nb }, λi is the ith non zero eigenvalue of Wb and x+ = max{0, x}. Further, µ is a constant chosen to
satisfy the power constraint tr{Λ} = Pth . Accordingly, the following rate is achievable over Alice to Bob channel:


N
+
X
(µ − σb2 λ−1
i (Wb )) λi (Wb )
Rpd (δ) =
log 1 +
σb2
i=1

+
N 
X
µλi (Wb )
=
log
.
(81)
σb2
i=1

21

However, it is technically difficult to expand (81) to check the applicability of the square-root law. Therefore, we obtain an
achievable rate assuming that Alice splits Pth equally across active eigenmodes of her channel to Bob. Note that, this rate is
indeed achievable since it is less than or equal to (81). Also note that, when Alice to Bob channel is well conditioned, the
power allocation in (80) turns into equal power allocation. Then, the following rate is achievable:
!
√ 2
N
X
2σw δλi (Wb )
√
R(δ) =
log 1 +
.
(82)
σb2 γw nNw
i=1
Now suppose that Alice uses a code of rate R̂ ≤ R(δ), then, Bob can obtain
!
√ 2
N
X
2σw δλi (Wb )
√
nR̂ ≤ n
log 1 +
σb2 γw nNw
i=1
√ 2
N p
(a) X
2σw δλi (Wb )
≤
n/Nw
σb2 γw ln 2
i=1

(83)

bits in n independent channel uses, where (a) follows from ln(1 + x) ≤ x, and note that the inequality is met with equality
for sufficiently large n. Now, assume that λi (Wb ) = λb for all 1 ≤ i ≤ N , i.e., Bob’s channel is well conditioned. Then, Bob
can obtain
√ 2
p
2σ δλb
(84)
nR(δ) ≤ N n/Nw 2 w
σb γw ln 2
bits in n independent channel uses since the inequality is met with equality for sufficiently large n. 
Converse. To show the converse, we assume the most favorable scenario for Alice to Bob channel when Hb is well
conditioned. That is because, the rate Alice can achieve over a well conditioned channel to Bob sets an upper bound to that
can be achieved over any other channel of the same Frobenius norm [17]. Then, Alice will split her power equally across
active eigenmodes of her channel to Bob. Now, Let us choose ξ ≥ 1 such that,


γw
γw
log INa + 2 Q − Nw ≥ tr
Q − ξNw
(85)
2
σw
σw
and note that for small D, ξ is a function of δ that approaches 1 as δ → 0. Hence, combining (71) with (85) we obtain:
(

−1 )

(a)
γw
γw
Q + tr
INa + 2 Q
− ξNw
D ≥ tr
2
σw
σw


(b)

=

 γw Pth
Nw 
 N σ2 +
w


1
− ξ

γw Pth
1+
2
N σw
(86)

following the same steps as in the achievability proof, we can insure that
√
2
2N ξσw
δ
√
,
Pth ≤
γw nNw

(87)

otherwise, Alice can not ensure that (8) is satisfied. Now let Alice set Pth equals to the RHS of (87). Then, we can verify
that:
!
√
N
2
X
2ξσw
δλb
Cpd (δ) ≤
log 1 + 2 √
.
(88)
σb γw nNw
i=1
Now suppose that Alice uses a code of rate R(δ) ≤ Cpd (δ), then, Bob can obtain
!
√
2
2ξσw
δλb
nR(δ) ≤ nN log 1 + 2 √
σb γw nNw
√
p
2ξσ 2 δλb
≤ N n/Nw 2 w
σb γw ln 2
bits in n independent channel uses since the inequality is met with equality for sufficiently large n. 

(89)

22

A PPENDIX G
P ROOF OF T HEOREM 5
∗
∗
We solve (34) for Ww = Ww
. Without loss of generality, assume that Nw ≥ Na , hence, Ww
= γw INw . For Nw < Na ,
we can set the γw = 0 for the Na − Nw minimum eigenvalues of Q∗ . Now observe that,
|I + Wb Q| ≤

Na
Y

(1 + λi (Wb )λi (Wb ))

(90)

i=1
∗
with equality if and only if Wb and Q have the same eigenvectors and note that this choice does not affect (71) since Ww
is
isotropic. Hence, the eigenvectors of Q∗ is the same as the eigenvectors of Wb which is the same as the right singular vectors
of Hb .
Now, we form the following Lagrange dual problem
1
L = log INa + 2 Wb Q + λ(tr(Q) − P ) − tr(MQ)
σb


1
2
+ η log INa + 2 Ww Q − 2δ /n ,
(91)
σw
where λ, η ≥ 0 are the Lagrange multipliers that penalize violating the power and LPD constraints, respectively, and M  0
penalizes the violation of the constraint Q  0. Where the associated KKT conditions can be expressed as:

λ, η ≥ 0,

λ(tr(Q) − P ) = 0, MQ = 0,

Q  0, M  0, tr(Q) ≤ P,

(92)

where the equality constraints in (92) are the complementary slackness conditions. Note that, (34) is not a concave problem
in general. Thus, KKT conditions are not sufficient for optimality. Yet, since the constraint set is compact and convex and the
objective function is continuous, KKT conditions are necessary for optimality. Hence, we proceed by finding the stationary
points of the gradient of the dual Lagrange problem in the direction of Q and obtain the stationary points that solve the KKT
conditions. By inspecting the objective function at these points, the global optimum can be identified. To identify the stationary
points of the Lagrangian (91), we get its gradient with respect to Q as follows:

−1
1
Wb
5Q L = − IN a + 2 W b Q
+ λINa − M
σb
σb2
!

−1
ηγw
γw
+ 2
INa + 2 Q
σw
σw

−1
1
Wb
= − INa + 2 Wb Q
+ λINa − M
σb
σb2
−1
 2
σ
(93)
+ η w INa + Q
γw
Assume, without loss of generality, that Q  0, then, from MQ = 0 it follows that M = 0. Now, from 5Q L = 0 we obtain
−1
 2

−1
σ
λINa = σb2 Wb−1 + Q
− η w INa + Q
(94)
γw
Since we know that Q∗ and Wb have the same eigenvectors, hence, the eigenvalues of Q∗ , Λii , can be found from
 2
−1
σw
2 −1
−1
λ =(σb λi (Wb ) + Λii ) − η
+ Λii
,
(95)
γw
as required. 
A PPENDIX H
P ROOF OF T HEOREM 6
Achievability. We observe that
log INa

1
+ 2 Ww Q
σw



(a)

1
Ww Q
2
σw

≤

tr

=

M
X
γw Λii
( 2 )
σw
i=1

(b)

≤

Pth M γw Λii
2
N σw



(96)

23

where (a) follows from the inequality log |A| ≤ tr {A − I}, (b) follows since the RHS is maximized for Λii = Pth /N for
all i. For Alice to ensure that the constraint in (34) is satisfied, she needs the RHS of (96(b)) to be less than or equal 2δ 2 /n.
Thus, Alice needs
Pth ≤

2 2
2N σw
δ
.
γw nM

(97)

Now let Alice set Pth for (97) to be met with equality. Given that choice of Pth , the LPD constraint is met and the rest of
the problem is that of choosing the input distribution to maximize the achievable rate. The solution of the problem is then the
SVD precoding with conventional water filling [12], [17] as follows:

+
(µ − σb2 λ−1
for 1 ≤ i ≤ N
i (Wb ))
Λii =
(98)
0
for N < i ≤ Na ,
where λi is the ith non zero eigenvalue of Wb . Further, µ is a constant chosen to satisfy the power constraint tr{Λ} = Pth .
Accordingly, the following rate is achievable over Alice to Bob channel:


N
+
X
(µ − σb2 λ−1
i (Wb )) λi (Wb )
Rpd (δ) =
log 1 +
σb2
i=1

+
N 
X
µλi (Wb )
=
log
.
(99)
σb2
i=1
However, it is technically difficult to expand (99) to check the applicability of the square-root law. Therefore, we obtain an
achievable rate assuming that Alice splits Pth equally across active eigenmodes of her channel to Bob. Note that, this rate is
indeed achievable since it is less than or equal to (99). Also note that, when Alice to Bob channel is well conditioned, the
power allocation in (98) turns into equal power allocation. Then, the following rate is achievable:


N
2
X
δλi (Wb )
2σw
R(δ) =
log 1 +
.
(100)
σb2 γw nM
i=1
Now suppose that Alice uses a code of rate R̂ ≤ R(δ), then, Bob can obtain


N
X
2σ 2 δ 2 λi (Wb )
nR̂ ≤ n
log 1 + w2
σb γw nM
i=1
√
N
2 2
(a) X
2σw
δ λi (Wb )
≤
M σb2 γw ln 2
i=1

(101)

bits in n independent channel uses, where (a) follows from ln(1 + x) ≤ x, and note that the inequality is met with equality
for sufficiently large n. Now, assume that λi (Wb ) = λb for all 1 ≤ i ≤ N , i.e., Bob’s channel is well conditioned. Then, Bob
can obtain
√ 2 2
N 2σw
δ λb
nR(δ) ≤
(102)
2
M σb γw ln 2
bits in n independent channel uses since the inequality is met with equality for sufficiently large n.
Converse. To show converse, we assume the most favorable scenario for Alice to Bob channel when Hb is well conditioned.
Then, Alice will split her power equally across active eigenmodes of her channel to Bob. Now,
(

−1 )
(a)
1
1
log INa + 2 Ww Q ≥ tr INa − INa + 2 Ww Q
σw
σw


=

=

Na
X

1 −



1

γw Pth 
i=1
1+
2
N σw


M γw Pth
2 +γ P
N σw
w th

(103)


where (a) follows from the inequality log |A| ≥ tr I − A−1 . Following the same steps as in the achievability proof, we can
insure that
2 2
2ξN ξσw
δ
Pth ≤
,
(104)
γw nM

24

nM
where ξ =
> 1, otherwise, Alice can not meet the LPD constraint. Now let Alice set Pth equals to the RHS of
nM − 2δ 2
(104). Then, we can verify that:


N
X
2ξσ 2 δ 2 λb
Cpd (δ) ≤
log 1 + 2 w
.
(105)
σb γw nM
i=1
Now suppose that Alice uses a code of rate R(δ) ≤ Cpd (δ), then, Bob can obtain


2ξσ 2 δ 2 λb
nR(δ) ≤ nN log 1 + 2 w
σb γw nM
2 2
2N ξσw
δ λb
≤
M σb2 γw ln 2
bits in n independent channel uses since the inequality is met with equality for sufficiently large n. 

(106)

