1

Degrees of Freedom of the Broadcast Channel
with Hybrid CSI at Transmitter and Receivers

arXiv:1709.02884v1 [] 9 Sep 2017

Mohamed Fadel, Student Member, IEEE, and Aria Nosratinia, Fellow, IEEE

Abstract
In general the different links of a broadcast channel may experience different fading dynamics
and, potentially, unequal or hybrid channel state information (CSI) conditions. The faster the fading
and the shorter the fading block length, the more often the link needs to be trained and estimated at
the receiver, and the more likely that CSI is stale or unavailable at the transmitter. Disparity of link
fading dynamics in the presence of CSI limitations can be modeled by a multi-user broadcast channel
with both non-identical link fading block lengths as well as dissimilar link CSIR/CSIT conditions.
This paper investigates a MISO broadcast channel where some receivers experience longer coherence
intervals (static receivers) and have CSIR, while some other receivers experience shorter coherence
intervals (dynamic receivers) and do not enjoy free CSIR. We consider a variety of CSIT conditions
for the above mentioned model, including no CSIT, delayed CSIT, or hybrid CSIT. To investigate the
degrees of freedom region, we employ interference alignment and beamforming along with a product
superposition that allows simultaneous but non-contaminating transmission of pilots and data to different
receivers. Outer bounds employ the extremal entropy inequality as well as a bounding of the performance
of a discrete memoryless multiuser multilevel broadcast channel. For several cases, inner and outer
bounds are established that either partially meet, or the gap diminishes with increasing coherence times.

Index Terms
Broadcast channel, Channel state information, Coherence time, Coherence diversity, Degrees of
freedom, Fading channel, Multilevel broadcast channel, Product superposition.
The authors are with the Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 750830688 USA, E-mail: mohamed.fadel@utdallas.edu;aria@utdallas.edu. This work was presented in part at the IEEE International
Symposium on Information Theory (ISIT), Germany, June 2017 [1].

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I. I NTRODUCTION
The performance of a broadcast channel depends on both the channel dynamics as well as
the availability and the quality of channel state information (CSI) on the two ends of each
link [2]–[5]. The two issues of CSI and the channel dynamics are practically related. The faster
the fading, the more often the channel needs training, thus consuming more channel resources,
while a very slow fading link requires infrequent training, therefore slow fading models often
assume that CSIR is available due to the cost of training being small when amortized over time.
In practice, in a broadcast channel some links may fade faster or slower than others. Recently,
it has been shown [6], [7], that the degrees of freedom of the broadcast channel are affected
by the disparity of link fading speeds, but existing studies have focused on a few simple and
uniform CSI conditions, e.g., neither CSIT nor CSIR were available in [6], [7] for any user. This
paper studies a broadcast channel where the links experience both disparate fading conditions
as well as non-uniform or hybrid CSI conditions.
A review of the relevant literature is as follows. Under perfect instantaneous CSI, the degrees
of freedom of a broadcast channel increase with the minimum of the transmit antennas and
the total number of receivers antennas [8], [9]. However, due to the time-varying nature of the
channel and feedback impairments, perfect instantaneous transmit-side CSI (CSIT) may not be
available, and also receive-side CSI (CSIR) can be assumed for slow-fading channels only.
Broadcast channel with perfect CSIR has been investigated under a variety of CSIT conditions,
including imperfect, delayed, or no CSIT [2]–[4], [10]–[12]. In the absence of CSIT, Huang et
al. [2] and Vaze and Varanasi [3] showed that the degrees of freedom collapse to that of the
single-receiver, since the receivers are stochastically equivalent with respect to the transmitter.
For a MISO broadcast channel Lapidoth et al. [4] conjectured that as long as the precision of
CSIT is finite, the degrees of freedom collapse to unity. This conjecture was recently settled in
the positive by Davoodi and Jafar in [10]. Moreover, for a MISO broadcast channel under perfect
delayed CSIT Maddah-Ali and Tse in [11] showed using retrospective interference alignment
that the degrees of freedom are

1
1
1+ 12 +...+ K

> 1, where K is the number of the transmit antennas

and also the number of receivers. A scenario of mixed CSIT was investigated in [12], where the
transmitter has partial knowledge on the current channel in addition to delayed CSI.
The potential variation between the quality of feedback links has led to the model of hybrid

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3

CSIT, where the CSIT with respect to different links may not be identical [10], [13]–[15]. A
MISO broadcast channel with perfect CSIT for some receivers and delayed for the others was
studied by Tandon et al. in [13] and Amuru et al. in [14]. Davoodi and Jafar in [10] showed
that for a MISO two-receiver broadcast channel under perfect CSIT for one user and no CSIT
for the other, the degrees of freedom collapse to unity. Tandon et al. in [15] considered a MISO
broadcast channel with alternating hybrid CSIT to be perfect, delayed, or no CSIT with respect
to different receivers.
As mentioned earlier, investigation of broadcast channels under unequal link fading dynamics
is fairly recent. An achievable degrees of freedom region for one static and one dynamic receiver
was given in [16]–[19] via product superposition, producing a gain that is now known as
coherence diversity. Coherence diversity gain was further investigated in [6], [7] for a K-receiver
broadcast channel with neither CSIT nor CSIR. Also, a broadcast channel was investigated in
[20], where the receivers MIMO fading links experience nonidentical spacial correlation.
In this paper, we consider a multiuser model under a hybrid CSIR scenario where a group of
receivers, denoted static receivers, are assumed to have CSIR, and another group with shorter
link coherence time, denoted dynamic receivers, do not have free CSIR. We consider this model
under a variety of CSIT conditions, including no CSIT, delayed CSIT, and two hybrid CSIT
scenarios. In each of these conditions, we analyze the degrees of freedom region. A few new
tools are introduced, and inner and outer bounds are derived that partially meet in some cases.
The results of this paper are cataloged as follows.
In the absence of CSIT, an outer bound on the degrees of freedom region is produced via
bounding the rates of a discrete memoryless multilevel broadcast channel [21], [22] and then
applying the extremal entropy inequality [23], [24]. Our achievable degrees of freedom region
meets the outer bound in the limiting case where the coherence times of the static and dynamic
receivers are the same.
For delayed CSIT, we use the outdated CSI model that was used by Maddah-Ali and Tse [11]
under i.i.d. fading and assuming global CSIR at all nodes. Noting that our model does not have
uniform CSIR, we produced a technique with alignment over super-symbols to utilize outdated
CSIT but merge it together with product superposition to reuse the pilots of the dynamic receivers
for the purpose of transmission to static receivers. Moreover, we develop an outer bound that
is suitable for block-fading channels with different coherence times, by appropriately enhancing
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4

static

Tx
dynamic

Fig. 1.

A broadcast channel with multiple static and multiple dynamic users

the channel to a physically-degraded broadcast channel and then applying the extremal entropy
inequality [23], [24]. For one static and one dynamic receiver, our achievable degrees of freedom
partially meet our outer bound, and furthermore the gap decreases with the dynamic receiver
coherence time T .
Under hybrid CSIT, we analyze two conditions: First, we consider perfect CSIT for the static
receivers and no CSIT with respect to the dynamic receivers. The achievable degrees of freedom
in this case are obtained using product superposition with the dynamic receiver’s pilots reused
and beamforming for the static receivers to avoid interference. Second, we consider perfect CSIT
with respect to the static receivers and delayed CSIT with respect to the dynamic receivers. An
achievable transmission scheme is proposed via a combination of beamforming, interference
alignment, and product superposition methodologies. The outer bounds for the two hybrid-CSIT
cases were based on constructing an enhanced physically degraded channel and then applying the
extremal entropy inequality. For one static receiver with perfect CSIT and one dynamic receiver
with delayed CSIT, the gap between the achievable and the outer sum degrees of freedom is

1
.
T

II. S YSTEM M ODEL
Consider a broadcast channel with multiple single-antenna receivers and the transmitter is
equipped with Nt antennas. The expressions “receiver” and “user” are employed without distinction throughout the paper, indicating the receiving terminals in the broadcast channel. The
channels of the users are modeled as Rayleigh block fading where the channel coefficients
remain constant over each block and change independently across blocks [25], [26]. As shown
in Fig. 1, the users are partitioned into two sets based on channel availability and the length of the
coherence interval: one set of dynamic users and another set of static users. The former contains
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TABLE I
N OTATION

Static Users

Dynamic Users

m′

m

MISO channel gains

g1 , . . . , gm′

h1 , . . . , hm

received signals (continuous)

′
y1′ , . . . , ym
′

y1 , . . . , ym

DMC receive variables

′
Y1′ , . . . , Ym
′

Y1 , . . . , Ym

transmission rates

′
R1′ , . . . , Rm
′

R1 . . . , Rm

′
M1′ , . . . , Mm
′

M1 , . . . , Mm

d′1 , . . . , d′m′

d1 , . . . , dm

number of users

messages
degrees of freedom
coherence time

T

′

T

General Variables
X

transmit signal

ρ

signal-to-noise ratio

Ui , Vj , W

auxiliary random variables

H

set of all channel gains

Dx

vertex of degrees of freedom region

ei

canonical coordinate vector

m dynamic users having coherence time T and no free CSIR1 , and the latter contains m′ static
users having coherence time T ′ and perfect instantaneous CSIR. We consider the transmitter is
equipped with more antennas than the number of dynamic and static users, i.e., Nt ≥ m′ + m.
The received signals yj′ (t), yi (t) at the static user j, and the dynamic user i, respectively, at
time instant t are
yj′ (t) = gj† (t)x(t) + zj′ (t),

j = 1, . . . , m′ ,

yi (t) = h†i (t)x(t) + zi (t),

i = 1, . . . , m,

(1)

where x(t) ∈ CNt is the transmitted signal, zj′ (t), zi (t) denote the corresponding additive i.i.d.
Gaussian noise of the users, and gj (t) ∈ CNt , hi (t) ∈ CNt denote the channels of the static
user j and the dynamic user i whose coefficients stay the same over T ′ and T time instances,
1

This means that the cost of knowing CSI at the receiver, e.g., by channel estimation, is not ignored.

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6

respectively. The distributions of gj and hi are globally known at the transmitter and at the users.2
Having CSIR, the value of gj (t) is available instantaneously and perfectly at the static user j.
Furthermore, the static user j obtains an outdated version of the dynamic users channels hi , and
also the dynamic user i obtains an outdated version of the static users channel gi (completely
stale) [11]. CSIT for each user can take one of the following:
•

Perfect CSIT: the channel vectors, gj (t), hi (t), are available at the transmitter instantaneously and perfectly.

•

Delayed CSIT: the channel vectors, gj (t), hi (t), are available at the transmitter after they
change independently in the following block (completely stale [11]).

•

No CSIT: the channel vectors, gj (t), hi (t), cannot not be known at the transmitter.

We consider the broadcast channel with private messages for all users and no common
′

messages. More specifically, we assume that the independent messages Mj′ ∈ [1 : 2nRi (ρ) ], Mi ∈
[1 : 2nRi (ρ) ] associated with rates Rj′ (ρ), Ri (ρ) are communicated from the transmitter to the
static user j and dynamic user i, respectively, at ρ signal-to-noise ratio. The degrees of freedom
of the static and dynamic users achieving rates Rj′ (ρ), Ri (ρ) can be defined as
Rj′ (ρ)
,
ρ→∞ log(ρ)
Ri (ρ)
di = lim
,
ρ→∞ log(ρ)

j = 1, . . . , m′ ,

d′j = lim

i = 1, . . . , m.

(2)

The degrees of freedom region is defined as

′ +m
′
∃(R1′ (ρ), . . . , Rm
D = (d′1 , . . . , d′m′ , d1 , . . . , dm ) ∈ Rm
′ (ρ), R1 (ρ), . . . , Rm (ρ)) ∈ C(ρ),
+
Rj′ (ρ)
Ri (ρ)
, di = lim
, j = 1, . . . , m′ , i = 1, . . . , m ,
ρ→∞ log(ρ)
ρ→∞ log(ρ)

d′j = lim

(3)

where C(ρ) is the capacity region at ρ signal-to-noise ratio. The sum degrees of freedom is
defined as
Csum (ρ)
,
ρ→∞ log(ρ)

dsum = lim
where

′

Csum (ρ) = max

m
X
j=1

2

Rj′ (ρ)

+

m
X

(4)

Ri (ρ).

(5)

i=1

Also, the coherence times of all channels are globally known at the transmitter and at the users.

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7

Y1’

X

Y2’

p(y3’|y2’)

p(ym’’|ym’-1’)

p(y3|y2)

p(ym|ym-1)

Ym’’

p(y1’,y1|x)
Y1

Fig. 2.

p(y2’|y1’)

p(y2|y1)

Y2

Ym

Discrete memoryless multiuser multilevel broadcast channel

In the sequel, we study the degrees of freedom of the above MISO broadcast channel under
different CSIT scenarios that could be perfect, delayed or no CSIT.
III. N O CSIT

FOR

A LL U SERS

In this section, we study the broadcast channel defined in Section II when there is no CSIT for
all users. In particular, we give outer and achievable degrees of freedom regions in Section III-B
and Section III-C, respectively. The outer degrees of freedom region is based on the construction
of an outer bound on the rates of a multiuser multilevel discrete memoryless channel that is given
in Section III-A.
A. Multiuser Multilevel Broadcast Channel
The multilevel broadcast channel was introduced by Borade et al. [21] as a three-user broadcast
discrete memoryless broadcast channel where two of the users are degraded with respect to each
other. The capacity of this channel under degraded message sets was established by Nair and El
Gamal [22]. Here, we study a multiuser multilevel broadcast channel with two sets of degraded
users (see Fig. 2). One set contains m′ users with Yj′ received signal at user j, and the other set
contains m users with Yi received signal at user i. Therefore,
X →Y1′ → Y2′ → · · · → Ym′ ′
X →Y1 → Y2 → · · · → Ym

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(6)

DRAFT

8

form two Markov chains. We consider a broadcast channel with (m′ + m) private messages and
no common message. An outer bound for the above multilevel broadcast channel is given in the
following theorem.
Theorem 1: The rate region of the multilevel broadcast channel with two sets of degraded
users (Eq. (6)) is outer bounded by the intersection of
R1 ≤I(Um′ , W ; Y1|V1 ) − I(W ; Ym′ ′ |Um′ ),

(7)

Ri ≤I(Vi−1 ; Yi |Vi ),

(8)

Rj′ ≤I(Uj−1 ; Yj′ |Uj ),

i = 2, . . . , m,
j = 1, . . . , m′ − 1,

′
′
′
′
Rm
′ ≤I(W ; Ym′ |Um′ ) + I(X; Ym′ |Um′ , W ) − I(X; Ym′ |Um′ −1 ),

(9)
(10)

and
Ri ≤I(Ũi−1 ; Yi |Ũi ),

i = 1, . . . , m − 1,

(11)

Rm ≤I(W̃ ; Ym |Ũm ) + I(X; Ym|Ũm , W̃ ) − I(X; Ym|Ũm−1 ),

(12)

R1′ ≤I(Ũm , W̃ ; Y1′ |Ṽ1 ) − I(W̃ ; Ym |Ũm ),

(13)

Rj′ ≤I(Ṽj−1; Yj′ |Ṽj ),

(14)

j = 2, . . . , m′ ,

for some pmf
p(u1 , . . . , um′ , ũ1 , . . . , ũm , v1 , . . . , vm , ṽ1 , . . . , ṽm′ , w, w̃, x),

(15)

where
Um′ → · · · → U1 →X → (Y1 , . . . , Ym, Y1′ , . . . , Ym′ ′ )
Vm → · · · → V1 → (W, Um′ ) →X → (Y1 , . . . , Ym, Y1′ , . . . Ym′ ′ )
Ũm → · · · → Ũ1 →X → (Y1 , . . . Ym , Y1′ , . . . , Ym′ ′ )
Ṽm′ → · · · → Ṽ1 → (W̃ , Ũm ) →X → (Y1 , . . . Ym , Y1′ , . . . , Ym′ ′ )

(16)

forms Markov chains and U0 = Ũ0 , X.
Proof: See Appendix I.
Remark 1: Theorem 1 is an extension of the Körner-Marton outer bound [27, Theorem 5] to
more than two users, and it recovers the Körner-Marton bound when m = m′ = 1.

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Remark 2: For the multiuser multilevel broadcast channel characterized by (6), we establish
the capacity for degraded message sets in Appendix II, where one common message is communicated to all receivers and one further private message is communicated to one receiver.
B. Outer Degrees of Freedom Region
In the sequel, we give an outer bound on the degrees of freedom of the broadcast channel
defined in Section II when there is no CSIT for all users. The outer bound development depends
on the results of Theorem 1 in Section III-A.
Theorem 2: An outer bound on the degrees of freedom region of the fading broadcast channel
characterized by Eq. (1) without CSIT is,
′

m
X

d′j ≤ 1,

m
X

di ≤ 1 −

(17)

j=1

i=1

m′
X
j=1

d′j

+

m
X
i=1

di ≤



1

4
3

1
,
T
T = T ′ , ∆T = 0

(18)

(19)

otherwise,

where ∆T is the offset between the two coherence intervals.
Proof: Equations (17) and (18) are outer bounds for a broadcast channel whose users are
either all homogeneously static or all homogeneously dynamic [7], [18]. The remainder of the
proof is dedicated to establishing (19). We enhance the channel by giving all users global CSIR.
When T ′ = T and ∆T = 0, (19) follows directly from [7], [18]. When T ′ 6= T or ∆T 6= 0,
having no CSIT, the channel belongs to the class of multiuser multilevel broadcast channels in
Section III-A. We then use the two outer bounds developed for the multilevel broadcast channels
to generate two degrees of freedom bounds, and merge them to get the desired result.
We begin with the outer bound described in (7)-(10); we combine these equations to obtain
P
P
partial sum-rate bounds on the static ( Rj′ ) and dynamic ( Ri ) receivers:
′

m
X

Rj′

j=1

≤

′ −1
m
X

′
′
I(Uj−1 ; yj′ |Uj , H) + I(W ; ym
′ |Um′ , H) + I(x; ym′ |Um′ , W, H)

j=1

′
− I(x; ym
′ |Um′ −1 , H)

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10

=

′ −1
m
X

′
′
h(yj′ |Uj , H) − h(yj′ |Uj−1 , H) + I(W ; ym
′ |Um′ , H) + h(ym′ |Um′ , W, H)

j=1

′
− h(ym
′ |Um′ −1 , H) + o(log(ρ))

(20)

′
′
=I(W ; ym
′ |Um′ , H) + h(ym′ |Um′ , W, H) + o(log(ρ)),

(21)

where H is the set of all channel vectors, (20) follows from the chain rule, h(yj′ |x, H) =
o(log(ρ)), and (21) follows since the received signals of all static users, yj′ , have the same
statistics [7], [18]. Also, using Theorem 1,
m
X

Rj ≤I(Um′ , W ; y1|V1 , H) −

′
I(W ; ym
′ |Um′ , H)

+

m
X

I(Vj−1; yj |Vj , H)

j=2

j=1

=h(y1 |V1 , H) − h(y1 |Um′ , W, H) −

′
I(W ; ym
′ |Um′ , H)

+

m
X

h(yj |Vj , H)

j=2

− h(yj |Vj−1, H)

(22)

′
= − h(y1 |Um′ , W, H) − I(W ; ym
′ |Um′ , H) + h(ym |Vm , H) + o(log(ρ))

(23)

′
≤ − h(y1 |Um′ , W, H) − I(W ; ym
′ |Um′ , H) + log(ρ) + o(log(ρ)),

(24)

where (22) follows from the chain rule, (23) follows since yj have the same statistics, and (24)
′
follows since h(ym |Vm , H) ≤ n log(ρ) + o(log(ρ)). Define Yj,k
to be the received signal of user

j at time instance k. From (21) and (24), we can obtain the bound (27) on the rates.
m′

m

1X ′ X
1
1
′
′
Rj +
Rj ≤ I(W ; ym
h(ym
′ |Um′ , H) +
′ |Um′ , W, H) − h(y1 |Um′ , W, H)
2 j=1
2
2
j=1
′
− I(W ; ym
′ |Um′ , H) + log(ρ) + o(log(ρ))

1
′
= h(ym
′ |Um′ , W, H) − h(y1 |Um′ , W, H) + log(ρ) + o(log(ρ))
2
1
′
≤ h(ym
′ , y1 |Um′ , W, H) − h(y1 |Um′ , W, H) + log(ρ) + o(log(ρ))
2
n
X
1
′
′
′
≤
h(ym
′ ,k , y1,k |Um′ , W, H, ym′ ,1 , . . . , ym′ ,k−1 , y1,1 , . . . , y1,k−1 )
2
k=1

(25)

′
′
− h(y1,k |Um′ , W, H, ym
′ ,1 , . . . , ym′ ,k−1 , y1,1 , . . . , y1,k−1 ) + log(ρ)

+ o(log(ρ))
≤

March 26, 2018

max

Tr{Σx }≤ρ,Σx <0

(26)
EH

1
log |I + HΣx H† | − log(1 + h†1 Σx h1 ) + log(ρ)
2
DRAFT

11

+ o(log(ρ)),

(27)

where (25) and (26) follow from the chain rule and that conditioning does not increase differential
entropy, and (27) follows from extremal entropy inequality [23], [24], [28]. In order to bound (27),
we use a specialization of [29, Lemma 3] as follows.
Lemma 1: Consider two random matrices H1 ∈ CN1 ×Nt and H2 ∈ CN2 ×Nt , where N1 ≥ N2 .
For a covariance matrix Σx , where Tr{Σx } ≤ ρ, we have
1
1
log |I + H1 Σx H†1 | −
log |I + H2 Σx H†2 | ≤ o(log(ρ)). (28)
Σx min{Nt , N1 }
min{Nt , N2 }
The proof of Lemma 1 is omitted as it directly follows from [29, Lemma 3]. Lemma 1 yields
max

the following outer bound on the degrees of freedom:
m′

m

1X ′ X
d +
di ≤ 1.
2 j=1 j i=1

(29)

We now repeat the exercise of bounding the sum rates and deriving degrees of freedom, this
time starting from (11)-(14). By following bounding steps parallel to (21), (24), (27),
′

m
X

m

d′j

j=1

1X
di ≤ 1.
+
2 i=1

(30)

Adding (29) and (30) yields the outer bound (19), completing the proof of Theorem 2.
C. Achievable Degrees of Freedom Region
Theorem 3: The fading broadcast channel described by Eq. (1) can achieve the following
degrees of freedom without CSIT:
m
X

di ≤ 1 −

m
X

di ≤ 1.

i=1

m′
X
j=1

d′j

+

1
,
T

(31)

(32)

i=1

Proof: The achievable scheme uses product superposition [17], [18], where the transmitter
uses one antenna to send the super symbol to two users: one dynamic and one static,
x† = xs x†d ,

(33)

where xs ∈ C is a symbol intended for the static user,
x†d = [xτ , x†δ ]
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(34)
DRAFT

12

where xτ ∈ C is a pilot and xδ ∈ CT −1 is a super symbol intended for the dynamic user. Since
degrees of freedom analysis is insensitive to the additive noise, we omit the noise component
in the following.
y† = hxs [xτ , x†δ ]
= [hxτ , hx†δ ],

(35)

where h = hxs . The dynamic user estimates the equivalent channel h during the first time
instance and then decodes xδ coherently based on the channel estimate. The static receiver only
utilizes the received signal during the first time instance:
y1′ = gxs .

(36)

Knowing its channel gain g, the static receiver can decode xs . The achievable degrees of freedom
of the two users are,
(d′ , d) =

1
1
.
,1 −
T
T

(37)

We now proceed to prove that the degrees of freedom region characterized by (31) and (32) can
be achieved via a combination of two-user product superposition strategies that were outlined
above, and single-user strategies. For clarity of exposition we refer to (31), which describes
the degrees of freedom constraints of the dynamic receivers, as the non-coherent bound, and
to (32) as the coherent bound. The non-negativity of degrees of freedom restricts them to the
′

m+m
. The intersection of the coherent bound and the non-negative orthant
non-negative orthant R+

is a (m′ + m)–simplex that has m + m′ + 1 vertices. The non-coherent bound is a hyperplane
that partitions the simplex with m′ + 1 vertices on one side of the non-coherent bound and m
on the other. Therefore the intersection of the simplex with the non-coherent bound produces
a polytope with (m′ + 1)(m + 1) vertices.3 For illustration, see Fig. 3 showing the three-user
degrees of freedom with two static users and Fig. 4 with one static user.
We now verify that each of the (m′ + 1)(m + 1) vertices can be achieved with either a
single-user strategy, or via a two-user product superposition strategy:
•

m′ vertices corresponding to single-user transmission to each static user j achieving one
degree of freedom.

3

This can be verified with a simple counting exercise involving the number of edges of the simplex that cross the non-coherent

bound.
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13

d

d1

Non-coherent bound
Non-coherent bound

d2
Coherent bound

d’2
d’

d’1
Fig. 3.

Achievable degrees of freedom region of one

dynamic and two static users

•

Coherent bound

Fig. 4. Achievable degrees of freedom region of one static
and two dynamic users

m vertices corresponding to single-user transmission to each dynamic user i achieving
(1 − T1 ) degrees of freedom.

•

m′ m vertices corresponding to product superposition applied to all possible pairs of static
and dynamic users, achieving

1
T

degrees of freedom for one static user and (1 − T1 ) degrees

of freedom for one dynamic user.
•

One trivial vertex at the origin, corresponding to no transmission achieving zero degrees of
freedom for all users.

Hence, the number of the vertices is m′ + m + m′ m + 1 = (m + 1)(m′ + 1). This completes the
achievability proof of Theorem 3.
Remark 3: When the static and dynamic users have the same coherence time, the inner and
outer bounds on degrees of freedom coincide. In this case it is degrees of freedom optimal to
serve two-users at a time (one dynamic and one static).
IV. D ELAYED CSIT

FOR ALL USERS

Under delayed CSIT, the transmitter knows each channel gain only after it is no longer valid.
This condition is also known as outdated CSIT. We begin by proving inner and outer bounds

March 26, 2018

DRAFT

14

when transmitting only to static users, only to dynamic users, and to one static and one dynamic
user. We then synthesize this collection of bounds into an overall degrees of freedom region.
A. Transmission to Static Users
Theorem 4: The degrees of freedom region of the fading broadcast channel characterized by
Eq. (1) with delayed CSIT and having m′ static users and no dynamic users is
1
j = 1, . . . , m′ .
(38)
1 ,
1 + + . . . + m′
Proof: The special case of fast fading (T ′ = 1) was discussed by Maddah-Ali and Tse
d′j ≤

1
2

in [11], where the achievability was established by retrospective interference alignment that
aligns the interference using the outdated CSIT, and the converse was proved by generating
an improved channel without CSIT having a tight degrees of freedom region against TDMA
according to the results in [2], [3]. For T ′ ≥ 1, the achievability is established by employing
retrospective interference alignment presented in [11] over super symbols each of length T ′ .
The converse is proved by following the same procedures in [11] to generate a block-fading
improved channel without CSIT and with identical coherence intervals of length T ′ . According
to the results of [7], [18], TDMA is tight against the degrees of freedom region of the improved
channel.
B. Transmission to Dynamic Users
Theorem 5: The fading broadcast channel characterized by Eq. (1), with delayed CSIT and
having m dynamic users and no static users, can achieve the degrees of freedom
di ≤

1
1 + + ...+
1
2

1 (1
m

−

m
),
T

i = 1, . . . , m.

(39)

An outer bound on the degrees of freedom region is
di ≤1 −
m
X

1
,
T

(40)

m
.
(41)
1 + + . . . + m1
i=1
Proof: The achievability part can be proved as follows. At the beginning of each super
di ≤

1
2

symbol, m pilots are sent for channel estimation. Then retrospective interference alignment
in [11] over super symbols is employed during the remaining (T − m) instances, to achieve (39).

March 26, 2018

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15

For the converse part, (41) is proved by giving the users global CSIR, and then applying
Theorem 4. Moreover, (40) is the single-user bound for each dynamic user that can be proved as
follows. For a single user with delayed CSIT, feedback does not increase the capacity [30], and
consequently the assumption of delayed CSIT can be removed. Hence, the single-user bound for
each dynamic user with delayed CSIT is the same as the single-user bound without CSIT [26].

C. Transmission to One Static and One Dynamic User
Theorem 6: The fading broadcast channel characterized by Eq. (1), with delayed CSIT and
having one static and one dynamic user, can achieve the following degrees of freedom
2
1 2
2 
(1 + ), (1 − ) ,
3
T 3
T
1
1
D2 : (d′ , d) = ( , 1 − ).
T
T
D1 : (d′ , d) =

(42)
(43)

Furthermore, the achievable degrees of freedom region is the convex hull of the above degrees
of freedom pairs.
Proof: From Section III-C, product superposition achieves the pair (43) that does not require
CSIT for any of the two users. The remainder of the proof is dedicated to the achievability of the
pair (42). We provide a transmission scheme based on retrospective interference alignment [11]
along with product superposition.
1) The transmitter first emits a super-symbol intended for the static user:
X1 = [X1,1 , · · · , X1,ℓ ],
where ℓ =

T′
,
T

(44)

and each X1,n ∈ C2×T occupies T time instances and has the following

structure:
X1,n = [Ūn , Ūn Un ],

n = 1, . . . , ℓ,

(45)

both the diagonal matrix Ūn ∈ C2×2 and Un ∈ C2×(T −2) contain symbols intended for the
′

′

′

†
†
static user. The components of y1† = [y1,1
, · · · , y1,ℓ
] are:
′

†
y1,n
=[g1† Ūn , g1† Ūn Un ],
†
†
=[g̃1,n
, g̃1,n
Un ],

March 26, 2018

n = 1, . . . , ℓ
(46)

DRAFT

16

†
= g1† Ūn . The static user by definition knows g1 so it can decode Ūn which
where g̃1,n
′

yields 2 TT degrees of freedom. The remaining

†
Un involve
(T − 2) observations in g̃1,n

T′
T

′

2 TT (T − 2) unknowns, so they require a further

T′
T

(T − 2) independent observations for

reliable decoding.
†
†
The components of y1† = [y1,1
, · · · , y1,ℓ
] are
†
=[h†1,n Ūn , h†1,n Ūn Un ],
y1,n

n = 1, . . . , ℓ

=[h̃†1,n , h̃†1,n Un ],

(47)

where h̃†1,n = h†1,n Ūn is the equivalent channel estimated by the dynamic user. The dynamic
user saves h̃†1,n Un for interference cancellation in the upcoming steps.
2) The transmitter sends a second super symbol intended for the dynamic user:
X2 = [X2,1 , · · · , X2,ℓ ],

(48)

where
X2,n = [Ũn , Ũn Vn ],

n = 1, . . . , ℓ,

(49)

Ũn ∈ C2×2 is diagonal and includes 2 independent symbols intended for the static user,
and Vn ∈ C2×(T −2) contains independent symbols intended for the dynamic user. The
†
†
components of y2† = [y2,1
, · · · , y2,ℓ
] are
†
y2,n
=[h†2,n Ũn , h†2,n Ũn Vn ],

n = 1, . . . , ℓ

=[h̃†2,n , h̃†2,n Vn ],

(50)

where h̃†2,n = h†2,n Ũn is the equivalent channel estimated by the dynamic user. The dynamic
user saves h̃†2,n Vn which includes

T′
T

T′
(T −
T
′†
y2,ℓ
] are

unknowns, and hence an additional
′

′

†
components of y2† = [y2,1
, ··· ,

′

(T − 2) independent observations about 2 TT (T − 2)
2) observations are needed to decode Vn . The

′
y2,n
=[g2† Ũn , g2† Ũn Vn ],

n = 1, . . . , ℓ

†
†
=[g̃2,n
, g̃2,n
Vn ],

(51)

†
where g̃2,n
= g2† Ũn is the equivalent channel estimated by the static user; the static user
†
saves g̃2,n
Vn for the upcoming steps. Knowing g2 , the static user achieves 2 TT further
′

degrees of freedom from decoding Ũn .
March 26, 2018

DRAFT

17

3) The transmitter emits a third super symbol consisting of a linear combination of the signals
generated from the first and the second super symbols.
X3 = [X3,1 , · · · , X3,ℓ ],

(52)

where
†
X3,n = [Ûn , Ûn (h̃†1,n Un + g̃2,n
Vn )],

n = 1, . . . , ℓ,

(53)

Ûn ∈ C2×2 is diagonal and contains 2 independent symbols intended for the static user,
′

and hence the static user achieves further 2 TT degrees of freedom.
†
The static user cancels g̃2,n
Vn saved during the second super symbol and obtains h̃†1,n Un

that includes the additional independent

T′
T

(T − 2) observations needed for decoding Un .

T′

Therefore, the static user achieves 2 T (T − 2) further degrees of freedom. The dynamic
user estimates the equivalent channel h̃†3,n = h†3,n Ûn , cancels h̃†1,n Un saved during the
†
Vn that contains the additional observations needed
first super symbol, and obtains g̃2,n
′

for decoding Vn . Hence, the dynamic user achieves 2 TT (T − 2) degrees of freedom.
In aggregate, over 3T ′ time instants, the static and dynamic user achieve the degrees of freedom
d′ = 6

T′
T′
+ 2 (T − 2),
T
T

d=2

T′
(T − 2).
T

(54)

This completes the proof of Theorem 6.
Theorem 7: An outer bound on the degrees of freedom region of the fading broadcast channel
characterized by Eq. (1), with one static and one dynamic user having delayed CSIT, is
d′
+ d ≤ 1,
2
d
d′ + ≤ 1,
2

(55)
(56)

1
.
(57)
T
Proof: The inequality (57) represents the single-user outer bound [26]. We prove the
d≤1−

bound (55) as follows. We enhance the original channel by giving both users global CSIR. In
addition, the channel output of the dynamic user, y(t), is given to the static user. Therefore,
the channel outputs at time instant t are (y ′(t), y(t), H) at the static user, and (y(t), H) at the
dynamic user. The enhanced channel is physically degraded [31], [32], hence, removing the
delayed CSIT does not reduce the capacity [33]. Also,
R′ ≤I(x(t); y ′(t), y(t)|U, H) = h(y ′(t), y(t)|U, H) − h(y ′ (t), y(t)|U, x(t), H)
March 26, 2018

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18

R ≤I(U; y(t)|H) = h(y(t)|H) − h(y(t)|U, H),

(58)

where U is an auxiliary random variable, and U → x → (y ′(t), y(t)) forms a Markov chain.
Therefore,
1
R′
+ R ≤h(y(t)|H) + h(y ′(t), y(t)|U, H) − h(y(t)|U, H) + o(log(ρ))
2
2
1
(59)
≤ log(ρ) + h(y ′(t), y(t)|U, H) − h(y(t)|U, H) + o(log(ρ))
2
1
≤ log(ρ) +
max
EH
log |I + HΣx H† | − log(1 + h† (t)Σx h(t)) + o(log(ρ))
Tr{Σx }≤ρ,Σx <0
2
(60)
≤ log(ρ) + o(log(ρ)),

(61)

where (59) follows since h(y(t)|H) ≤ log(ρ)+o(log(ρ)) [34], (60) follows from extremal entropy
inequality [23], [24], [29], and (61) follows from Lemma 1. Hence, the bound (55) is proved.
A similar argument, with the role of the two users reversed, leads to the bound (56).
Remark 4: The inner and outer bounds obtained for the two-user case partially meet, with
the gap diminishing with the coherence time of the dynamic user as shown in Fig. 5 and Fig. 6
for T = 15 and T = 30, respectively.
D. Transmission to arbitrary number of static and dynamic users
Theorem 8: The fading broadcast channel characterized by Eq. (1), with delayed CSIT, can
achieve the multiuser degrees of freedom characterized by vectors Di ,
1
D1 :
1
1 + 2 + ...+

′

m
X

1
m′ i=1

e†i ,

1
2
2
2
D2 , . . . , Dmm′ +1 : (1 + )e†j + (1 − )e†m′ +i , j = 1, . . . , m′ , i = 1, . . . , m,
3
T
3
T
m
m †
1
m X †
Dmm′ +2 , . . . , Dmm′ +m′ +2 : ej +
e , j = 1, . . . , m′ ,
(1 − )
T
T i=1 i
1 + 21 + . . . + m1

(62)
(63)
(64)

where ej is the canonical coordinate vector. Their convex hull characterized an achievable degrees
of freedom region.
Proof: The achievability of (62) was proved in Section IV-A via multiuser transmission to
static users. The achievability of (63) was proved in Section IV-C, via a two-user transmission
to a dynamic-static pair.
March 26, 2018

DRAFT

19

1.5

Achievable region
Outer region
1

0.5

0
0

Fig. 5.

0.2

0.4

0.6

0.8

1

1.2

One static and one dynamic with delayed CSIT and T = 15

1.5

Achievable region
Outer region
1

0.5

0
0

Fig. 6.

0.2

0.4

0.6

0.8

1

1.2

One static and one dynamic with delayed CSIT and T = 30

March 26, 2018

DRAFT

20

We now show the achievability of (64) via retrospective interference alignment [11] along with
product superposition. Over a super symbol of length T , consider the following transmission:
X = [U, UV],

(65)

where U ∈ Cm×m is diagonal and includes m independent symbols intended for the static user j,
and V ∈ Cm×(T −m) is a super symbol containing independent symbols intended for the dynamic
users according to retrospective interference alignment [11]. Therefore, the static user decodes
U. Thus, over T time instants, the static user achieves m degrees of freedom and the dynamic
users achieve

1
1 (T
1+ 12 +...+ m

− m), hence (64) is achieved.

Theorem 9: An outer bound on the degrees of freedom of the fading broadcast channel
characterized by Eq. (1), with delayed CSIT, is
′

m
X
d′j
di
+
≤ 1,
′
m + m i=1 m

(66)

m
m
X
d′j X
di
+
≤ 1,
′
′
m
m
+
m
j=1
i=1

(67)

m
X
j=1
′

d′j ≤ 1,

∀j = 1, . . . , m′ ,

(68)

1
, ∀i = 1, . . . , m.
(69)
T
Proof: The inequalities (68), (69) represent the single-user bounds on the static and the
di ≤ 1 −

dynamic users, respectively [26], [34]. The remainder of the proof is dedicated to establishing
the bounds (66) and (67).
We enhance the channel by providing global CSIR as well as allowing full cooperation among
static users and full cooperation among dynamic users. The enhanced channel is equivalent to
a broadcast channel with two users: one static equipped with m′ antennas and one dynamic
′

equipped with m antennas. Define Y ′ ∈ Cm and Y ∈ Cm to be the received signals of the
static and the dynamic super-user, respectively, in the enhanced channel. We further enhance
the channel by giving Y to the static user, generating a physically degraded channel since
X → (Y ′ , Y) → Y forms a Markov chain. Feedback including delayed CSIT has no effect
on capacity [33], therefore we remove it from consideration. Subsequently, we can utilize the
Körner-Marton outer bound [27],
′

m
X

Rj′ ≤I(X; Y ′, Y|U, H)

j=1

March 26, 2018

DRAFT

21
m
X

Ri ≤I(U; Y|H).

(70)

i=1

Therefore, from applying extremal entropy inequality [23], [29], [35] and Lemma 1,
′

m
X
j=1

m
X
Rj′
1
1
Ri
′
+
≤
I(X;
Y
,
Y|U,
H)
+
I(U; Y|H)
m′ + m i=1 m m′ + m
m

=

m′

1
1
1
h(Y ′, Y|U, H) + o(log(ρ)) + h(Y|H) − h(Y|U, H)
+m
m
m

≤ log(ρ) + o(log(ρ)).

(71)

Therefore, the bound (66) is proved. Similarly, we can prove the bound (67) using the same
steps after switching the roles of the two users in the enhanced channel.
V. H YBRID CSIT: P ERFECT CSIT

FOR THE

S TATIC U SERS

AND

N O CSIT

FOR THE

DYNAMIC U SERS
Theorem 10: The fading broadcast channel characterized by Eq. (1), with perfect CSIT for
the static users and no CSIT for the dynamic users, can achieve the following multiuser degrees
of freedom,
′

D1 :

m
X

e†j ,

(72)

j=1

′

D2 , . . . , Dm+1

m
1
1X †
ej + (1 − )e†i ,
:
T j=1
T

i = 1, . . . , m.

(73)

Therefore, their convex hull is also achievable.
Proof: D1 is achieved by inverting the channels of the static users at the transmitter and
then every static user achieves one degree of freedom. D2 , . . . , Dm+1 in (73) are achieved using
product superposition along with channel inversion as follows. The transmitted signal over T
instants is,
X = [u, uv† ],
where u =

Pm′

j=1 bj uj ,

(74)

uj is a symbol intended for the static user j, gj† bj = 0, and v ∈ CT −1

contain independent symbols intended for the dynamic user i. Each of the static users receive
an interference-free signal during the first time instant achieving one degrees of freedom. The
dynamic user estimates its equivalent channel during the first time instant and decodes v during
the remaining (T − 1) time instants.
March 26, 2018

DRAFT

22

Theorem 11: An outer bound on the degrees of freedom of the fading broadcast channel
characterized by Eq. (1), with perfect CSIT for the static users and no CSIT for the dynamic
users, is
′

m
X
j=1

m
X
d′j
+
di ≤ 1,
m′ + 1 i=1

d′j ≤ 1,
m
X

(75)
∀j = 1, . . . , m′ ,

(76)

1
.
(77)
T
i=1
Proof: The inequalities (76) represent single-user bounds for the static users [34], and (77)
di ≤ 1 −

is a time-sharing outer bound for the dynamic users that was established in [7], [18]. It remains
to prove (75), as follows.
We enhance the channel by giving global CSIR to all users and allowing full cooperation
between the static users. This gives rise to an equivalent static user with m′ antennas receiving
Y ′ over an equivalent channel G and noise Z′ . At this point, we have a multi-user system where
CSIT is available with respect to one user, but not others. We then bound the performance
of this system with that of another (similar) system that has no CSIT. To do so, we use the
local statistical equivalence property developed and used in [13], [15], [36]. First, we draw
G̃, Z̃ according to the distribution of G, Z′ and independent of them. We enhance the channel
by providing Ỹ = G̃X + Z̃ to the static receiver and G̃ to all receivers. Because we do not
provide G̃ to the transmitter, there is no CSIT with respect to Ỹ. According to [36], we have
h(Ỹ, Y ′|H) = h(Y ′ |H) + o(log(ρ)), where H = (G, G̃, h1 , . . . , hm ), therefore we can remove
Y ′ from the enhanced channel without reducing its degrees of freedom. This new equivalent
channel has one user with m′ antennas receiving (Ỹ, H), m single-antenna users receiving
(yi , H), and no CSIT.4 Having no CSIT, the enhanced channel is in the form of a multilevel
broadcast channel studied in Section III-A, and hence using Theorem 1,
′

m
X

Rj′ ≤I(W ; Ỹ|U, H) + I(X; Ỹ|U, W, H)

j=1

4

In the enhanced channel after removal of Y ′ , the transmitter and receivers still share information about G, but this random

variable is now independent of all (remaining) transmit and receive variables.

March 26, 2018

DRAFT

23

R1 ≤I(U, W ; y1|V1 , H) − I(W ; Ỹ|U, H)
Ri ≤I(Vi−1 ; yi |Vi , H),

i = 2, . . . , m.

(78)

The dynamic receiver received signals have the same distribution. By following bounding steps
parallel to (22), (23), (24),
m
X

Ri ≤ log(ρ) + o(log(ρ)) − I(W ; Ỹ|U, H) − h(y1 |U, W, H).

(79)

j=1

Therefore,
′

m
X
j=1

m
X
Rj′
1
h(Ỹ|U, W, H)
+
R
≤
log(ρ)
+
o(log(ρ))
+
(
−
1)I(W
;
Ỹ|U,
H)
+
i
m′ + 1 j=1
m′ + 1
m′ + 1

− h(y1 |U, W, H),
≤ log(ρ) + o(log(ρ)) +

(80)
h(Ỹ, y1 |U, W, H)
− h(y1 |U, W, H)
m′ + 1

≤ log(ρ) + o(log(ρ)),

(81)
(82)

where the last inequality follows from applying the extremal entropy inequality [23], [29], [35]
and Lemma 1. This concludes the proof of the bound (75).
VI. H YBRID CSIT: P ERFECT CSIT

FOR THE

S TATIC U SERS

AND

D ELAYED CSIT

FOR THE

DYNAMIC U SERS
We begin with inner and outer bounds for one static and one dynamic user, then extend the
result to multiple users. The transmitter knows the channel of the static users perfectly and
instantaneously, and an outdated version of the channel of the dynamic users.
A. Transmitting to One Static and One Dynamic User
Theorem 12: For the fading broadcast channel characterized by Eq. (1) with one static and
one dynamic user, with perfect CSIT for the static user and delayed CSIT for the dynamic user,
the achievable degrees of freedom region is the convex hull of the vectors,
1
1 1
, −
),
2T 2 2T
1
1
D2 :(d′, d) = ( , 1 − ).
T
T
D1 :(d′, d) = (1 −

March 26, 2018

(83)
(84)

DRAFT

24

Proof: The degrees of freedom (84) can be achieved by product superposition as discussed
in Section III, without CSIT. We proceed to prove the achievability of (83).
1) Consider [u1 ,

··· ,

uT −1 ] to be a complex 2 × (T − 1) matrix containing symbols

intended for the static user, [v1 , · · · , vT −1 ] intended for the dynamic user, and b ∈ C is
a beamforming vector so that g† b = 0. In addition we define u0 = 0, v0 = 1. Using these
components, the transmitter constructs and transmits a super-symbol of length T , whose
value at time t is:
x†1 (t) = ut + b vt .

(85)

Note that x1 (0) = b does not carry any information for either user, and serves as a pilot.
The received super symbol at the static user is:
′

y1† = [0, g† u1 , · · · , g† uT −1 ].

(86)

The received super symbol at the dynamic user
y1† =[h†1 b, (h†1 u1 + h†1 bv1 ), · · · , (h†1 uT −1 + h†1 bvT −1 )].

(87)

The dynamic user estimates its equivalent channel h†1 b from the received value in the first
time instant. The remaining terms include symbols intended for the dynamic user plus
some interference, whose cancellation is the subject of the next step.
2) The transmitter next sends a second super symbol of length T ,
x2 = [ū, ū(h†1 u1 ), · · · , ū(h†1 uT −1 )],

(88)

where ū ∈ C is a symbol intended for the static user. Hence,
y2† = [h2 ū, h2 ū(h†1 u1 ), · · · , h2 ū(h†1 uT −1 )].

(89)

The dynamic user estimates the equivalent channel h2 ū during the first time instant and
then acquires h†1 ut , the interference in (87). Therefore, using y1 , y2 , the dynamic user
solves for vt achieving (T − 1) degrees of freedom. Furthermore,
′

y2† = [g1 ū, g1 ū(h†1 u1 ), · · · , g1 ū(h†1 uT −1 )].

(90)

The static user solves for ū achieving one degree of freedom and also uses h†1 ut to solve
for ut achieving further 2 (T − 1) degrees of freedom.

March 26, 2018

DRAFT

25

In summary, during 2T instants, the static user achieves (2T − 1) degrees of freedom and the
dynamic user achieves (T − 1) degrees of freedom. This shows the achievability of (83).
Theorem 13: For the fading broadcast channel characterized by Eq. (1) with one static and
one dynamic user, where there is perfect CSIT for the static user and delayed CSIT for the
dynamic user, an outer bound on the degrees of freedom region is,
d′
+ d ≤1,
2
d′ ≤1,

(91)
(92)

1
.
(93)
T
Proof: The inequalities (92) and (93) represent the single-user outer bounds [26], [34]. It
d ≤1 −

only remains to prove the outer bound (91), as follows.
1) We enhance the channel by giving global CSIR to both users and also give y to the static
user. The enhanced channel is physically degraded having (Y ′ , G) at the static user and
(y, G) at the dynamic user, where Y ′ , (y ′, y) and G , (h, g). In a physically degraded
channel, causal feedback (including delayed CSIT) does not affect capacity [33], so we
can remove the delayed CSIT with respect to the dynamic user.
2) We now use another enhancement with the motivation to remove the remaining CSIT (noncausal, with respect to the static user). This is accomplished, similar to Theorem 11, via
local statistical equivalence property [13], [15], [36] in the following manner. We create a
channel G̃, and noise Z̃ with the same distribution but independently of the true channel
and noise, and a signal Ỹ = G̃X + Z̃. A genie will give Ỹ to the static receiver and G̃
to both receivers. It has been shown [36] that h(Ỹ, Y ′ |H) = h(Y ′ |H) + o(log ρ), where
H = (G, G̃), therefore we can remove Y ′ from the enhanced channel without reducing
its degrees of freedom.
3) The enhanced channel is still physically degraded, therefore [31], [32]
R′ ≤I(x; Ỹ|U, H) = h(Ỹ|U, H) + o(log(ρ))
R ≤I(U; y|H) = h(y|H) − h(y|U, H),

(94)

where U is an auxiliary random variable, and U → x → (y ′ , y) forms a Markov chain.
Therefore,
1
1 ′
R + R ≤h(y|H) + h(Ỹ|U, H) − h(y|U, H) + o(log(ρ))
2
2
March 26, 2018

DRAFT

26

1.5

Achievable region
Outer region
1

0.5

0
0

Fig. 7.

0.2

0.4

0.6

0.8

1

1.2

One static and one dynamic user with hybrid CSIT and T = 15

≤ log(ρ) + o(log(ρ)),

(95)

where the last inequality follows from extremal entropy inequality and Lemma 1 [23],
[29], [35]. This concludes the proof of the bound (91).

Remark 5: For the above broadcast channel with hybrid CSIT, the achievable sum degrees of
freedom is dsum =

3
2

− T1 , and the outer bound on the sum degrees of freedom is dsum ≤ 32 . The

gap decreases with the dynamic user coherence time (see Fig. 7 and 8).
B. Multiple Static and Dynamic Users
Theorem 14: The fading broadcast channel characterized by Eq. (1), with perfect CSIT for
the static users and delayed CSIT for the dynamic users, can achieve the following degrees of
freedom,
′

D1 :

m
X

e†j ,

(96)

j=1

D2 , . . . , Dmm′ +1 : (1 −
March 26, 2018

1
1 †
1 †
)ej + ( −
)e ,
2T
2 2T i

j = 1, . . . , m′ , i = 1, . . . , m,

(97)
DRAFT

27

1.5

Achievable region
Outer region
1

0.5

0
0

Fig. 8.

0.2

0.4

0.6

0.8

1

1.2

One static and one dynamic user with hybrid CSIT and T = 30

′

Dmm′ +2 , . . . , Dmm′ +m+2
m′

Dmm′ +m+3

m
1
1X †
ej + (1 − )e†i ,
:
T j=1
T

mX †
1
:
ej + (
1
T j=1
1 + 2 + ...+

i = 1, . . . , m,

(98)

m

m X †
(1
−
))
e.
1
T i=1 i
m

(99)

The achievable region consists of the convex hull of the above vectors.
Proof: D1 is achieved by inverting the channel of the static users at the transmitter, providing
one degree of freedom per static user. The achievability of D2 , . . . , Dmm′ +1 was established in
Section VI-A, and that of Dmm′ +2 , . . . , Dmm′ +m+2 was proved in Section V without CSIT for the
dynamic user, so it remains achievable with delayed CSIT. Dmm′ +m+3 is achieved by retrospective
interference alignment [11] along with product superposition, as follows. The transmitted signal
over T instants is
X = [Ū, ŪV],

(100)

where Ū ∈ Cm×m contains independent symbols intended for the static users sent by inverting the
channels of the static users. Therefore, during the first m time instants, each static user receives
an interference-free signal and achieves m degree of freedom, and furthermore the dynamic users
estimate their equivalent channels. During the remaining time instants, each dynamic receiver
March 26, 2018

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28

obtains coherent observations of (T − m) transmit symbols, which are pre-processed, combined
and interference-aligned into super-symbols V according to retrospective interference alignment
techniques of [11]. Accordingly, each dynamic receiver achieves

1
1 (1
1+ 12 +...+ m

−

m
)
T

degrees of

freedom.
Theorem 15: An outer bound on the degrees of freedom region of the fading broadcast channel
characterized by Eq. (1), with perfect CSIT for the static users and delayed CSIT for the dynamic
users, is
′

m
X
j=1

m
X
d′j
di
+
≤ 1,
′
m + m i=1 m
m
X
i=1

di ≤

1+

d′j ≤ 1,

(101)

1
2

m
+ ...+

1
m

,

j = 1, . . . , m′ ,

(102)
(103)

1
,
i = 1, . . . , m.
(104)
T
Proof: The inequalities (103) and (104) represent the single-user outer bounds for the static
di ≤ 1 −

and dynamic users, respectively [26], [34]. According to Theorem 5, (102) represents an outer
bound for the dynamic users. It only remains to prove (101) as follows.
1) The original channel is enhanced by giving the users global CSIR. Furthermore, we assume
full cooperation between the static users and between the dynamic users. The resulting
enhanced channel is a broadcast channel with two users: one static user equipped with
m′ antennas, received signal Y ′ , channel G, and noise noise Z′ , and one dynamic user
equipped with m antennas, received signal Y, channel H, and noise Z.
2) We further enhance the channel by giving Y to the static user, constructing a physically
degraded channel. For the enhanced channel, the static receiver is equipped with m′ + m
′

antennas and has received signal Ŷ = [Y † , Y † ]† , channel Ĝ = [G† , H† ]† , and noise
′

Ẑ = [Z† , Z † ]† . Since any causal feedback (including delayed CSIT) does not affect the
capacity of a physically degraded channel [33], the delayed CSIT for the dynamic receiver
can be removed.
3) We now use another enhancement with the motivation to remove the remaining CSIT
(non-causal, with respect to the static user). We create an artificial channel and noise, G̃,
Z̃, with the same distribution but independent of Ĝ, Ẑ, and a signal Ỹ = G̃X + Z̃. A
March 26, 2018

DRAFT

29

genie will give Ỹ to the static receiver and G̃ to both receivers. It has been shown [36]
that h(Ỹ, Ŷ|H) = h(Ŷ|H) + o(log ρ), where H = (Ĝ, G̃), therefore we can remove Ŷ
from the enhanced channel without reducing its degrees of freedom.
4) The enhanced channel is physically degraded without CSIT, therefore [31], [32],
′

m
X

Rj′ ≤ I(X; Ỹ|U, H)

m
X

Ri ≤ I(U; Y|H).

j=1

(105)

i=1

Hence,
′

m
X
j=1

m
X
Rj′
1
1
1
Ri
+
≤ ′
h(Ỹ|U, H) + h(Y|H) − h(Y|U, H) + o(log(ρ))
′
m + m j=1 m m + m
m
m

≤ log(ρ) + o(log(ρ)),

(106)

where the last inequality follows from the extremal entropy inequality [23], [29], [35] and
Lemma 1 and since h(Y|H) ≤ m log(ρ) + o(log(ρ)) [34]. This concludes the proof of the
bound (101).

VII. C ONCLUSION
A multiuser broadcast channel was studied where some receivers experience longer coherence
intervals and have CSIR while other receivers experience a shorter coherence interval and do not
enjoy free CSIR. The degrees of freedom were studied under delayed CSIT, hybrid CSIT, and
no CSIT. Among the techniques employed were interference alignment and beamforming along
with product superposition for the inner bounds. The outer bounds involved a bounding of the
rate region of the multiuser (discrete memoryless) multilevel broadcast channel. Some highlights
of the results are: for one static and one dynamic user with delayed CSIT, the achievable degrees
of freedom region partially meets the outer bound. For one static user with perfect CSIT and one
dynamic user with delayed CSIT, the gap between the achievable and the outer sum degrees of
freedom is inversely proportional to the dynamic user coherence time. For each of the considered
CSI conditions, inner and outer bounds were also found for arbitrary number of users.
From these results we conclude that in the broadcast channel, coherence diversity delivers gains
that are distinct from, and augment, the gains from beamforming and interference alignment.
March 26, 2018

DRAFT

30

The authors anticipate that the tools and results of this paper can be helpful for future studies
of hybrid CSIT/CSIR in other multi-terminal networks.
A PPENDIX I
P ROOF

OF

T HEOREM 1

Recall, Mj′ , Mi are the messages of users j = 1, . . . , m′ and i = 1, . . . , m, respectively. We
′
′
enhance the channel by assuming that user j = 1, . . . , m′ knows the messages Mj+1
, . . . , Mm
′

and M1 , . . . , Mm and user i = 1, . . . , m knows the messages Mi+1 , . . . , Mm . Using Fano’s
inequality, chain rule, and data processing inequality we can bound the rates of the static user
j = 1, . . . , m′ ,
′
′
′
′
nRj′ ≤ I(Mj′ ; Yj,1
, . . . , Yj,n
|Mj+1
, . . . , Mm
′ , M1 , . . . , Mm )

(107)

=

n
X

′
I(Mj′ ; Yj,k
|Uj,k )

(108)

≤

n
X

′
′
′
I(Mj′ , Uj,k , Yj−1,1
, . . . , Yj−1,k−1
; Yj,k
|Uj,k )

(109)

=

n
X

′
I(Uj−1,k ; Yj,k
|Uj,k )

(110)

k=1

k=1

k=1

where

′
′
′
′
, . . . , Mm
Uj,k = Mj+1
′ , M1 , . . . , Mm , Yj,1 , . . . , Yj,k−1 ,
′
denotes the received signal of user j at time instant k,
Yj,k

Um′ → · · · → U1 → X → (Y1′ , . . . , Ym′ ′ , Y1 , . . . , Ym)
forms a Markov chain, and U0 = X. The rate of static user m′ can be bounded as
n
X
′
I(Um′ −1,k ; Ym′ ′ ,k |Um′ ,k )
nRm′ ≤
=
≤
=

k=1
n
X

k=1
n
X

k=1
n
X
k=1

March 26, 2018

I(Xk ; Ym′ ′ ,k |Um′ ,k )

−

n
X

I(Xk ; Ym′ ′ ,k |Um′ −1,k )

(111)
(112)

k=1

I(Xk , Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k )

−

n
X

I(Xk ; Ym′ ′ ,k |Um′ −1,k )

(113)

k=1

I(Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k )

+

n
X

I(Xk ; Ym′ ′ ,k |Um′ ,k , Y1,k+1, . . . , Y1,n )

k=1

DRAFT

31

−

n
X

I(Xk ; Ym′ ′ ,k |Um′ −1,k )

(114)

k=1

=

n
X

I(Wk ; Ym′ ′ ,k |Um′ ,k )

+

n
X

I(Xk ; Ym′ ′ ,k |Um′ ,k , Wk )

I(Xk ; Ym′ ′ ,k |Um′ −1,k ),

k=1

k=1

k=1

−

n
X

(115)
n
where Wk = Y1,k+1
. Similarly,

nRi ≤I(Mi ; Yi,1, . . . , Yi,n |Mi+1 , . . . , Mm )
=
=
=

n
X

k=1
n
X

k=1
n
X

(116)

I(Mi ; Yi,k |Vi,k )

(117)

I(Mi , Vi,k , Yi−1,k+1, . . . , Yi−1,n ; Yi,k |Vi,k )

(118)

I(Vi−1,k ; Yi,k |Vi,k ),

(119)

k=1

where we define Vi,k , (Mi+1 , . . . , Mm , Yi,k+1, . . . , Yi,n ), which leads to the Markov chain Vm →
· · · → V1 → (Um′ , W ) → X → (Y1′ , . . . , Ym′ ′ , Y1 , . . . , Ym ) . Using the chain rule and Csiszár
sum identity [37], we obtain the bound (125).

R1 ≤

n
X

I(M1 , . . . , Mm ; Y1,k |V1,k )

(120)

≤

n
X

I(M1 , . . . , Mm , Y1,k+1, . . . , Y1,n ; Y1,k |V1,k )

(121)

=

n
X

I(M1 , . . . , Mm , Y1,k+1, . . . , Y1,n , Ym′ ′ ,1 , . . . , Ym′ ′ ,k−1 ; Y1,k |V1,k )

(122)

k=1

k=1

k=1

−

n
X

I(Ym′ ′ ,1 , . . . , Ym′ ′ ,k−1; Y1,k |M1 , . . . , Mm , Y1,k+1, . . . , Y1,n )

(123)

k=1

=

n
X

=

n
X

I(Um′ ,k , Wk ; Y1,k |V1,k ) −

n
X

I(Y1,k+1, . . . , Y1,n ; Ym′ ′ ,k |Um′ ,k )

(124)

I(Um′ ,k , Wk ; Y1,k |V1,k ) −

n
X

I(Wk ; Ym′ ′ ,k |Um′ ,k ).

(125)

k=1

k=1

k=1

k=1

By introducing a time-sharing auxiliary random variable, Q, [38] and defining
X ,(XQ , Q),
March 26, 2018

′
Yj′ , (Yj,Q
, Q)

DRAFT

32

Yi ,(Yi,Q , Q),

Ui , (Ui,Q , Q)

Vj ,(Vj,Q , Q),

W , (WQ , Q),

(126)

we establish (7)-(10). Similarly, we can follow the same steps to prove (11)-(14) after switching
the role of the two sets of variables Y1′ , . . . , Ym′ ′ and Y1 , . . . , Ym . This completes the proof of
Theorem 1.
A PPENDIX II
M ULTILEVEL B ROADCAST C HANNEL

WITH

D EGRADED M ESSAGE S ETS

Here, we study the capacity of the multiuser multilevel broadcast channel that is characterized


by (6) with degraded message sets. In particular, M0 ∈ 1 : 2nR0 is to be communicated to


all receivers, and furthermore M1 ∈ 1 : 2nR1 is to be communicated to receiver Y1 .5 A threereceiver special case was studied by Nair and El Gamal [22] where the idea of indirect decoding
was introduced, and the capacity is the set of rate pairs (R1 , R0 ) such that

R0 ≤ min I(U; Y2 ), I(V ; Y1′ ) ,
R1 ≤I(X; Y1|U),
R0 + R1 ≤I(V ; Y1′ ) + I(X; Y1|V ),

(127)

for some pmf p(u, v)p(x|v). In the sequel, we give a generalization of Nair and El Gamal for
multiuser multilevel broadcast channel.
Theorem 16: The capacity of multiuser multilevel broadcast channel characterized by (6),
with degraded message sets, is the set of rate pairs (R1 , R0 ) such that

R0 ≤ min I(U; Ym ), I(V ; Ym′ ′ ) ,
R1 ≤I(X; Y1|U),
R0 + R1 ≤I(V ; Ym′ ′ ) + I(X; Y1 |V ),

(128)

for some pmf p(u, v)p(x|v).
Proof: The converse parallels the proof of the converse of the three-receiver case studied
by Nair and El Gamal in [22] after replacing Y2 , Y1′ with Ym , Ym′ ′ , respectively. In particular, U
5

For compactness of expression, here we refer to each receiver by the variable denoting its received signal.

March 26, 2018

DRAFT

33

and V are defined as follows.
Uk , (M0 , Y1,1, . . . , Y1,k−1, Ym,k+1, . . . , Ym,n ),
Vk , (M0 , Y1,1, . . . , Y1,k−1, Ym′ ′ ,k+1, . . . , Ym′ ′ ,n ),
k = 1, . . . , n, and let Q be a time-sharing random variable uniformly distributed over the
set {1, . . . , n} and independent of X n , Y1n , Ym,1 , . . . , Ym,n , Ym′ ′ ,1 , . . . , Ym′ ′ ,n . We then set U =
(UQ , Q), V = (VQ , Q), X = XQ , Y1 = Y1,Q , Ym = Ym,Q , and Ym′ ′ = Ym′ ′ ,Q . This completes the
converse part of the proof.
The achievability part uses superposition coding and indirect decoding as follows.
•

Rate splitting: divide the private message M1 into two independent messages M10 at rate
R10 and M11 at rate R11 , where R1 = R10 + R11 .

•

Codebook generation: fix a pmf p(u, v)p(x|v) and randomly and independently generate


Q
2nR0 sequences un (m0 ), m0 ∈ 1 : 2nR0 , each according to nk=1 pU (uk ). For each m0 ,

randomly and conditionally independently generate 2nR10 sequences v n (m0 , m10 ), m10 ∈ [1 :
Q
2nR10 ], each according to nk=1 pV |U (vk |uk (m0 )). For each pair (m0 , m10 ), randomly and
conditionally independently generate 2nR11 sequences xn (m0 , m10 , m11 ), m11 ∈ [1 : 2nR11 ],
Q
each according to nk=1 pX|V (xk |vk (m0 , m10 )).

•

Encoding: to send the message pair (m0 , m1 ) = (m0 , m10 , m11 ), the encoder transmits

xn (m0 , m10 , m11 ).
•

Decoding at the users Y2 , . . . , Ym : decoder i declares that m̂0i ∈ [1 : 2nR0 ] is sent if it is
(n)

the unique message such that (un (m̂0i ), yin) ∈ Tǫ . Hence, by law of large numbers and
the packing lemma [38], the probability of error tends to zero as n → ∞ if
R0 <

min {I(U; Yi ) − δ(ǫ)},
2≤i≤m

= I(U; Ym ) − δ(ǫ),

(129)

where the last equality follows from applying data processing inequality on the Markov
chain U → X → Y1 → Y2 → · · · → Ym .
•

Decoding at Y1 : decoder 1 declares that (m̂01 , m̂10 , m̂11 ) is sent if it is the unique message

triple such that un (m̂01 ), v n (m̂01 , m̂10 ), xn (m̂01 , m̂10 , m̂11 ), y1n ∈ [1 : 2nR0 ]. Hence, by law
of large numbers and the packing lemma [38], the probability of error tends to zero as

March 26, 2018

DRAFT

34

n → ∞ if
R11 < I(X; Y1|V ) − δ(ǫ),
R10 + R11 < I(X; Y1|U) − δ(ǫ),
R0 + R10 + R11 < I(X; Y1) − δ(ǫ).
•

(130)

Decoding at users Y1′ , . . . , Ym′ ′ : decoder j decodes m0 indirectly by declaring m̃0j is sent
(n)

if it is the unique message such that (un (m̃0j ), v n (m̃0j , m10 ), zjn ) ∈ Tǫ

for some m10 ∈

[1 : 2nR0 ]. Hence, by law of large numbers and packing lemma, the probability of error
tends to zero as n → ∞ if
R0 + R10 <

min

{I(U, V ; Yj′ ) − δ(ǫ)},

min

{I(V ; Yj′ ) − δ(ǫ)},

1≤j≤m′

=

1≤j≤m′

= I(V ; Ym′ ′ ) − δ(ǫ),

(131)

where the last two equalities follow from applying the chain rule and data processing
inequality on the Markov chain U → V → X → Y1′ → Y2′ → · · · → Ym′ ′ .
By combining the bounds in (129), (130), (131), substituting R10 + R11 = R1 , and eliminating
R10 and R11 by the Fourier-Motzkin procedure [22], the proof of the achievability is completed.

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DRAFT

