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Real-time Broadcasting over Block-Fading Channels

G. Cocco†, D. G¨und¨uz and C. Ibars

Centre Tecnol`ogic de Telecomunicacions de Catalunya – CTTC

Parc Mediterrani de la Tecnologia, Av. Carl Friedrich Gauss 7 08860, Castelldefels – Spain

{giuseppe.cocco, dgunduz, christian.ibars}@cttc.es

I. ABSTRACT

Broadcast transmission from a base station (BS) to a group

of users is studied. It is assumed that the BS receives data at a

constant rate and transmits these messages to the whole set of

users within a certain deadline. The channels are assumed to

be block fading and independent over blocks and users. Our

performance measure is the total rate of received information

at the users within the transmission deadline. Three different

encoding schemes are proposed, and they are compared with

an informed transmitter upper bound in terms of the average

total reception rate for a set of users with varying channel

qualities. It is shown that no single transmission strategy

dominates for all channel setups, and the best broadcasting

technique depends on the distribution of the average channel

conditions over the users.

II. INTRODUCTION

Consider a satellite or a base station (BS) broadcasting

to a set of users distributed over a geographical area. We

assume a block fading channel model in which the channel

state information (CSI) is available only at the receiver. At

the beginning of each channel block the transmitter is provided

with an independent message whose rate is controlled by an

external source. We assume for simplicity that all the messages

have the same ﬁxed rate. For example, these messages might

correspond to the video packets of a live event whose rate is

ﬁxed by the recording unit, and cannot be changed.

The goal of the BS is to broadcast these data packets to all

the users in the system. Each user wants to receive as many

packets as possible. We further assume a delay constraint on

the transmission, that is, Mmessages that arrive gradually over

Mchannel blocks need to be transmitted by the end of the

last channel block. Hence, the last message sees only a single

channel realization, while the ﬁrst packet can be transmitted

over the whole span of Mchannel blocks.

Performance measure is the total decoded rate at the users.

Note that, for a ﬁnite number of Mpackets and Mchannel

blocks, it is not possible to average out the effect of fading

due to the delay constraint, and there is always a non-zero

outage probability for any message [1]. Hence, we cannot talk

This work was partially supported by the European Commission under

project ICT-FP7-258512 (EXALTED), by the Spanish Government under

project TEC2010-17816 (JUNTOS) and by the Generalitat de Catalunya under

grant 2009-SGR-940. †G. Coccois partially supported bythe European Space

Agency under the Networking/Partnering Initiative.

about a capacity region in the Shannon sense. We will study

the cumulative mass function (c.m.f) of the total decoded rate

as well as the behavior of the average total decoded rate over

a set of users with varying average channel quality.

It is important to identify a transmission scheme that per-

forms well over the whole set of users. In a narrow-beam

satellite system, for instance, the average signal-to-noise ratio

(SNR) experienced by users in different parts of the beam

footprint changes little (in clear sky conditions), while in a

cell-based broadcasting system the SNR experienced by users

in different parts of the cell may vary signiﬁcantly with the

distance from the BS. Hence, it is important for the BS to adapt

the encoding technique to the speciﬁc channel characteristics.

The simplest transmission scheme is to transmit each mes-

sage only over the following channel block. In this scheme,

for any given user each packet will be received with equal

probability. However, for the users with low SNR, this scheme

might lead to a very low average rate. Instead, on the other

extreme, BS can transmit only the ﬁrst message over all chan-

nel blocks, increasing the probability of its correct decoding at

the users that are located at the cell boundary. In general, the

resources for each channel block can be distributed among all

the available messages. This can be achieved in various ways.

In particular we will consider time-division, superposition and

joint encoding schemes, and compare numerically the c.m.f. of

the number of successfully decoded packets for each of these

schemes. We also introduce an upper bound considering the

availability of the CSI at the transmitter.

III. SYSTEM MODEL

We consider broadcasting over a block fading channel that

is constant for a block of nchannel uses. We assume that

the BS receives one new message at the beginning of each

channel block. We consider broadcasting of Mmessages over

Mchannel blocks. Assume that message Wtis available at the

beginning of channel block t,t= 1,...,M. Each message is

chosen randomly with uniform distribution from the set Wt∈

{1,...,2nR}. Equivalently, each message Wthas rate R. All

the messages are addressed to a population of Nusers.

The channel from the BS to user jin block tis given by

yj[t] = hj[t]x[t] + zj[t],

where hj[t]is the channel state, x[t]is the length-nchannel

input vector of BS, zj[t]is the vector of independent and

identically distributed (i.i.d.) unit-variance Gaussian noise, and

W1

W2

W3

WM

X[1]

X[2]

X[3]

X[M]

hi[1]

hi[2]

hi[3]

hi[M]

zi[1]

zi[2]

zi[3]

zi[M]

yi[1]

yi[2]

yi[3]

yi[M]

(ˆ

W1,ˆ

W2,..., ˆ

WM)

Fig. 1. Equivalent channel model for the sequential transmission of M

messages over Mblocks of the fading channel to a single receiver.

yj[t]is the length-nchannel output vector of user i. We

assume that the channel coefﬁcients hj[t]are i.i.d. with zero-

mean unit variance complex Gaussian. These instantaneous

channel gains are known at the receiving end of each link,

while the BS only has a statistical information. We have a

short-term average power constraint of P, i.e., E[x[t]x[t]†]≤

nP for t= 1,...,M.

The channel from the source to each receiver can be seen

as a multiple access channel (MAC) with a special message

hierarchy [2], in which the encoder at each channel block acts

as a separate transmitter and each user tries to decode as many

of the messages as possible. See Fig. 1 for an illustration

of this channel model. We denote the instantaneous channel

capacity to user jover channel block tby Cj

t:

Cj

t,log2(1 + φj[t]P).(1)

Note that Cj

tis a random variable, and due to the random

nature of the channel, it is not possible to guarantee any non-

zero rate to any user at any channel block. We consider the

c.m.f. of the total average decoded rate at each user as our

performance measure.

IV. SINGLE USER SCENARIO

In this section we focus on a single user. For simplicity of

notation, we drop the subscripts indicating the user index to

simplify the notation. We introduce three different transmis-

sion schemes.

A. Time Sharing Transmission

One of the resources that the encoder can allocate among

different messages is the total number of channel uses within

each channel block. While the whole of the ﬁrst time slot has

to be dedicated for message W1, as it is the only available

message, the second time slot can be divided among the

messages W1and W2, and so on so forth. Assume that the

encoder divides the channel block tinto tportions α1t,...,αtt

such that αit ≥0and Pt

i=1 αit = 1. In channel block t,

αitnchannel uses is allocated for the transmission of message

Wi. A constant power Pis used throughout the block. Then

the total amount of received mutual information relative to

message Wiis:

Itot

i,

M

X

t=i

αit log2(1 + φ[t]P).(2)

Different time allocations among the messages lead to different

c.m.f. for the total decoded rate. For simplicity, we assume

equal time allocation among all available messages, that is,

for i= 1,...,M, we have αit =1

tfor t=i, i + 1,...,M,

and αit = 0 for t= 1,...,i. Hence,

Itot

i=

M

X

t=i

1

tlog2(1 + φ[t]P).(3)

In this scheme the messages that arrive earlier are allocated

more resources, and hence, are more likely to be decoded. We

have Itot

i> Itot

jfor 1< i < j < M . Hence, the probability

of decoding exactly mmessages is:

η(m),P r{Itot

m+1 < R < Itot

m}.(4)

B. Superposition Transmission

Next we consider superposition encoding (SE). In SE the

source generates a Gaussian codebook of size 2nR for each

message to be transmitted in each block. In channel block t, it

transmits the superposition of the tcodewords, chosen from t

different codebooks generated independently, corresponding to

messages {W1,...,Wt}. The codewords are scaled such that

the average total transmit power is Pin each block. In the ﬁrst

block, only information about message W1is transmitted with

average power P11 =P; in the second block we divide the

total power Pamong the two messages, allocating P12 and P22

for the codewords corresponding to W1and W2, respectively.

In general, over channel block twe allocate average power

Pit for the codeword corresponding to message Wi, while

Pt

i=1 Pit =P. We let Pdenote the M×Mupper triangular

power allocation matrix such that Pi,t =Pit.

Let Sbe any subset of the set of messages M=

{1,...,M}. We deﬁne C(S)as follows:

C(S),

M

X

t=1

log2 1 + φ[t]Ps∈S Pst

1 + φ[t]Ps∈M\S Pst !.(5)

This provides an upper bound on the total rate of messages in

set Sthat can be decoded jointly at the user considering the

codewords corresponding to the remaining messages as noise.

The receiver ﬁrst checks if any of the messages can be

decoded alone by considering the other transmissions as noise.

If a message can be decoded, the corresponding signal is sub-

tracted and the algorithm is run over the remaining signal. If

no message can be decoded alone, then the receiver considers

joint decoding of message pairs, followed by triplets, and so

on so forth. This optimal decoding algorithm for superposition

coding to ﬁnd the total decoded rate at the receiver is outlined

in Algorithm 1 below. The user calls the algorithm with

Rate = 0 and M={1,...,M}initially. While Algorithm 1

Algorithm 1 Total Decoded Rate (Rate,M,P)

Old Rate = 0

for i= 1 to |M| do

if iR ≤maxS:S⊆M,|S |=iC(S)then

Rate =Rate +iR

M=M\S

end if

end for

if (M 6=∅)&(Old Rate < Rate)then

Total_Decoded_Rate (Rate,M,P)

else

Output= Rate

end if

gives us the maximum decoded total rate, it is hard in general

to ﬁnd a closed form expression for the average decoded total

rate, and optimize it over power allocation matrices. Hence, we

focus here on two special cases. In memoryless transmission

(MT) scheme, we consider a diagonal power allocation matrix

P, that is, each message is transmitted over a single channel

block. In equal power allocation (EPA) scheme, we divide

the total average power Pamong all the available messages

at each channel block. The power allocation matrix Ptakes

the following form:

PEP A =

PP

2

P

3... P

M

0P

2

P

3... P

M

.

.

.0P

3... P

M

.

.

..

.

.0... P

M

.

.

..

.

..

.

..

.

..

.

.

0... ... 0P

M

(6)

where PEP A

j,t is the power allocated to message jin block t.

In MT, messages can be decoded independently, and joint

decoding is not needed. Wtcan be decoded if and only if

log2(1 + φ[t]P)≥R. (7)

Due to the i.i.d. nature of the channel over blocks, successful

decoding probability is constant over messages. We deﬁne

p,P r φ[t]>2R−1

P=Z∞

2R−1

P

fΦ(φ)dφ =e−2R−1

P,(8)

where fΦ(φ)is the p.d.f. of φ[t]. The probability that exactly

mmessages are decoded is given by

η(m) = M

mpm(1 −p)M−m.(9)

Note that, we have a closed-form expression for η(m)in MT

for any regime of the channel SNR. If we let the number of

R

R

R

R

R

R

R

2R

2R

2R

2R

2R

2RC1

C1

C2

C2

0

0

Fig. 2. Total decoded rate regions in the (C1, C2)domain in the case of

M= 2 messages for independent encoding (on the left) and joint encoding

(on the right) schemes.

messages Mgo to inﬁnity, then (9) can be approximated with

a Gaussian distribution, i.e.,

η(m)≃1

p2πM p(1 −p)e−(m−M p)2

2Mp(1−p).(10)

Then the average achievable rate is

R=RE[m] = R

M

X

m=0

mη(m)≃RM p, (11)

where the approximation is tighter for higher values of M.

C. Joint Encoding Transmission

In the superposition scheme, we generate independent code-

books for each message available at the BS at each channel

block and transmit the superposition of the corresponding

codewords. Another possibility is to generate a single multiple-

index codebook for each channel block. We call this the joint

encoding (JE) scheme.

In the JE scheme, the transmitter generates a tdimensional

codebook to be used in channel block tfor t= 1,...,M.

That is, for channel block t, we generate a codebook of

size s1× · · · × stsuch that si= 2nR ,∀i∈ {1,...,t},

with Gaussian distribution, and index them as xn

t(m1,...,mt)

where mi∈[1,2nR]for i= 1,...t. The receiver uses joint

typicality decoder and tries to estimate as many messages as

possible at the end of block M. With high probability, it will

be able to decode the ﬁrst mmessages correctly if,

(m−j+ 1)R≤

m

X

t=j

Ct(12)

for all j= 1,2,...,m.

As a comparison, we illustrate the achievable rate regions

for the MT and JE schemes in the case of M= 2 in Fig. 2.

In the case of memoryless transmission, a total rate of 2Ris

achieved if both capacities C1and C2are above R. We achieve

a total rate of Rif only one of the capacities is above R. On

the other hand, in the case of joint encoding, we tradeoff a

part of the region of rate Rfor rate 2R, that is, we achieve a

rate of 2Rinstead of rate R, while rate 0is achieved rather

than rate Rin the remaining part.

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of decoded packets

CMF

Time−sharing

Memoryless

Superposition

Joint encoding

Upper bound

Fig. 3. CMF of the number of decoded packets for the different techniques

considered. In the simulations we set R= 1 bit/sec/Hz, M= 50 and P=

1.44 dB.

We deﬁne functions fm(R), for m= 0,1,...,M, as

follows:

fm(R) = (1,if (m−j+ 1)R≤Pm

t=jCt, j = 1,...,m

0,otherwise.

Exactly mmessages, m= 0,1,...,M, can be decoded if,

Cm≥R(13)

Cm−1+Cm≥2R(14)

···

C1+···+Cm≥mR, (15)

and

Cm+1 < R (16)

Cm+1 +Cm+2 <2R(17)

···

Cm+1 +···+CM<(M−m)R. (18)

Then the probability of decoding exactly mpackets can be

written as,

η(m) = P r fm(R) = 1 and fm+1(R) = 0.(19)

Then (19) can be calculated as in Eqn. (13) at the bottom

of the page, where we have deﬁned x+= max{0, x}, and

fC1···Cm(c1···cm)is the joint p.d.f. of C1,...,Cm, equal to

the product of the marginal p.d.f.’s due to independence.

D. Informed Transmitter Upper Bound

We study an upper bound on the average decoded rate

obtained by assuming the availability of the CSI at the BS.

20 40 60 80 100

0

10

20

30

40

50

60

Total number of messages (M)

Average total number of decoded packets

Time−sharing

Memoryless

Superposition

Joint encoding

Upper bound

Fig. 4. Average total rate achieved plotted against the total number of packets

Mfor a transmission rate R= 1 bit/s/Hz, P=−3 dB.

There will still be a non-zero probability of outage for each

message due to the short-time power constraint which prevents

power allocation among time slots; but the CSI allows the BS

to transmit over the channel blocks in a manner to maximize

the total decoded rate at each channel realization.

Assume that a total of m≤Mpackets can be decoded

at some channel realization. We can always have the ﬁrst m

messages to be the successfully decoded ones by reordering.

At any channel realization, the ﬁrst mmessages can be

decoded successfully if and only if [2],

R≤Cm+Cm+1 +···+CM,

2R≤CM−1+Cm+···+CM,

···

mR ≤C1+C2+···+CM.

We can equivalently write these conditions as

R≤min

i∈{1,...,m}

1

m−i+ 1

M

X

j=i

Cj

.(20)

Then, for each channel realization, the upper bound on the

total decoded rate is given by m∗R, where m∗is the greatest

mvalue that satisﬁes (20). We obtain the upper bound on the

average total decoded rate by averaging m∗Rover the channel

realizations.

E. Numerical Results for Single User Scenario

In this subsection we provide several numerical results

comparing the proposed transmission schemes and the upper

bound. In Fig. 3 the c.m.f. of the number of decoded packets is

shown for the different techniques for M= 50 and P= 1.44

η(m) = Z∞

RZ∞

(2R−xm)+

···Z∞

(mR−xm−···−x2)+

fC1···Cm(x1,...,xm)dx1···dxm

×ZR

0Z2R−xm+1

0

· · · Z(M−m)R−xm+1−···−xM−1

0

fCm+1···CM−m(xm+1 ,...,xM−m)dxm+1 ···dxM(13)

20 40 60 80 100

0

10

20

30

40

50

60

70

80

90

100

Total number of messages (M)

Average total number of decoded packets

Time−sharing

Memoryless

Superposition

Joint encoding

Upper bound

Fig. 5. Average total rate achieved plotted against the total number of packets

Mfor a transmission rate R= 1 bit/s/Hz, P= 2 dB.

dB. From the ﬁgure it is evident that MT outperforms SE and

TS, as its c.m.f. lays below the other two. On the other hand,

the improvement of the JE scheme with respect to the other

methods depends on the performance metric we choose. For

instance, JE has the lowest probability to decode more than m

packets, for m≤15, while the same scheme has the highest

probability to decode more than mpackets for m≥22.

In Fig. 4 and 5 the total average rate is plotted against the

total number of messages Mfor channel SNR values equal

to −3and 2 dB, respectively, and a message rate of R= 1.

While JE outperforms other schemes at SNR = 2 dB, it has

the poorest performance at S N R =−3 dB. The opposite

applies to the TS scheme.

V. BROADCAST SCENARIO

In Section IV we have focused on the average rate decoded

by a single user. We now consider the broadcasting scenario

in which the BS wants to broadcast Mmessages to a group

of users which are located at different distances from the

BS. In this case we model the average channel power at

node ias d−α

i, where diis the distance from BS to node

iand αis the path loss exponent. Note that each proposed

transmission scheme has a different behavior in terms of the

c.m.f. of the received messages at different channel SNR

values. A technique that may perform well at a given channel

SNR, may perform poorly, compared to other schemes, at

another SNR value. In the broadcast scenario, what becomes

important is the range of average channel SNR values of the

receivers, and to use a transmission scheme that performs

well over this range. For instance, in a system in which all

users have the same average SNR, which is the case for a

narrow-beam satellite system where the SNR within the beam

footprint has variations of at most a few dB on average, the

transmission scheme should perform well around the average

SNR of the beam. A similar situation may occur in a microcell,

where the relatively small radius of the cell implies a limited

variation in the average SNR range experienced by the users

at different distances from the BS instead. Instead, in the case

of a macrocell, in which the SNR may vary signiﬁcantly from

2 4 6 8 10

10−3

10−2

10−1

100

101

102

Distance

Average total number of decoded packets

Time−sharing

Memoryless

Superposition

Joint encoding

Upper bound

Fig. 6. The average total number of decoded messages against distance for

the proposed schemes and the upper bound. In the simulations we set R= 1

bit/sec/Hz, M= 100 and P= 20 dB.

the proximity of the BS to the edge of the cell, the BS should

adopt a scheme which performs well over a larger range of

SNR values. For a given scenario the transmitter can choose

the transmission scheme based on this average behavior.

A. Numerical Results for Broadcast Scenario

We present numerical results assuming that the users are

placed at increasing distances from the BS. The average

number of decoded messages is plotted against the distance

from the base station in Fig. 6. We see that there is no scheme

that outperforms the others in the whole range of distances

considered. In the range up to d= 4 the JE scheme achieves

the highest total number of decoded packets while for d≥6

the TS scheme outperforms the others. We see how the upper

bound is tighter at smaller values of d. This is because the

channel knowledge at the BS becomes more important as the

SNR decreases.

VI. CONCLUSIONS

We have considered a BS broadcasting to a set of users,

with the BS being provided with an independent message at a

ﬁxed rate at the beginning of each channel block. We have used

the average total decoded rate as our performance metric. We

have considered time-division, superposition and joint encod-

ing schemes, and compared numerically their performances.

An upper bound has also been introduced considering the

availability of the CSI at the transmitter. We have showed

that no single transmission strategy dominates for all channel

setups, and the best broadcasting technique depends on the

distribution of the average channel conditions over the users.

REFERENCES

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theoretic and communications aspects,” IEEE Transactions on Informa-

tion Theory, vol. 44, no. 6, pp. 2619–2692, October 1998.

[2] V. V. Prelov, “Transmission over a multiple access channel with a special

source hierarchy,” Probl. Peredachi Inf., vol. 20, no. 4, pp. 3–10, 1984.