Real-time broadcasting over block-fading channels

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 tries to broadcast 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 settings, and the best broadcasting technique depends on the distribution of the average channel conditions over the users.

Full text

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
mpm(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.
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