Content uploaded by Giuseppe Cocco
Author content
All content in this area was uploaded by Giuseppe Cocco
Content may be subject to copyright.
1
Streaming Transmission over Block Fading
Channels with Delay Constraint
Giuseppe Cocco†, Deniz G¨und¨uz∗and Christian Ibars†
†CTTC, Barcelona, Spain
∗Imperial College London, London, UK
giuseppe.cocco@cttc.es, d.gunduz@imperial.ac.uk, christian.ibars@cttc.es
Abstract—Streaming transmission over a block fading channel
is studied assuming that the transmitter receives a new message
at each channel block at a constant rate, which is fixed by an
underlying application. A common deadline is assumed for all
the messages, at which point the receiver tries to decode as many
messages as possible. Various achievable schemes are proposed
and compared with an informed transmitter upper bound in
terms of average throughput. It is shown that the adaptive
joint encoding (aJE) scheme is asymptotically optimal; that is, it
achieves the ergodic capacity as the transmission deadline goes
to infinity; and it closely follows the upper bound in the case of a
finite transmission deadline. On the other hand, in the presence
of multiple receivers with different signal-to-noise ratios (SNR),
memoryless transmission (MT), generalized time-sharing (gTS)
and superposition transmission (ST) schemes are shown to be
more robust than the joint encoding (JE) scheme as they have
gradual performance degradation with the decreasing SNR.
Index Terms—Block-fading; Delay-constrained transmission;
Multimedia streaming; Multiple access channel; Ergodic capac-
ity; Satellite broadcasting
I. INTRODUCTION
In a streaming transmitter data becomes available at the
transmitter over time rather than being available at the be-
ginning of transmission. Consider, for example, digital TV
satellite broadcasting. The satellite receives video packets
from a gateway on Earth at a fixed data rate and has to
forward the received packets to the users within a certain
deadline. Hence, the transmission of the first packet starts
before the following packets arrive at the transmitter. We
consider streaming transmission over a block fading channel
with channel state information (CSI) available only at the
receiver. This assumption results from practical constraints
when the receiver belongs to a large population of terminals,
or when the transmission delay is significantly larger than
This is the pre-peer reviewed version of the following article: Giuseppe
Cocco, Deniz G¨und¨uz and Christian Ibars, “Streaming Transmission over
Block Fading Channels with Delay Constraint”, IEEE Transactions on
Wireless Communications, Vol. 12, Issue 9, September 2013. This work
was partially supported by the European Commission under project ICT-FP7-
258512 (EXALTED), by the Spanish Government under projects TEC2010-
17816 (JUNTOS) and TEC2010-21100 (SOFOCLES), and by the Govern-
ment of Catalonia under grant 2009-SGR-940. Giuseppe Cocco is partially
supported by the European Space Agency under the Networking/Partnering
Initiative. Deniz G¨und¨uz was supported by the European Commission’s Marie
Curie IRG Fellowship with reference number 256410 (COOPMEDIA) under
the Seventh Framework Programme.
the channel coherence time1[2]. The data that arrives at
the transmitter over a channel block can be modeled as an
independent message whose rate is fixed by the quality of
the gateway-satellite link and the video encoding scheme
used for recording the event. We assume that the transmitter
cannot modify the contents of the packets to change the data
rate, as the satellite transmitter is oblivious to the underlying
video coding scheme adopted by the source, and considers
the accumulated data over each channel block as a single data
packet that can be either transmitted or dropped.
We further impose a delay constraint on the transmission
such that the receiver buffers the received messages for M
channel blocks before displaying the content, which is typical
of multimedia streaming applications (see Fig. I). As the
messages arrive at the transmitter gradually over Mchannel
blocks, the last message sees only a single channel realization,
while the first message can be transmitted over the whole span
of Mchannel blocks. For a finite number Mof messages
and Mchannel blocks, it is not possible to average out the
effect of fading in the absence of CSI at the transmitter, and
there is always a non-zero outage probability [3]. Hence, the
performance measure we study is the average throughput, that
is, the average decoded data rate by the user.
Communication over fading channels has been extensively
studied [4]. The capacity of a fading channel depends on the
available information about the channel behavior [5]. When
both the transmitter and the receiver have CSI, the capacity
is achieved though waterfilling [6]. This is called the ergodic
capacity as the capacity is averaged over the fading distribu-
tion. In the case of a fast fading channel without CSI at the
transmitter ergodic capacity is achieved with constant power
transmission [4]. However, when there is a delay requirement
on the transmission as in our model, and the delay constraint is
short compared to the channel coherence time, we have a slow
fading channel. In a slow-fading channel, if only the receiver
can track the channel realization, outage becomes unavoidable
[4]. An alternative performance measure in this case is the ǫ-
1Transmission rate can be adjusted to the channel state through adaptive
coding and modulation (ACM) driven by a feedback channel. However,
in real-time broadcast systems with large delays and many receivers, such
as satellite systems, this is not practical. For instance, in real-time video
transmission the ACM bit-rate control-loop may drive the source bit-rate (e.g.,
variable bit rate video encoder), but this may lead to a large delay (hundreds
of milliseconds) in executing rate variation commands. In such cases the total
control loop delay is too large to allow real time compensation of fading [1,
Section 4.5.2.1].
2
W1W2W3WM
t
Deadline
0n2n(M−1)nMn
Ch. block 1 Ch. block 2 Ch. block M
Fig. 1. The transmitter receives message Wiof rate Rat the beginning of channel block i. All the Mmessages need to be transmitted to the receiver by
the end of channel block M.
outage capacity [7]. In general it is hard to characterize the
outage capacity exactly; hence, many works have focused on
either the high signal-to-noise ratio (SNR) [8] or low SNR [9]
asymptotic regimes. Another approach, which is also adopted
in this work, is to study the average transmission rate, i.e.,
average throughput, as in [10] and [11]. Outages may occur
even if the transmitter has access to CSI when the system is
required to sustain a constant transmission rate at all channel
states, called the delay-limited capacity [12], [13]. Due to the
constant rate of the arriving messages at all channel blocks,
our problem is similar to the delay-limited capacity concept.
However, here we neither assume CSI at the transmitter nor
require all arriving messages to be transmitted. Our work also
differs from the average rate optimization in [10] since the
transmitter in [10] can adapt the transmission rate based on the
channel characteristics and the delay constraint, whereas in our
model the message rate is fixed by the underlying application.
The only degree-of-freedom the transmitter has in our setting
is the multiple channel blocks it can use for transmitting the
messages while being constrained by the causal arrival of the
messages and the total delay constraint of Mchannel blocks.
Streaming transmission has received significant attention
recently especially with the increasing demand for multimedia
streaming applications [14]. Most of the work in this area
focus on practical code construction [15], [16], [17]. The
diversity-multiplexing tradeoff in a streaming transmission
system with a maximum delay constraint for each message
is studied in [18]. Unlike in [18], we assume that the whole
set of messages has a common deadline; hence, in our setting
the degree-of-freedom available to the first message is higher
than the one available to the last.
In the present paper we extend our work in [19] by
presenting analytical results and introducing more effective
transmission schemes. We study joint encoding (JE), which
encodes all the available messages into a single codeword at
each channel block, and the more classical time-sharing (TS)
and superposition (ST) coding schemes. The main contribu-
tions of the present work can be summarized as follows:
1) We introduce a channel model for streaming transmission
over block fading channels with a common decoding
deadline to study real-time multimedia streaming in net-
works with large delays, such as digital satellite broad-
casting systems.
2) We introduce an informed transmitter upper bound on the
performance assuming the availability of perfect CSI at
the transmitter.
3) We find a simplified expression for the average through-
put of the JE scheme, and use this expression to show
that, in the limit of infinite channel blocks, the JE scheme
has a threshold behavior that depends on the average
channel quality.
4) We propose the adaptive JE (aJE) scheme, which drops
certain packets depending on the average channel SNR,
and show that it performs very close to the informed
transmitter upper bound for a finite number of messages,
and approaches the ergodic capacity as the number of
channel blocks goes to infinity.
5) We propose a generalized time-sharing (gTS) scheme,
in which each message is transmitted over a window of
Wchannel blocks through time-sharing. We show that
optimizing the window size Wsignificantly improves the
average throughput in the high SNR regime.
6) We show that the gTS and the ST schemes provide
gradual performance improvement with increasing aver-
age SNR as opposed to the threshold behavior of the
JE scheme. Focusing on the gTS scheme with equal
time allocation and the ST scheme with equal power
allocation, we show, through numerical simulations, that
either scheme can outperform the other depending on the
average SNR. Both schemes can be further improved by
optimizing the time or power allocation, respectively.
7) We show that the aJE scheme is advantageous in a single
receiver system whereas the simple gTS and ST schemes
can be attractive when broadcasting to multiple users with
a wide range of SNR values, or in a point-to-point system
with inaccurate CSI.
The rest of the paper is organized as follows. In Section II we
describe the system model. In Section III we provide an upper
bound on the average throughput. In Section IV we describe
the proposed transmission schemes in detail. Section V is
devoted to the numerical results. Finally, Section VI contains
the conclusions.
II. SYSTEM MODEL
We consider streaming transmission over a block fading
channel. The channel is constant for a block of nchannel
uses and changes in an independent and identically distributed
(i.i.d.) manner from one block to the next. We assume that the
transmitter accumulates the data that arrives at a fixed rate
during a channel block, and considers the accumulated data
as a single message to be transmitted during the following
channel blocks. We consider streaming of Mmessages over
Mchannel blocks, such that message Wtbecomes available
at the beginning of channel block t, for t= 1,...,M (see Fig.
I). Each message Wthas rate Rbits per channel use (bpcu),
i.e., Wtis chosen randomly with uniform distribution from
the set Wt={1,...,2nR}, where nis the number of channel
uses per channel block. Following a typical assumption in the
3
W1
W2
W3
WM
x[1]
x[2]
x[3]
x[M]
h[1]
h[2]
h[3]
h[M]
z[1]
z[2]
z[3]
z[M]
y[1]
y[2]
y[3]
y[M]
(ˆ
W1,ˆ
W2,..., ˆ
WM)
Fig. 2. Equivalent channel model for the sequential transmission of M
messages over Mchannel blocks to a single receiver.
literature (see, e.g., [10]), we assume that n, though still large
(as to give rise to the notion of reliable communication [20]),
is much shorter than the dynamics of the slow fading process.
The channel in block tis given by
y[t] = h[t]x[t] + z[t],(1)
where h[t]∈Cis the channel state, x[t]∈Cnis the channel
input, z[t]∈Cnis the i.i.d. unit-variance Gaussian noise, and
y[t]∈Cnis the channel output. The instantaneous channel
gains are known at the receiver, while the transmitter only
has knowledge of the statistics of the process h[t]. We have a
short-term average power constraint of P, i.e., E[x[t]x[t]†]≤
nP for t= 1,...,M, where x[t]†represents the Hermitian
transpose of x[t], and E[x]is the mean value of x. As we
assume a unitary noise power, in the following we will use
interchangeably the quantities Pand SN R.
This point to point channel can be seen as an orthogonal
multiple access channel (MAC) with a special message hier-
archy [21], in which the encoder at each channel block acts
as a separate virtual transmitter (see Fig. 2). The receiver
tries to decode as many of the messages as possible, and
the performance measure is the average throughput, denoted
by Rand defined as the normalized average number of
received messages multiplied by the transmission rate. We
denote the instantaneous channel capacity over channel block
tby Ct,log2(1 + φ[t]P), where φ[t]is a random variable
distributed according to a generic probability density function
(pdf) fΦ(φ). Note that Ctis also a random variable. We
define C,E[log2(1 + φP )], where the expectation is taken
over fΦ(φ).Cis the ergodic capacity when there is no delay
constraint on the transmission.
III. INFORMED TRANSMITTER UPPER BOUND
We first provide an upper bound on the performance by as-
suming that the transmitter is informed about the exact channel
realizations at the beginning of the transmission. This allows
the transmitter to optimally allocate the resources among
messages so that the average throughput Ris maximized.
Assume that C1,...,CMare known by the transmitter and
the maximum number of messages that can be decoded is
m≤M. We can always have the first mmessages to be the
successfully decoded ones by reordering. When the channel
state is known at the transmitter, the first mmessages can be
decoded successfully if and only if [21]:
iR ≤Cm−i+1 +Cm−i+2 +···+CM,for i= 1,...,m.
We can equivalently write these conditions as
R≤min
i∈{1,...,m}
1
m−i+ 1
M
X
j=i
Cj
.(2)
Then, for each channel realization {h[1],...,h[M]}, the upper
bound on the average throughput is given by m∗
MR, where m∗
is the greatest mvalue that satisfies (2). This is an upper
bound for each specific channel realization. An upper bound
on Rcan be obtained by averaging this upper bound over the
distribution of channel realizations.
Another upper bound on Rcan be found from the ergodic
capacity assuming all messages are available at the encoder at
the beginning, and letting Mgo to infinity. Finally, the bound
R≤Rfollows naturally from the data arrival rate. Thus, R
is bounded above by min R, C .
IV. TRANSMISSION SCHEMES
The most straightforward transmission scheme consists in
transmitting each message only within the channel block
following its arrival, and discard it. This is called memoryless
transmission (MT). Due to the i.i.d. nature of the channel,
successful decoding probability is constant over messages. De-
noting this probability by p,P r {Ct≥R}, the probability
that exactly mmessages are decoded is
η(m),M
mpm(1 −p)M−m.(3)
The average throughput of the MT scheme RMT is
R
MPM
m=1 mη(m) = Rp. The MT scheme treats all messages
equally. However, depending on the average channel condi-
tions, it might be more beneficial to allocate more resources
to some of the messages in order to increase the average
throughput. In the following, we will consider three transmis-
sion schemes based on different types of resource allocation.
We will find the average throughput for these schemes and
compare them with the upper bound introduced in Section III.
A. Joint Encoding (JE) Transmission
In the joint encoding (JE) scheme we generate a single
multiple-index codebook for each channel block. JE is also
studied in [18] in terms of the diversity-multiplexing tradeoff
(DMT) with a per-message delay constraint rather than the
common deadline constraint we consider. Moreover, unlike in
[18], here we study the JE scheme in terms of its average
throughput for a fixed message rate R.
For channel block t, we generate a tdimensional codebook
of size s1×···×st,si= 2nR,∀i∈ {1,...,t}, with Gaussian
distribution, and index the codewords as xt(W1,...,Wt),
where Wi∈ W ={1,...,2nR}for i= 1,...,t. The receiver
4
R
R
R
R
R
R
R
2R
2R
2R
2R
2R
2RC1
C1
C2
C2
0
0
Fig. 3. Total decoded rate regions illustrated on the (C1, C2)plane with
M= 2 messages for MT (on the left) and JE (on the right) schemes.
uses joint typicality decoder and tries to estimate as many
messages as possible at the end of block M. The decoder will
be able to decode the first mmessages correctly if [21]:
(m−j+ 1)R≤
m
X
t=j
Ct,∀j= 1,2,...,m. (4)
As a comparison, we illustrate the achievable rate regions
for MT and JE schemes for M= 2 in Fig. 3. In the case of
MT, a total rate of 2Rcan be decoded successfully if both
C1and C2are above R. We achieve a total rate of Rif only
C1or C2is above R. On the other hand, in the case of joint
encoding, we tradeoff a part of the region of rate Rfor rate
2R.
Using the conditions in Eqn. (4) we define functions gm(R),
for m= 0,1,...,M, as follows:
gm(R) = (1,if (m−j+ 1)R≤Pm
t=jCt, j = 1,...,m,
0,otherwise.
Then the probability of decoding exactly mmessages can be
written as,
η(m) = P r gm(R) = 1 and gm+1(R) = 0.(5)
After some manipulation, it is possible to prove that exactly
mmessages, m= 0,1,...,M, can be decoded if:
Cm−i+1 +···+Cm≥iR, i = 1,...,m, (6)
Cm+1 +···+Cm+i< iR, i = 1,...,M −m. (7)
Then η(m)can be calculated as in Eqn. (8) at the bottom of
the next page, where we have defined x+= max{0, x}, and
fC1···Cm(c1,...,cm)as the joint pdf of C1,...,Cm, which is
equal to the product of the marginal pdf’s due to independence.
The probability in Eqn. (8) cannot be easily evaluated for
a generic M. However, we provide a much simpler way to
calculate the average throughput RJ E . This simplification is
valid not only for i.i.d. but also for conditionally i.i.d. channels.
Random variables {C1,···, CM}are said to be conditionally
i.i.d. given a random variable Uif the joint distribution is of
the form
fC1,···,CM,U (c1,...,cM, u)
=fC1|U(c1|u)× · ·· × fCM|U(cM|u)fU(u),
(9)
where
fCi|U(ci|u) = fCj|U(cj|u),∀i, j ∈ {1,...M}.(10)
Note that i.i.d. channels is a particular case of conditionally
i.i.d. channels where Uis a constant.
Theorem 1: The average throughput for the JE scheme in
the case of conditionally i.i.d. channel capacities is given by:
RJE =R
M
M
X
m=1
P r{C1+···+Cm≥mR}.(11)
Proof: See Appendix A.
In general it is still difficult to find an exact expression for
RJE , but Theorem 1simplifies the numerical analysis signif-
icantly. Moreover, it is possible to show that RJ E approaches
Rfor large Mif C > R. To prove this, we rewrite Eqn. (11)
as:
RJE =R−R
M
M
X
m=1
am,(12)
where we have defined
am,P r C1+···+Cm
m< R.(13)
It is sufficient to prove that, if C > R, then
limM→∞ PM
m=1 am=c, for some 0< c < ∞. We start
by noting that limm→+∞am= 0, since, by the law of large
numbers, C1+···+Cm
mconverges to Cin probability as mgoes
to infinity. To prove the convergence of the series sum we
show that
lim
m→+∞
am+1
am
=λ, (14)
with 0< λ < 1.
We define:
lm,√mC−C1+···+Cm
m, m = 1,2,...,M, (15)
where each lmis a random variable with zero mean and
variance σ2
c, which corresponds to the variance of the channel.
From the central limit theorem we can write:
lim
m→+∞
am+1
am
= lim
m→+∞
P r nlm+1 >C−r
1/√m+1 o
P r nlm>C−r
1/√mo(16)
= lim
m→+∞
QC−R
σc/√m+1
QC−R
σc/√m(17)
≤lim
m→+∞
σc/√m+1
(C−R)√2πe−1
2C−R
σc/√m+1 2
C−R
σc/√m
1+C−R
σc/√m21
√2πe−1
2C−R
σc/√m2(18)
= lim
m→+∞
σ2
c+m(C−R)2
pm(m+ 1)(C−R)2e−(C−R)2
2m+1
σ2
c−m
σ2
c(19)
=e−(C−R)2
2σ2
c<1,(20)
5
where equality (17) follows from the fact that lmconverges in
distribution to a Gaussian random variable with zero mean and
variance σ2
c, while inequality (18) follows from the bounds on
the Q-function:
x
(1 + x2)√2πe−x2
2< Q(x)<1
x√2πe−x2
2,(21)
for x > 0. Similarly, we prove that if C < R, the average
rate tends to zero asymptotically with M. To see this, we
consider the series in Eqn. (11) defining bm=P r{C1+···+
Cm≥mR}. We want to prove that RJ E =M−1PM
m=1 bm
converges to zero. It is sufficient to prove that PM
m=1 bm
converges to a constant. We first notice that limm→+∞bm= 0
by the law of large numbers, as M−1PM
m=1 Cmconverges in
probability to C, which is lower than Rby hypothesis. One
can also show that limm→+∞bm+1
bm= 0; and hence, RJ E goes
to zero with M. Overall we see that the average rate of the
JE scheme shows a threshold behavior, i.e., we have:
lim
M→∞ RJE =(R, if R < C
0,if R > C. (22)
1) Adaptive Joint Encoding (aJE) Transmission: Eqn. (22)
indicates a phase transition behavior such that RJ E is zero
even for large Mif R > C. While the transmission rate R
cannot be modified by the transmitter, it may choose to trans-
mit only a fraction α=M′
M<1of the messages, allocating the
extra M−M′channel blocks to the M′messages, effectively
controlling the transmission rate. In other words, the M′
messages are encoded and transmitted in M′channel blocks
as described above, while each of the remaining M−M′
blocks is divided into M′equal parts, and the encoding process
used for the first M′blocks is repeated, using independent
codewords, across the M′parts of each block. For instance,
if M= 3 and M′= 2,x1(W1)and x2(W1, W2)are
transmitted in the first and second channel blocks, respectively.
The third channel block is divided into M′= 2 equal parts
and the independent codewords x31 (W1)and x32(W1, W2)
are transmitted in the first and in the second half of the block,
respectively. We call this variant of the JE scheme adaptive
JE (aJE) scheme. The conditions for decoding exactly m
messages, m= 0,1,...,M′, in aJE can be obtained from
those given in Eqn. (6) and Eqn. (7) by replacing Ciwith
C∗
i=Ci+1
M′PM
j=M′+1 Cj,i∈ {1, . . . , M′}. Note that the
random variables C∗
i,i∈ {1,...,M′}, are conditionally i.i.d.,
i.e., they are i.i.d. once the variable U=1
M′PM
j=M′+1 Cjis
fixed. This implies that Theorem 1 holds.
In Appendix B we prove that, by choosing αsuch that αR ≤
C, we can have:
lim
M→∞ RaJE = min{R, C }.(23)
Eqn. (23) suggests that the average transmission rate can be
adapted at the message level while keeping a fixed rate at the
physical layer. Comparing the bound min R, Cand Eqn.
(23) we see that the aJE scheme achieves the optimal average
throughput in the limit of infinite M; hence, as the number
of messages and the channel blocks go to infinity, the aJE
scheme achieves the optimal performance. We will show in
Section V through numerical analysis that near optimality of
the aJE scheme is valid even for finite M. However, when
there are multiple users or inaccuracy in the channel statistics
information at the transmitter, aJE performs very poorly for
users whose average received SNR is below the target value.
In the following we propose alternative transmission schemes
providing gradual performance degradation with decreasing
SNR.
B. Time-Sharing (TS) 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 first channel block has
to be dedicated to message W1(the only available message),
the second channel block 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,αitn
channel uses are allocated to message Wi. A constant power
Pis used throughout the block. Then the total amount of
received mutual information (MI) relative to message Wiis
Itot
i,PM
t=iαitCt. Letting αit = 1 if t=iand αit = 0
otherwise, we obtain the MT scheme.
For simplicity, in the time-sharing (TS) scheme we assume
equal time allocation among all the 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. 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,
RT S =R
M
M
X
m=1
P r Cm
m+Cm+1
m+ 1 +···+CM
M≥R.
(24)
1) Generalized Time-Sharing (gTS) Transmission: Note
that, in TS transmission, message Wiis transmitted over
M−i+ 1 channel blocks, which allocates significantly more
resources to the earlier messages. To balance the resource
allocation between the messages, we consider transmitting
each message over a limited window of channel blocks.
In generalized time-sharing transmission each message is
encoded with equal time allocation over Wconsecutive blocks
as long as the total deadline of Mchannel blocks is not met.
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(xm+1 ,...,xM)dxm+1 ···dxM(8)
6
20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
W
Average throughput
P= 2 dB (C > R)
P= 0 dB (C < R)
Fig. 4. Average throughput for the gTS scheme plotted against the window
size Wfor M= 104messages and R= 1 bpcu for two different average
SNR values.
Messages from W1to WM−W+1 are encoded over a window
of Wblocks, while messages Wi, for i∈ {M−W+ 2, M −
W+3,...,M}are encoded over M−i+1 blocks. In particular
we focus on the effect of variable Won the average throughput
RgT S . In case W≪M, most of the messages are transmitted
over Wslots together with W−1other messages. In this case
the MI accumulated for a generic message Wiis:
Itot
i=1
W
i+W−1
X
t=i
Ct.(25)
By the law of large numbers, Eqn. (25) converges in proba-
bility to the average channel capacity Cas W→ ∞. Thus,
we expect that, when the transmission rate Ris above C, the
gTS scheme shows poor performance for large W(and hence,
large M), while almost all messages are received successfully
if R < C. We confirm this by analyzing the effect of Won
RgT S numerically in Fig. 4 for M= 104and R= 1 bpcu. For
P= 0 dB we have C < R, which leads to a decreasing RgT S
with increasing window size W. On the other hand, for P= 2
dB, we have C > R, and accordingly Rg T S approaches 1as
Wincreases.
The same reasoning cannot be applied if the window size
is on the order of the number of messages, as the number of
initial messages which share the channel with less than W−1
other messages and the number of final messages which share
the channel with more than W−1messages are no longer
negligible with respect to M. In Fig. 5(a), we plot RgT S as
a function of Wfor relatively small number of messages and
C≥R. As seen in the figure, for a given Man optimal value
of Wcan be chosen to maximize RgT S . Optimal Wincreases
with Mwhen R < ¯
C. We plot RgT S for C < R in Fig. 5(b).
From the figure we see that RgT S decreases monotonically
with Wup to a minimum, after which it increases almost
linearly. The initial decrease in the average throughput is due
to the averaging effect described above, which is relevant for
W≪M. The following increase in RgT S is because the
messages which are transmitted earlier (i.e., Wiwith i≪W)
get an increasing amount of resources as Wincreases, and so
the probability to be decoded increases. As a matter of fact,
for each finite i, the average MI accumulated for message Wi
grows indefinitely with W, i.e.:
lim
W→∞ E(i+W−1
X
t=i
Ct
min{t, W })
=Clim
W→∞
i+W−1
X
t=i
1
min{t, W }= +∞.(26)
Thus, for a fixed i, letting Wgo to infinity leads to an infinite
average MI, which translates into a higher RgT S . Note that
this is valid only for relatively small iand large W, i.e., only
messages transmitted earlier benefit from increasing W, while
the rest of the messages are penalized. If Wis small compared
to M, as in the plot of Fig. 4for P= 0 dB, the fraction of
messages which benefit from the increasing W(i.e., messages
W1,...,WLwith L≪W) remains small compared to M.
For this reason RgT S does not increase with Wfor the range
of Wconsidered in Fig. 4, while it does for the same range
in Fig. 5(b).
Although the idea of encoding a message over a fraction
of the available consecutive slots (e.g., W < M for message
W1in gTS) can be applied to all the schemes considered in
this paper, the analysis becomes quite cumbersome. Hence,
we restrict our analysis to the TS scheme as explained above.
C. Superposition Transmission (ST)
In superposition transmission (ST) the superposition of
tcodewords, chosen from tindependent Gaussian code-
books of size 2nR , corresponding to the available mes-
sages {W1,...,Wt}is transmitted in channel block t,t∈
{1,...,M}. The codewords are scaled such that the average
total transmit power in each block is P. 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 W1and
W2, respectively. In general, over channel block twe allocate
an average power Pit for Wi, while Pt
i=1 Pit =P.
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 !.(27)
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 subtracted and the process is repeated 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 algorithm gives us the maximum
throughput. However, it is challenging in general to find
a closed form expression for the average throughput, and
optimize the power allocation. Hence, we focus here on the
special case of equal power allocation, where we divide the
total average power Pamong all the available messages at
each channel block. The performance of the ST scheme will
7
0 50 100 150 200 250 300
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
W
Average throughput
M= 500
M= 200
M= 100
M= 50
M= 20
(a) R= 1 bpcu, P= 5 dB (C > R).
1 50 100 150 200 250 300 350 400 450 500
0
0.05
0.1
0.15
W
Average throughput
M=500
M=200
M=100
M=50
M=20
(b) R= 1 bpcu, P=−3 dB (C < R).
Fig. 5. Average throughput for the gTS scheme plotted against the window size Wfor different values of M.
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
Number of decoded messages
CMF
Joint encoding (JE)
Time−sharing (TS)
Generalized TS (gTS)
Memoryless (MT)
Superposition (ST)
Upper bound
(a) P= 1.44 dB (C > R).
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
Number of decoded messages
CMF
Joint encoding (JE)
Time−sharing (TS)
Generalized TS (gTS)
Memoryless (MT)
Superposition (ST)
Upper bound
(b) P= 0 dB (C < R).
Fig. 6. The cumulative mass function (cmf) of the number of decoded messages for R= 1 bpcu and M= 50.
be studied in Section V numerically and compared with the
other transmission schemes and the upper bound.
V. NUMERICAL RESULTS
In this section we provide numerical results comparing
the proposed transmission schemes. For the simulations we
assume that the channel is Rayleigh fading, i.e., the channel
gain φ(t)is exponentially distributed with parameter 1, i.e.,
fΦ(φ) = e−φfor φ > 0, and zero otherwise.
In Fig. 6(a) the cumulative mass function (cmf) of the
number of decoded messages is shown for the proposed
transmission techniques for R= 1,M= 50 and P= 1.44
dB, which correspond to an average channel capacity C≃
1.07 > R. We see that MT outperforms ST and TS schemes,
as its cmf lays below the other two. The gTS scheme improves
significantly compared to the ordinary TS scheme. On the
other hand, the comparison with the JE scheme depends on the
performance metric we choose. For instance, JE has the lowest
probability to decode more than mmessages, for m≤15,
while it has the highest probability for m≥22.
In Fig. 6(b) the cmf’s for P= 0 dB are shown. In this
case the average capacity is C≃0.86. Comparing Fig. 6(b)
and Fig. 6(a), we see how the cmf of the JE scheme has
different behaviors depending on whether Cis above or below
R. We see from Fig. 6(b) that for the JE scheme there is a
probability of about 0.3not to decode any message, while in
all the other schemes such probability is zero. However, the JE
scheme also has the highest probability to decode more than
23 messages. Furthermore, we note that the cmf of the gTS
scheme converges to the cmf of the TS scheme at low SNR.
This is because, as shown in Section IV-B1, when C < R, the
optimal window size Wis equal to M, which is nothing but
the TS scheme.
In Fig. 7(a) and Fig. 7(b) the average throughput is plotted
against the delay constraint Mfor SNR values of −3 dB and
2 dB, respectively, and a message rate of R= 1 bpcu. While
JE outperforms the other schemes at SNR = 2 dB, it has
the poorest performance at SNR =−3 dB. This behavior is
expected based on the threshold behavior of the JE scheme
that we have outlined in Section IV-A. Note that the average
capacity corresponding to SN R =−3dB and 2dB are C=
0.522 < R and C= 1.158 > R, respectively.
When C > R the average throughput of the JE scheme gets
close to the upper bound as the delay constraint Mincreases.
It can be seen from Fig. 7(b) that, with a relatively short delay
constraint of M= 100, the JE scheme achieves 90% of the
average throughput of the upper bound. Also the gTS scheme
improves its performance as the delay constraint gets larger,
8
10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
Total number of messages (M)
Average throughput
Joint encoding (JE)
Time−sharing (TS)
Generalized TS (gTS)
Memoryless (MT)
Superposition (ST)
Upper bound
(a) P=−3 dB (C < R).
10 20 30 40 50 60 70 80 90 100
0.4
0.5
0.6
0.7
0.8
0.9
1
Total number of messages (M)
Average throughput
Joint encoding (JE)
Time−sharing (TS)
Generalized TS (gTS)
Memoryless (MT)
Superposition (ST)
Upper bound
(b) P= 2 dB (C > R).
Fig. 7. Average throughput vs delay constraint (total number of messages) Mfor R= 1 bpcu.
while the other schemes achieve an average throughput that is
almost constant with M.
We observe that, in the low SNR regime, i.e., when C < R,
all the proposed schemes other than JE perform very close to
each other, and significantly below the upper bound. The large
gap to the informed transmitter upper bound is mostly due to
the looseness of this bound in the low SNR regime.
Although none of the schemes dominates the others at all
SNR values, it is interesting to note how, for a given SNR,
the best scheme does not depend on the delay constraint. This
may be useful in case of a practical implementation of the
proposed schemes for the case of a single receiver or multiple
receivers with similar average SNRs, as indicates that the SNR,
rather than the delay, is the main design parameter. Further in
this section we consider the case in which the receivers have
different average SNR values.
In Fig. 8 the average throughput Ris plotted against the
transmission rate Rfor the case of M= 100 and P= 20 dB.
The aJE scheme outperforms all the other schemes, performing
very close to the upper bound, illustrating its rate adaptation
capability. The number M′of messages transmitted in the
aJE scheme is chosen so that M′
M= 0.95 C
R. In the figure we
also show the upper bound obtained from the ergodic capacity
min(R, C). It can be seen how it closely approximates the
informed transmitter upper bound for R < 6. The JE scheme
performs better than the others up to a certain transmission
rate, beyond which rapidly becomes the worst one. This is
due to the phase transition behavior exposed in Section IV-A
in the case of asymptotically large delay, and observed here
even for a relatively small M. Among the other schemes, MT
achieves the highest average throughput in the region R < 6.8,
while TS has the worst performance. The opposite is true in
the region R > 6.8, where the curve of ST scheme is upper
and lower bounded by the curves of the MT and TS schemes.
We have repeated the simulations with different parameters
(i.e., changing Pand M) with similar results, that is, MT, TS,
and ST schemes meet approximately at the same point, below
which MT has the best performance of the three while above
the intersection TS has the best performance. At the moment
we have no analytical explanation for this observation, which
would mean that there is always a scheme outperforming ST.
We next study the performance of the considered schemes as
a function of the distance from the transmitter.
We scale the average received power at the receiver with
d−αpl , where dis the distance from the transmitter to the
receiver and αpl is the path loss exponent. The results are
shown in Fig. 9 for P= 20 dB, M= 100,R= 1 bpcu and
a path loss exponent αpl = 3. The dependence of Ron the
distance is important, for instance, in the context of broadcast
transmission in cellular networks, in which case the receiving
terminals may have different distances from the transmitter.
In such a scenario the range of the average channel SNR
values at the receivers becomes important, and the transmitter
should use a transmission scheme that performs well over this
range. For instance, in a system in which all users have the
same average SNR, such as a narrow-beam satellite system
where the SNR within the beam footprint has variations of
at most a few dB’s on average [22], 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. Instead, in the case of a macrocell, in which
the received SNR may vary significantly from the proximity
of the transmitter to the edge of the cell, the transmitter should
adopt a scheme which performs well over a larger range
of SNR values. In the range up to d= 4 the JE scheme
achieves the highest average throughput while for d≥6the
TS scheme outperforms the others. The drop in the average
throughput in the JE scheme when passing from d= 4 to
d= 5 is similar to what we observe in Fig. 8 when the rate
increases beyond R= 6 bpcu. In both cases the transition
takes place as the transmission rate surpasses the average
channel capacity. The aJE scheme, which selects the fraction
of messages to transmit based on C, outperforms all other
schemes and gets relatively close to the informed transmitter
upper bound and the ergodic capacity. It is interesting to
observe that the behavior of the JE and the aJE schemes in
case of finite delay constraint (M= 100) closely follows the
results shown for the asymptotic case in Section IV-A. The
aJE scheme adapts the average transmission rate at message
9
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
R (bpcu)
Average throughput
Joint encoding (JE)
Adaptive JE (aJE)
Time-sharing (TS)
Memoryless (MT)
Superpositio n (ST)
Upper bound
min(R, C)
Fig. 8. Average throughput vs Rfor P= 20 dB and M= 100 messages.
The upper bound min(R, C)is also shown.
level to the average channel capacity. We recall that, in the
aJE scheme, the transmitter only has a statistical knowledge
of the channel, and yet gets pretty close to the performance of
a genie-aided transmitter even for a reasonably low number of
channel blocks. We further notice how the adaptive JE scheme
closely approaches the ergodic capacity, even though data
arrives gradually at the transmitter during the transmission,
instead of being available at the beginning, which is generally
assumed for the achievability of the ergodic capacity [6].
We should note that in Fig. 9 the average transmission rate
is optimized for each given distance for the aJE scheme,
while such optimization is not done for the other schemes.
Thus, when two (or more) terminals have different distances
from the transmitter, the performance for the receivers might
not be optimized simultaneously by the same scheme, and a
tradeoff between the average throughputs of the two nodes is
required. The performance can be improved by considering a
combination of the aJE scheme with the TS or ST schemes.
The plots in Fig. 9 show how TS, MT and ST schemes are
more robust compared to the JE scheme, as their average
throughput decreases smoothly with the distance, unlike the
JE scheme, which has a sudden drop. This provides robustness
in the case of multiple receivers with different average SNRs
or when the channel statistics information at the transmitter is
not accurate.
VI. CONCLUSIONS
We have considered a transmitter streaming data to a
receiver over a block fading channel, such that the transmitter
is provided with an independent message at a fixed rate at the
beginning of each channel block. We have used the average
throughput as our performance metric. We have proposed
several new transmission schemes based on joint encoding,
time-division and superposition encoding. A general upper
bound on the average throughput has also been introduced
assuming the availability of CSI at the transmitter.
We have shown analytically that the joint encoding (JE)
scheme has a threshold behavior. It performs well when the
target rate is below the average channel capacity C, while
its performance drops sharply when the target rate surpasses
C. To adapt to an average channel capacity that is below
the fixed message rate R, the adaptive joint encoding (aJE)
scheme transmits only some of the messages. We have proved
analytically that the aJE scheme is asymptotically optimal as
the number of channel blocks goes to infinity, even though data
arrives gradually over time at a fixed rate, rather than being
available initially. We have also shown numerically that, even
for a finite number of messages, the aJE scheme outperforms
other schemes in all the considered settings and performs close
to the upper bound.
The JE and the aJE schemes create an M-block long
concatenated code, and imposes a certain structure on the way
messages can be decoded. For example, message Wkcan be
decoded only if message Wk−1can also be decoded. This is
useful when the underlying application has a minimum rate re-
quirement that needs to be satisfied over Mchannel blocks, or
when the average SNRs of the users vary over a limited range
of SNR values. Independent encoding used in time-sharing
based schemes (TS, gTS, MT), instead, makes messages less
dependent on the decoding of the other messages. The ST
scheme, based on message superposition, collocates itself
between JE- and TS-based schemes, as messages are encoded
independently, but the probability of correctly decoding each
one is affected by decoding of others.
We conclude that the aJE scheme is advantageous in systems
with a single receiver or with multiple receivers having similar
average SNR values, as the performance of the user with
the highest average SNR is limited by the user with the
lowest average SNR. On contrast, the gTS and ST schemes
can be attractive when broadcasting to multiple users with a
wide range of SNR values, or in a point-to-point system with
inaccurate CSI, as the average throughput decreases gradually
with decreasing SNR in these transmission schemes.
APPENDIX A
Proof of Theorem 1
Let Bkdenote the event “the first kmessages can be
decoded at the end of channel block k”, while Bc
kdenotes
the complementary event. The event Bkholds if and only if
Ck−i+1 +Ck−i+2 +···+Ck≥iR (28)
is satisfied for all i= 1,...,k. Let Ek,j denote the event
“the j-th inequality needed to decode the first kmessages in
kchannel blocks is satisfied”, that is:
Ek,j ,{Ck−j+1 +···+Ck≥jR},(29)
for j= 1,...,k, while Ec
k,j denotes the complementary event.
Note that in the JE scheme if mmessages are decoded these
are the first mmessages. Let nddenote the number of decoded
messages at the end of channel block M. Then the average
throughput is
RJE =R[P r {nd≥1}+P r{nd≥2}+
···+ Pr{nd≥M−1}+ Pr{nd≥M}].(30)
The k-th term in the sum of Eqn. (30) is the probability of
decoding at least k(i.e. kor more) messages. Each term in
Eqn. (30) can be expressed as the sum of two terms as:
P r{nd≥k}=P r{Bk, nd≥k}+P r{Bc
k, nd≥k}(31)
10
3 4 5 6 7 8 9 10
10−4
10−3
10−2
10−1
100
d
Average throughput
Joint encoding (JE)
Adaptive JE (aJE)
Time -sharing (TS)
Memoryless (MT)
Superposition (ST)
Upper bound
Ergodic capacity (C)
Fig. 9. Average throughput Rvs distance from the transmitter for R= 1
bpcu, M= 100,P= 20 dB and αpl = 3.
The first term in Eqn. (31) is the probability of “decoding k
messages at the end of channel block kand decoding at least
kmessages at the end of Mchannel blocks”. If Bkholds, the
event “decode at least kmessages at the end of channel block
M” is satisfied; hence, we have:
P r{Bk, nd≥k}=P r{Bk}=P r{Ek,1,...,Ek,k }.(32)
As for the second term in Eqn. (31), it is the probability of
decoding at least kmessages but not kat the end of channel
block k. It can be further decomposed into the sum of two
terms:
P r{Bc
k, nd≥k}=P r{Bc
k, Bk+1, nd≥k}
+P r{Bc
k, Bc
k+1, nd≥k}.(33)
The event nd≥kholds if the condition Bk+1 is satisfied (i.e.,
if k+ 1 messages are decoded at the end of block k+ 1, then
more than kmessages are decoded at the end of channel block
M); hence, we have:
P r{Bc
k, Bk+1, nd≥k}=P r {Bc
k, Bk+1}.
Plugging these into Eqn. (31), we obtain
P r{nd≥k}=P r{Bk}+P r{Bc
k, Bk+1}
+P r{Bc
k, Bc
k+1, nd≥k}.(34)
We can continue in a similar fashion, so that, the event “at
least kmessages are decoded” can be written as the union of
the disjoint events (“kmessages are decoded in kslots”) S
(“kmessages are not decoded in kslots but k+ 1 messages
are decoded in k+ 1 slots”) S··· S(“no message can be
decoded before slot Mbut Mmessages are decoded in slot
M”). Hence, by the law of total probability, we have:
P r{nd≥k}=
M
X
j=k
P r{Bc
k, Bc
k+1, . . . , B c
j−1, Bj}.(35)
Note that each term of the sum in Eqn. (35) says nothing about
what happens to messages beyond the j-th, which can either
be decoded or not. Plugging Eqn. (35) in Eqn. (30) we find:
E[m] =
M
X
k=1
M
X
j=k
P r{Bc
k, Bc
k+1,...,Bc
j−1, Bj}
=
M
X
j=1
j
X
k=1
P r{Bc
k, Bc
k+1,...,Bc
j−1, Bj}.(36)
We can rewrite each of these events as the intersection of
events of the kind Ek,i and Ec
k,i. Each term of the sum in
Eqn. (36) can be written as:
P r{Bc
k, Bc
k+1,...,Bc
j−1, Bj}
=P r{Ek,1, Bc
k, Bk+1,...,Bc
j−1, Bj}
+P r{Ec
k,1, Bc
k, Bc
k+1,...,Bc
j−1, Bj}.(37)
As the event Ec
k,1implies the event Bc
k, this can be removed
from the second term on the right hand side of Eqn. (37).
Note that, in general, the event Ec
k,i,i∈ {1,··· , k }implies
the event Bc
k. In order to remove the event Bc
kfrom the first
term as well, we write it as the sum of probabilities of two
disjoint events: one intersecting with Ek,2and the other with
Ec
k,2. Then we get:
P r{Bc
k,Bc
k+1,...,Bc
j−1, Bj}
=P r{Ek,1, Ek,2, Bc
k,...,Bc
j−1, Bj}
+P r{Ek,1, Ec
k,2, Bc
k,...,Bc
j−1, Bj}
+P r{Ec
k,1, Bc
k+1,...,Bc
j−1, Bj}.(38)
Now Bc
kcan be removed from the second term of the sum
thanks to the presence of Ec
k,2. Each of the terms in the right
hand side of Eqn. (38) can be further written as the sum of the
probabilities of two disjoint events, and so on so forth. The
process is iterated until all the Bc
d,d < j events are eliminated
and we are left with the intersections of events only of the type
Ep,q and Ec
p,q, for some p, q ∈ {k, k +1,...,M}and Bj. The
iteration is done as follows:
For each term of the summation, we take the Bc
levent
with the lowest index. If any Ec
l,j event is present, then Bc
l
can be eliminated. If not, we write the term as the sum of
the two probabilities corresponding to the events which are
the intersections of the Bc
levent with El,d+1 and Ec
l,d+1,
respectively, where dis the highest index jamong the events
in which El,j is already present. The iteration process stops
when l=j.
At the end of the process all the probabilities involving
events Bc
k,...,Bc
j−1will be removed and replaced by se-
quences of the kind:
{Ek,1, Ek,2,...,Ec
k,ik, Ek+1,ik+1,...,
Ec
k+1,ik+1 ,...,Ej−1,ij−2+1, Ec
j−1,ij−1, Bj},
(39)
where ij−1∈ {j−1−k,...,j−1}is the index corresponding
to the last inequality required to decode j−1messages, which
is not satisfied. Note that exactly one Ec
l,r event for each Bc
l
is present after the iteration.
For Bjto hold, all the events Ej,1,...,Ej,j must hold. It is
easy to show that, after the iterative process used to remove the
11
Bc
l’s, the event Ej,ij−1+1 ensures that all the events required
for Bjwith indices lower than or equal to ij−1automatically
hold. Thus, we can add the events {Ej,ij−1+1,...,Ej,j }to
guarantee that Bjholds, and remove it from the list. It is
important to notice that the term Ej,j is always present. At
this point we are left with the sum of probabilities of events,
which we call E-events, each of which is the intersection of
events of the form Ei,j and Ec
i,j . Thus, an E-event Sj
khas the
following form:
Sj
k,{Ek,1, Ek,2,...,Ec
k,ik, Ek+1,ik+1,...,Ec
k+1,ik+1 ,
...,Ej−1,ij−2+1, Ec
j−1,ij−1, Ej,ij−1+1,...,Ej,j }.(40)
By construction, the number of E-events for the generic term
jof the sum in Eqn. (36) is equal to the number of possible
dispositions of j−k Ec’s over j−1positions. As the number
of events of type Ecis different for the E-events of different
terms in Eqn. (36), the E-events relative to two different terms
of Eqn. (36) are different. We define Sjas the set of all
E-events which contain the event Ej,j . The elements of Sj
correspond to all the possible ways in which jmessages can
be decoded at the end of block number j. The cardinality of
Sjis equal to:
|Sj|=
j
X
k=1
(j−1)!
(k−1)!(j−k)! = 2j−1.(41)
Now we want to prove that
X
Sj
k∈Sj
P r{Sj
k}=P r{Ej,j }.(42)
Note that each Ek,l corresponds to a different event if the
index kis different, even for the same index l; thus, the law
of total probability can not be directly applied to prove Eqn.
(42). However, we will prove in the following preposition that
P r{Ek1,l}=P r {Ek2,l},∀k1, k2.
Proposition 1: Let us consider a set of random variables
C1,···, Cjthat are conditionally i.i.d. given U. Given any
two ordering vectors i=i1, i2,···, ijand l=l1, l2,···, lj,
we have
P r{Ci1≷R,...,Ci1+···+Cij≷jR}
=P r{Cl1≷R,...,Cl1+···+Clj≷jR}.(43)
Proof: The left hand side of Eqn. (43) can be rewritten as:
P r{Ci1≷R,...,Ci1+···+Cij≷jR}
=Z+∞
−∞
du Zθup
1
θlow
1
dci1...Zθup
j
θlow
j
dcijfCi|U(ci|u)fU(u),
(44)
where Ci=Ci1,...,Cijand ci=ci1,...,cij, while θlow
h
and θup
hare the lower and upper extremes of the integration
interval. θlow
his either equal to −∞ or to hR−ci1−· ··−cih−1,
∀h∈ {1,...,j}, depending on whether there is a <or a ≥in
the h-th inequality within brackets in Eqn. (44), respectively,
while θup
his either equal to hR −ci1−· ·· −cih−1or to +∞
depending on whether there is a <or a ≥in the h-th inequality
of Eqn. (44), respectively. By using Eqn. (9) and Eqn. (10) we
can write:
P r{Ci1≷R,...,Ci1+···+Cij≷jR}
=Z+∞
−∞
dufU(u)Zθup
1
θlow
1
dci1...Zθup
j
θlow
j
dcijfCi|U
=Z+∞
−∞
dufU(u)Zθup
1
θlow
1
dcl1...Zθup
j
θlow
j
dcljfCl|U
=P r{Cl1≷R,...,Cl1+···+Clj≷jR},(45)
where we defined fCi|U,fCi1|U(ci1|u)···fCij|U(cij|u)and
fCl|U,fCl1|U(cl1|u)× ·· · × fClj|U(clj|u)The proposition
above guarantees that, although these events do not partition
the whole probability space of Ej,j, their probabilities add up
to that of Ej,j , i.e.:
2j−1
X
k=1
P r{Sj
k}=P r{Ej,j }
=P r{C1+···+Cj≥jR}.(46)
Finally, plugging Eqn. (46) into Eqn. (36) we can write:
E[m] =
M
X
j=1
j
X
k=1
P r{Bc
k, Bc
k+1,...,Bc
j−1, Bj}(47)
=
M
X
j=1 X
Sj
k∈Sj
P r{Sj
k}
=
M
X
j=1
P r{C1+···+Cj≥jR}.(48)
APPENDIX B
In the following we prove that the average throughput of the
aJE scheme RaJE approaches αR for large Mif C > αR.
Similarly to the JE scheme, it is sufficient to prove that, if
C > αR,
lim
M→∞
Mα
X
m=1
a∗
m=c, (49)
for some 0< c < ∞, where a∗
m,P r nC∗
1+···+C∗
m
m< Ro.
We can rewrite a∗
mas:
a∗
m=P r
lm>C/α −R
σcq1
m+1−α
Mα2
,(50)
where
lm,C/α −C1+···+Cm
m−(1−α)
α
1
M(1−α)PM
j=Mα+1 Cj
σcq1
m+1−α
Mα2
(51)
is a random variable with zero mean and unit variance. From
the law of large numbers applied to Eqn. (50), we have
limm→+∞a∗
m= 0. First we show that
lim
m→+∞a∗
m
dm=c′,(52)
12
for some 0< c′<+∞where we have defined:
dm,P r
l′
m>C/α −R
σcq1
m+1−α
mα2
,(53)
and
l′
m,C/α −C1+···+Cm
m−(1−α)
α
1
m(1−α)PMα+m
j=Mα+1 Cj
σcq1
m+1−α
mα2
(54)
such that l′
mis a random variable with zero mean and unit
variance. We have
lim
m→+∞a∗
m
dm= lim
m→+∞
P r (lm>C /α−R
σcq(1
m+1−α
Mα2))
P r (l′
m>C/α−R
σcq(1
m+1−α
mα2))
= lim
m→+∞
Q C/α−R
σcq(1
m+1−α
Mα2)!
Q C/α−R
σcq(1
m+1−α
mα2)!(55)
≤1,(56)
where inequality (56) follows from the fact that m < M and
Q(x)is monotonically decreasing in x. Then we show that
lim
M→∞
Mα
X
m=1
dm=c′′,(57)
for some 0< c′′ <+∞. To prove the convergence of the
series sum we show that limm→+∞dm+1
dm=λ′, for some 0<
λ′<1. From the central limit theorem we can write:
lim
m→+∞
dm+1
dm
= lim
m→+∞
P r
lm+1 >C/α−R
σcr1
m+1 +1−α
(m+1)α2
P r (lm>C /α−R
σcq(1
m+1−α
mα2))
= lim
m→+∞
Q
C/α−R
σcr1
m+1 +1−α
(m+1)α2
Q C/α−R
σcq(1
m+1−α
mα2)!
≤lim
m→+∞
σcr1
m+1 +1−α
(m+1)α2
(C/α−R)√2πe
−1
2
C/α−R
σcs1
m+1 +1−α
(m+1)α2
2
C/α−R
σcr(1
m+1−α
mα2)
1+
C/α−R
σcr(1
m+1−α
mα2)
21
√2πe−1
2
C/α−R
σcr(1
m+1−α
mα2)
2
(58)
= lim
m→+∞r1
m+1 +1−α
(m+1)α2
(C/α −R)2
·σ2
c1
m+1−α
mα2+ (C/α −R)2
q1
m+1−α
mα2
e−(C/α−R)2
2σ2
cα2
α2−α+1
(59)
=e−(C/α−R)2
2σ2
cα2
α2−α+1 <1,(60)
where inequality (58) follows from Eqn. (21).
From Eqn. (60) it follows that limM→∞ RaJ E =Rif αR <
C. Similarly, it can be easily shown that limM→∞ RaJE = 0
if αR > C.
REFERENCES
[1] European Telecommunications Standards Institute, “ETSI TR 102 376
V1.1.1 (2005-02) Digital Video Broadcasting (DVB) user guidelines for
the second generation system for broadcasting, interactive services, news
gathering and other broadband satellite applications (DVB-S2),” Feb.
2005.
[2] D. Tse, R. Yates, and L. Zang, “Fading broadcast channels with state
information at the receivers,” in IEEE Allerton Conf. on Comm., Control,
and Computing, Monticello, IL, U.S.A., Sep. 2008.
[3] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information-
theoretic and communications aspects,” IEEE Trans. on Info. Theory,
vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[4] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.
Cambridge University Press, 2005.
[5] A. Goldsmith, Wireless Communication. Cambridge University Press,
2005.
[6] A. Goldsmith and P. P. Varaiya, “Capacity of fading channels with
channel side information,” IEEE Trans. on Info. Theory, vol. 43, no. 6,
pp. 1986–1992, Nov. 1997.
[7] G. J. Foschini and M. J. Gans, “On limits of wireless communications in
a fading environment when using multiple antennas,” Wireless Personal
Comm., vol. 6, no. 3, pp. 311–335, Mar. 1998.
[8] L. Zheng and D. Tse, “Diversity and multiplexing: A fundamental
tradeoff in multiple antenna channels,” IEEE Trans. on Info. Theory,
vol. 49, no. 5, pp. 1073–1096, May 2003.
[9] A. S. Avestimehr and D. N. C. Tse, “Outage capacity of the fading relay
channel in the low snr regime,” IEEE Trans. on Info. Theory, vol. 53,
no. 4, pp. 1401–1415, Apr. 2007.
[10] S. Shamai, “A broadcast strategy for the Gaussian slowly fading chan-
nel,” in IEEE Int’l Symp. on Info. Theory, Ulm, Germany, June-July
1997.
[11] S. Shamai and A. Steiner, “A broadcast approach for a single user slowly
fading mimo channel,” IEEE Trans. on Info. Theory, vol. 49, no. 10, pp.
2617–2635, Oct. 2003.
[12] D. G¨und¨uz and E. Erkip, “Opportunistic cooperation by dynamic re-
source allocation,” IEEE Trans. on Wireless Comm., vol. 6, no. 4, pp.
1446–1454, Apr. 2007.
[13] S. V. Hanly and D. N. C. Tse, “Multiaccess fading channels. II. Delay-
limited capacities,” IEEE Trans. on Info. Theory, vol. 44, no. 7, pp.
2816–2831, Nov. 1998.
[14] O. Oyman and S. Singh, “Quality of experience for HTTP adaptive
streaming services,” IEEE Comm. Magazine, vol. 50, no. 4, pp. 20–27,
Apr. 2012.
[15] M. C. O. Bogino, P. Cataldi, M. Grangetto, E. Magli, and G. Olmo,
“Sliding-window digital fountain codes for streaming multimedia con-
tents,” in Int’l Symp. on Circuits Systems, New Orleans, LA, U.S.A.,
May 2007.
[16] A. Badr, A. Khisti, and E. Martinian, “Diversity embedded streaming
erasure codes (DE-SCo): Constructions and optimalty,” in IEEE Global
Telecomm. Conf., Miami, FL, U.S.A., Dec. 2010.
[17] D. Leong and T. Ho, “Erasure coding for real-time streaming,” in IEEE
Int’l Symp. Info. Theory (ISIT), Boston, MA, U.S.A., July 2012.
[18] A. Khisti and S. C. Draper, “Streaming data over fading wireless
channels: The diversity-multiplexing tradeoff,” in IEEE Int’l Symp. on
Info. Theory, St. Petersburg, Russia, Aug. 2011.
13
[19] G. Cocco, D. Gunduz, and C. Ibars, “Real-time broadcasting over block-
fading channels,” in Int’l Symp. on Wireless Comm. Systems, Aachen,
Germany, Nov. 2011.
[20] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoretic
considerations for cellular mobile radio,” IEEE Trans. on Vehicular
Technology, vol. 43, no. 2, pp. 359–378, May 1994.
[21] V. V. Prelov, “Transmission over a multiple access channel with a special
source hierarchy,” Problemy Peredachi Informatsii, vol. 20, no. 4, pp.
3–10, Oct.-Dec. 1984.
[22] G. Maral and M. Bousquet, Satellite Communications Systems - systems,
techniques and technologies. John Wiley & Sons Ltd, 2009.
