It is well known that for the Gaussian broadcast channel, timeshared coding is not as efficient as more sophisticated broadcast coding strategies. However, the relative advantage of broadcast coding over timeshared coding is shown to be small if the signal-to-noise ratios of both receivers are small. One surprising consequence of this is that for the wideband Gaussian broadcast channel, which is defined, broadcast coding offers no advantage over timeshared coding at all, and so timeshared coding is optimal

Mceliece, R. J.

Posner, E. C.

Swanson, L.

English

NASA Technical Reports Server

TDA Progress Report 42-84 October-December 1985
N 8 6 - 2 2 7 8 9 1
A Note on the Wideband Gaussian Broadcast Channel
R. J. McEliece
California Institute of Technology
E. C. Posner
Office of Telecommunications and Data Acquisition
L. Swanson
Communications Systems Research Section
It is well known that for the Gaussian broadcast channel, timeshared coding is not as
efficient as more sophisticated "broadcast" coding strategies. However, in this article we
will show that the relative advantage of broadcast coding over timeshared coding is small
if the signal-to-noise ratios of both receivers are small. One surprising consequence of this
is that for the wideband Gaussian broadcast channel, which we shall define, broadcast
coding offers no advantage over timeshared coding at all, and so timeshared coding is
optimal.
I. Introduction
T. M. Cover (Ref. 1) introduced the "broadcast channel"
with one transmitter and two (or more) different receivers.
Following Ref. 4, we ask the following question about a
broadcast channel: certain common information is to be com-
municated simultaneously to both receivers. How much addi-
tional information can be communicated to the better receiver
at the same time9
For channels like Gaussian channels, where one receiver is
just a degraded version of the other, one obvious approach is
timeshared coding, devote a fixed fraction of the total trans-
mission time to sending the common information, coded for
the weaker channel. This information will be comprehensible
to both receivers. During the remaining time, transmit addi-
tional information.coded for the stronger receiver. This infor-
mation will not be comprehensible to the weaker receiver.
But in Ref. 1, Cover introduced a technique called broad-
cast coding, and showed that, in general, broadcast coding
achieves greater, often much greater, data rates than time
sharing. Later El Gamal and Cover (Ref. 2) found that broad-
cast coding cannot be further improved upon.
In this article we will discuss the Gaussian broadcast chan-
nel. For this channel, we will show that the relative advantage
of broadcast coding over timeshared coding is small if the
signal-to-noise ratios of both receivers are small. One surprising
consequence of this is that for the wideband Gaussian broad-
cast channel, which we shall define, broadcast coding offers no
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https://ntrs.nasa.gov/search.jsp?R=19860013318 2020-03-20T15:04:22+00:00Z
advantage over timeshared coding at all, and so timeshared
coding is optimal.
II. The Gaussian Channel: A Review
A Guassian channel is a discrete-time memoryless channel
model whose input X and output Y are related by Y = X + Z,
where Z is a mean zero Gaussian random variable independent
from X. If the input is constrained by E(X2) < S, and if the
variance of Z is denoted by a2, it is well known that the chan-
nel capacity depends only on the ratio x = S/o2, which is
called the signal-to-noise ratio, and is given by the formula
(1)
In Eq. (1), C(x) represents the maximum possible amount of
information (measured in bits, nats, etc., depending on the
base of the logarithm) which can be reliably transmitted per
channel use; in the usual physical sense, C(x) is dimensionless.
Equation (1) can be used to derive the following formula
for the capacity of a continuous-time, band- and power-
limited Guassian channel model:
C = B log (2)
where B is the channel bandwidth in Hertz, P is the average
transmitter power in Watts, and N0 is the noise spectral den-
sity in Watts per Hertz. The transition from Eq. (1) to Eq. (2)
is explained in Ref. 3 (Chapter 4), for example. In Eq. (2),C
represents the maximum possible information which can be
reliably transmitted per unit of time; the physical dimensions
of Care see"1.
If in Eq. (2) we assume natural logarithms and pass to the
limit as the bandwidth B approaches infinity, we obtain
(3)
which is the well-known formula for the capacity of the infinite
bandwidth white Gaussian channel. The units in Eq. (3) are
nats per second.
III. The Gaussian Broadcast Channel
In Ref. 1, Cover introduced a discrete-time memoryless
channel model with one transmitter and two receivers, which
he called a Gaussian broadcast channel. This channel has one
input X, and two outputs Y^ and Y2, related by
where now Zl and Z2 are mean zero Gaussian random vari-
ables, and X, Zj, and Z2, are independent. Let us denote by
a\ and a?, the variances of Zj and Z2, respectively, and assume
that a\ < a2, so that Y{ is received more reliably than Y2. If
the channel input X is constrained as in Section I by E(X2 ) < S,
then separately channels 1 and 2 have capacity C(xl) and
C(x2), respectively, where Xj = S/a2 andx2 =5/a2.
In Ref. 4 a Gaussian broadcast channel was used to model
deep-space communications in the presence of weather uncer-
tainties; the high signal-to-noise ratio corresponds to good
weather, and the low signal-to-noise ratio, to bad weather. The
problem posed there was the following Suppose the weather
on earth is unknown to a distant spacecraft, and that data
must be sent to earth so that even in bad weather, certain
minimal but critical information will get through; but if the
weather is good, additional information will be received.
Motivated by this point of view, we state the fundamental
question about broadcast channels in the following somewhat
unusual way. Suppose we wish to send certain information,
called the common information, simultaneously to both
receivers. If we do this, how much extra information, called
bonus information, can we send to the better receiver at the
same time?
One approach to this problem is the timesharing approach,
in which the transmitter devotes a fixed fraction 1 - p (where
0 < p < 1) of the total transmission time to sending the com-
mon information. During this time the information is coded
for the weaker receiver. This information will also be compre-
hensible to the stronger receiver. By Eq. (1), during this com-
mon time, information can be transmitted at a maximum rate
of C(jc2). During the remaining fraction p of the transmission
time, the transmitter sends bonus information to the stronger
channel, at the rate C(xl). This will not be comprehensible to
the weaker receiver.
It follows that for the timesharing strategy, the data rates
will be
Common Rate = ( l -p)C(x 2 )
Bonus Rate = pC(x l)
(4)
and the parameter p can be selected arbitrarily by the trans-
mitter.
Cover showed, however, that it is possible to do better
than timesharing. Using a technique called broadcast coding,
61
he showed that for any choice of the parameter a, 0 < a < 1,
the following rates are achieveable:
(5)
Common Rate = C(x2) -C(ax2)
Bonus Rate = C(axl)
(Actually Cover gave the common rate in the form
C{[(1-«)*,]/(!+«, )},
but it is an easy exercise to show that this is the same as we
have given in Eq. [5].) Later El Gamal and Cover (Ref. 2)
showed that in fact no improvement over Eq. (5) is possible,
so that the region of the first quadrant bounded by the curve
given parametrically by Eq. (5) is now called the capacity
region of the Gaussian Broadcast Channel (see Fig. 1).
Motivated by the discussion in Section I, let us pass from the
discrete-time Gaussian broadcast channel to the continuous-
time band- and power-limited Gaussian broadcast channel. The
resulting expressions are for timesharing.
Common Rate = (1 -p)B log (1 + N2B
(6)
Bonus Rate = pB log
and for broadcast coding:
/ p \ /Common Rate = B log 1 + -;—= I - B log 1 + a •
\ N2BI \ 1
I P \Bonus Rate = B log II + a^-g I
where P is the transmitter power, B is the transmission band-
width, and Nl, N2, are the noise spectral densities for the two
receivers In Eqs. (6) and (7) the units are nats per second.
To investigate wideband Gaussian broadcast channels, we
pass as before to the limit as B -> °° The results follow easily
from Eqs (6) and (7) for wideband timesharing:
Common Rate = (1 -p) —
Bonus Rate = p —
(8)
and for wideband broadcast coding:
Common Rate = (1 -a) —
Bonus Rate = a —
(9)
We thus reach the surprising conclusion that for wideband
Gaussian broadcast channels, broadcast coding offers no
advantage over timesharing (Actually, this was mentioned but
not further investigated in Ref. 4.) We investigate this interest-
ing phenomenon more closely in the next section.
IV. A More Detailed Analysis
In this section we will see that the reason wideband broad-
cast coding offers no advantage over wideband timesharing is
that, for a given common rate, the bonus rates in Eqs. (4) and
(5) are nearly equal, when the "good" SNR*j is small. More
precisely, we have the following'
Theorem' If a and p are chosen so that the common rates
in Eqs. (4) and (5) are equal, then
Broadcast bonus rate (BBR)
Timesharing bonus rate (TBR)
log(l
Corollary 1 Since l og ( l+ jc 2 )<x 2 , we also have
BBR
TBR
independent of x2. Thus also
hm
x^O
BBR
again independent of x2.
62
Corollary 2: For the continuous time channel, the corre- by dividing the common rate by its maximum value C(x2),
spending result is and the bonus rate by its maximum value C(xa)'
BBR
TBR log(l
" log(l +P/N1B)
-+ I as B -*• °°
Proof of Theorem: For the two common rates to be equal,
we have, from Eqs (4) and (5), that
(10)C(cue2) =
On the other hand, the ratio of the bonus rates is
Normalized Common Rate (NCR) = 1 - •
(12)
Normalized Bonus Rate (NBR) =
For a given value of Xj, the parametric curves described by
Eq. (12) vary monotomcally outward from x2 = xl, in which
case they reduce to
NCR = 1 -
NBR =
(11)
Combining Eqs. (10) and (11), we see that, for a fixed com-
mon rate, the ratio of the bonus rates is
C(x)
C(x,)
The desired result now follows from the fact that the function
€(01X^)1 C(ax2) is a decreasing function of a, and approaches
X j / X j as a-> 0. •
We conclude with a brief discussion of the shape of the
broadcast capacity regions as a function of xl and x2. It is
useful to normalize the parametric curves described by Eq. (5)
which is just the timesharing straight line, to *2 = 0+, in which
case they reduce to
NCR = 1 - a
NBR = Ox,)
Thus for a given good SNR Xj, broadcast coding offers the
largest relative advantage over timesharing when x2 -*• 0, and
the smallest relative advantage (none at all) when x2 =xl .Of
course, as we have seen, when xl is small, even the largest
relative advantage is quite small. In Fig. 2, we have graphed
the outer (x2 = 0) and inner (x2 = xt) envelopes for several
values
References
1. Cover, T. M., "Broadcast Channels," IEEE Trans. Inform Theory IT-11, pp 2-14,
1972.
2. El Gamal, A., and Cover, T. M., "Multiple User Information Theory " Proc. IEEE 68,
pp.1466-1483, 1980.
3. McEhece, R. J., The Theory of Information and Coding, Reading, Mass.. Addison-
Wesley, 1977.
4. Posner, E. C., "Strategies for Weather-Dependent Data Acquisition," IEEE Trans.
Communications, COM-31, pp. 509-417,1983
63
0 I I I I I I I I I
COMMON RATE
Fig. 1. The capacity region of some Gaussian broadcast channels
64
<
a:
i i i i i i i i r
I I I I 1 I I I I0 I I I I I I I I I
I I I I I I I I II I I I I I I I I
I I I I I I0 I I I I I I I I I
I I I I I I I I II I I I I I I
x, =1000
I I I I I I I I I0 I I I I I I I I _J
COMMON RATE
Fig. 2. The extreme capacity regions (x2 = 0 and x2 = x,) for several values of x1
65


A note on the wideband Gaussian broadcast channel

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