Uncoded transmission in MAC channels
achieves arbitrarily small error probability
Mainak Chowdhury, Andrea Goldsmith and Tsachy Weissman
Department of Electrical Engineering,
Stanford University, Stanford, CA 94305, USA
Email: {mainakch, andrea, tsachy}@stanford.edu
Abstract—We consider an uplink with a large number of
uncoded non cooperating transmitters and joint processing
at the multi-antenna receiver. We investigate the minimum
number of receiver antennas that are needed for perfect
recovery of the transmitted signals with asymptotically high
probability. We find that in the limit of a large number
of users, and in a rich scattering environment, the per
user number of receive antennas can be arbitrarily small.
Comparison with the ergodic capacity of the resulting
channel in the limit of a large number of users suggests that
even with uncoded transmissions we can achieve optimal
per user number of receiver antennas.
I. I NTRODUCTION
A common paradigm to manage interference in multiuser wireless systems is to orthogonalize transmissions
in some domain (e.g. in time, frequency or space). While
this leads to simple low complexity decoders, the need
to accommodate a large number of users within the
limited available spectrum makes the inefficiency of this
approach increasingly undesirable. The optimal schemes
for achieving capacity in multiple access channels have
been shown to be based on superposition coding schemes
at the transmitting side and joint decoding at the receiving side [1]. In particular, time division strategies
have been shown to be suboptimal in many regimes.
Although there are results to show that rates achievable
by TDMA schedules converge to the capacity region in
the low power regime for static channels, [2] identifies
different conditions under which TDMA will fail to
be optimal. Moreover, for fading channels orthogonal
schemes without transmitter channel state information
(CSI) are strictly suboptimal for achieving sum capacity
[3], although in the presence of a large number of users
(i.e. sufficient diversity) the penalty decreases.
These prior results motivate the search for practical
transmit schemes that are influenced by the capacityachieving strategies of superimposing users within the
same signaling dimensions and jointly decoding them
at the receiver. A seminal work in this direction can be
found in [4], which builds and analyses the optimal (minimum error probability) decoder for superimposed users
in an asynchronous Gaussian multiple access channel.
In this work, we focus on uplink transmissions in our
model and investigate the minimum number of uncoded
transmitters that the receiver can support with a certain
number of antennas. We show that, in a rich scattering
environment, we can support a number of users which is
much larger than the number of receiver antennas using
ML or minimum distance decoding across simultaneous
user transmissions. This number turns out to be the
minimum number of receiver antennas required to ensure
that the capacity of the underlying channel matrix scales
at least as the number of transmitters as the latter
becomes large.
An unusual feature of our work is the use of uncoded
transmissions. In fact, we show that while coding does
provide coding gain as expected, it does not change the
scaling behaviour of the minimum per user number of
antennas required for the system to have a vanishing
error probability. However, we do use multiple antennas
at the receiver, so we are exploiting spatial diversity.
This diversity in the system, in fact, is crucial in our
subsequent derivation.
The rest of this paper is organized as follows. We
present the system model first and then describe an upper
bound on the decoding error probability. This bound
is then analysed to derive the fundamental limits of
underdetermined measurements possible for the uplink
system. In other words, we investigate the limits of the
number of receiver antennas per transmitter antenna. We
subsequently point out why the above mentioned limit
possible with this scheme is the best that we can achieve
under our system model. We conclude with some future
directions of investigation.
II. S YSTEM M ODEL
We consider an uplink system with n users and m
receiver antennas. The channel matrix H ∈ Rm×n has
been assumed to have its entries drawn iid from a
Gaussian N (0, 1) distribution. Let hk ∈ Rm denote the
k th column of H. Thus
H = h1
h2
...
hn−1
hn
Here Pe is the probability of decoding a codeword in
error, and Si refers to the set of all vectors representing
subsets of size i from {1, . . . , n}. Thus
n
|Si | =
i
We further assume that the users are noncooperating
and transmit from the standard BPSK constellation with
unit energy. We assume that the noise ν at the receiver is
distributed iid N (0, σ 2 ). Then the signal at the receiver
y ∈ Rm is given by
y = Hx + ν
We now look at expected symbol errors with respect
to the noise distribution. We use Ex to denote the
expectation with respect to the distribution of x. The
expected number of total symbol errors in a codeword
(across all users) is thus
(1)
Here x ∈ {−1, +1}n is the transmitted signal from the
n users. It is henceforth referred to as a codeword to
indicate that the receiver decodes all user transmissions
simultaneously. Note, however, that while codewords are
often defined over time, these codewords are defined
over users. Each individual constellation point is referred
to as a symbol. Thus a codeword is a vector made up
of n symbols. We assume no channel state information (CSI) at the transmitter, but perfect receiver CSI.
Throughout the discussion we use maximum likelihood
(ML) decoding at the receiver.
Eν (SE) ≤
j=1
−m/2
i
= 0.5 1 + 2
σ
R∞
2
Here Q(x) = √12π x e−x /2 dx, bi ∈ {1, . . . , n} is a
vector of size i whose entries are positions where the
codewords differ, and bi (j) is the j th symbol position
where the codewords differ. hk refers to the k th column
of H. We now use the fact that Q(x) ≤ 0.5 exp(−x2 /2)
to arrive at
j=1
(6)
Note in particular that it is independent of bi and depends
only on i. So summing this over all codewords differing
from the transmitted codeword in all symbol positions,
we get the following bound for the expected probability
of codeword decoding error over the statistics of the
channel realizations
−m/2
n
X
n
i
(7)
EH (Pe ) ≤
0.5
1+ 2
σ
i
i=1
(2)
Here bi is a vector representing a size i subset of
{1, . . . , n}. Summing over all positions where codewords may differ, we have the result that
Pe ≤
1≤i≤n b∈Si
i
X
X X 1i
2hbi (j) ||/2σ)2 /2)
exp(−(||
Eν (SEav ) ≤
2n
j=1
1≤i≤n b∈Si
(4)
Until now we have assumed a fixed channel realization
and upper bounded the expected number of symbol
errors over the noise distribution. We now take the
expectation with respect to channel realizations. Using
the moment generating function for the chi squared
distribution we note that
i
X
EH (0.5 exp(−(||
2hbi (j) ||/2σ)2 /2))
(5)
j=1
2hbi (j) ||/2σ)2 /2)
i
X X 1
X
exp(−(||
2hb(j) ||/2σ)2 /2)
2
j=1
In particular the expected number of symbol errors for
each user is upper bounded by the RHS above. By a
similar argument, the expected number of symbol errors
averaged over all users is given by
A bound on the probability of decoding error for each
codeword can be derived from the union bound by looking at the pairwise error probability of the transmitted
codeword. Since the receiver uses minimum distance
decoding, the probability of mistaking a codeword for
another codeword differing in i symbol positions is
exactly
i
X
2hbi (j) ||/2σ)
Pe,bi = Q(||
i
X
1≤i≤n b∈Si
≤n
III. A N U PPER B OUND ON THE D ECODING E RROR
Pe,bi ≤ 0.5 exp(−(||
i
X
X X 1
i exp(−(||
2hb(j) ||/2σ)2 /2)
2
j=1
This is a bound on the expected probability of codeword
error. A bound on the expected symbol error rate for each
user is
−m/2
n
X
n
i
EH (Eν (SE)) ≤ n
0.5
1+ 2
(8)
i
σ
i=1
i
X
X X 1
exp(−(||
2hb(j) ||/2σ)2 /2) (3)
2
j=1
1≤i≤n b∈Si
2
104
m=14
m=12
m=8
Number of nodes visited
A bound on the expected symbol error rate averaged over
all users is
−m/2
n
X
i n
i
(9)
EH (Eν (SEav )) ≤
1+ 2
2n i
σ
i=1
The last expression (9) is compared to empirically computed quantities in Section IV, whereas (7) has been used
to derive asymptotic bounds. But the arguments in the
asymptotic analysis of (7) go through for (8) and (9)
also.
103
102
4
6
IV. T IGHTNESS OF THE DERIVED BOUND
In this section, we present simulation results illustrating the tightness of the bounds derived in Section III.
We use the sphere decoding approach to ML decoding
(see e.g. [5]). There are variants of the sphere decoder
for underdetermined systems (see e.g. [6],[7]). We use
the depth-first tree-search-based decoder, as described in
Algorithm 1 in Appendix A, in our numerical experiments. We give plots for number of transmit antennas
n = 16 and different numbers of antennas at the
receiver. Figure 1 shows that the upper bounds are
12
Receiver SNR (dB)
14
16
18
20
drastically. Thus the sphere decoding ML decoder may
work efficiently even in regimes where the number of
receiver measurements is less than the number of transmit symbols to be decoded. One can therefore expect
similar BER/SER (symbol error rate) performances at
a still lower complexity from approximate ML sphere
decoders.
ML BER for m=14
Bound for m=14
ML BER for m=12
Bound for m=12
ML BER for m=8
Bound for m=8
101
10
Fig. 2. The number of nodes visited for n = 16 and different m.
Note that for n = 16 a trivial ML implementation would involve
search over 216 = 65536 nodes. The sphere decoder is much faster.
Note that a node refers to an entry in the stack S described in the
algorithm 1
103
102
8
100
V. A SYMPTOTIC BEHAVIOUR OF THE UPPER BOUND
Bit error rate
10−1
10−2
In this section, we describe conditions under which
the upper bound on the codeword error probability (7)
goes to zero. We bound the number of receiver antennas
per transmitting antenna for which the error is “small”
in some sense. More specifically we define α = m
n . We
then have the following result:
10−3
10−4
10−5
10−6
10−7
4
6
8
10
12
Receiver SNR (dB)
14
16
18
20
Theorem 1. Fix α > 0. Then for m = αn there exists a
c(α) < 0 and a n0 , such that for all n > n0 ,
Fig. 1. The BER curves for n = 16 and different m. The dotted
lines represent the bound in (9).
EH (Pe ) ≤ 2nc(α) .
increasingly tight at higher SNRs. Figure 2 shows the
complexity of the sphere decoding algorithm for the
SNRs of interest. The ML decoder would require visits to
all of 2n nodes. However, the sphere decoder can prune
off several trees based on what it has already decoded
so far. The plot demonstrates that with higher number of
receiver antennas and/or with lower noise, the number
of nodes visited in the sphere decoding algorithm falls
Note that the c(α) and n0 in the above theorem
depends also on the σ 2 , the noise variance at the receiver
antennas.
Proof: We consider the expression:
Pe ≤
3
−αn/2
n X
n
i
1+ 2
σ
i
i=1
(10)
Here λmax is the largest eigenvalue of HH T . We now
use results about the distribution of the largest eigenvalue of the Wishart matrices, also known as Laguerre
ensembles. More specifically we use the result that the
expected value of λmax is linear in n [8] . This may
also be proved for example by using the bounds on the
tail probability of the maximum eigenvalue of Laguerre
ensembles presented in [9],[10]. This implies that
We find the error exponent of the expression. Clearly
−αn/2
n
i
(11)
EH (Pe ) ≤ n max
1+ 2
1≤i≤n i
σ
−αn/2
i
≤ n max 2nH2 (i/n) 1 + 2
(12)
1≤i≤n
σ
−αn/2
nδ
(13)
≤ n max 2nH2 (δ) 1 + 2
σ
1/n≤δ≤1
α
= 2n(max1/n≤δ≤1 H2 (δ)− 2
log n
nδ
log(1+ σ
2 )+ n )
EH (λmax ) ≤ C1 n
(14)
α
= 2n(max1≤y≤n H2 (y/n)− 2
for some C1 > 0 . C1 is independent of m as long as m
grows at most linearly with n. Plugging this bound into
the expression for ergodic capacity, we get the following:
n
log(1+ σy2 )+ log
n )
(15)
nc(α)
=2
→0
where c(α) < 0 (Appendix B) (16)
as n → ∞
EH (C) ≤ 0.5m log2 (1 + C1 n/σ 2 ) ≤ 0.5m log2 n + Cm
(17)
for some C > 0 independent of m, n. The scaling is
dominated by the first term. Thus, in order to transmit
one bit per user, m needs to grow at least like
Note that the result in Theorem 1 holds for any
positive α. It thus seems plausible that α can even be
allowed to vanish with increasing n, provided the decay
is slow enough. while maintaining a regime of reliable
communication. How slow can α decrease for this to
hold ? The following result addresses this question.
m ≥ 2n/ log2 (n)
Hence we see that asymptotically (in the limit of large
transmitters and receivers), one shot uncoded communication in a fading channel comes very close to the limit
achievable with coding across time and fading states.
Another way to view our result is as follows. EH (C),
the ergodic capacity of the system, can scale at most like
1/2m log n + cm where c is a constant. This implies
that the best scheme employing coding across time
and fading states can achieve an average (in time) rate
of at most 1/2m log n across all users or equivalently
1/2(m/n) log n per user. This suggests that we cannot
hope to push m/n down below 2/(log n) even if we
could code across time and fading states.
On the other hand, we have the result that the average
(across fading states) probability of error in the transmission of BPSK symbols scales down exponentially with
increasing number of transmitters and receivers. But the
simple BPSK scheme has a fixed rate of n with some
error probability. The result shown here implies that we
can provably push the number of receivers all the way
down to the minimum required to maintain a capacity
of greater than or equal to n asymptotically and still
have a vanishing error probability without employing any
transmitter side coding.
Theorem 2. For m = logkn n for any k > 2 and a large
2
enough n, there exists a d < 0 such that the expected
probability of block error goes to zero exponentially fast
with n, i.e.
dn
EH (Pe ) ≤ 2 (log2 n)
We thus have the result that we can decode perfectly
even with Θ(1/ log2 n) number of measurements per
transmitter. The proof, which follows in a very similar
manner as the earlier ones, is presented in Appendix D.
We show in a subsequent section that in the limit
of large n, the above is the best number we can push
the minimum number of receiver antennas such that the
capacity of the equivalent MIMO channel is at least
the number of transmitters. That way the channel could
support 1 bit per transmitter on an average even if coding
across time were employed.
VI. C OMPARISON WITH OPTIMAL SCHEMES
In this section we show that we cannot achieve a
smaller ratio for m
n even if we employ coding across
time. We bound the ergodic capacity of the equivalent
channel matrix H ∈ Rm×n as
EH (C) = EH (0.5 log2 |I + HH T /σ 2 |)
≤ mEH (0.5 log2 (1 + λmax /σ 2 ))
≤ 0.5m log2 (1 + EH (λmax /σ 2 ))
VII. E XTENSION TO ARBITRARY FADING
DISTRIBUTIONS
(18)
We note that once we have the results about reliable
uncoded transmission in the previous sections, we can
extend them to arbitrary fading distributions by a central
(19)
(20)
4
limit theorem type of argument. In particular, we start
from (2)
Pe,i ≤ 0.5 exp(−(||
i
X
j=1
achieve a performance close to that possible with coding
across time. Moreover, the number of receive antennas per transmitting user needed for perfect decoding
vanishes in the limit of large transmitters. It would
be interesting to also extend our performance analysis
to include the effects of imperfect channel knowledge,
suboptimal decoders, and correlated channels, in order
to get more realistic estimates on the underdeterminism
achievable in practice.
2hb(j) ||/2σ)2 /2)
Under the independent fading assumption, we note that
the following holds
EH (Pe,i ) ≤ 0.5EH (exp(−||
i
X
j=1
2H1,b(j) ||2 /8σ 2 ))m
IX. ACKNOWLEDGEMENTS
We now argue that under arbitrary fading, the expectation
above is “close” to that of a Rayleigh fading scenario.
To see this, we use the Berry Esseen bound for the rate
of convergence of the distribution functions of normalized sums of random variables to a normal distribution
function (see e.g. [11]) We state this for completeness’
sake.
The first author would like to thank Yash Deshpande
and Kartik Venkat for enlightening discussions and Boon
Sim Thian for help with his implementation of the
sphere decoder. This work is supported by the 3Com
Corporation Stanford Graduate Fellowship and by the
NSF Center for Science of Information (CSoI) grant
NSF-CCF-0939370 .
Theorem
3. If, for a distribution function F (.),
R 3
|x| dF (x) is bounded, then the deviation
of the disPn
tribution function G of y = √1n i=1 xi ( where
Ry
2
xi iid, ∼ F ) from Φ(y) = √12π −∞ exp−s /2 ds is
bounded above, i.e.
√
sup |G(y) − Φ(y)| ≤ C0 / n
A PPENDIX
A. Description of the sphere decoding algorithm
The details are presented in Algorithm 1 for the sake
of concreteness.
y
Algorithm 1 ML detection using sphere decoder (depth
first search)
Require: y ∈ Rm , H ∈ Rm×n
[Q, R]
⇐
qr(H), Q
∈
Rm×m , R
∈
m×n
R
QR decomposition step
ỹ ⇐ QT y
Push 2n−m length n − m codewords from alphabet
{−1, +1} onto stack S
r=∞
while S is not empty do
x ⇐ Pop(S)
d ⇐ length(x)
if ||(ỹ − Rx)(n − d + 1 : m)||2 < r then
if length(x) < n then
Push 2 codewords of length d + 1 from
{−1, +1} ending with x onto stack S
else
if r > ||ỹ − Rx||2 then
r = ||ỹ − Rx||2
xM L = x
end if
end if
end if
end whilereturn xM L
Using this result we can come up with an upper bound
for a sufficiently large i > i0 (say) as follows
!m
−1/2
i
−1/2
+ Ci
EH (Pe,i ) ≤ 0.5
1+ 2
σ
The details will be presented in the Appendix E. Here C
depends only on the distribution, and is independent of
n, m. We can choose i0 to be the minimum i for which
−1/2
i
1+ 2
+ Ci−1/2 < 1
σ
Note that this is also independent of m, n and depends
only on the fading distribution. For all i < i0 we
note that the expectation is always less than one (by
inspection) and vanishes in the limit as n, m → ∞. With
this, the arguments of the previous sections go through
and we have the same limiting behaviour.
VIII. C ONCLUSION
We have derived the minimum number of receivers in
an underdetermined uplink system which would ensure
perfect decodability for uncoded transmissions over fading multiple access channels. The results suggest that in
the limit of large number of users, a simple constellation
and joint ML decoding at the receiver is sufficient to
5
0.2
B. Derivation of the c(α)
g(n,y)
We define
0.0
g(n) = max g(n, y)
−0.2
1≤y≤n
where
−0.4
log2 n
α
y
g(n, y) = H2 (y/n) − log2 (1 + 2 ) +
2
σ
n
−0.6
We now show that g(n) is bounded above by a negative constant for a sufficiently large n. We note that
g(∞, 1) = − α2 log(1 + σ12 ) < 0. Define
−0.8
−1.0
0
100
200
300
400
y
c(α) = g(∞, 1) + s.t. c(α) < 0. We show that for any > 0, there exists
a n0 such that
Fig. 3.
g(n) < c(α) ∀n > n0
This n2 can be one possible n0 that we were searching
for earlier.
A more precise characterization of the behaviour of
g(n) can be had by differentiating g(n, y) respect to
y. We get that the maximizing y should satisfy the
following if it lies in the interior of [1, n]
n
n−y
y
800
900
Behaviour of g(n, y) with y, σ 2 = 10, α = 0.3
−k1 ≤ g(n) ≤ −k2
∀n large enough
C. Derivation for Θ(1/(log2 n) ) receivers per transmitter
We define
g(n) = max g(n, y)
n−y
α
)−
=0
y
2(log 2)(y + σ 2 )
(log 2) log2
700
Thus the local maximum at β = β2 (α) would eventually
(with larger n) become more negative than the value at
β = 1/n (note that y = βn needs to be ∈ [1, n]). But
the value at β = 1/n is Θ(−1). In other words there
exists constants k1 > 0 and k2 > 0 such that
for all y > n1 . Then there exists an n2 such that for all
n > n2 ,
log2 n
H2 (n1 /n) +
< −c(α)
n
or
600
is governed for large n, by the values of g(n, y) at two
points, y = 1 and y = β2 (α)n But we know that
α
lim H2 (β)− log2 (1+βn/σ 2 ) → −∞ < 0, β = β2 (α)
n→∞
2
This observation follows from the fact that there exists
a n1 such that
α
y
− log2 (1 + 2 ) < −1 + 2c(α)
2
σ
1/n(log2
500
1≤y≤n
where
g(n, y) = H2 (y/n) −
α
−
=0
2(y + σ 2 )
k/2
y
log2 n
log2 (1 + 2 ) +
(log2 n)
σ
n
Differentiating g(n, y) respect to y we get that the
optimal y should satisfy the following if it lies in the
interior of [1, n]
We now study the behaviour of the optimal y for large
n→∞
n. We note that if the optimal y −→ βn, 0 < β < 1
then the following has to hold
1−β
α
log2
−
=0
β
2(log 2)β
1/n(log2
k
n−y
)−
=0
y
2(log 2)(log2 n) (y + σ 2 )
or
The above equation has two solutions in general (it may
also have no solution in which case we are done). Let
us call the solutions β1 , β2 such that β1 (α) < β2 (α). By
inspection, the second extremum must be a maximum.
Please see figure 3. Thus the maximizing value of g(n)
(log 2)(log2 n) log2
n
n−y
y
−
k
=0
2(y + σ 2 )
We now study the behaviour of the optimal y for large
n→∞
n. We note that if the optimal y −→ βn, 0 < β < 1
6
for large n, by the values of g(n, y) at two points, y = 1
and y = βn ≈ n/2 But we know that
then the following has to hold
k
1−β
−
=0
(log2 n) (log 2) log2
β
2β
k
log2 (1 + βn/σ 2 )
n→∞
2 log2 n
1
→ c0 < 0 (when β = , k > 2)
2
lim H2 (β) −
The above equation has two solutions in general (it may
also have no solution in which case we are done). In
particular, as n becomes large, we see that one of the
solutions is at β ≈ 12 and another is at β = (n) (very
small). Thus the maximizing value of g(n) is governed
for large n, by the values of g(n, y) at two points, y = 1
and y = βn ≈ n/2 But we know that
lim H2 (β)−
n→∞
Thus the local maximum at β = 21 would eventually
(with larger n) become more negative than the value at
β = 1/n (note that y = βn needs to be ∈ [1, n]). But
the value at β = 1/n is Θ(−1/ log2 n). In other words
there exists constants k1 and k2 such that
1
k
log2 (1+βn/σ 2 ) → −∞, β =
2(log2 n)
2
−
Thus the local maximum at β = 21 would eventually
(with larger n) become more negative than the value at
β = 1/n (note that y = βn needs to be ∈ [1, n]). But the
value at β = 1/n is Θ(−1/(log2 n) ). In other words
there exists constants k1 and k2 such that
k2
k1
≤ g(n) ≤ −
−
(log2 n)
(log2 n)
k1
k2
≤ g(n) ≤ −
log2 n
log2 n
E. Derivation of the bound for arbitrary fading statistics
We start from the following expression for the upper
bound on the error probability of mistaking a codeword
with another differing in i symbol positions
i
X
2H1,b(j) )2 /8σ 2 ))m
EH (Pe,i ) ≤ 0.5(EH exp(−(
∀n large enough
j=1
D. Derivation for Θ(1/ log2 n) receivers per transmitter
Defining
Pi
We define
x=
g(n) = max g(n, y)
where
EH (Pe,i ) ≤ 0.5(Ex exp(−ix2 /8σ 2 ))m
k/2
y
log2 n
log2 (1 + 2 ) +
log2 n
σ
n
We now use the Berry-Esseen bound for the distribution
function F (x) of x, the normalized sum of zero mean,
unit variance random variables, which reads
√
sup |F (x) − Φ(x)| < C0 / i
Differentiating g(n, y) respect to y we get that the
optimal y should satisfy the following if it lies in the
interior of [1, n]
x
Rx
Here Φ(x) = √12π −∞ exp(−x2 /2)dx is the distribution function of standard Gaussian r.v. Using this we can
proceed to find an upper bound for the expected error
term under arbitrary fading as follows:
Z ∞
2
2
Ex exp(−ix /8σ ) =
exp(−ix2 /8σ 2 )f (x)dx
−∞
Z ∞
(a)
=
2ix exp(−ix2 /8σ 2 )F (x)dx
−∞
Z ∞
(b)
≤
2ix exp(−ix2 /8σ 2 )Φ(x)dx
−∞
Z
∞
C0
+2
2ix exp(−ix2 /8σ 2 ) √ dx
i
0
2 −1/2
−1/2
= (1 + i/σ )
+ Ci
n−y
k
1/n(log2
)−
=0
y
2(log 2) log2 n(y + σ 2 )
or
log2 n log2
n
n−y
y
−
2H1,b(j)
√
i
j=1
we get that
1≤y≤n
g(n, y) = H2 (y/n) −
∀n large enough
k
=0
2(log 2)(y + σ 2 )
We now study the behaviour of the optimal y for large
n→∞
n. We note that if the optimal y −→ βn, 0 < β < 1
then the following has to hold
1−β
k
log2 n(log 2) log2
−
=0
β
2β
The above equation has two solutions in general (it may
also have no solution in which case we are done). In
particular, as n becomes large, we see that one of the
solutions is at β ≈ 21 and another is at β = (n) (very
small). Thus the maximizing value of g(n) is governed
In
R x the above f (x) is the density function of x, F (x) =
f (s)ds is the distribution function of x, Φ(x) is the
−∞
7
distribution function for the standard normal distribution.
(a) is by integration by parts. (b) is by application of the
Berry Esseen bound. The constant C is defined as
Z ∞
2ix exp(−ix2 /8σ 2 )C0 dx
C=2
R EFERENCES
[1] T. Cover and J. Thomas, Elements of Information Theory. Wiley
Online Library, 1991, vol. 6.
[2] G. Caire, D. Tuninetti, and S. Verdú, “Suboptimality of TDMA in
the low-power regime,” Information Theory, IEEE Transactions
on, vol. 50, no. 4, pp. 608–620, 2004.
[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge Univ Pr, 2005.
[4] S. Verdu, “Minimum probability of error for asynchronous
Gaussian multiple-access channels,” Information Theory, IEEE
Transactions on, vol. 32, no. 1, pp. 85 – 96, jan 1986.
[5] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I.
Expected complexity,” Signal Processing, IEEE Transactions on,
vol. 53, no. 8, pp. 2806–2818, 2005.
[6] K. Wong and A. Paulraj, “Efficient near maximum-likelihood
detection for underdetermined mimo antenna systems using a
geometrical approach,” EURASIP Journal on Wireless Communications and Networking, vol. 2007, p. 10, 2007.
[7] G. Romano, F. Palmieri, P. Rossi, and D. Mattera, “A tree-search
algorithm for ML decoding in underdetermined mimo systems,”
in Wireless Communication Systems, 2009. ISWCS 2009. 6th
International Symposium on. IEEE, 2009, pp. 662–666.
[8] I. Johnstone, “On the distribution of the largest eigenvalue in
principal components analysis,” The Annals of statistics, vol. 29,
no. 2, pp. 295–327, 2001.
[9] M. Ledoux, B. Rider et al., “Small deviations for beta ensembles,” Electronic Journal of Probability, vol. 15, pp. 1319–1343,
2010.
[10] K. Johansson, “Shape fluctuations and random matrices,” Communications in mathematical physics, vol. 209, no. 2, pp. 437–
476, 2000.
0
Although the above bound holds for all i, it is useful for
proving perfect decoding only when it is less than one.
We note that there exists an i0 dependent only on C,
such that for all i ≥ i0 ,
C
g(i) = (1 + i/σ 2 )−1/2 + √ < 1
i
(21)
For i < i0 we note by inspection that
Ex exp(−ix2 /8σ 2 ) ≤ c < 1
We now note that results like Theorems 1, 2 continue to
hold since for all three cases
i0 i
X
X
n
2H1,b(j) )2 /8σ 2 )))m (22)
(EH (exp(−(
i
j=1
i=0
/ ni0 cm → 0 as c < 1
(23)
for m, n scaling as in the Theorems. Thus the behaviour
of the upper bound on the error probability is determined
by terms in the regime i > i0 where (21) holds. Thus
the results of Section V continue to hold even when the
fading distribution is not Gaussian.
[11] I. Shevtsova, “Sharpening of the Upper Bound of the
Absolute Constant in the Berry Esseen Inequality,” Theory of
Probability and Its Applications, vol. 51, no. 3, pp. 549–553,
2007. [Online]. Available: http://epubs.siam.org/doi/abs/10.1137/
S0040585X97982591
8