ISIT 2008, Toronto, Canada, July 6 - 11, 2008
Finite Signal-set Capacity of Two-user Gaussian
Multiple Access Channel
Harshan J.
B. Sundar Rajan
Dept of ECE, Indian Institute of science
Bangalore 560012, India
Email:[email protected]
Dept of ECE, Indian Institute of science
Bangalore 560012, India
Email:[email protected]
Abstract— The capacity region of a two-user Gaussian Multiple
Access Channel (GMAC) with complex finite input alphabets and
continuous output alphabet is studied. When both the users are
equipped with the same code alphabet, it is shown that, rotation
of one of the user’s alphabets by an appropriate angle can make
the new pair of alphabets not only uniquely decodable, but will
result in enlargement of the capacity region. For this set-up, we
identify the primary problem to be finding appropriate angle(s)
of rotation between the alphabets such that the capacity region
is maximally enlarged. It is shown that the angle of rotation
which provides maximum enlargement of the capacity region also
minimizes the union bound on the probability of error of the sumalphabet and vice-verse. The optimum angle(s) of rotation varies
with the SNR. Through simulations, optimal angle(s) of rotation
that gives maximum enlargement of the capacity region of GMAC
with some well known alphabets such as M -QAM and M -PSK
for some M are presented for several values of SNR. It is shown
that for large number of points in the alphabets, capacity gains
due to rotations progressively reduce. As the number of points
N tends to infinity, our results match the results in the literature
wherein the capacity region of the Gaussian code alphabet doesn’t
change with rotation for any SNR.
and the number of points in the code alphabets, we address
the problems related to designing alphabet pairs, (S1 , S2 ) for
a two - user GMAC which maximally enlarges the capacity
region. Throughout the paper, the terms alphabet and signal
set are used interchangeably. The contributions of the paper
may be summarized as below :
•
•
I. I NTRODUCTION AND P RELIMINARIES
Capacity region of the two-user Gaussian Multiple Access
Channels (GMAC) with continuous input alphabet and continuous output alphabet is well known [1] - [6]. Such a model
assumes the users to have Gaussian code alphabets and the
additive noise to be Gaussian distributed. Though, the capacity
region of such a channel provides insights in to the achievable
rate pairs (R1 , R2 ) in an information theoretic sense, it fails
to provide information on the achievable rate pairs when we
consider finitary restrictions on the input alphabets and analyze
some real world practical signal constellations like QAM and
PSK etc.
In this paper, we obtain the capacity region of a two-user
GMAC with finite complex input alphabets and continuous
output alphabet. Throughout the paper, the mutual information
of the GMAC when the symbols from the input alphabets
are chosen with uniform distribution is referred to as the
Constellation Constrained (CC) capacity of the GMAC [7].
Henceforth, unless specified, (i) input alphabet refers to a finite
complex alphabet and a GMAC refers to a Gaussian MAC with
finite complex input alphabets and continuous output alphabet
and (ii) capacity (capacity region) refers to the CC capacity
(capacity region). For the given transmit power of each user
978-1-4244-2571-6/08/$25.00 ©2008 IEEE
•
We identify the primary problem for a GMAC when both
users use the same alphabet is to find angle(s) of rotation
between the two alphabets such that the capacity region
enlargement is maximum.
It is shown that the optimal angle(s) of rotation between
the alphabets which gives maximum enlargement of the
capacity region (i) depends on the operating SNR and
(ii) also minimizes the union bound on the probability
of error of the sum-alphabet and vice-verse. (Section
III). Through simulations, we provide optimal angles
of rotation that provides maximum enlargement of the
capacity region for some well known alphabets such as
M-PSK, M-QAM etc. for some M at some fixed SNR
values. (Table I, Section IV).
Furthermore, it is shown that (i) capacity gains due to
rotations are significant when the number of points in
the alphabet is small. In particular, for a given SNR, as
the number of points in the alphabet, N increases, the
difference in the capacity region between the alphabet
pair with optimal rotation and without rotation becomes
progressively smaller; (ii) Also, for a fixed number of
points, as SNR increases the significance of rotation
increases. However, for very large values of N , capacity
gains due to rotations progressively reduce at all SNR,
and (iii) As N tends to infinity, these observations match
the results in the literature wherein the capacity region
of the Gaussian codebook doesn’t change with rotation
at any SNR. (Section IV)
Notations: For a variable x which takes values from the set
S, we always assume some ordering of its elements and use
xi to represent the ith element of S. Cardinality of the set S
is denoted by |S|. Absolute value of a complex number x is
denoted by |x| and E [x] denotes the expectation of the random
variable x. A circularly symmetric complex Gaussian random
vector, x with mean μ and covariance matrix Γ is denoted by
x ∼ CG (μ, Γ). For a random variable x, H(x) represents the
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ISIT 2008, Toronto, Canada, July 6 - 11, 2008
i
User 1
w3
y
x1 + x2
−1
x2
Adder
AWGN
1
Destination
w4
w2
−i
User 2
Signal set − 1
Fig. 1.
entropy of x. Also, the set of all real and complex numbers
are denoted by R and C respectively.
The remaining content of the paper is organized as follows:
In Section II, the signal model of the two user GMAC is
presented along with its capacity region. In particular, the
capacity region of Uniquely Decodable (UD) (see Definition
1) alphabet pairs are discussed in the high SNR regime.
In Section III, for a GMAC, where both users uses the
same alphabet, we consider the problem of finding angles of
rotation between the alphabets so as to enlarge the capacity
region. Conditions on the angle of rotations in terms of the
distance distribution of the sum-alphabet is provided which
gives maximum enlargement of the capacity region of GMAC.
In Section IV, using computer search, optimal angle(s) of
rotation are presented for some well known alphabets at fixed
SNR values. Concluding remarks and possible directions for
further work constitute Section V.
II. S IGNAL MODEL , M UTUAL INFORMATION AND UD
SIGNAL SET PAIRS
The model of a two-user Gaussian MAC is as shown
in Figure 1 consisting of two users which need to convey
information to a single destination. It is assumed that User-1
and User-2 communicate to the destination at the same time
and in the same frequency band. Symbol level synchronization
is assumed between the two users. The two users are equipped
with alphabets S1 and S2 each of size N1 and N2 respectively.
When User-1 and User-2 transmit symbols x1 and x2 from S1
and S2 respectively, the destination receives a symbol y given
by,
y = x1 + x2 + z where x1 ∈ S1 , x2 ∈ S2 , z ∼ CG 0, σ 2 .
(1)
We compute the mutual information I(x2 : y) for User-2 and
I(x1 : y | x2 ) for User-1 when symbols x1 and x2 are assumed
to be uniformly distributed. By symmetry, I(x1 : y) and I(x2 :
y | x1 ) can also be found. Considering x1 + z as the additive
noise, I(x2 : y) is given below [1],
I(x2 : y) = H(y) − H(y|x2 ).
(2)
Since x1 is assumed to be uniformly distributed and z is
Gaussian distributed, p(y | x2 = xi2 ) and H(y | x2 ) are given
by
xi2 )
Signal set − 2
Two-user Gaussian MAC model
Fig. 2.
p(y | x2 =
w1
z
x1
N1 −1
1 =
p(y | x1 = xk1 , x2 = xi2 ).
N1
k=0
(3)
H(y | x2 ) =
Two 4 - point signal sets
N2 −1
1 H y | x2 = xi2 .
N2 i=0
(4)
Since x2 is also uniformly distributed,
|y − xk1 − xi2 |2
1
exp(−
p(y | x1 = xk1 , x2 = xi2 ) = √
) and
2σ 2
2πσ
(5)
N
1 −1 N
2 −1
1
k
i
p(y) =
p(y | x1 = x1 , x2 = x2 ).
(6)
N1 N2
i=0
k=0
Using (2) - (6), the mutual information, I(x2 : y) is given in
(7) at the top of the next page, where the expectation is with
respect the distribution of z. The mutual information I(x1 :
y | x2 ) is defined by
I(x1 : y | x2 ) = H(y | x2 ) − H(y | x2 , x1 ).
(8)
Using (3), (4), (5) and (8), I(x1 : y | x2 ) is given in (9)
at the top of the next page, where the expectation is with
respect to the distribution of z. Therefore, using (7) and (9),
the sum mutual information of both the users at the destination
is I(x2 : y) + I(x1 : y | x2 ).
The upper-bounds on the rate-pair (R1 , R2 ) is given by [1],
R1 < I(x1 : y | x2 ),
R1 + R2
R2 < I(x2 : y | x1 ) and
< I(x1 : y | x2 ) + I(x2 : y),
(10)
where the terms I(x2 : y) and I(x1 : y | x2 ) are given in (7)
and (9) respectively.
A. Uniquely decodable alphabet pairs for GMAC
We formally define a UD alphabet pair and discuss the
advantages of UD alphabet pair over a non-UD pair in terms
of their capacity region for GMAC at high SNRs. Given two
alphabets S1 and S2 , we denote the sum-alphabet of S1 and
S2 by Ssum where
Ssum = {x1 + x2 | ∀ x1 ∈ S1 , x2 ∈ S2 }
and the adder channel in a two-user Gaussian MAC in Figure
1 can be viewed as a mapping φ given by
φ : S1 × S2 −→ Ssum where φ((x1 , x2 )) = x1 + x2 .
(11)
Definition 1: An alphabet pair (S1 , S2 ) is said to be
Uniquely Decodable (UD) if the mapping φ in (11) is oneone.
1204
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
⎤⎤
⎡
⎡ N −1 N −1
k1
k2
i1
i2
1
2
2
2
N
1 −1 N
2 −1
i1 =0
i2 =0 exp −|x1 + x2 − x1 − x2 + z| /2σ
1
⎦⎦ .
I(x2 : y) = log2 (N2 ) −
E ⎣log2 ⎣
N1 −1
N1 N2
exp −|xk1 − xi1 + z|2 /2σ 2
k1 =0 k2 =0
1
i1 =0
(7)
1
⎤⎤
⎡ N −1
⎡
k2
i2
2
2
N2 −1
i2 =0 exp −|x1 − x1 + z| /2
1 ⎦⎦ .
I(x1 : y | x2 ) = log2 (N1 ) −
E ⎣log2 ⎣
N1
exp (−|z|2 /2σ 2 )
(9)
k2 =0
Example for a UD alphabet pair is shown in Figure 2.
Example for a non - UD alphabet pair is given in (12).
S1 = S2 = {(x, x), (−x, x), (−x, −x), (x, −x) | x = 0.707} .
(12)
It can be easily seen that if two signal sets S1 and S2 have
more than one signal point common then the pair (S1 , S2 )
is necessarily non-UD. However, not having more than one
common signal point is not sufficient for a pair to be UD, as
exemplified by the pair S1 = {1, ω, ω 2} and S2 = {−1, 1 +
ω, 1 + ω 2 } where ω is a complex cube root of unity.
In (11), for some x ∈ Ssum , let
φ−1 (x) = {(x1 , x2 ) ∈ S1 × S2 | φ(x1 , x2 ) = x} ,
∗
and let Mx = |φ−1 (x)|. Also, let Ssum
⊆ Ssum be given by
∗
Ssum
= {x ∈ Ssum |Mx = 1}
(13)
∗
|. As SNR tends to infinity, the maximum
and let M = |Ssum
achievable sum-rate of a alphabet pair (S1 , S2 ) is log2 (M )
bits per channel use. It is straightforward to verify that, if the
mapping φ is one-one, then M = N1 N2 . If the alphabet pair
is non-UD, then M < N1 N2 and therefore, at large SNRs,
maximum achievable sum rate of a UD pair is more than
that of a non-UD pair provided the alphabet pairs have same
number of points and same average power. It can be verified
that the maximum achievable sum rate for the signal set pairs
in (12) and Figure 2 are 3 and 4 bits respectively.
From the above arguments, it is clear that, at high SNR, a
UD alphabet pair provides larger capacity-region than a nonUD pair when both the alphabet pairs have same number of
points and equal average power. At low SNR, the capacity
region of a non-UD alphabet pair can be larger than a UD
alphabet pair, one such example is discussed in Section IV
(see Table I, 8-QAM at 6 dB). Characterizing the differences
in the capacity region between UD and non-UD alphabets at
low SNR seems difficult due to its dependence on the chosen
alphabets. So, hence forth, in the rest of the paper, we restrict
ourselves to designing UD alphabet pairs that enlarges the
capacity region of the two-user GMAC for different values of
SNR.
other, then UD property is attainable. Moving one step further,
we consider the problem of finding the optimal angle(s) of
rotation between the alphabet pair such that the capacity region
is maximally enlarged at different values of SNR.
Let S1 be a finite alphabet and S2 be a set of symbols
obtained by rotating all symbols of S1 by an angle Θ. From
(10), the upper-bounds on the rate region depends on the
mutual information values I(x1 : y | x2 ), I(x2 : y | x1 )
and I(x2 : y). The terms I(x1 : y | x2 ) and I(x2 : y | x1 ) are
some functions of the Distance Distribution (DD) of S1 and
S2 respectively. Since, we start with a known S1 and S2 is a
rotated version of S1 , for any Θ, the DD of S1 and S2 are the
same. Hence, the values I(x1 : y | x2 ) and I(x2 : y | x1 ) do
not change for different values of Θ. However, from (7), the
term I(x2 : y) is a function of the DD of the sum-alphabet,
Ssum of S1 and S2 . The DD of Ssum changes for different
values of Θ and hence the term I(x2 : y) changes for different
values of Θ.
In the following theorem, we provide conditions on the
angle(s) of rotation, Θ such that I(x2 : y) is maximized which
in-turn maximally enlarges the capacity region in (10).
Theorem 1: Let (S1 , S2 ) be an alphabet pair such that S2
is a rotated version of S1 by an angle Θ. Let N be the
cardinality of S1 . The mutual information I(x2 : y) in (7)
can be maximized by choosing the angle of rotation, Θ such
that the following expression in (14) is minimized
N
−1 N
−1
−1 N
−1 N
exp −|xk11 + xk22 − xi11 − xi22 |2 /4σ 2 .
k1 =0 k2 =0 i1 =0 i2 =0
(14)
Proof: Since N1 = N2 = N and it is fixed, I(x2 : y) in
(7) can be maximized by minimizing the term in (15) shown
at the top of the next page. Since the denominator term inside
the logarithm of (15) doesn’t change for different values of Θ,
minimizing the term in (15) and minimizing the term given by
(16) (also shown at the top of the next page) are equivalent.
Since (16) is a sum of several expectations, the rotation Θ
needs to be chosen to minimize each of the expectations.
Therefore, for every k1 and k2 , the term in (17) needs to be
minimized.
"
III. C APACITY MAXIMIZING ALPHABET PAIRS FROM
E log2
ROTATIONS
For a GMAC with S1 = S2 , it is clear that if one of the users
uses an appropriate rotated version of the alphabet used by the
1205
"N −1 N −1
1
2
X
X
i1 =0 i2 =0
“
”
exp −|xk1 1 + xk2 2 − xi11 − xi22 + z|2 /2
##
.
(17)
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
⎤⎤
exp −|xk11 + xk22 − xi11 − xi22 + z|2 /2
⎦⎦ .
E ⎣log2 ⎣
N1 −1
k1
i1
2 /2
exp
−|x
−
x
+
z|
k1 =0 k2 =0
1
1
i1 =0
N
N
1 −1 N
2 −1
1 −1 N
2 −1
k1
k2
i1
i2
2
E log2
exp −|x1 + x2 − x1 − x2 + z| /2
.
N
1 −1 N
2 −1
⎡
k1 =0 k2 =0
⎡ N
1 −1
i1 =0
N2 −1
i2 =0
"N −1 N −1
1
2
X
X
#
“
”
exp −|xk1 1 + xk2 2 − xi11 − xi22 + z|2 /2 .
(18)
i1 =0 i2 =0
Evaluating the expectations in (18) with respect to z, for every
k1 and k2 , (18) reduces to
N1 −1 N2 −1
X X
(16)
i1 =0 i2 =0
As log2 (.) is an monotonic increasing function of its argument,
the rotation, Θ that minimizes the term (17) also minimizes the
term in (18) shown below, for every k1 and k2 and vice-verse.
E
(15)
“
”
exp −|xk1 1 + xk2 2 − xi11 − xi22 |2 /4 .
alphabets, will cause perturbations in the sum-alphabet and
hence the points in Br gets rearranged. For large values of N ,
even though the points in Br rearrange themselves as a result
of rotation, the density of Br is so large that the distance
distribution of the points inside the ball changes negligibly
and as a result of Theorem 1, there is no gain in the capacity
due to rotation. When the number of points tends to infinity,
these observations match the results in the literature, where
the Gaussian code book (which is of infinite cardinality) is
invariant to rotations as far as capacity gains are concerned.
(19)
i1 =0 i2 =0
IV. O PTIMAL ROTATIONS FOR SOME KNOWN ALPHABETS
In this section, for a given alphabet, S1 and for a given value
of P1 = P2 , through computer search (using the metric in (14)),
we find optimal angle(s) of rotation, Θ (in degrees) that results
in a DD of the sum-alphabet which maximizes I(x2 : y). For
the simulation results, additive noise, z is assumed to have
unit variance per dimension. i.e. σ 2 = 2. The optimal values
of Θ are calculated by varying the angle of rotation from 0
to 180 in steps of 0.25 degrees. In Table I, for various values
N−1
N−1
N−1
N−1
”
“
of P1 /σ 2 = SNR, optimal values of Θ are presented for some
X
X
X
X
1
P ≤ 2
exp −|xk1 1 + xk2 2 − xi11 − xi22 |2 /4σ 2 . well known alphabets such as M -QAM, M -PSK for M =
N k =0 k =0 i =0 i =0
1
2
1
2
(20) 2, 4, 8 and 16. Against every signal set, a 3-tuple (a, b, c) is
presented where a denotes the optimal value of Θ, b represents
From (20) and (14), it can be observed that the alphabet pair the multiplicity of optimal angle and c denotes the difference
(S1 , S2 ) that minimizes the union bound on the probability of between the metric values in (14), Δm when the alphabets are
error for the sum-alphabet, Ssum also maximizes I(x2 : y) and used with zero rotation and with optimal angles of rotation.
vice-verse. Therefore, the rotation Θ has to be chosen such that In Table I, observe that there are multiple angles of rotation
the distance distribution of Ssum minimizes the union bound that enlarges the capacity region for certain alphabets at some
SNRs (Example : QPSK at SNR = 8 db, 8-PSK at SNR =
on the probability of error of Ssum .
From the above theorem, since the value of I(x2 : y) 10 db). In general, if the optimal values of Θ are calculated
depends on the DD of Ssum , optimal value of Θ depends by varying the angle of rotations with different intervals, then
on the average transmit powers P1 and P2 of the alphabets S1 the optimal Θ and the multiplicity of the optimal Θ will also
change. When there are multiple optimal angles of rotation for
and S2 respectively.
a signal set, only one of them is provided in the Table. Among
A. Relationship with Gaussian codebook
the several optimal angles, the one presented in the Table
For a particular SNR, as the number of points in the reduces the complexity at the transmitters compared to the
input alphabet increases, capacity advantage due to rotation rest of the angles (Example : for BPSK, 90 degrees is chosen
progressively reduces. This result follows from the following over other angles of rotation at SNR = 10 db). However, when
sphere packing argument : A fixed SNR can correspond to a there is not much difference between the complexity among
fixed radius, r of a two dimensional ball, Br and the signal several optimal angles, we present the one with the least value
points in the sum-alphabet can correspond to points inside Br . (Example : for QPSK at SNR = 8 db, 10 db).
As the number of points in the input alphabets increases, the
number of points, N in Br increases and hence the density of A. Capacity region of GMAC with S1 = S2 =QPSK
points in Br increases. From Theorem 1, it is known that the
In Figure 3, capacity regions using QPSK alphabet pair is
capacity of the GMAC depends on the distance distribution shown with optimal rotation and without rotation at different
of the points in Br . It is clear that rotation of one of the SNR values. It can be observed that rotation provides enlarged
Therefore, in order to maximize I(x2 : y), the angle of rotation
Θ must be such that the metric in (14) is minimized.
The well known union bound on the probability of error
for the sum-alphabet, Ssum generated by the pair (S1 , S2 ) is
given in (20) which is a sum of several terms for indices k1 ,
k2 , i1 and i2 such that the vectors (k1 , k2 ) = (i1 , i2 ).
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ISIT 2008, Toronto, Canada, July 6 - 11, 2008
TABLE I
3- TUPLES , (a, b, c) FOR M -PSK AND M -QAM ALPHABETS FOR SOME M : a - OPTIMAL Θ. b - MULTIPLICITY OF OPTIMAL Θ. c - DIFFERENCE
BETWEEN THE METRIC VALUES IN (14), WHEN THE ALPHABETS ARE USED WITH ZERO ROTATION AND WITH OPTIMAL ANGLES OF ROTATION ( FROM
SNR
= -2 DB TO SNR = 16 DB )
SNR in db
-2
0
2
4
6
8
10
12
14
16
BPSK
(90, 1, 1.69)
(90, 1, 1.92)
(90, 1, 1.99)
(90, 1, 1.99)
(90, 1, 1.99)
(90, 1, 2.00)
(90, 21, 2.00)
(90, 107, 2.0)
(90, 160, 2.0)
(90, 203, 2.0)
QPSK
(45.0, 1, 0.065)
(45.0, 1, 0.280)
(45.0, 1, 1.450)
(45.0, 1, 4.610)
(47.5, 1, 9.340)
(34.0, 2, 14.18)
(31.5, 2, 17.83)
(31.0, 2, 19.54)
(31.5, 2, 19.96)
(30.5, 2, 19.99)
8-QAM
(90, 1, 166.9)
(90, 1 183.4)
(90, 1, 175.7)
(90, 1, 147.8)
(90, 1, 114.8)
(72, 1, 92.00)
(116.5, 1, 104)
(62, 1, 139)
(62, 1, 172)
(118, 1, 191)
with rotation
2.2
without rotation
rate region of
Guassian
codebook at 6 db
1.8
R2
1.6
1.4
snr = 6 db
This work was partly supported by the DRDO-IISc Program
on Advanced Research in Mathematical Engineering, partly by
the Council of Scientific & Industrial Research (CSIR), India,
through Research Grant (22(0365)/04/EMR-II) to B.S. Rajan.
1
2 db
0.8
0.6
0.4
0.4
snr = 0 db
0.6
0.8
16- QAM
(45, 1, 5.70)
(45, 1, 9.20)
(45, 1, 13.5)
(45, 1, 18.9)
(45, 1, 24.1)
(45, 1, 37.6)
(45, 1, 126)
(45, 1, 391)
(45, 1, 785)
(45, 1, 103 )
ACKNOWLEDGMENT
4 db
1.2
16-PSK
(15.75, 1, 10−9 )
(0. 1, 0)
(12.5, 1, 10−9 )
(11.25, 1, 10−9 )
(11.25, 1, 10−6 )
(11.25, 1, 10−4 )
(11.25, 1, 0.03)
(9.50, 1, 0.99)
(14.75, 1, 11.89)
(7.250, 1, 59.58)
be studied. For such a set up, optimal angles of rotations
correspond to the optimal choice of unitary matrices. Since
every symbol of the alphabet undergoes rotation by unitary
matrices, the trade-off between the encoding complexity of
signal sets and maximum achievable capacity region can be
studied.
2.4
2
8-PSK
(22.5, 1, 10−6 )
(22.5, 1, 10−4 )
(22.5, 1, 10−3 )
(22.5, 1, 0.04)
(22.5, 1, 0.57)
(22.5, 1, 3.69)
(16.5, 2, 13.8)
(29.75, 1, 34.4)
(15.0, 2, 59.2)
(15.0, 1, 79.4)
1
1.2
1.4
R1
1.6
1.8
2
2.2
R EFERENCES
Fig. 3. Capacity region of QPSK alphabet pair with optimal rotation and
without rotation at SNR = 0, 2, 4 and 6 db
capacity region from SNR value of 2 db onwards. However,
at SNR = 0 db, the capacity regions with optimal rotation
and without rotation coincides. The increase in the capacity
region is due to the increase in the value of I(x2 : y) with
rotation. The percentage increase in I(x2 : y) ranges from 3.8
percent at 2 db to 100 percent asymptotically. At SNR = 6 db,
capacity region of a GMAC with Gaussian alphabets is shown
to explicitly point out the difference between the rate regions
of a GMAC and a GMAC with Gaussian alphabets.
V. D ISCUSSION
We have obtained the capacity region of a two-user Gaussian Multiple Access Channel (GMAC) with input alphabets
being finite subsets of C and continuous output alphabets. We
considered a GMAC where User-1 and User-2 use the same
alphabet and for such a set-up, we identified the problem of
finding optimal angles of rotation between the alphabets such
that the capacity region is enlarged.
One of the possible directions for future work is as follows:
As mentioned in Section I, capacity regions of GMAC with
input alphabets from subsets of Rn for some n > 2 can
[1] Thomas M Cover and J. A. Thomas, ”Elements of information theory”,
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