BlackSea_Presentation

On the Capacity of MIMO Gaussian Channels with
Amplitude-Limited Inputs
Ahmad ElMoslimany1 Tolga M. Duman1 2
1
School of Electrical, Computer and Energy Engineering
Arizona State University, Tempe, AZ
2
Department of Electrical and Electronics Engineering
Bilkent University, Ankara, Turkey
IEEE BlackSeaCom, 2014
Ahmad ElMoslimany, Tolga M. Duman MIMO Channel with Amplitude-Limited Inputs

Outline
• Introduction
• MIMO Systems with Amplitude Limited Inputs
• Upper and Lower Bounds on the Capacity of MIMO Systems
• Asymptotic Bounds on the Capacity of MIMO Systems
• Numerical Simulations
• Conclusions

Introduction
The Capacity of Scalar Gaussian Channel
Channel Model
Y = X +Z
where |X| ≤ A, and z ∼ N (0,σ2), FX is the probability distribution
function of the channel input X in the class of probability distribution
functions F.
• Smith 1
proved that there is a unique distribution that maximizes
the mutual information function between the transmitted signal X
and the received signal Y.
• Also, he proved optimal distribution is discrete.
• Numerical optimization technique is needed to ﬁnd the optimal
distribution.
1Smith, Joel G. "The information capacity of amplitude-and variance-constrained
scalar Gaussian channels." Information and Control 18.3 (1971): 203-219.

Introduction
Theorem
C sup
X:|X|≤AX
I(X;Y)
There is a unique input distribution FX that maximizes the mutual
information function I(X;Y) (sometimes it is written as I(FX )). The
proof is based on the following arguments
1 The space of probability distribution functions F is convex and
compact.
2 The mutual information function I(X;Y) is a continuous function of
the distribution FX (using the Helly-Bray Theorem).
3 The mutual information function I(X;Y) is weakly diﬀerentiable with
respect to the probability distribution function FX .

Introdcution
4 The function I(X;Y) is a strictly concave function of the distribution
FX .
• Writing the mutual information as I(X;Y) = HFX (Y)−HFX (Y|X).
• Using Ash’s Lemma 2
HF1 (Y) = −
ˆ
P(y;F1)logP(y;F1)dy ≤ −
ˆ
P(y;F1)logP(y;F2)dy
it is shown that the entropy HFX (Y) is a concave function of the
distribution FX .
5 The entropy density h(x;Fx) is an analytic function.
6 Let E0 denote the points of increase of FX on [−AX ,AX ]. Using
Bolzano-Weierstrass Theorem and the Identity Theorem, Smith
showed that the optimal probability distribution function F0 has a
ﬁnite number of points of increase.
2B. Ash, Robert, Information Theory. Dover, New York, 1965

Introdcution
7 The discreteness of the capacity-achieving distribution is based on
the following arguments (proof by contradiction)
• If E0 is not finite then h(x;F0) = I(F0)+D ∀x ∈ R.
• The Bolzano-Weierstrass Theorem states that an infinite set of
points on a finite interval must have a limit point.
• Thus E0 is an infinite set of points on [−AX ,AX ] with a limit point in
[−AX ,AX ].
• The Identity Theorem states that if two functions in some region
agree on an infinite set of points in that region and the set of points
has a limit point in this region. Then these functions are equal in the
region.
• If h(x;F0) = I(F0)+D ∀x ∈ R then
PY (y;F0) = exp(−I(F0)−D) ∀y ∈ R, which can not be true.

Introduction
The Capacity of Two Users Multiple Access Gaussian Channel
• Recently 3 4
the multiple access channel (MAC) with
amplitude-limited inputs is studied.
• It has been shown that the sum-capacity achieving distribution is
discrete, and this distribution achieves rates at any of the corner
points of the capacity region.
Channel Mode
Y = X1 +X2 +Z
where |X1| ≤ A1 and |X1| ≤ A2
3B. Mamandipoor, K. Moshksar, and A. K. Khandani, "On the sum capacity of
Gaussian MAC with peak constraint," in IEEE International Symposium on
Information Theory Proceedings, July 2012, pp. 26-30.
4O. Ozel and S. Ulukus, "On the capacity region of the Gaussian MAC with
batteryless energy harvesting transmitters," in IEEE Global Communications
Conference, Dec. 2012, pp. 2385-2390.

Introduction
The Capacity of Two Users Multiple Access Gaussian Channel
1 Fix X1 = X∗
1 then the new problem
˜X2 = arg sup
X2:|X2|≤A2
h(X∗
1 +X2 +Z) (1)
It is shows that ˜X2 is a discrete random variable.
2 Fix X2 = X∗
2 then the new problem
˜X1 = arg sup
X1:|X1|≤A1
h( ˜X2 +X1 +Z) (2)
It is shown that ˜X1 is a discrete random variable.
The proof of (1) and (2) follow similar lines of arguments with those of
Smith.

Introduction
Channel Model
We consider a MIMO system where the received signal y is written as
MIMO System
y = Hx+z,
• H is an Nr ×Nt channel matrix, Nr is the number of receive
elements, and Nt is the number of transmit elements.
• The channel matrix H is assumed to be deterministic
• The vector z denotes AWGN such that z ∼ N (0,Σ), where Σ is the
covariance.
Capacity of 2×2 MIMO System
C = max
f(x1,x2):|x1|≤Ax1
,|x2|≤Ax2
I(y1,y2;x1,x2).

Introduction
Why Can We Not Use the Same Arguments for the Capacity of the MIMO Systems?!
• Extending Smith results for MIMO systems is not possible.
• The Identity Theorem is available only for single-dimensional
functions.
• Instead we compute the capacity of the MISO systems.
• And we derive an upper and lower bounds by constraining the inputs
to rectangular regions that inscribe the feasible region (upper
bound) and inscribed by the feasible region (lower bound).

The Capacity of Gaussian MISO Systems
The Received Signal
y = h1x1 +h2x2 +z.
Deﬁne an auxiliary variable u such that u = h1x1 +h2x2. Since x1 and x2
are amplitude-limited, u will also be amplitude limited, i.e.,
−|h1|Ax1 −|h2|Ax2 ≤ u ≤ |h1|Ax1 +|h2|Ax2 .
• The received signal y can be written as y = u+z,
• The problem boils down to the classical point-to-point scalar
problem that has been investigated by Smith.
• The distribution of the auxiliary random variable U that achieves the
capacity is discrete, i.e.,
fU (u) =
N−1
∑
i=0
p(ui)δ(u−ui)

The Capacity of Gaussian MIMO Systems
Channel Model
We consider the 2×2 MIMO system.
channel model
The received signal y can be written as,
y = Hx+z.
The singular value decomposition of the channel matrix H is
H = UΣWH
= UΣV.
• The channel inputs are amplitude limited as |x1| ≤ Ax1 and |x2| ≤ Ax2 .
• These constraints form a square of region Ax1 ×Ax2 .

The Feasible Region (1/2)
Then the region deﬁnes the limits
on the equivalent input is,
−
1
a
x+
b
a
y ≤ Ax1
1
a
x−
b
a
y ≤ Ax1
−
1
c
x+
d
c
y ≤ Ax2
1
c
x−
d
c
y ≤ Ax2
−1
ax+b
ay=
A
x1
1
ax−b
ay=Ax1
− 1
c x + d
c y =
Ax2
1
c x − d
c y =
Ax2

The Feasible Region (2/2)
a, b, c, and d are the channel ratios deﬁned as,
a =
det(V)
v22
,
b =
v12
v22
,
c = −
det(V)
v21
,
d =
v11
v21
.

Bounds on the Capacity of the Gaussian MIMO Systems
Strategy for Finding the Upper and Lower Bounds (1/2)
• We ﬁnd bounds on the channel capacity by solving the capacity
optimization problem in diﬀerent regions other than the original one.
• Solving for rectangles decompose the capacity problem into two
one-dimensional problems other than the two-dimensional one.
• For the lower bound we solve the capacity problem for a smaller
rectangle inside the actual parallelogram.
• For the upper bound we look for the smallest square that surrounds
the actual region.

Strategy for Finding the Upper and Lower Bounds (2/2)

The Upper Bound (1/2)
• The upper bound is derived by optimizing the mutual information
over the smallest rectangle that inscribe the feasible region.
• This rectangle is constructed by connecting the intersection of every
pair of lines of the original region.
x =
bc
d −b
Ax2 −
ad
d −b
Ax1 , y =
c
d −b
Ax2 −
a
d −b
Ax1 ,
the second point is
x = −
bc
d −b
Ax2 −
ad
d −b
Ax1 , y = −
c
d −b
Ax2 −
a
d −b
Ax1 ,
the third point is
x =
bc
d −b
Ax2 +
ad
d −b
Ax1 , y =
c
d −b
Ax2 +
a
d −b
Ax1 ,
and the forth point is
x = −
bc
d −b
Ax2 +
ad
d −b
Ax1 , y = −
c
d −b
Ax2 +
a
d −b
Ax1 .

The Upper Bound (2/2)
Thus the amplitudes corresponding to the sides of the square are,
∆xupp =
bc
d −b
Ax2 +
ad
d −b
Ax1 ,
∆yupp =
c
d −b
Ax2 +
a
d −b
Ax1 ,
can be used to compute an upper bound on the channel capacity of the
original MIMO system.
The Upper Bound
C ≤ C0(∆xupp)+C0(∆yupp),
where C0(A) is the capacity of the point-to-point AWGN channel for a
given amplitude constraint A (computed using Smith’s approach).

The Lower Bound (1/2)
• A lower bound on the
capacity of the channel can
be found by optimizing the
mutual information over a
smaller rectangular region
inside the feasible region.
• To ﬁnd such a rectangle, we
determine the intersection of
a straight line, y = mx, that
passes through the origin
and the boundary of the
feasible region.

The Lower Bound (2/2)
Lower Bound on the Capacity
∆xlow = min
aAx1
1+bm
,
aAx1
1−bm
,
cAx2
1+dm
,
cAx2
1−dm
, (3)
∆ylow = min
amAx1
1+bm
,
amAx1
1−bm
,
cmAx2
1+dm
,
cmAx2
1−dm
, (4)
for some arbitrary values for the slope m such that the set of points
{(l∆xlow,k∆ylow) ∈ R : l,k ∈ {1,−1}},
where R is the feasible region. Thus, the lower bound on the channel
capacity is given by
C ≥ C0(∆xlow)+C0(∆ylow). (5)

Asymptotic Bounds on the Capacity of MIMO Systems
At Low Noise Variance
The entropy of the noise is very small compared to the entropy of the
input.
h(y) h(y|x) and h(x) h(x|y).
As a result,
h(x) = I(x;y)+h(x|y) = h(y)−h(y|x)+h(x|y) ≈ h(y).
That is, the capacity can be approximated as
C = maxI(x;y) ≈ maxh(x)−h(y|x) = log(2A)−
1
2
log(2πeσ2
).
Asymptotic Bounds on the Capacity
C ≤ log(4∆xupp∆yupp)−
1
2
log(2πeσ2
1 )−
1
2
log(2πeσ2
2 ),
C ≥ log(4∆xlow∆ylow)−
1
2
log(2πeσ2
1 )−
1
2
log(2πeσ2
2 ).

An Asymptotic Results
At High Noise Variance
• The optimal distribution is discrete and consists of only two mass
points with equal probabilities.
• The capacity of this discrete-time binary-input AWGN is well known.
Asymptotic Bounds on the Capacity
C ≤ g
∆xupp
σ1
+g
∆yupp
σ2
,
C ≥ g
∆xlow
σ1
+g
∆ylow
σ2
,
where g(x) = 1−
´ ∞
−∞
1√
2π
e−
(u−x)2
2 log2 1+e−2ux du.

The Gap Between the Upper and Lower Bounds
For very low noise variances, assume that Ax1 = Ax2 .
From the Upper Bound
From the upper bounds, and if Ax1 = Ax2 = A0 we have,
∆xupp = GuppA0, ∆yupp = HuppA0,
where Gupp and Hupp are only function of the channel coeﬃcients.
From the Lower Bound
∆xlow = GlowA0, ∆yupp = HlowA0,
where Glow and Hlow are only functions of the channel coeﬃcients.
The gap Between the Upper and Lower Bounds ∆C
∆C = log 4GuppHuppA2
0 −log 4GlowHlowA2
0 = log
GuppHupp
GlowHlow
,
which is independent of the amplitude constraints imposed on the inputs.

Numerical Results
Simulation Parameters
We consider two arbitrarily picked channel matrices given by
H1 =
0.177 0.28
1 0.31
, H2 =
0.997 0.295
1 0.232
.
py(y) =
ˆ A
−A
PN(y−x)dF(x)
I( ˜y1, ˜y2; ˜x1, ˜x2) = I( ˜y1; ˜x1)+I( ˜y2; ˜x2),
= h( ˜y1)+h( ˜y2)−D1 −D2,
where Di = 1
2 log(2πeσ2
i ) is the entropy of the Gaussian noise with
variance equals to σ2
i , i = 1,2.

Numerical Results
Bounds on the Capacity of H1, A = 2
−25 −20 −15 −10 −5 0 5 10
0
1
2
3
4
5
6
7
8
9
10 log1 0(σ 2
)
C(bits/channeluse)
Lower bound
Upper bound
Asymptotic upper
bound
Asymptotic lower
bound
Asymptotic results
coincide with the
upper and lower
bounds

Numerical Results
Bounds on the Capacity of H2, A = 2
−25 −20 −15 −10 −5 0 5 10
0
1
2
3
4
5
6
7
8
9
10 log1 0(σ 2
)
C(bits/channeluse)
Lower bound
Upper bound
Asymptotic lower bound
Asymptotic results
coincide with the upper
and lower bounds
Asymptotic upper
bound

Numerical Results
Bounds on the Capacity of H2, A = 2,3,4
−15 −10 −5 0 5 10 15
0
1
2
3
4
5
6
7
10 log1 0(σ 2
)
C(bits/channeluse)
Lower bound, A=2
Upper bound, A=2
Lower bound, A=3
Upper bound, A=3
Upper bound, A=4
Lower bound, A=4

Numerical Results
Another Bounds on the Capacity
• Another lowerbound on the
capacity can be attained by
choosing a discrete input
distribution, the support of this
distribution is the feasible region of
the capacity optimization problem.
• Another upperbound on the
capacity can be found by relaxing
the constraints on the input. This
can be done by replacing the
peak-power constraint on the input
with an average-power constraint.
Figure: An input distribution
that achieves a better lower
bound on the capacity.

Numerical Results
Another Bounds on the Capacity
−4 −2 0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
3
3.5
4
10 log10(σ2
)
C(bits/channeluse) Lower bound
Upper bound
Better lowerbound
Relaxed upperbound

Conclusions
• We consider capacity of MIMO systems with amplitude constrained
inputs
• MISO systems can be handled in a straightforward way
• We derive upper and lower bounds on the capacity of MIMO systems
by solving the capacity optimization problems over diﬀerent regions
• We study asymptotic bounds on the capacity at low and high noise
variances

BlackSea_Presentation

Recommended

Recommended

More Related Content

Similar to BlackSea_Presentation

Similar to BlackSea_Presentation (20)

BlackSea_Presentation