A New Communication Scheme Implying Amplitude-Limited Inputs and Signal-Dependent Noise: System Design, Information Theoretic Analysis and Channel Coding

Scheme Fading Channels Signal-Dependent Noise MIMO Bounds Parallel Channels Coding Summary
A New Communication Scheme Implying
Amplitude-Limited Inputs and Signal-Dependent
Noise: System Design, Information Theoretic
Analysis and Channel Coding
Ahmad ElMoslimany
Adviser: Prof. Tolga M. Duman
Ahmad ElMoslimany Proposed Communication Scheme 1/80

Outline
1 A Novel Communication Scheme
2 Capacity of Fading Channels with Amplitude-Limited Inputs
3 Capacity of Signal-Dependent Additive Noise Channels
4 Bounds on the Capacity of MIMO Channels with Amplitude-Limited
Inputs
5 Capacity of Independent Parallel Gaussian Channels
6 Coding for Signal-Dependent Noise Channels
7 Summary

Outline
Inputs
7 Summary

System Model
• We exploit signatures signals to carry the digital information by
modulating the parameters of these signatures with the transmitted
bits.
• One possible application for the proposed communication scheme is
underwater acoustic communications
• We utilize analytical models for certain biomimetic signals
characterized by certain parameters
The NFM signal
s(t;c) = Aα(t)exp(j2πcξ(t/tr)), ∆t < t ≤ (Td +∆t),
• In our proposed scheme, the signal parameters, i.e., the amplitude,
the frequency, and the chirp rate, etc, carry information bits.

System Model

Receiver Structure
The discrete time version of a real generalized chirp is deﬁned as
s[n] = A ν[n]cos(2πcξ[n]), n = 0,1,...,M −1
The received signal can be written as,
x[n] =
s[n]+w[n] n = 0,1,...,M −1
w[n] n = M,...,N −1
where w[n] is the AWGN noise
MLE for the signal parameters


ˆc
ˆM
ˆA

 = arg min
c,M,A
M−1
∑
n=0
x[n]−A ν[n]cos(2πcξ[n])
2
+
N−1
∑
n=M
(x[n])2

Assymptotic MLE
Asymptotic ML estimator
θ = θ0 +z
where z ∼ N (0,σ2(θ0))
The Fisher information matrix which is deﬁned as
I(θ) = E
∂
∂θ
logP(x;θ)
2
|θ
the i jth element of this matrix is
[I (θ)]ij = E
∂ lnP
∂θi
∂ lnP
∂θj
|θ
The distribution of the asymptotic ML is Gaussian with mean θ0 and
variance I(θ)−1, i.e., θ ∼ N θ0,I(θ)−1 .

Decoding of KAM11 Data
Experiment Setup
The transmitted signal is a linear phase chirp signal x(t), given by
x(t) = Acos 2π f0t +2πct2
, 0 < t < T
• A, the amplitude of the chirp signal, A ∈ [0.5,1]
• T, the signal duration, T ∈ [100,200]ms
• f0, the center frequency, f0 ∈ [22,26]kHz
• c, the chirp rate, c ∈ [2,10]kHz
• Each parameter is quantized into four to ten bits to obtain diﬀerent
transmission rates.

Probability of Error at rate 107bps
0 5 10 15 20
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
frame index
BEP(%)

Probability of Error for Diﬀerent Transmission Rates

Communication Theoretic Study of the Proposed Scheme
• This approximate channel model of the proposed scheme motivates
the study of amplitude-limited inputs channels and channels with
signal-dependent noise
• We study the capacity of fading channels with amplitude limited
inputs
• We study the capacity of signal-dependent noise channels
• We study the capacity of MIMO systems and parallel Gaussian
channels
• We propose an upper bound on the error probability
• This bounds inspire code design approach for Z-channels that can be
generalized to signal-dependent Gaussian noise channels

Outline
Inputs
7 Summary

Fading Channels Under Peak Power Constraints
Fading channel model
Y = αX +Z
Setup
• X is the channel input that is amplitude-constrained such that
|X| ≤ A with a probability distribution function FX (x) ∈ FX
• Z is an AWGN such that Z ∼ N (0,σ2)
• α is the fading channel coefficient that has a probability density
function fα (u)
• We assume that the channel coefficient has a finite support, i.e.,
α ∈ [0,u0]
• We assume that the channel state information is available at the
receiver
• The objective is to find the capacity of the given channel

Our Results Compared to Existing Results
Existing Results
• The capacity-achieving distribution was shown to be discrete for
• Noncoherent Rayleigh fading channels with peak and average power
limited inputs.1
• Rician fading channels with inputs having constraints on the second
and the fourth moments has been studied.2
• Conditionally Gaussian channels with amplitude-limited inputs.3
Our Results
We show that the capacity-achieving distribution is discrete for
"arbitrary" fading distribution, with ﬁnite support.
1 Perera, Rasika R., Tony S. Pollock, and Thushara D. Abhayapala. "On non-coherent Rician fading channels with average and peak power
limited input."
2 Gursoy, Mustafa Cenk, H. Vincent Poor, and Sergio Verdu. "The noncoherent Rician fading channel-part I: structure of the capacity-achieving
input." IEEE Transactions on Wireless Communications, 4.5 (2005): 2193-2206.
3 Chan, Terence H., Steve Hranilovic, and Frank R. Kschischang. "Capacity-achieving probability measure for conditionally Gaussian channels
with bounded inputs." IEEE Transactions on Information Theory, 51.6 (2005): 2073-2088.

Capacity Theorem
Theorem
C, the capacity of the channel, is achieved by a unique probability
distribution function F0 in FX , i.e.,
C max
FX inFX
I(X;Y|α)
for some unique F0 in FX . Furthermore a necessary and suﬃcient
condition for F0 to achieve capacity is for all FX in FX
iF0 (x|α) ≤ IF0 (X;Y|α), ∀x ∈ [−A,A]
iF0 (x|α) = IF0 (X;Y|α), ∀x ∈ E0
where E0 is the set of points of increase of the probability distribution
function FX .

Capacity Optimization Problem
Concavity of the Mutual Information Function
Lemma
The conditional mutual information is strictly concave function
IFX (Y;X|α) = HFX (Y|α)−D
So, it is enough to show that the conditional entropy HFX (Y|α) is strictly
concave function to conclude the concavity of the mutual information
function.
HFX (Y|α) =
ˆ ∞
−∞
HFX (Y|α = u)fα (u)du
The function HFX (Y|α = u) is a strictly concave function in the
distribution and since fα (u) ≥ 0, the conditional entropy function
HFX (Y|α) is strictly concave. The concavity of HFX (Y|α = u) is shown
using Ash’s lemma.

Capacity Optimization Problem
Weak Differentiability and Continuity of the Conditional Mutual Information
Lemma
The mutual information function I(X;Y|α) is a weakly differentiable.
The weak derivative is defined as
IF1,F2
(X;Y|α) = lim
θ→0
I(1−θ)F1+θF2
(X;Y|α)−IF1
θ
=
ˆ A
−A
iF1 (x|α)dF2(x)−IF1 (X;Y|α).
Lemma
The mutual information I(X;Y|α) is a continuous function of distribution.
Let us fix a sequence {F
(n)
X (x)}n≥1 in FX such that F
(n)
X (x) → FX for
some FX ∈ FX . Then we use the Dominated Convergence Theorem and
Helly-Bray Theorem to show the continuity.
lim
n→∞
ˆ ∞
−∞
fY|α (y|u;F
(n)
X )log fY|α (y|u;F
(n)
X ) dy =
ˆ ∞
−∞
fY|α (y|u;FX )log fY|α (y|u;FX ) dy

Discreteness of the Optimal Distribution
Contradiction Arguments
• We assume that the set E0 has inﬁnite points of increase.
• The set E0 is bounded then it has a limit point (Bolzano-Weierstrass
Theorem)
• The conditional mutual information density can be extendable to an
open connected set D ∈ C
• Using Morera’s Theorem we can show that mutual information
density is an analytic function on an open connected set D
• Using the Identity Theorem, we establish that the optimality
condition holds on the whole real line
iF0 (x|α) = IF0 (X;Y|α), ∀x ∈ R
• We show that this does not hold for very large values of x

Numerical Results
The locations of mass points of the optimal input distributions
• Truncated Rayleigh fading with variance 1/2
• Noise variance 0.1
• Amplitude constraint 3
x
-3 -2 -1 0 1 2 3
fX
(x)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

Numerical Results
The capacity of the Rayleigh Fading Channel
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
A
I(Y;X|α)
E[X
2
]<A
2
|X|<A

Outline
Inputs
7 Summary

Signal-Dependent Noise Channels
Channel Modell
Y = X +N(X)+Z
Setup
• X has an amplitude-constrained such that |X| ≤ A
• Z is an independent additive Gaussian noise with zero mean and
variance σ2
z
• N(X) is an additive Gaussian noise, that depends on the transmitted
signal X with zero mean and variance σ2
n (x) when X = x
• We deﬁne σ2
n (x) = σ2
n (A) for all x ≥ A, and σ2
n (x) = σ2
n (−A) for all
x ≤ −A
• Our objective is to ﬁnd the capacity of this channel under the given
constraints

Existing Results and Our Results
Existing Results
• The capacity-achieving distribution of well-behaved independent
additive channels was shown to be discrete4
• The capacity of linearly dependent AWGN was shown to be discrete5
• Also there are some bounds on the capacity of linearly dependent
AWGN6
Our Results
We show that the capacity-achieving distribution of an arbitrary
well-behaved signal-dependent noise function is discrete
4 Tchamkerten, Aslan. "On the discreteness of capacity-achieving distributions." IEEE Transactions on Information Theory 50.11 (2004):
2773-2778.
5 Chan, Terence H., Steve Hranilovic, and Frank R. Kschischang. "Capacity-achieving probability measure for conditionally Gaussian channels
with bounded inputs." IEEE Transactions on Information Theory, 51.6 (2005): 2073-2088.
6 Lapidoth, Amos, Stefan M. Moser, and Michele Wigger. "On the capacity of free-space optical intensity channels."IEEE Transactions on
Information Theory, 55.10 (2009): 4449-4461.

Deﬁnitions
fY (y;FX ) =
ˆ A
−A
PN(y−x,x)dFX (x).
HF (Y|X) =
ˆ A
−A
H(Y|X = x)dF(x)
=
1
2
log(2πeσ2
z )+
1
2
E[log(σ2
(X))],
where σ2(x) = 1+
σ2
n (x)
σ2
z
iFX (x)
ˆ ∞
−∞
PN(y−x,x)log
PN(y−x,x)
fY (y;Fx)
dy,
hFX (x) −
ˆ ∞
−∞
PN(y−x,x)log fY (y;Fx)dy.
I(FX ) = H(FX )−D−
1
2
EF [log(σ2
(X))],
where D = 1
2 log(2πeσ2
z ).
iFX (x) = hFX (x)−
1
2
log(σ2
(x))−D.

Capacity Theorem
Theorem
C is achieved by a random variable, denoted by X0 with probability
distribution function F0 ∈ FX , i.e.,
C = max
FX ∈FX
I(FX ) = I(F0)
for some F0 ∈ FX . A necessary and suﬃcient condition for F0 to achieve
capacity is
i(x;F0)−I(F0) ≤ 0, ∀x ∈ [−A,A],
i(x;F0)−I(F0) = 0, ∀x ∈ E0,
Furthermore, this distribution is discrete and consists of ﬁnite number of
mass points if some technical conditions on σ2(X) hold.

Proof Outline
Lemma
The mutual information function given by
I(FX ) = H(FX )−D−
1
2
EF [log(σ2
(x))],
is a concave and continuous function of the distribution
The concavity can be shown using Ash’s lemma, the continuity is shown
using Dominated Convergence Theorem and Helly-Bray Theorem
Lemma
The mutual information function I(FX ) is a weakly diﬀerentiable function
and
IF1
(F2) =
ˆ
i(x;F1)dFX −I(F1)

Technical Conditions on the Noise Variance Function σ2(x)
We have some technical conditions on the noise variance function σ2(x)
which can be summarized as following:
• The noise variance function σ2(x) can be extended to an open
connected set in the complex plane containing the real line
• The function log(σ2(z)) is deﬁned over an open connected set that
includes the real line except the branch points
• We also assume that the function log(σ2(z)) is analytic over some
open connected set on the complex domain.
These technical conditions are needed to extend the mutual information
density to the complex plane and also needed to show the analyticity of
the information density.

Discreteness of the Capacity-Achieving Distribution
The discreteness can be shown following similar arguments as described
before
• We assume that the set E0 has inﬁnite points of increase.
• The set E0 is bounded then it has a limit point (Bolzano-Weierstrass
Theorem)
• The conditional mutual information density can be extendable to an
open connected set D ∈ C
• Using Morera’s Theorem we can show that mutual information
density is an analytic function on an open connected set D
• Using the Identity Theorem, we establish that the optimality
condition holds on the whole real line except the branch points
i(x;F0) = I(F0), ∀x ∈ R
• We show that this does not hold for very large values of x

Numerical Example
We consider the optical communication channel with intensity modulated
inputs. The received signal Y is given by,
Y = x+
√
xZ1 +Z0,
The parameter σ2 > 0 describes the strength of the input-independent
noise, while ς > 0 is the ratio of the input-dependent noise variance to
the input-independent noise. Thus, σ2(x) = 1+x.
Before applying our results, we need to verify the following:
• The branch point of log(σ2(x)) is the point (−1,0)
• The extension of the function σ2(x) to an open connected set D,
excluding the branch cut, is well deﬁned
• The function log(σ2(z)) is analytic on the complex plane excluding
the branch cut

Numerical Example
A/σ2
(dB)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
I(X;Y)(bitsperchanneluse)
10-3
10-2
10-1
10
0
Asymptotic capacity at low SNR
Exact capacity
1
1Lapidoth, Amos, Stefan M. Moser, and Michele Wigger. "On the capacity of
free-space optical intensity channels."IEEE Transactions on Information Theory, 55.10
(2009): 4449-4461.

Numerical Example
A/σ2
(dB)
10 11 12 13 14 15 16 17 18 19 20
0
0.5
1
1.5
2
2.5
Upper bound
Lower bound
Exact capacity

Outline
Inputs
7 Summary

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).

Existing Results and Diﬃculties
The Capacity of Two Users Multiple Access Gaussian Channel
• The literature does not include results on the MIMO channels with
amplitude-limited inputs
• Recently, the MAC channel has been 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
• The approach that has been used is similar to Smith’s one
• Their results are possible because of the availability of the Identity
Theorem
• Extending the previous arguments to the MIMO case is not viable
due to some technical diﬃculties

Bounds on the Capacity of MIMO Channels
Strategy for Finding the Upper and Lower Bounds

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
Alternative Bound on the Capacity
• An alternative lower bound 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 upper bound 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.

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

Outline
Inputs
7 Summary

Parallel Gaussian channel Model
• We consider N parallel Gaussian channels with independent inputs
under peak and average power constraints.
• The capacity-achieving distribution of Parallel Gaussian channels is
known to be discrete.
• If we know the power assigned for each channel then the problem is
solved.
• Assigning the optimal power for each channel is not feasible since
the problem dimension is very large and there is no closed form
expressions for the capacity.
• We consider bounds on the channel capacity at diﬀerent SNR
regimes
• The advantage of these bounds is that we are able to write closed
form expressions and hence we can develop power assignment policy
with low computational complexity compared to the original problem

Approximation of the Capacity at High Values of Noise
Variance
• When the noise variance is high, the capacity achieving distribution
consists of two points
• The channel can be approximated as binary symmetric channel
where the power represents the probability of error, i.e.,
CBSC = 1−H(p)
p = Q
P
σ2
,
Deﬁne a function J(Pi), which is basically the binary entropy function, as
J(Pi) = −Q
Pi
σ2
i
log Q
Pi
σ2
i
− 1−Q
Pi
σ2
i
log 1−Q
Pi
σ2
i
.
Then, the channel capacity of the parallel Gaussian channel is lower
bounded by,
C ≥ max
Pi, ∀i=1,2,···,N, 0≤Pi≤A2
i
1T P≤P0
N −
N
∑
i=1
J(Pi).

Approximation of the Capacity at Low Values of Noise
Variance
The capacity-achieving distribution
Consider a random variable X with a probability density function
fX (x) ∈ FX where |X| ≤ A, E[X2] ≤ P, and FX denotes the corresponding
class of probability density functions such that P(X > A) = 0 and
P(X < −A) = 0. The probability density function that maximizes its
entropy is fX (x) = c1 exp(−c2x2), where c1 and c2 are the solutions of
c1 =
1−2c2P
2Aexp(−c2A2)
,
and
1−2c2P
2Aexp(−c2A2)
π
c2
erf(
√
c2A) = 1.

Capacity of 2 parallel Gaussian channels, A = [0.1, 10]
P/σ2
(dB)
-20 -15 -10 -5 0 5 10 15 20 25 35
I(X:Y)(bitsperchanneluse)
10
-3
10
-2
10
-1
10
0
10
1
P/σ
2
(dB)
10 15 20 25 30 35
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Channel capacity using low SNR power policy
Channel capacity using high SNR power policy
Exact capacity
High SNR asymptote
Lower bound on the capacity

Capacity of 6 parallel Gaussian channels,
A = [0.1, 0.1, 1, 1, 10, 10]
P/σ2
(dB)
0 5 10 15 20 25 30 32
10
-1
10
0
10
1
10
2
0 1 2 3 4
0.4
0.6
0.8
1
1.2
24 26 28 30 32
6
7
8
9
10
11
12
Capacity evaluated using high SNR policy
Capacity evaluated using low SNR policy
Capacity evaluated using uniform power assignment

Outline
Inputs
7 Summary

Performance Bound with Coding
• We propose a tight upper bound on the probability of error
• In our study we consider two diﬀerent models
• Gaussian channels with signal-dependent variance
• Z-channels
• We use these bounds to develop a code design process
• On our study we concentrate on ultra-small block codes
• Ultra-small codes are codes with small number of messages and
ﬁnite block length
• There are various applications that require codes with small number
of messages.
• In the initiation of a wireless communication link as there is no much
information to be transmitted

Introduction
Consider a binary code C = {c0,c1,··· ,c2k−1,} with parameters (n,k), to
be used along BPSK modulation on an AWGN channel. The resulting
signal set is
S = {s0,s1,··· ,s2k−1}
The received signal vector r is
r = su +n
given that su is transmitted.
P(E|su) = Pr
i=u
Eui|su
where,
Eui = {||r−si|| ≤ ||r−su||}
For equiprobable signal set, we have
P(E) =
1
M ∑
u
P(E|su)

Bonferroni Type Lower Bound on P(E) of AWGN
P(∪m
i=1Ai) ≥ ∑
i=u
P(Ai)− ∑
j,i=u
P(Ai ∪Aj)
P(E|su) ≥ ∑
i=u
P(Eui|su)− ∑
j=u
P(Eui ∩Euj|su)
P(Eui|su) = Q
||su −si||
√
2N0
P(Eui ∩Euj|su) = Pr ||r−si||2
< ||r−su||2
,||r−sj||2
< ||r−su||2
|su
which easily reduces to
Pr Xi ≥
dui
√
2N0
,Xj ≥
duj
√
2N0

Bonferroni Type Lower Bound on P(E) of AWGN
Xi =
√
2
√
N0||si −su||
< n,si −su >, i = u
Xi,Xj are jointly Gaussian random variables with 0-mean, unite variance
and with correlation
ρij = E[XiXj] =
< si −su,sj −su >
||si −su||||sj −su||
P(Eui ∩Euj|su) =
1
2π 1−ρ2
ij
ˆ ∞
dui/
√
2N0
ˆ ∞
duj/
√
2N0
exp −
(x2 −2ρijxy+y2)
2(1−ρ2
ij)
dxdy

Proposed Upper Bound on P(E) of AWGN
We propose an upper bound on the error probability by considering the
probability of intersection of triplet error events. This can be expressed as
follow,
P(E|su) ≤ ∑
i=u
Pr(Eiu|su)− ∑
i,k=u
Pr(Eiu ∩Eku|su)+ ∑
i,k,j=u
Pr(Eiu ∩Eku ∩Eju|su).
P(Eiu ∩Eju ∩Eku|su) = Pr[||r−si|| < ||r−su||,||r−sj|| < ||r−su||,||r−sk|| < ||r−su|| |su]
which easily reduces to
Pr Xi ≥
diu
√
2N0
,Xj ≥
dju
√
2N0
,Xk ≥
dku
√
2N0

Proposed Upper Bound on P(E) of AWGN
where diu is the Euclidean distance between the codeword xi and xu, and
Xi =
√
2
√
N0||si −su||
< n,si −su >
We deﬁne the mutual correlation coeﬃcients ρij,ρik,ρjk as
ρij = E[XiXj] =
< si −su,sj −su >
||si −su||||sj −su||
,
ρik = E[XiXk] =
< si −su,sk −su >
||si −su||||sk −su||
,
ρjk = E[XjXk] =
< sj −su,sk −su >
||sj −su||||sk −xu||
.
P(Eui ∩Euj ∩Euk|su) =
1
(2π)3|ρ|
ˆ ∞
dui/
√
2N0
ˆ ∞
duj/
√
2N0
ˆ ∞
duk/
√
2N0
exp −
1
2
[x y z]ρ−1
[x y z]T
dx dy dz

Proposed Upper Bound on P(E) of Signal-Dependent
Gaussian Noise
y = x+n(x),
where x is the transmitted message, and n = [n0,n1,··· ,nN−1] is an
AWGN vector, the elements of this vector has a Gaussian distribution
such that the variance corresponding to transmission of zeros is diﬀerent
than the variance corresponding to transmission of ones such that
ni ∼ N (0,N
(i)
0 (x)/2) where N
(i)
0 ∈ {N
(0)
0 ,N
(1)
0 } and
N
(i)
0 (x) =
N
(0)
0 ∀i|x(i) = 0
N
(1)
0 ∀i|x(i) = 1

Proposed Upper Bound on P(E) of Signal-Dependent
Gaussian Noise
The pairwise error event in this case is
εiu = {||y−xi|| < ||y−xu||xu},
and the pairwise error probability is
P(εiu|xu) = Q

 ||xu −xi||2
2 ˆN0
(ij)

.
where ˆN0
(ij)
= ∑m N
(m)
0 (xi(m)−xu(m))2.
Pr Xi ≥
d2
iu
2
,Xj ≥
d2
ju
2
,Xk ≥
d2
ku
2
We deﬁne the mutual correlation coeﬃcients ρij,ρik,ρjk as
ρij = E[XiXj] =
ˆN0
(iju)
ˆN0
(iu)
ˆN0
(ju)
,
where ˆN0
(iju)
= 1
2 ∑m N
(m)
0 (xi(m)−xu(m))(xj(m)−xu(m)),

Z Channel
The conditional probability of the received vector
given the sent codeword can now be written as
PY|X (y|x) = (1−ε)d00(x,y)
.εd01(x,y)
where y is the received codeword, x = [x0,x1,...,xn−1]
is the transmitted codeword. The ML decoder
ˆx = argmax
xi
P(y|xi).
εiu = {P(y|xi) < P(y|xu)}.

Definitions
PY|X (y|x) = (1−ε0)d00(x,y)
.ε
d01(x,y)
0 .
We define dα,β (x,y) to be the number of positions m at which xm = α
and ym = β.
d10(xj,y) = wH(I{xj −y > 0})
d11(xj,y) =
1
2
wH(xj +y−|xj −y|)
d01(xj,y) = wH(I(y−xi < 0))
d00(xj,y) = n−d11(xi,y)−d01(xi,y)−d10(xj,y)
We define Di as
Di = (1−ε0)d00(xi,y)
.ε
d01(xi,y)
0
and Du
Du = (1−ε0)d00(xu,y)
.ε
d01(xu,y)
0

Proposed Upper Bound on P(E) of Z-Channels
The pairwise error event εiu is
εiu = {Du < Di}.
The pairwise error probability is
P(εiu|xu) = ∑
y
I{Du < Di}P(y).
The probability of intersection of two error events is given by,
P(εiu ∩εju|xi) = ∑
y
I{Du < Di}I{Du < Dj}P(y)
where I{ } is the indicator function. The probability of intersection of
triplet error events is,
P(εiu ∩εju ∩εku|xi) = ∑
y
I{Du < Di}I{Du < Dj}I{Du < Dk}P(y)

Proposed Upper Bound on P(E) of Z-Channels
A classical upper bound on the error probability is the union bound that
bounds the error probability by summing up the pairwise error
probabilities such that
P(ε|xu) ≤ ∑
j=u
Pr[εju].
P(ε|xu) ≥ ∑
i=u
Pr[εiu]− ∑
i,k=u
Pr[εiu ∩εku]
Based on the previous inequalities, we propose an upper bound on the
error probability by considering the probability of intersection of triplet
error events. Thus,
P(ε|xu) ≤ ∑
i=u
Pr[εiu]− ∑
i,k=u
Pr[εiu ∩εku]+ ∑
i,k,j=u
Pr[εiu ∩εku ∩εju].
An upper bound on the average error probability is,

Numerical Results
ǫ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Pe
10-3
10-2
10-1
Union bound
Two events based lower bound
Proposed upper bound
Monte Carlo simulation
0.37 0.38 0.39 0.4
0.055
0.06
0.065
0.07
0.075

Numerical Results
Eb
/N0
-4 -2 0 2 4 6 8
Pe
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Union bound
Seguin bound
-4 -3.5 -3 -2.5 -2
0.2
0.25
0.3
0.35
0.4
0.45

Numerical Results
Eb
/N0
-4 -2 0 2 4 6 8
Pe
10
-4
10-3
10-2
10
-1
10
0
Union bound
Youssefi bound
-4 -3.5 -3 -2.5
0.54
0.56
0.58
0.6
0.62
0.64

Proposed Code Design Approach
Pairwise Error Probability
The pairwise error probability is
Pr(xi → xj) Pr[g(y) = j|x = xi]
= ∑
y
P(y|xi)I(g(y) = j, j = i)
P(y|xi) = (1−ε)d00(xi,y)
.εd01(xi,y)
= (1−ε)n−wH (y)
.εwH (y)−wH (xi)
= (1−ε)n
ε−wH (xi) ε
1−ε
wH (y)
I(g(y) = j) = I PY|X (y|xj)) > PY|X (y|xi)
I(g(y) = j) = I (1−ε)d00(xj,y)
.εd01(xj,y)
> (1−ε)d00(xi,y)
.εd01(xi,y)
,

Pairwise Error Probability
The probability of receiving y given that xi is transmitted can be written
as
P(y|xi) = (1−ε)n
ε−wH (xi) ε
1−ε
w(y)
= (1−ε)n
ε−wH (xi) ε
1−ε
d1o(xj,xi)
ε
1−ε
k
ε
1−ε
w(xi)
= (1−ε)n−wH (xi) ε
1−ε
d10(xj,xi)
ε
1−ε
k
Hence, the pairwise error probability is
P(xi → xj) = (1−ε)n−wH (xi) ε
1−ε
d10(xj,xi)
I (1−ε)d00(xj,xi)−n+wH (xi)
εd01(xj,xi)
> 1
d00(xj,xi)
∑
k=0
ε
1−ε
k
d00
k

Design Principles
We deﬁne the weighted sum Hamming distance as
dα (i, j) w0d01(xj,xi)+w1d10(xj,xi)+w2d00(xj,xi).
If we have a pool of codewords C , then an optimal codebook C0 are
chosen such that
C0 = argmax
C
min
code words pairs
dα (i, j).

Designed Code, M = 4, n = 15
C4 codewords =




0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 0 0 0 0 0 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0





Code Performance
ǫ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Pe
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
Union bound

C6 codewords =








0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1









Comparison Between Designed Code and Another Code
ǫ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Pe
10
-4
10
-3
10
-2
10
-1
Lower bound
Upper bound
Another code
Designed
code

C8 codewords =












0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 0 0 0 0 0 0 0 1 1 1
0 0 0 0 1 1 0 1 0 1 0 1 1 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0













Comparison Between Designed Code and Another Code
ǫ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Pe
10
-4
10
-3
10
-2
10
-1
10
0
Designed code
Another code

Using Designed Codes Over Signal-Dependent Noise
Channels
Eb
/N0
-4 -2 0 2 4 6 8
Pe
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Union bound
Another code
Code designed for
Z-channels

Outline
Inputs
7 Summary

Conclusions
• We propose a novel communication scheme
• We made communication theoretic study of the proposed
communications scheme that includes
• Capacity of fading channels with amplitude-limited inputs which can
be achieved by a discrete distribution
• Capacity of signal-dependent noise channels with amplitude-limited
inputs which can be achieved by a discrete distribution
• Bounds on the capacity of MIMO systems and parallel Gaussian
channels
• Code Design for signal-dependent Gaussian channels and Z-channels

Any Questions ?

T hank Y ou!

A New Communication Scheme Implying Amplitude-Limited Inputs and Signal-Dependent Noise: System Design, Information Theoretic Analysis and Channel Coding

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