a new communication scheme implying amplitude-limited inputs and signal-dependent noise: system...

Scheme Fading Channels Signal-Dependent Noise MIMO Bounds Parallel Channels Coding Summary

A New Communication Scheme ImplyingAmplitude-Limited Inputs and Signal-DependentNoise: 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-LimitedInputs

5 Capacity of Independent Parallel Gaussian Channels

6 Coding for Signal-Dependent Noise Channels

7 Summary



Outline







7 Summary



System Model

• We exploit signatures signals to carry the digital information bymodulating the parameters of these signatures with the transmittedbits.

• One possible application for the proposed communication scheme isunderwater acoustic communications

• We utilize analytical models for certain biomimetic signalscharacterized 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−1w[n] n = M, . . . ,N−1

where w[n] is the AWGN noise

MLE for the signal parameters

cMA

= arg minc,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

[(∂

∂θlogP(x;θ)

)2

|θ

]

the i jth element of this matrix is

[I (θ)]i j = E[

∂ lnP∂θi

∂ lnP∂θ j|θ]

The distribution of the asymptotic ML is Gaussian with mean θ0 andvariance I(θ)−1, i.e., θ ∼N

(θ0, I(θ)−1

).



Decoding of KAM11 DataExperiment 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 differenttransmission rates.



Decoding of KAM11 DataProbability of Error at rate 107bps

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

frame index

BE

P (

%)



Decoding of KAM11 DataProbability of Error for Different Transmission Rates



Communication Theoretic Study of the Proposed Scheme

• This approximate channel model of the proposed scheme motivatesthe study of amplitude-limited inputs channels and channels withsignal-dependent noise

• We study the capacity of fading channels with amplitude limitedinputs

• 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







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 densityfunction 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 thereceiver

• 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 powerlimited inputs.1

• Rician fading channels with inputs having constraints on the secondand the fourth moments has been studied.2

• Conditionally Gaussian channels with amplitude-limited inputs.3

Our ResultsWe show that the capacity-achieving distribution is discrete for"arbitrary" fading distribution, with finite support.

1Perera, Rasika R., Tony S. Pollock, and Thushara D. Abhayapala. "On non-coherent Rician fading channels with average and peak power

limited input."

2Gursoy, 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.

3Chan, 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.Ahmad ElMoslimany Proposed Communication Scheme 14/80


Definitions

The average conditional mutual information function is defined as,

IFX (X ;Y |α),ˆ u0

0IFX (X ;Y |α = u)dFα(u)

and

IFX (X ;Y |α = u),ˆ

∞

−∞

ˆ A

−APN(y−ux) log

(PN(y−ux)fY (y;FX )

)dFX (x)dy

where PN(y−ux) = fY |X ,α(y|x,u).We define the average conditional entropy HFX (Y |α) as

HFX (Y |α),−ˆ u0

0

ˆ∞

−∞

fY,α(y,u) log fY |α(y|u)dydu

and the noise entropy is defined as

D,−ˆ

∞

−∞

PN(z) logPN(z)dz



Capacity Theorem

TheoremC, the capacity of the channel, is achieved by a unique probabilitydistribution function F0 in FX , i.e.,

C , maxFX inFX

I(X ;Y |α)

for some unique F0 in FX . Furthermore a necessary and sufficientcondition 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 distributionfunction FX .



Capacity Optimization ProblemConcavity of the Mutual Information Function

LemmaThe 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 strictlyconcave function to conclude the concavity of the mutual informationfunction.

HFX (Y |α) =

ˆ∞

−∞

HFX (Y |α = u) fα(u)du

The function HFX (Y |α = u) is a strictly concave function in thedistribution and since fα(u)≥ 0, the conditional entropy functionHFX (Y |α) is strictly concave. The concavity of HFX (Y |α = u) is shownusing Ash’s lemma.



Capacity Optimization ProblemWeak 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

I′F1 ,F2(X ;Y |α) = lim

θ→0

I(1−θ)F1+θF2(X ;Y |α)− IF1

θ

=

ˆ A

−AiF1 (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 forsome FX ∈FX . Then we use the Dominated Convergence Theorem andHelly-Bray Theorem to show the continuity.

limn→∞

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 DistributionContradiction Arguments

• We assume that the set E0 has infinite points of increase.• The set E0 is bounded then it has a limit point (Bolzano-WeierstrassTheorem)

• The conditional mutual information density can be extendable to anopen connected set D ∈ C

• Using Morera’s Theorem we can show that mutual informationdensity is an analytic function on an open connected set D

• Using the Identity Theorem, we establish that the optimalitycondition 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 ResultsThe 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

f X(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35



Numerical ResultsThe capacity of the Rayleigh Fading Channel

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

A

I(Y

;X|α

)

E[X2]<A

2

|X|<A



Outline







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 andvariance σ2

z

• N(X) is an additive Gaussian noise, that depends on the transmittedsignal X with zero mean and variance σ2

n (x) when X = x• We define σ2

n (x) = σ2n (A) for all x≥ A, and σ2

n (x) = σ2n (−A) for all

x≤−A• Our objective is to find the capacity of this channel under the givenconstraints



Existing Results and Our Results

Existing Results• The capacity-achieving distribution of well-behaved independentadditive 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 dependentAWGN6

Our ResultsWe show that the capacity-achieving distribution of an arbitrarywell-behaved signal-dependent noise function is discrete

4Tchamkerten, Aslan. "On the discreteness of capacity-achieving distributions." IEEE Transactions on Information Theory 50.11 (2004):

2773-2778.

5Chan, 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.

6Lapidoth, 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

−APN(y− x,x)dFX (x).

HF (Y |X) =

ˆ A

−AH(Y |X = x)dF(x)

=12

log(2πeσ2z )+

12

E[log(σ2(X))],

where σ2(x) = 1+ σ2n (x)σ2

z

iFX (x) ,ˆ

∞

−∞

PN(y− x,x) logPN(y− x,x)

fY (y;Fx)dy,

hFX (x) , −ˆ

∞

−∞

PN(y− x,x) log fY (y;Fx)dy.

I(FX ) = H(FX )−D− 12

EF [log(σ2(X))],

where D = 12 log(2πeσ2

z ).

iFX (x) = hFX (x)−12

log(σ2(x))−D.



Capacity Theorem

TheoremC is achieved by a random variable, denoted by X0 with probabilitydistribution function F0 ∈FX , i.e.,

C = maxFX∈FX

I(FX ) = I(F0)

for some F0 ∈FX . A necessary and sufficient condition for F0 to achievecapacity 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 ofmass points if some technical conditions on σ2(X) hold.



Proof Outline

LemmaThe mutual information function given by

I(FX ) = H(FX )−D− 12

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 shownusing Dominated Convergence Theorem and Helly-Bray Theorem

Lemma

The mutual information function I(FX ) is a weakly differentiable functionand

I′F1(F2) =

ˆi(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 openconnected set in the complex plane containing the real line

• The function log(σ2(z)) is defined over an open connected set thatincludes the real line except the branch points

• We also assume that the function log(σ2(z)) is analytic over someopen connected set on the complex domain.

These technical conditions are needed to extend the mutual informationdensity to the complex plane and also needed to show the analyticity ofthe information density.



Discreteness of the Capacity-Achieving Distribution

The discreteness can be shown following similar arguments as describedbefore

• We assume that the set E0 has infinite points of increase.• The set E0 is bounded then it has a limit point (Bolzano-WeierstrassTheorem)

• The conditional mutual information density can be extendable to anopen connected set D ∈ C

• Using Morera’s Theorem we can show that mutual informationdensity is an analytic function on an open connected set D

• Using the Identity Theorem, we establish that the optimalitycondition 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 modulatedinputs. The received signal Y is given by,

Y = x+√

xZ1 +Z0,

The parameter σ2 > 0 describes the strength of the input-independentnoise, while ς > 0 is the ratio of the input-dependent noise variance tothe 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 excludingthe branch cut



Numerical Example

A/σ2 (dB)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

I(X

;Y)

(bits p

er

ch

an

ne

l u

se

)

10-3

10-2

10-1

100

Asymptotic capacity at low SNRExact capacity

11Lapidoth, 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

I(X

;Y)

(bits p

er

channel use)

0

0.5

1

1.5

2

2.5

Upper boundLower boundExact capacity



Outline







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 receiveelements, 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 thecovariance.

Capacity of 2×2 MIMO System

C = maxf (x1,x2):|x1|≤Ax1 ,|x2|≤Ax2

I(y1,y2;x1,x2).



Existing Results and DifficultiesThe Capacity of Two Users Multiple Access Gaussian Channel

• The literature does not include results on the MIMO channels withamplitude-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 cornerpoints 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 viabledue to some technical difficulties



Bounds on the Capacity of MIMO ChannelsStrategy for Finding the Upper and Lower Bounds



Numerical ResultsSimulation 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

−APN(y− x)dF(x)

I(y1, y2; x1, x2) = I(y1; x1)+ I(y2; x2),

= h(y1)+h(y2)−D1−D2,

where Di =12 log(2πeσ2

i ) is the entropy of the Gaussian noise withvariance equals to σ2

i , i = 1,2.



Numerical ResultsBounds on the Capacity of H1, A = 2

−25 −20 −15 −10 −5 0 5 100

1

2

3

4

5

6

7

8

9

10 log1 0(σ2)

C (

bits/c

hannel use)

Lower bound

Upper bound

Asymptotic upperbound

Asymptotic lowerbound

Asymptotic resultscoincide with theupper and lowerbounds



Numerical ResultsBounds on the Capacity of H2, A = 2

−25 −20 −15 −10 −5 0 5 100

1

2

3

4

5

6

7

8

9

10 log1 0(σ2)

C (

bits/c

hannel use)

Lower bound

Upper bound

Asymptotic lower bound

Asymptotic resultscoincide with the upperand lower bounds

Asymptotic upperbound



Numerical ResultsBounds on the Capacity of H2, A = 2,3,4

−15 −10 −5 0 5 10 150

1

2

3

4

5

6

7

10 log1 0(σ2)

C (

bits/c

hannel use)

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 ResultsAlternative Bound on the Capacity

• An alternative lower bound on thecapacity can be attained bychoosing a discrete inputdistribution, the support of thisdistribution is the feasible region ofthe capacity optimization problem.

• Another upper bound on thecapacity can be found by relaxingthe constraints on the input. Thiscan be done by replacing thepeak-power constraint on the inputwith an average-power constraint.



Numerical ResultsAnother Bounds on the Capacity

−4 −2 0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3

3.5

4

10 log10(σ2)

C (

bits/c

hannel use)

Lower bound

Upper bound

Better lowerbound

Relaxed upperbound



Outline







7 Summary



Parallel Gaussian channel Model

• We consider N parallel Gaussian channels with independent inputsunder peak and average power constraints.

• The capacity-achieving distribution of Parallel Gaussian channels isknown to be discrete.

• If we know the power assigned for each channel then the problem issolved.

• Assigning the optimal power for each channel is not feasible sincethe problem dimension is very large and there is no closed formexpressions for the capacity.

• We consider bounds on the channel capacity at different SNRregimes

• The advantage of these bounds is that we are able to write closedform expressions and hence we can develop power assignment policywith low computational complexity compared to the original problem



Approximation of the Capacity at High Values of NoiseVariance

• When the noise variance is high, the capacity achieving distributionconsists of two points

• The channel can be approximated as binary symmetric channelwhere 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

σ2i

)log

(Q

(√Pi

σ2i

))−

(1−Q

(√Pi

σ2i

))log

(1−Q

(√Pi

σ2i

)).

Then, the channel capacity of the parallel Gaussian channel is lowerbounded by,

C ≥ maxPi , ∀i=1,2,··· ,N, 0≤Pi≤A2

i1T P≤P0

N−N

∑i=1

J(Pi).



Approximation of the Capacity at Low Values of NoiseVariance

The capacity-achieving distribution

Consider a random variable X with a probability density functionfX (x) ∈FX where |X | ≤ A, E[X2]≤ P, and FX denotes the correspondingclass of probability density functions such that P(X > A) = 0 andP(X <−A) = 0. The probability density function that maximizes itsentropy is fX (x) = c1 exp(−c2x2), where c1 and c2 are the solutions of

c1 =1−2c2P

2Aexp(−c2A2),

and1−2c2P

2Aexp(−c2A2)

[√π

c2erf(√

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)

(bits p

er

channel use)

10-3

10-2

10-1

100

101

P/σ2 (dB)

10 15 20 25 30 35

I(X

;Y)

(bits p

er

channel use)

2.5

3

3.5

4

4.5

5

5.5

66.5

77.5

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

I(X

;Y)

(bits p

er

channel use)

10-1

100

101

102

0 1 2 3 4

0.4

0.6

0.8

1

1.2

24 26 28 30 326

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







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 andfinite block length

• There are various applications that require codes with small numberof messages.

• In the initiation of a wireless communication link as there is no muchinformation to be transmitted



Introduction

Consider a binary code C = {c0,c1, · · · ,c2k−1,} with parameters (n,k), tobe used along BPSK modulation on an AWGN channel. The resultingsignal 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 6=u

Eui|su

]where,

Eui = {||r− si|| ≤ ||r− su||}For equiprobable signal set, we have

P(E) =1M ∑

uP(E|su)



Bonferroni Type Lower Bound on P(E) of AWGN

P(∪mi=1Ai)≥∑

i6=uP(Ai)− ∑

j,i 6=uP(Ai∪A j)

P(E|su)≥∑i6=u

P(Eui|su)−∑j 6=u

P(Eui∩Eu j|su)

P(Eui|su) = Q(||su− si||√

2N0

)P(Eui∩Eu j|su) = Pr

[||r− si||2 < ||r− su||2, ||r− s j||2 < ||r− su||2|su

]which easily reduces to

Pr[

Xi ≥dui√2N0

,X j ≥du j√2N0

]



Bonferroni Type Lower Bound on P(E) of AWGN

Xi =

√2√

N0||si− su||< n,si− su >, i 6= u

Xi,X j are jointly Gaussian random variables with 0-mean, unite varianceand with correlation

ρi j = E[XiX j] =< si− su,s j− su >

||si− su||||s j− su||

P(Eui∩Eu j|su)=1

2π

√1−ρ2

i j

ˆ∞

dui/√

2N0

ˆ∞

du j/√

2N0

exp

(−(x2−2ρi jxy+ y2)

2(1−ρ2i j)

)dxdy



Proposed Upper Bound on P(E) of AWGN

We propose an upper bound on the error probability by considering theprobability of intersection of triplet error events. This can be expressed asfollow,

P(E|su)≤∑i6=u

Pr(Eiu|su)− ∑i,k 6=u

Pr(Eiu∩Eku|su)+ ∑i,k, j 6=u

Pr(Eiu∩Eku∩E ju|su).

P(Eiu ∩E ju ∩Eku|su) = Pr[||r− si||< ||r− su||, ||r− s j||< ||r− su||, ||r− sk||< ||r− su|| |su]

which easily reduces to

Pr[

Xi ≥diu√2N0

,X j ≥d ju√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 ρi j,ρik,ρ jk as

ρi j = E[XiX j] =< si− su,s j− su >

||si− su||||s j− su||,

ρik = E[XiXk] =< si− su,sk− su >

||si− su||||sk− su||,

ρ jk = E[X jXk] =< s j− su,sk− su >

||s j− su||||sk−xu||.

P(Eui∩Eu j∩Euk|su)=1√

(2π)3|ρ|

ˆ∞

dui/√

2N0

ˆ∞

du j/√

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-DependentGaussian Noise

y = x+n(x),

where x is the transmitted message, and n = [n0,n1, · · · ,nN−1] is anAWGN vector, the elements of this vector has a Gaussian distributionsuch that the variance corresponding to transmission of zeros is differentthan the variance corresponding to transmission of ones such thatni ∼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-DependentGaussian Noise

The pairwise error event in this case isεiu = {||y−xi||< ||y−xu||xu},

and the pairwise error probability is

P(εiu|xu) = Q

||xu−xi||2√2N0

(i j)

.

where N0(i j)

= ∑m N(m)0 (xi(m)−xu(m))2.

Pr

[Xi ≥

d2iu2,X j ≥

d2ju

2,Xk ≥

d2ku2

]

We define the mutual correlation coefficients ρi j,ρik,ρ jk as

ρi j = E[XiX j] =N0

(i ju)

N0(iu)N0

( ju),

where N0(i ju)

= 12 ∑m N(m)

0 (xi(m)−xu(m))(x j(m)−xu(m)),Ahmad ElMoslimany Proposed Communication Scheme 59/80


Z Channel

The conditional probability of the received vectorgiven 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

x = argmaxxi

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(x j,y) = wH(I{x j−y > 0})

d11(x j,y) =12

wH(x j +y−|x j−y|)

d01(x j,y) = wH(I(y−xi < 0))d00(x j,y) = n−d11(xi,y)−d01(xi,y)−d10(x j,y)

We define Di asDi = (1− ε0)

d00(xi,y).εd01(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∩ ε ju|xi) = ∑y

I{Du < Di}I{Du < D j}P(y)

where I{�} is the indicator function. The probability of intersection oftriplet error events is,

P(εiu∩ ε ju∩ εku|xi) = ∑y

I{Du < Di}I{Du < D j}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 thatbounds the error probability by summing up the pairwise errorprobabilities such that

P(ε|xu)≤ ∑j 6=u

Pr[ε ju].

P(ε|xu)≥∑i6=u

Pr[εiu]− ∑i,k 6=u

Pr[εiu∩ εku]

Based on the previous inequalities, we propose an upper bound on theerror probability by considering the probability of intersection of tripleterror events. Thus,

P(ε|xu)≤∑i 6=u

Pr[εiu]− ∑i,k 6=u

Pr[εiu∩ εku]+ ∑i,k, j 6=u

Pr[εiu∩ εku∩ ε ju].

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/N

0

-4 -2 0 2 4 6 8

Pe

10-5

10-4

10-3

10-2

10-1

100

Union boundSeguin boundProposed upper boundMonte Carlo simulation

-4 -3.5 -3 -2.5 -20.2

0.25

0.3

0.35

0.4

0.45



Numerical Results

Eb/N

0

-4 -2 0 2 4 6 8

Pe

10-4

10-3

10-2

10-1

100

Union boundProposed upper boundMonte Carlo simulationTwo events based lower boundYoussefi bound

-4 -3.5 -3 -2.5

0.54

0.56

0.58

0.6

0.62

0.64



Proposed Code Design ApproachPairwise Error Probability

The pairwise error probability is

Pr(xi→ x j) , Pr[g(y) = j|x = xi]

= ∑y

P(y|xi)I(g(y) = j, j 6= i)

P(y|xi) = (1− ε)d00(xi,y).εd01(xi,y)

= (1− ε)n−wH (y).εwH (y)−wH (xi)

= (1− ε)nε−wH (xi)

(ε

1− ε

)wH (y)

I(g(y) = j) = I(PY |X (y|x j))> PY |X (y|xi)

)I(g(y) = j) = I

((1− ε)d00(x j ,y).εd01(x j ,y) > (1− ε)d00(xi,y).εd01(xi,y),

)Ahmad ElMoslimany Proposed Communication Scheme 67/80


Proposed Code Design ApproachPairwise Error Probability

The probability of receiving y given that xi is transmitted can be writtenas

P(y|xi) = (1− ε)nε−wH (xi)

(ε

1− ε

)w(y)

= (1− ε)nε−wH (xi)

(ε

1− ε

)d1o(x j ,xi)(

ε

1− ε

)k(ε

1− ε

)w(xi)

= (1− ε)n−wH (xi)

(ε

1− ε

)d10(x j ,xi)(

ε

1− ε

)k

Hence, the pairwise error probability is

P(xi→ x j) = (1− ε)n−wH (xi)

(ε

1− ε

)d10(x j ,xi)

I((1− ε)d00(x j ,xi)−n+wH (xi)ε

d01(x j ,xi) > 1)

d00(x j ,xi)

∑k=0

(ε

1− ε

)k( d00k

)



Proposed Code Design Approach

Design Principles

We define the weighted sum Hamming distance as

dα(i, j), w0d01(x j,xi)+w1d10(x j,xi)+w2d00(x j,xi).

If we have a pool of codewords C , then an optimal codebook C0 arechosen such that

C0 = argmaxC

mincode 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 00 0 0 0 0 1 1 1 1 1 1 1 1 1 11 1 1 1 1 0 0 0 0 0 1 1 1 1 11 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 boundTwo events based lower boundProposed upper bound




C6 codewords =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 1 1 1 1 1 1 10 1 0 1 0 1 0 1 0 1 0 1 0 1 01 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 boundUpper bound

Another code

Designedcode




C8 codewords =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 0 0 0 0 0 0 0 0 1 1 10 0 0 0 1 1 0 1 0 1 0 1 1 0 01 1 1 1 1 1 1 0 0 0 0 0 0 0 00 1 1 1 1 1 1 1 1 1 1 1 0 0 00 0 0 0 0 0 0 1 1 1 1 1 1 1 10 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

100

Designed codeAnother code



Using Designed Codes Over Signal-Dependent NoiseChannels

Eb/N

0

-4 -2 0 2 4 6 8

Pe

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Union boundTwo events based lower boundProposed upper boundMonte Carlo simulation

Another code

Code designed forZ-channels



Outline







7 Summary



Conclusions

• We propose a novel communication scheme• We made communication theoretic study of the proposedcommunications scheme that includes

• Capacity of fading channels with amplitude-limited inputs which canbe achieved by a discrete distribution

• Capacity of signal-dependent noise channels with amplitude-limitedinputs which can be achieved by a discrete distribution

• Bounds on the capacity of MIMO systems and parallel Gaussianchannels

• 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...

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