capacity for communication channels

Capacity for Communication Channels

The material is presented at a workshop tutorial at the 43rd Conference on Decision and Control, Bahamas, December 2004.

2

Overview

Importance of Uncertainty in CommunicationsShannon’s Definition of Capacity Capacity of Additive Gaussian Channels

Random Variable CaseRandom Process Case

Capacity of MIMO Gaussian ChannelsReview of Maximin Capacity

Saddle Point Solutions

3

Overview

Maximin Capacity Subject to NormedUncertainties

Uncertain ChannelUncertain NoiseUncertain Channel and Noise

Coding TheoremExamples and ConclusionsReferences

4

Importance of Communication Subject to Uncertainties

5


Channel measurement errorsNetwork operating conditionsChannel modelingCommunication in presence of jammingSensor networks Teleoperations

6


Channel measurement errorsProblem of allocation of power and bandwidth for channel soundingAllocation depends on the benefit that can be obtained from more accurate measurement (e.g., when channel changes too rapidly)Feedback (allocation of bandwidth to provide accurate channel state information, robustness of feedback to channel error)Effect on new wireless systems (higher carrier frequency, ultra-wide bandwidth)

7


Network operating conditionsUncertainty can impair interference cancellation in multiple access systemsInterplay between physical layer and higher layers (e.g., effect of adaptive coding on congestion)

Channel modeling Models used for developing communication schemes are simple, but the gap between real channel and model can be big

8


Communication in presence of jammingUseful transmitted signal is accompanied with adversary signal making communication channel uncertain

Sensor networksTeleoperations

9

Shannon’s Definition of Capacity

10


Model of communication system

wSource Encoder Decoder SinkChannel +

xn

y wf g

{ }M,...,1 { }M,...,1( )wf ( )( )wfh ( )( )( )wfhg( )nM ,

( )( )( ){ }( ) ∑

=

=

=≠=

=

M

ii

ne

i

MP

Misentiiifhg

nMR

1

1,...,1,|Pr

log

λ

λ

11


R is achievable rate if there exists a sequence of codes (M=2nR,n) such that the probability of error tends to zero as n tends to infinity.

The operational capacity is the supremum of all achievable rates.

12


Discrete memoryless channel

Channel capacity depends on channel transition matrix Q(y|x) that is known

( )( )

( ) ( ) ( ) ( )( ) ( )∑∑ ∑∈ ∈

∈

∈

=

=

Xx YyXx

XP

xyQxPxyQxyQxPQPI

QPIC

''|'

|log|,

,maxπ

13


What if Q(y|x) is unknown ?Example: compound BSC

What is the channel capacity ?

θ

θ

1- θ

1- θ0 0

1 1

{ } [ ]

( )⎩⎨⎧

=−≠

=

⊂Θ==

xyxy

xyQ

YX

,1,

;|

1,0,1,0

θθ

θ

14


Model of communication system with CSI

Source Encoder Decoder SinkChannel +x

ny x

CSIu v

15


Examples: arbitrary varying channels (AVC), arbitrary varying finite state channels (AVFSC), and others.Discrete memoryless AVC, - the state space

What is the channel capacity ?

( ) ( )∏=t

ttt sxyQsxyQ ;|;|

Sst ∈

16


Additive Gaussian ChannelsRandom Variable Case

( )[ ]

⎟⎠⎞

⎜⎝⎛ +=

≤+=

2

2

1log21

,0~,

σ

σ

PC

PxVarNnnxy

( )( ) [ ] PxVaryxIC

XP≤=

∈,;max

π

17


Random variable case derivation

( )( ) ( ) ( )

( ) ( )

( )

⎟⎠⎞

⎜⎝⎛ +=

−+=

−=

−=

2

22

1log21

2log212log

21

|;max

σ

σπσπ

π

P

ePe

nHyH

xyHyHyxIX

18


What is the capacity if noise is unknown?Gaussian AVC

Statistic of is unknown

( )2,0~, σNnsnxy ttttt ++=

ts

19


Additive Gaussian ChannelsRandom Process Case

+

( )fW

( )fH

n

yx

20


Random process case derivation

{ }

( ) ( )PdfSdfWS

HSSJ

PdfSSA

dfWS

HSC

xn

xx

xx

n

x

ASx

−+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

≤=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

∫∫

∫

∫∈

λ2

2

2

2

1log21

;

1log21sup

21


Necessary conditions

( )∫

∫

=−

>−=

>=−=+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

+

∈

PdfHWS

HWSS

HWSS

dfWS

HSC

n

nx

nx

n

x

ASx

22*

22**

**

22*

2

2

/)2

0/

01/

1)1

1log21sup

ν

ν

νλ

22


Capacity of continuous time additive Gaussian channel

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛= df

WS

HC

n2

2*log

21 ν

PdfH

WSn =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−∫

+

2

2

*ν

23


Range of integration

What is the capacity if the frequency response of the channel H, or psd of the noise Sn belong to certain sets?

0*,0* 2

2

>≥− ννH

WSn

24


Water-filling

*ν2

2

H

WSn

fBfBf−

psd

25

MIMO Channels

26

MIMO Channels

Static MIMO Gaussian Channel

n is zero-mean complex Gaussian noise with independent, equal variance real, and imaginary parts,

[ ][ ] [ ]( ) PxxEtracexxE

InnEnHxy r

≤=

=+=**

*,

trrt CHCyCx ×∈∈∈ ,,

27

MIMO Channels

Capacity of static MIMO channel

are the eigenvalues of HH*, ν is Lagrange multiplierWhat is the capacity if the matrix H belongs to a certain set?

( )( )( )∑

∑+

+−

=

=−

ii

ii

C

P

νλ

λν

log

1

{ }iλ

28

Review of MaximinCapacity

29

Review of Minimax Capacity

Example: compound DMC

This result is due to Blackwell et. al. [6]. Also look at Csiszar [8], and Wolfowitz [21] Blachman [5], and Dobrushin [12] were first to apply game theoretic approach in computing the channel capacity with mutual information as a pay-off function for discrete channels

( )( )( )θ

θπ;|,infmax ⋅⋅=

Θ∈∈QPIC

XP

30


The existence of saddle point ?

For further references see Lapidoth, Narayan [18]

( )( )( )

( )( )( )

( )( ) ( )( ) ( )( )θθθ

θθπθθπ

;|,;|,;|,

;|,maxinf;|,infmax

**** ⋅⋅≤⋅⋅≤⋅⋅

⋅⋅=⋅⋅=∈Θ∈Θ∈∈

QPIQPIQPI

QPIQPICXPXP

31


Gaussian AVC (GAVC) ChannelsHughes, and Narayan [16] determined the λ-capacity of discrete time GAVC for averaged, and peak power constraints imposed on channel, and transmitted sequence for random codes Hughes, and Narayan [17] determined the capacity of vector discrete time GAVC accompanied with water-filling equation, and proved the saddle pointCsiszar, and Narayan [9] determined the capacity of GAVC for deterministic codesAhlswede [1] computed the channel capacity of GAVC when the noise variance varies but does not exceed certain bound

32


Gaussian AVC (GAVC) ChannelsMcEliece [22] was first to apply game theoretic approach on continuous channels with mutual information as pay-off functionBasar, and Wu [3] considered jamming Gaussian channels with mean-square error pay-off functionDiggavi, and Cover [23] used the game theoretic approach to find the worst case noise when the covariance matrix constraint is imposed on the noiseBaker [2] computed the channel capacity of M-dimensional Gaussian continuous time channels with energy constraints posed on the transmitted signal, and jammerRoot, and Varaiya [20] considered the capacity of white noise Gaussian continuous time compound channel

33


Gaussian AVC (GAVC) ChannelsVishwanath, et. al. [24], computed the worst case capacity of Gaussian MIMO channels by applying duality principle

34

Maximin Capacity Subject to NormedUncertainties

35

Motivation

We deal with one class of channels, time continuous Gaussian compound channelsThe choice of Gaussian noise does not present a limitation because it is known that Gaussian noise is the worst case noise in terms of mutual information (see [15])As opposed to GAVC that models the uncertainty through the additive (jamming) signal with unknown statistics (see slide 11), we model the uncertainty through the uncertainty set of all possible frequency responses of the channel (see [13], [20])

36

Motivation

This approach is appealing because the uncertainty set is well defined in the normedlinear spaces H∞, and L1

It enables the treatment of both parametric, and non-parametric uncertaintiesIt is practical because the uncertainty set can be extracted from Nyquist, or Bode plotsThe obtained channel capacity formula explicitly depends on the size of the uncertainty set

37

Motivation

It gives the optimal transmitted power in the form of water-filling that depends on the size of uncertainty setOur computation does not require the saddle point because it relies on the work of Root, and Varaiya [20], and Gallager [15]It enables us to deal in the same time with two types of uncertainty (frequency response of the channel, and noise uncertainty) that has not been done until now (to our best knowledge)

38

Motivation

Our approach gives the solution to the jamming problem for continuous time channels accompanied with optimal transmitter and jammer strategies in terms of optimal PSD. We show that the optimal PSD of the signal is proportional to the optimal PSD of the noise Although, we consider similar problem to that of Baker’s paper [2], his constraints on the signal, and noise are in terms of the energy of the signal while we deal with power constraints

39

Communication system model

Most of the models that are used are probabilistic (compound channel, and arbitrary varying channel with and without memory) and finite dimensional channelsPresent model employs unknown transfer functions; transmitted signal is constrained in power with known PSD, noise is Gaussian whose PSD belongs to a certain set

40


Modeln

+

( )fW

( )fHyx

41


Both could be unknownUnknown transfer function can be model by using additive uncertainty

and where

Other models are possible: multiplicative

( ) ( ) ( ) ( )fWffGfG nom 1∆+=

( ) ( )fWfH ,( )fG

( ) ( ) ( ) ( ) ( ) 1,,, 1 ≤∆∈∆∞

∞ fHfWffGfG nom

( ) ( ) ( ) ( )( )fWffGfG nom 11 ∆+=

( ) ( )fGfGf

sup=∞

42


Uncertainty models: additive and multiplicative

( ) ( )fWf 1∆

( )fGnom +( ) ( )fWf 1∆

( )fGnom +

43


Example

α/β(1-δ)

Re

Im

α/β(1+δ)

α/β

( ) βαξβπ

α /,2

=+

=fj

fGnom

( ) ( )

( ) ( ) 10,

1/2

<<∆+=

+=

δδξξξ

βπξ

ff

fjf

fG

p

p

44


The uncertainty set is described by the ball in frequency domain centered at and with radius of

( ) ( ) ( )βπ

ξδ+

=≤−fj

fWfGfG nom 21

( )fW1

( )fGnom

45

Channel capacity with uncertainty

Define four sets

{ ,; 222 WWWHWA nom ∆+=∈= ∞

}1,,, 222 ≤∆∈∈∆∈∞

∞∞∞ HWHHWnom

( ) ( ){ }∫ ≤= xxx PdffSfSA ;1

{ ,; 113 WHHHHA nom ∆+=∈= ∞

}1,,, 111 ≤∆∈∈∆∈∞

∞∞∞ HWHHHnom

46


Overall PSD of noise is and uncertainty is modeled by uncertainty of filter

or by the set A4

( ) ( ){ }∫ ≤= nnn PdffSfSA ;4

( ) ( ) 2fWfSn

( ) ( ) ( ) ( )fWffWfW nom 22∆+=

47


Mutual information rate is pay-off function

( ) ∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= df

WS

HSyxI

n

x2

2

1log21;

48

Channel capacity with uncertainty I

Three problems could be definedNoise uncertainty

Channel uncertainty

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆++=

∈∈df

WWS

HSC

nomn

x

AWASNU

x2

22

2

1log21infsup

21

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∆++=

∈∈df

WS

WHSC

n

nomx

AHASCU

x2

2111log

21infsup

31

49


Channel – noise uncertainty

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆+

∆++=

∈∈∈df

WWS

WHSC

nomn

nomx

AHAWASCNU

x2

22

2111log

21infinfsup

321

50


Channel capacity with channel-noise uncertainty

Theorem 1. Assume that

is bounded and integrable.

( )( )22

21

WWS

WH

nomn

nom

−

+

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆+

∆++=

∈∈∈df

WWS

WHSC

nomn

nomx

AHAWASCU

x2

22

2111log

21infinfsup

321

51


Channel capacity is given parametrically

( )( )∫ ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−= df

WWS

WHC

nomn

nomNU 2

2

21*

log21 ν

( )( ) x

nom

nomn PdfWH

WWS=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

+−∫

+

21

22*ν

52


Such that

where ν* is related to Lagrange multiplier, and is obtained from constraint equation.Infimum over noise uncertainty is achieved at

( )( ) 0*,0* 2

1

22 >≥

−

+− νν

WH

WWS

nom

nomn

( ) ( )( ) ( )( )( ) ( ) 1,argargexp *22

*2 =∆+−=∆

∞ffWjfWjf nom

53


Mutual information rate after minimization is give as

( )∫

∫

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆++

≤∆ ∞

dfWWS

HS

dfWWS

HS

nomn

x

nomn

x

22

2

222

2

1

1log

1loginf2

54


Infimum over channel uncertainty is achieved at

( ) ( )( ) ( )( )( )( ) 1

,argargexp*1

1*1

=∆

++−=∆

∞f

jfHjfWjf nom π

55


Mutual information rate after second minimization is give as

( )

( )( )∫

∫

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆+

−+

≤∆ ∞

dfWWS

WHS

dfWWS

WHS

nomn

nomx

nomn

nomx

22

21

222

21

1

1log

1loginf1

56


Maximization gives water – filling equation

( )( ) *2

1

22* ν=

−

++

WH

WWSS

nom

nomnx

57


Water – filling

*ν

( )( )21

22

WH

WWSn

−

+

fBfBf−

psd

58


Channel capacity with channel uncertainty 02 =W

( )∫ ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −= df

WS

WHC

nomn

nomNU 2

21*

log21 ν

( ) xnom

nomn PdfWH

WS=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−∫

+

21

2

*ν

59


Channel capacity with noise uncertainty 01 =W

( )∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+= df

WWS

HC

nomn

nomNU 2

2

2*log

21 ν

( )x

nom

nomn PdfH

WWS=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +−∫

+

2

22*ν

60

Channel capacity with uncertainty II

Second approachNoise uncertainty

Channel – noise uncertainty

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

∈∈df

WS

HSC

n

x

ASASNU

nx2

2

1log21infsup

41

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∆++=

∈∈∈df

WS

WHSC

n

nomx

AHASASCNU

nx2

2111log

21infinfsup

341

61


Channel capacity with channel-noise uncertainty


is bounded and integrable,

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∆++=

∈∈∈df

WS

WHSC

n

nomx

AHASASCNU

nx2

2111log

21infinfsup

341

2

2

WS

HS

n

x

WWH

R nom 1−=

62


Channel capacity is given as

where are Lagrange multipliers

WWH

RdfRCCNU12

*2

*1 ;1log

21 −

=⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∫ λλ

( )*

*2

*1

*2

2*1

*2

2*1*

2

1

2*2*

2

021

nx

xxnn

SRRS

RSRSSS

λλ

λλλλ

λ

=+

=

=−+

0, *2

*1 >λλ

63


Integral constraints should be satisfied with equality

Power spectral densities satisfy water-filling

nn

xx

PdfS

PdfS

=

=

∫∫

*

*

*2

*2*

21λ

=+−xn SRS

64


Capacity for noise uncertainty case is obtained for


is bounded, and integrable. Define sets

01 =W

2

2

WS

HS

n

x

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤= ∫ xxx PdffSfSA ;1 ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤= ∫ nnn PdffSfSA ;4

01 =W

65


The lower value C- of pay-off function (mutual information rate, slide 40) is defined as

and is given by Theorem 2. The upper value C+

is defined by

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+==

∈∈

− dfWS

HSCC

n

x

ASASNU

nx2

2

1log21infsup

41

∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

∈∈

+ dfWS

HSC

n

x

ASASxn

2

2

1log21supinf

14

66


The lower value C- is equal to the upper value C+ implying that the saddle point existsThe optimal PSD of the noise is proportional to the optimal PSD of the transmitter, which can be explained such that the players in the game try to match each other

67

Channel Coding Theorem

Define the frequency response of equivalent channel

with impulse response and ten sets

( )2/1

2

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

WS

HSfG

n

x

( )tg

{ }3211 ,,; AHAWASGB x ∈∈∈=

68


{ }0,,; 2312 =∆∈∈= AHASGB x

{ }0,,; 1213 =∆∈∈= AWASGB x

{ }0,,,; 23414 =∆∈∈∈= AHASASGB nx

{ }0,0,,; 21415 =∆=∆∈∈= ASASGB nx

( ) ( ) ( ){ } 5,...,1,)3),2),1,; =∈= isatisfiestgBfGtgK ii

69


1) has finite duration δ2)

3)

Sets Ki are conditionally compact sets.

( )tg( ) 2Ltg ∈

( ) ( ) +∞→→+ ∫∫+∞−

∞−

AdffGdffGA

A

,022

70


Positive number Ri is called attainable rate for the set of channels Ki if there exists a sequence of codes such that when then uniformly over set Ki.

Theorem 4. The operational capacities Ci (supremum of all attainable rates Ri) for the sets of communication channels with the uncertainties Ki ,(i=1,…,5) are given by corresponding formulas in Theorem 1, and Theorem 2.

Proof. Follows from [15], and [20] (see [11])

( ){ }nnRT Te n ,,ε

+∞→nT 0→nε

71

Example 1

Uncertain channel, white noiseTransfer function

( )

( ) ( )

( ) ( ) βαξδδξξξβπ

ξβπβα

/,10,1/2

1/2/

1 =<≤∆+=+

=

+=

fffj

ffH

fjfH

p

p

nom

( ) ( ) ( ) ( ) ( )1/2111 +

=≤∆=−βπξδ

fjfWfWffHfH nom

72

Example 1

Channel capacity

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛= −

β

π

βππ

3/16/1

13/16/1

49

tan491 c

P

cPCCU

( ) ( )⎪⎩

⎪⎨

⎧⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛≤−⎟

⎠⎞

⎜⎝⎛

==

otherwisecPcPcfS fx

,049,

49

|

3/16/12

3/12

2

πωωππω

( )220

12 δα −=

Nc

73

Example 1

δ

P = 10-2 WN0 = 10-8 W/Hzβ = 1000 rad/s

⎪⎩

⎪⎨

⎧=

1000500250

α

74

Example 2

Uncertain noiseTransfer function

Noise uncertainty description

( )( ) ( ) 22

2

222 nn

n

fjfjfH

ωπξωπω

++=

( )βπ

ξδ/22 fj

fW =( ) ( )

( ) ( ) βαξδδξξξβπ

ξ

/,10,1/2

=<≤∆+=+

=

fffj

ffW p

2p

75

Example 2

δ

( )sradn /700=ω

( )sradn /1000=ω

( )sradn /1300=ω

( )srad

WP

/10001

2.001.0

====

βαξ

76

Example 2

( )( )srad

srad

WP

n /1300/1000

12.001.0

=====

ωβαξ

0=δ

2.0=δ

1.0=δ

77

Example 3

Uncertain channel, uncertain noise

Damping ration is uncertain

The noise uncertainty is modelled as in the Example 2

( )( ) ( ) 22

2

222 nn

n

fjfjfH

ωπξωπω

++=

ξ

78

Example 3

79

Example 3

80

References

[1] Ahlswede, R., “The capacity of a channel with arbitrary varying Gaussian channel probability functions”, Trans. 6th Prague Conf. Information Theory, Statistical Decision Functions, and Random Processes, pp. 13-31, Sept. 1971.

[2] Baker, C. R., Chao, I.-F., “Information capacity of channels with partially unknown noise. I. Finite dimensional channels”, SIAM J. Appl. Math., vol. 56, no. 3, pp. 946-963, June 1996.

[3] Basar, T., “A complete characterization of minimax, and maximinencoder-decoder policies for communication channels with incomplete statistical description”, IEEE Transactions on Information Theory, vol. 31, pp. 482-489, Jan., 1985.

81

References

[4] Biglieri, E., Proakis, J., Shamai, S., “Fading channels: information-theoretic and communications aspects,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2619-2692, October, 1998.

[5] Blachman, N. M., “Communication as a game”, IRE Wescon 1957 Conference Record, vol. 2, pp. 61-66, 1957.

[6] Blackwell, D., Breiman, L., Thomasian, A. J., “The capacity of a class of channels”, Ann. Math. Stat., vol. 30, pp. 1229-1241, 1959.

[7] Charalambous, C. D., Denic, S. Z., Djouadi, S. M. "Robust Capacity of White Gaussian Noise Channels with Uncertainty", accepted for43th IEEE Conference on Decision and Control.

82

References

[8] Csiszar, I., Korner, J., Information theory: Coding theorems for discrete memoryless systems. New York: Academic Press, 1981.

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