communication under normed uncertainties
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Communication Under Normed Uncertainties
S. Z. DenicSchool of Information Technology and EngineeringUniversity of Ottawa, Ottawa, Canada
C. D. CharalambousDepartment of Electrical and Computer EngineeringUniversity of Cyprus, Nicosia, Cyprus
S. M. DjouadiDepartment of Electrical and Computer EngineeringUniversity of Tennessee, Knoxville, USA
Dec 9, 2004
Robust Capacity of White Gaussian Noise with Uncertainty
2
Overview
Importance of Uncertainty in Communications
Shannon’s Definition of Capacity
Review of Maximin Capacity
Paper Contributions
3
Overview
Main Results
Examples
Conclusion and Future Work
4
Importance of Communication Subject to Uncertainties
Channel measurement errors Network operating conditions Channel modeling Communication in presence of jamming Sensor networks Teleoperations
5
Shannon’s Definition of Capacity
6
Shannon’s Definition of Capacity
Model of communication system
wSource Encoder Decoder SinkChannel +
x
ny w
f g
M,...,1 M,...,1 wf wfh wfhg
M
ii
ne
i
MP
Misentiiifhg
n
MR
1
1
,...,1,|Pr
log
nM ,
8
Shannon’s Definition of Capacity
Discrete memoryless channel
Channel capacity depends on channel transition matrix Q(y|x) that is known
Xx YyXx
XP
xyQxP
xyQxyQxPQPI
QPIC
'
'|'
|log|,
,max
9
What if Q(y|x) is unknown ? Example: compound BSC
What is the channel capacity ?
Shannon’s Definition of Capacity
1-
1- 0 0
1 1
xy
xyxyQ
YX
,1
,;|
1,0,1,0
12
Additive Gaussian Channels Random Process Case
Shannon’s Definition of Capacity
+x y
fW
fH
n
13
Random process case derivation
Shannon’s Definition of Capacity
2
2
1sup log 1
2
;
x
x
S An
x x
S HC df
S W
A S S df P
14
Capacity of continuous time additive Gaussian channel
Shannon’s Definition of Capacity
df
WS
HC
n
2
2*
log2
1
PdfH
WSn
2
2
*
16
Water-filling
Shannon’s Definition of Capacity
*2
2
H
WSn
fBfBf
psd
17
Review of Maximin Capacity
18
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
Review of Minimax Capacity
;|,infmax
QPIC
XP
19
Review of Minimax Capacity
The existence of saddle point ?
For further references see Lapidoth, Narayan [18]
;|,;|,;|,
;|,maxinf;|,infmax
****
QPIQPIQPI
QPIQPICXPXP
23
Paper Contributions
24
Modeling of uncertainties in the normed linear spaces H∞, and L1
Explicit channel capacity formulas for SISO communication channels that depend on the sizes of uncertainty sets for uncertain channel, uncertain noise, and uncertain channel, and noise
Explicit water-filling formulas that describe optimal transmitted powers for all derived channel capacities formulas depending on the size of uncertainty sets
Paper Contributions
28
Main Results
29
Model
Communication system model
+x
n
y
fW
fH
31
Communication system model
Uncertainty models: additive and multiplicative
fWf 1
fGnom + fWf 1
fGnom +
32
Example
Communication system model
/(1-)
Re
Im
/(1+)
/
/,
2
fjfGnom
10,
1/2
ff
fj
ffG
p
p
33
The uncertainty set is described by the ball in frequency domain centered at and with radius of
Communication system model
fj
fWfGfG nom 21
fW1
fGnom
34
Channel capacity with uncertainty
Define four sets
,; 222 WWWHWA nom
1,,, 222
HWHHWnom
xxx PdffSfSA ;1
,; 113 WHHHHA nom
1,,, 111
HWHHHnom
35
Overall PSD of noise is and uncertainty is modeled by uncertainty of filter
or by the set A4
nnn PdffSfSA ;4
Channel capacity with uncertainty
2fWfSn
fWffWfW nom 22
37
Three problems could be defined Noise uncertainty
Channel uncertainty
df
WWS
HSC
nomn
x
AWASNU
x
2
22
2
1log2
1infsup
21
df
WS
WHSC
n
nomx
AHASCU
x
2
2
111log2
1infsup
31
Channel capacity with uncertainty I
38
Channel – noise uncertainty
df
WWS
WHSC
nomn
nomx
AHAWASCNU
x
2
22
2
111log2
1infinfsup
321
Channel capacity with uncertainty I
40
Channel capacity is given parametrically
df
WWS
WHC
nomn
nomCNU 2
2
21*
log2
1
x
nom
nomn PdfWH
WWS
2
1
2
2*
Channel capacity with uncertainty I
45
Maximization gives water – filling equation
*2
1
2
2*
WH
WWSS
nom
nomnx
Channel capacity with uncertainty I
46
Channel capacity with uncertainty I
Water – filling
*
21
2
2
WH
WWSn
fBfBf
psd
47
Jamming Noise uncertainty
Channel – noise uncertainty
Channel capacity with uncertainty II
df
WS
HSC
n
x
ASASNU
nx
2
2
1log2
1infsup
41
df
WS
WHSC
n
nomx
AHASASCNU
nx
2
2
111log2
1infinfsup
341
49
The lower value C- of pay-off function is defined as
and is given by Theorem 2. The upper value C+ is defined by
Channel capacity with uncertainty II
dfWS
HSCC
n
x
ASASNU
nx
2
2
1log2
1infsup
41
dfWS
HSC
n
x
ASASxn
2
2
1log2
1supinf
14
50
Channel capacity is given as
where are Lagrange multipliers
Channel capacity with uncertainty II
W
HRdfRCNU
;1log
2
1 2*2
*1
*
*2
*1
*2
2*1
*2
2*1*
2
1
2*2*
2
02
1
nx
xxnn
SR
RS
RSRSSS
0, *2
*1
52
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
55
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 1. The operational capacities Ci (supremum of all attainable rates Ri) for the sets of communication channels with the uncertainties Ki are given by corresponding computed capacity formulas.
Proof. Follows from [15], and [20] (see [11])
nnRT Te n ,,
nT 0n
Channel coding theorem
62
Uncertain channel, white noise Transfer function
Example 1
/,10,
1/2
1/2
/
1
ff
fj
ffH
fjfH
p
p
nom
1/2111
fjfWfWffHfH nom
63
Channel capacity
Example 1
3/16/1
13/16/1
49
tan4
91 cP
c
PCCU
otherwise
c
PcPc
fS fx
,0
4
9,
4
9|
3/16/12
3/12
2
220
12
Nc
64
P = 10-2 WN0 = 10-8 W/Hz = 1000 rad/s
1000
500
250
Example 1
65
Example 2
Uncertain noise Transfer function
Noise uncertainty description
22
2
222 nn
n
fjfjfH
/22 fj
fW
/,10,
1/2
2
ff
fj
ffW
p
p
66
sradn /700
sradn /1000
sradn /1300
srad
WP
/1000
1
2.0
01.0
Example 2
67
srad
srad
WP
n /1300
/1000
1
2.0
01.0
0
2.0
1.0
Example 2
68
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
69
Example 3
70
Example 3
71
Currently generalizing these results to uncertain MIMO channels
Future work
72
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.
73
[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 for 43th IEEE Conference on Decision and Control.
References
74
References
[8] Csiszar, I., Korner, J., Information theory: Coding theorems for discrete memoryless systems. New York: Academic Press, 1981.
[9] Csiszar, I., Narayan P., “Capacity of the Gaussian arbitrary varying channels”, IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 18-26, Jan., 1991.
[10] Denic, S. Z., Charalambous, C. D., Djouadi, S.M., “Capacity of Gaussian channels with noise uncertainty”, Proceedings of IEEE CCECE 2004, Canada.
[11] Denic, S.Z., Charalambous, C.D., Djouadi, S.M., “Robust capacity for additive colored Gaussian uncertain channels,” preprint.
75
[12] Dobrushin, L. “Optimal information transmission through a channel with unknown parameters”, Radiotekhnika i Electronika, vol. 4, pp. 1951-1956, 1959.
[13] Doyle, J.C., Francis, B.A., Tannenbaum, A.R., Feedback control theory, New York: McMillan Publishing Company, 1992.
[14] Forys, L.J., Varaiya, P.P., “The -capacity of classes of unknown channels,” Information and control, vol. 44, pp. 376-406, 1969.
[15] Gallager, G.R., Information theory and reliable communication. New York: Wiley, 1968.
References
76
[16] Hughes, B., Narayan P., “Gaussian arbitrary varying channels”, IEEE Transactions on Information Theory, vol. 33, no. 2, pp. 267-284, Mar., 1987.
[17] Hughes, B., Narayan P., “The capacity of vector Gaussian arbitrary varying channel”, IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 995-1003, Sep., 1988.
[18] Lapidoth, A., Narayan, P., “Reliable communication under channel uncertainty,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2148-2177, October, 1998.
[19] Medard, M., “Channel uncertainty in communications,” IEEE Information Theory Society Newsletters, vol. 53, no. 2, p. 1, pp. 10-12, June, 2003.
References
77
[20] Root, W.L., Varaiya, P.P., “Capacity of classes of Gaussian channels,” SIAM J. Appl. Math., vol. 16, no. 6, pp. 1350-1353, November, 1968.
[21] Wolfowitz, Coding Theorems of Information Theory, Springer – Verlang, Belin Heildelberg, 1978.
[22] McElice, R. J., “Communications in the presence of jamming – An information theoretic approach, in Secure Digital Communications, G. Longo, ed., Springer-Verlang, New York, 1983, pp. 127-166.
[23] Diggavi, S. N., Cover, T. M., “The worst additive noise under a covariance constraint”, IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 3072-3081, November, 2001.
References
78
[24] Vishwanath, S., Boyd, S., Goldsmith, A., “Worst-case capacity of Gaussian vector channels”, Proceedings of 2003 Canadian Workshop on Information Theory.
[25] Shannon, C.E., “Mathematical theory of communication”, Bell Sys. Tech. J., vol. 27, pp. 379-423, pp. 623-656,July, Oct, 1948
References
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