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Opportunistic Orthogonal Writing on DirtyPaper
Pramod Viswanath
Joint work withTie Liu
University of Illinois at Urbana-Champaign
March 1, 2005
The Dirty Paper Channel
y(t) = x(t) + s(t) + w(t)
WGN
p.s.d.N0
2
The Dirty Paper Channel
y(t) = x(t) + s(t) + w(t)
p.s.d.N0
2
WGNWG Interference
p.s.d.Ns
2
The Dirty Paper Channel
y(t) = x(t) + s(t) + w(t)
p.s.d.N0
2
WGNWG Interference
p.s.d.Ns
2
PowerP
BandwidthW
¦ Observation (Cos81): Capacity of the channel is
W log(
1 +P
N0W
)
¦ An abstractbinningscheme
Focus of this talk
y(t) = x(t) + s(t) + w(t)
p.s.d.N0
2
WGNWG Interference
p.s.d.Ns
2
PowerP
Wideband
¦ Wideband dirty paper channel
Main Result
¦ Setting: Wideband channel
– Criterion: Minimum Energy per reliable BitEb
¦ Main Result:
– An explicit binning scheme
– Ramifications:
∗ Typical error events
∗ Low rate VQ for wideband Gaussian source
∗ Generalization to abstract alphabets
PPM and Wideband AWGN Channel
... ...
M Pulses
¦ Each message corresponds to a pulse
Opportunistic PPM
M Pulses
K Sub-pulses
¦ Each message corresponds toK sub-pulses
Encoding Opportunistic PPM
����������
����������
s(t)
¦ Transmit sub-pulse (amongK possible choices) wheres(t) is largest
Decoding Opportunistic PPM
���������
���������
y(t)
¦ Choose sub-pulse (amongMK possible choices) wherey(t) is
largest
Relieving the Tension
¦ Tranmit signal amplitude
√Es =
√Eb log M +
√Ns log K
¦ √Ns log K is the opportunistic amplitude gain
¦ LargerK means larger signal amplitude
¦ But decoder has more choices to get confused (MK sub-pulses)
Relieving the Tension
¦ Tranmit signal amplitude
√Es =
√Eb log M +
√Ns log K
¦ Error probability
P [Error] ≤ MK P[s + w ≥
√Es
]
Relieving the Tension
¦ Tranmit signal amplitude
√Es =
√Eb log M +
√Ns log K
¦ Error probability
P [Error] ≤ MK P[s + w ≥
√Es
]
≈ MK P[s ≥ Ns
Ns + N0
√Es
]P
[w ≥ N0
Ns + N0
√Es
]
Relieving the Tension
¦ Tranmit signal amplitude
√Es =
√Eb log M +
√Ns log K
¦ Error probability
P [Error] ≤ MK P[s + w ≥
√Es
]
≈ MK P[s ≥ Ns
Ns + N0
√Es
]P
[w ≥ N0
Ns + N0
√Es
]
?≈ M P[w ≥
√Eb log M
]
¦ If possible, then desired result follows
Choice ofK
¦ Transmit signal amplitude
√Es =
√Eb log M +
√Ns log K
¦ Chooselog K = Ns
N0log M
¦ Then
Ns
Ns + N0
√Es =
√Ns log K,
N0
Ns + N0
√Es =
√Eb log M
Tension at correct choice ofK
¦ Tranmit signal amplitude√Es =
√Eb log M +√
Ns log K
¦ Chooselog K = Ns
N0log M
Ns
Ns + N0
√Es =
√Ns log K,
N0
Ns + N0
√Es =
√Eb log M
¦ Error probability
P [Error] ≤ MK P[s ≥ Ns
Ns + N0
√Es
]P
[w ≥ N0
Ns + N0
√Es
]
= MK P[s ≥
√Ns log K
]P
[w ≥
√Eb log M
]
The AWGN Performance
¦ Tranmit signal amplitude√Es =
√Eb log M +√
Ns log K
¦ Chooselog K = Ns
N0log M
Ns
Ns + N0
√Es =
√Ns log K,
N0
Ns + N0
√Es =
√Eb log M
¦ Error probability
P [Error] ≤ MK P[s ≥ Ns
Ns + N0
√Es
]P
[w ≥ N0
Ns + N0
√Es
]
= M K P[s ≥
√Ns log K
]P
[w ≥
√Eb log M
]
≈ M 1 P[w ≥
√Eb log M
]
¦ AWGN performance is achieved
Non-Gaussian Dirt
¦ Interferences(t) is not AWG
¦ Make lengthN of each pulselong enough:
∑Ni=1 si√N
∼ N
¦ Essentially convert dirt into Gaussian
¦ Robustness: OPPM works over a (restricted) arbitrarily varying dirt
Story So Far
¦ OPPM is a simple concrete scheme to achieve AWGN performance
over wideband Dirty Paper channel
Pedagogical Utility
¦ OPPM is a simple concrete scheme to achieve AWGN performance
over wideband Dirty Paper channel
A text book example
Utility of OPPM
¦ OPPM is a simple concrete scheme to achieve AWGN performance
over wideband Dirty Paper channel
A text book example
¦ OPPM is also useful to prove new results
Typical Error Events
¦ Error occurs in two ways:
Typical Error Events
¦ Error occurs in two ways:
¦ Opportunistic gain is not large enough
P[
maxk=1...K
sk <√
Ns log K
]
Typical Error Events
¦ Error occurs in two ways:
¦ Opportunistic gain is not large enough
P[
maxk=1...K
sk <√
Ns log K
]
¦ Interference plus Noise is too large
P[s + w >
√Es
]
Typical Error Events
¦ Error occurs in two ways:
¦ Opportunistic gain is not large enough
P[
maxk=1...K
sk <√
Ns log K
]≈ exp (exp (−ε1 log K))
¦ Interference plus Noise is too large
P[s + w >
√Es
]≈ exp (−ε2 log K)
Typical Error Events
¦ Error occurs in two ways:
¦ Opportunistic gain is not large enough
P[
maxk=1...K
sk <√
Ns log K
]≈ exp (− exp (ε1 log K))
¦ Interference plus Noise is too large
P[s + w >
√Es
]≈ exp (−ε2 log K)
¦ Typical error event same as in wideband AWGN channel
Error exponent of wideband dirty paper channel same as that of
AWGN channel
OPPM is Structured Binning
¦ Interpretation of Costa’s binning scheme
– Each bin is a goodquantizer
– All bins put together,good channel code
OPPM is Structured Binning
¦ Interpretation of Costa’s binning scheme
– Each bin is a goodquantizer
– All bins put together,good channel code
¦ OPPM fits this theme
– Overall it is PPM - a good channel code
¦ Where is the quantizer?
Pulse Position Quantization
Reconstruction
Source
Optimality of PPQ
¦ Consider performance criterion:
maxQuantization Rate
Reduction in Distortion
Optimality of PPQ
¦ Consider performance criterion:
maxQuantization Rate
Reduction in Distortion
¦ An optimality property:
PPQ is a best low rate VQ of a wideband Gaussian source
A Deeper Understanding
¦ Opportunistic PPM is
a combination ofPPM(a good wideband channel code)
andPPQ(a good wideband VQ)
A Deeper Understanding
¦ Opportunistic PPM is
a combination ofPPM(a good wideband channel code)
andPPQ(a good wideband VQ)
¦ Generalization:
– Abstract alphabet Gelfand-Pinsker channel
– Abstract cost measure over input alphabet
– Performance measure:
Capacity per unit cost of an abstract Gelfand-Pinsker channel
Gelfand-Pinsker Channel
ENC DEC
Xn Y n
Pn(Y |XS)
Sn
¦ Memoryless Channel
¦ Alphabets:x ∈ X , s ∈ S, y ∈ Y
¦ Cost:b(x) for x ∈ X
Capacity per Unit Cost
¦ Largest rate of reliable communication per unit cost:
maxPXU|S
I(U ; Y )− I(U ; S)E [b(X)]
.
¦ U is an auxiliary random variable
¦ Achieved by an abstract binning scheme
Capacity per Unit Cost
¦ Largest rate of reliable communication per unit cost:
maxPXU|S
I(U ; Y )− I(U ; S)E [b(X)]
.
¦ U is an auxiliary random variable
¦ Achieved by an abstract binning scheme
¦ Main result:
Generalized Opportunistic PPM achieves the capacity per unit cost
¦ Need a zero cost input letter
Generalized PPM
N is length of each pulse
0 0 0 0 00000 00 0 0 0 0 00 0 0 0
M Pulses
x x x x x xx x
¦ Achieves capacity per unit cost of a classical channel
S. Verd́u, “On Capacity per Unit Cost”,IEEE Transactions on
Information Theory, Vol. 36(5), pp. 1019-1030, September 1990.
Generalized Opportunistic PPM
K Sub-Pulses
...
length N
...
¦ Each message corresponds toK sub-pulses
Encoding
... ................... ........ ............. .. .........
Type V
... ...Sn
K Sub-Pulses
¦ Transmitx = f(s) over sub-pulse that hastypeV
¦ Deterministic functionf : S → X
Encoding
... ................... ........ ............. .. .........
Type V
... ...Sn
K Sub-Pulses
¦ Transmitx = f(s) over sub-pulse that hastypeV
¦ Deterministic functionf : S → X
¦ NeedK = exp (D(V ||S))
Decoding
¦ Binary Hypothesis Testat any sub-pulse
H0 : PY |X=f(V ),V
H1 : PY |X=0,S
Decoding
¦ Binary Hypothesis Testat any sub-pulse
H0 : PY |X=f(V ),V
H1 : PY |X=0,S
¦ Fix P [H0 → H1] = ε.
¦ Stein’s Lemma (almost)
P [H1 → H0]·= exp
(−N D(PY |X=f(V ),V ||PY |X=0,S
))
Decoding
¦ Binary Hypothesis Testat any sub-pulse
H0 : PY |X=f(V ),V
H1 : PY |X=0,S
¦ Fix P [H0 → H1] = ε.
¦ Stein’s Lemma (almost)
P [H1 → H0]·= exp
(−N D(PY |X=f(V ),V ||PY |X=0,S
))
¦ Independent tests for each sub-pulse
– MK tests
Rate per Unit Cost
¦ Error Probability
P [Error] ≤ M K exp(−N D
(PY |X=f(V ),V ||PY |X=0,S
))
= M exp (N D (V ||S)) exp(−N D
(PY |X=f(V ),V ||PY |X=0,S
))
= M exp(−N
(D
(PY |X=f(V ),V ||PY |X=0,S
)−D (V ||S)))
Rate per Unit Cost
¦ Error Probability
P [Error] ≤ M K exp(−N D
(PY |X=f(V ),V ||PY |X=0,S
))
= M exp(−N
(D
(PY |X=f(V ),V ||PY |X=0,S
)−D (V ||S)))
¦ Cost incurred
N E [b(f(V ))]
Rate per Unit Cost
¦ Error Probability
P [Error] ≤ M K exp(−N D
(PY |X=f(V ),V ||PY |X=0,S
))
= M exp(−N
(D
(PY |X=f(V ),V ||PY |X=0,S
)−D (V ||S)))
¦ Cost incurred
NE [b(f(V ))]
¦ Conclusion: achievable rate per unit cost
log M
NE [b(f(V ))]=
D(PY |X=f(V ),V ||PY |X=0,S
)−D (V ||S)E [b(f(V ))]
Main Result
¦ Theorem: The capacity per unit cost is
maxPV ¿PS , f :S→X
D(PY |X=f(V ),V ||PY |X=0,S
)−D (V ||S)E [b(f(V ))]
¦ Converse follows by calculus on
maxPXU|S
I(U ; Y )− I(U ; S)E [b(X)]
¦ Key idea: identify
V (the prefered type) from
U (the auxiliary random variable)
Hindsight
¦ Return to Dirty Paper Channel:
y[m] = x[m] + s[m] + w[m]
¦ Cost functionb(x) = x2
¦ ChooseX = x andV = Ns
N0x + S
¦ ThenD
(PY |X=x,V ||PY |X=0,S
)−D (V ||S)x2
=1
N0
A Summary
¦ Capacity of Gelfand-Pinsker Channels achieved by binning scheme
¦ Binning scheme
– Each bin a good quantizer
– Overall good channel code
¦ Consider cost-efficient quantization and channel coding
– S. Verd́u, “Capacity per Unit Cost”
¦ Combination of cost-efficient quantizers and channel codes
achieves capacity per unit cost of Gelfand-Pinsker channel
Another Ramification
¦ Consider low-rate VQ of wideband Gaussian sourcex1(t)
¦ Decoder has access tocorrelatedwideband Gaussian sourcex2(t)
¦ A type ofWyner-Ziv lossy compression
– A binning scheme suggests no “rate loss”
¦ Ideas developed here suggest an explicit scheme: OPPQ
OPPQ: Encoding
Source
Reconstructionwithout side information
OPPQ: Reconstruction
Source
with side informationReconstruction
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