fundamental bounds for power consumption at the physical layer: 'waterslide curves...

Fundamental bounds for power consumption at thephysical layer: "waterslide curves" and the price of

certainty

Anant Sahaibased on joint work with student

Pulkit Grover

Wireless FoundationsDepartment of Electrical Engineering and Computer Sciences

University of California at Berkeley

Support from NSF and Sumitomo Electric

LIDS Seminar: Apr 28th, 2008

Anant Sahai (UC Berkeley) Low-Power Communication MIT LIDS 1 / 31

Shannon tells us:


Shannon tells us:

Delay: needed for laws of large numbers to apply


Shannon tells us:

Delay: needed for laws of large numbers to apply

Power: needed to apply the laws of large numbers


The promise of the waterfall curve

0 0.5 1 1.5 2 2.5 3 3.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Power

log

10(⟨

Pe ⟩

)Uncoded transmission BSCShannon Waterfall BSCShannon Waterfall AWGN


The new problem

Classical goal: arbitrarily low probability of error.Classical assumption: not delay sensitive at all.New twist: minimizetotal power consumption


The new problem

Classical goal: arbitrarily low probability of error.Classical assumption: not delay sensitive at all.New twist: minimizetotal power consumptionImportant technology trends


The new problem


“Moore’s law” allows billions of transistors, and but only mildly reducespower-consumption per transistor.


The new problem


“Moore’s law” allows billions of transistors, and but only mildly reducespower-consumption per transistor.

New short-range applications: in-home networks, dense meshes,personal-area networks, UWB, between-chip communication,on-chipcommunication, etc.


Review of key prior work

Ephremides, Wireless Communications Magazine, 2002 Power consumption crucial across networking layers Many related papers.




Howard, Schlegel, and Iniewski, EUARASIP Wireless Comm/Net, 2006 Empirical study Found uncoded is often better





Cui, Goldsmith, Bahai, Journal of Wireless Comm, 2005 Semi-empirical with uncoded modulation Found larger-constellations and higher rates better

Massad, Medard, and Zheng, ISITA 2004 Information-theoretic with constant decoder power Found better to use higher rates





Cui, Goldsmith, Bahai, Journal of Wireless Comm, 2005 Semi-empirical with uncoded modulation Found larger-constellations and higher rates better

Massad, Medard, and Zheng, ISITA 2004 Information-theoretic with constant decoder power Found better to use higher rates

Bhardwaj and Chandrakasan, Allerton 2007 Focus on receiver sampling cost in UWB Found lower-rates and adaptive sampling is better


Outline

1 Motivation and introduction2 Classical results revisited3 A model for decoder power consumption4 General lower bounds5 Asymptotic behavior near capacity6 Optimal choice of rate


What should capacity-achieving mean?

minξTPT + ξCPC + ξDPD

ξT : net path loss (e.g.≈ 86dB for short-range)

ξC, ξD choice of weights for encoder and decoder power.






AssumePe → 0What happens to optimizingPT , PC, PD?







PT stays bounded:weakly certainty achieving







PT stays bounded:weakly certainty achieving PT , PC, PD all stay bounded:strongly certainty achieving








What happens asξC, ξD → 0? Computationasymptotically free, but not actually free.








What happens asξC, ξD → 0? Computationasymptotically free, but not actually free. PT → C−1(R): capacity achieving.


Decoding power vs communication range

1mm 10mm 1m 100m 10km−120

−100

−80

−60

−40

−20

0

20

40

60

Distance

γ (d

B)


Dense linear codes with brute-force decoding

0 10 20 30 40 50 60 70 80

−2

−4

−6

−8

−10

−12

−14

−16

−18

−20

−22

−24

Power

log 10

(⟨ P

e ⟩)Total powerOptimal transmit powerDecoding powerShannon waterfall

Decoding PowernR2nR, Error Prob 2−Esp(R,P)n


Convolutional codes with Viterbi decoding

0 10 20 30 40 50 60 70 80

−2

−4

−6

−8

−10

−12

−14

−16

−18

−20

−22

−24

Power

log 10

(⟨ P

e ⟩)Total powerOptimal transmit powerDecoding powerShannon waterfall

Decoding PowerLcR2LcR, Error Prob 2−Econv(R,P)Lc


Convolutional with “magical” sequential decoding

0 5 10 15

−2

−4

−6

−8

−10

−12

−14

−16

−18

−20

−22

−24

Power

log 10

(⟨ P

e ⟩)Shannon waterfallOptimal transmit powerDecoding powerTotal power

Decoding PowerLcR, Error Prob 2−Econv(R,P)Lc


Dense linear codes with “magical” syndrome decoding

0 5 10 15

−2

−4

−6

−8

−10

−12

−14

−16

−18

−20

−22

−24

Power

log 10

(⟨ P

e ⟩)Shannon waterfallOptimal transmit powerTotal powerDecoding power

Decoding Power(1− R)nR, Error Prob 2−Esp(R,P)n


Outline



A new hope: iterative decoding

Make assumptions about the decoder rather than the code.

ComputationalNode

Y

Bj

i

Rich enough to capture LDPC, RA, Turbo, etc. codes.




ComputationalNode

Y

Bj

i

Each consumesEnode energy per iteration and can sendarbitrary messagesto itsα + 1 neighbors.





ComputationalNode

Y

Bj

i


Run for a fixed number of iterationsi.





ComputationalNode

Y

Bj

i


Run for a fixed number of iterationsi.

Rich enough to capture LDPC, RA, Turbo, etc. codes.Power-consumption≥ iEnode per received sample.


How to lower-bound the number of iterations?

X XY iB i

7 8R o o t b i tN o d e

YY 721B B1 2 YY 8B X4 5 YY 4 5 B X9 1 0 YY 4 1 0Key concept:decodingneighborhood



X XY iB i

7 8R o o t b i tN o d e


Decoding neighborhood sizen ≤ 1 + (α + 1)αi−1 ≈ αi.



X XY iB i

7 8R o o t 5 b i tN o d e



Need to lower-bound averageprobability of bit error in termsof n.



X XY iB i

7 8R o o t K b i tN o d e



Need to lower-bound averageprobability of bit error in termsof n.

Key insight:n is playing a roleanalogous to delay.


Outline



A local “sphere-packing” bound for the AWGN


〈Pe〉 ≥ supσ2G>σ2

Pµ(n): C(G)<R

h−1b (δ(G))

2exp

(

−nD(σ2G||σ

2P) −

12φ(n, h−1

b (δ(G)))

(

σ2G

σ2P

− 1

))

C(G) = 12 log2(1 + PT

σ2G), δ(G): 1− C(G)

R





Pµ(n): C(G)<R

h−1b (δ(G))

2exp

(

−nD(σ2G||σ

2P) −

12φ(n, h−1

b (δ(G)))

(

σ2G

σ2P

− 1

))

C(G) = 12 log2(1 + PT

σ2G), δ(G): 1− C(G)

R

µ(n) = 12(1 + 1

T(n)+1 + 4T(n)+2nT(n)(1+T(n)))

T(n) = −WL(−exp(−1)(1/4)1/n)

WL(x) solvesx = WL(x) exp(WL(x))

φ(n, y) = −n(WL

(

−exp(−1)( y2)

2n

)

+ 1)





Pµ(n): C(G)<R

h−1b (δ(G))

2exp

(

−nD(σ2G||σ

2P) −

12φ(n, h−1

b (δ(G)))

(

σ2G

σ2P

− 1

))

C(G) = 12 log2(1 + PT

σ2G), δ(G): 1− C(G)

R

µ(n) = 12(1 + 1

T(n)+1 + 4T(n)+2nT(n)(1+T(n)))

T(n) = −WL(−exp(−1)(1/4)1/n)

WL(x) solvesx = WL(x) exp(WL(x))

φ(n, y) = −n(WL

(

−exp(−1)( y2)

2n

)

+ 1)

Double-exponential potential return on investments in decoding power!


Waterslide curves for general AWGN case

0 1 2 3 4

−2

−4

−8

−16

−32

−64

Power

log 10

(⟨ P

e ⟩)γ = 0.4γ = 0.3γ = 0.2Shannon limit


A local “sphere-packing” bound for the BSC

〈Pe〉 ≥ supC−1(R)<g≤ 1

2

h−1b (δ(g))

22−nD(g||p)

(

p(1− g)

g(1− p)

)ǫ√

n

hb(p): Binary entropy function

δ(g): 1− C(g)R

D(g||p): KL Divergence

ǫ:√

1K(g) log( 2

h−1b (δ(G))

)

K(g): inf0<η<1−gD(g+η||g)

η2 .


A local “sphere-packing” bound for the BSC

〈Pe〉 ≥ supC−1(R)<g≤ 1

2

h−1b (δ(g))

22−nD(g||p)

(

p(1− g)

g(1− p)

)ǫ√

n

hb(p): Binary entropy function

δ(g): 1− C(g)R

D(g||p): KL Divergence

ǫ:√

1K(g) log( 2

h−1b (δ(G))

)

K(g): inf0<η<1−gD(g+η||g)

η2 .

Double-exponentialactual returns observed byLentmaier, et al. 2005 forregular LDPC codes with iterative decoding!


Detailed look at BPSK case

1 2 3 4 5 6

−1

−2

−4

−8

−16

−32

Power

log(

⟨ Pe ⟩

)

γ = 0.4γ = 0.3γ = 0.2Shannon Waterfall


Detailed look at BPSK case

1 2 3 4 5 6 7 8

−2

−4

−8

−16

−32

Power

log

(⟨ P

e ⟩ )

Upper bound on total power

Lower bound on total power

Optimal transmit power

Shannon Waterfall

Optimal transmit powerat low ⟨ P

e ⟩


Deviations from Shannon

−3 −2 −1 0 1 2 30

5

10

15

20

25

30

log10

(γ)

Pop

t/C−

1 (R)

in d

B



0 0.2 0.4 0.6 0.8 1−30

−20

−10

0

10

20

30

Rate

Pow

er (

dB)

Normalized power gapOptimal transmit powerShannon limit



−3 −2 −1 0 1 2 3

−2

−4

−8

−16

−32

log10

(γ)

log 10

(⟨ P

e ⟩)


Sketch of proof

Pretend the code runs over channelG instead ofP.


Sketch of proof


δ(G) > 0 implies average probability of error ish−1b (δ(G)).


Sketch of proof



Channel needs only to misbehave for the decoding neighborhoods tocause a bit-error.


Sketch of proof




ǫ/φ assure that the misbehavior is “typical” (even if n is small)


Sketch of proof





Probability of an error event underP is lower-bounded by a convex-∪functionf of the probability underG.


Sketch of proof





Probability of an error event underP is lower-bounded by a convex-∪functionf of the probability underG.

Average probability underP is minimized byf of the average probabilityunder G.


The low Pe limit

Dominant term:D(G||P) term in exponent.

Pe ≈ exp(−D(C−1(R)||P)n) = exp(−D(C−1(R)||P)αi)


The low Pe limit



Take double logs of both sides.

ln ln1Pe

≈ ln D(C−1(R)||P) + i ln α


The low Pe limit




ln ln1Pe

≈ ln D(C−1(R)||P) + i ln α

Minimize sumζ + γi whereγ = ξDEnode

σ2PξT ln α

andζ = PTσ2

P.


The low Pe limit




ln ln1Pe

≈ ln D(C−1(R)||P) + i ln α

Minimize sumζ + γi whereγ = ξDEnode

σ2PξT ln α

andζ = PTσ2

P.

Solved by:

f (R, ζ)/∂f (R, ζ)

∂ζ= γ

wheref (R, PTσ2

P) = D(C−1(R)||P).


Outline



The waterfall curve revisited

0 0.5 1 1.5 2 2.5 3 3.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Power

log

10(⟨

Pe ⟩

)

Uncoded transmission BSCShannon Waterfall BSCShannon Waterfall AWGN

Two gaps:(

C1−hb(Pe)

− C)

andgap = C − R.



0 0.5 1 1.5 2 2.5 3 3.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Power

log

10(⟨

Pe ⟩

)


Two gaps:(

C1−hb(Pe)

− C)

andgap = C − R.

Pick a path to certainty:Pe → 0 soi ≈ln ln 1

Peln α +

ln 1D(C−1(R)||P)

ln α



0 0.5 1 1.5 2 2.5 3 3.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Power

log

10(⟨

Pe ⟩

)


Two gaps:(

C1−hb(Pe)

− C)

andgap = C − R.

Pick a path to certainty:Pe → 0 soi ≈ln ln 1

Peln α +

ln 1D(C−1(R)||P)

ln α

Let R → C and soi = logα ln 1Pe

+ 2 logα1

gap+ o(· · · ).



0 0.5 1 1.5 2 2.5 3 3.5−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Power

log

10(⟨

Pe ⟩

)


Two gaps:(

C1−hb(Pe)

− C)

andgap = C − R.

Pick a joint path to certainty:Pe = gapβ .

Let R → C . . .


Zoom into the neighborhood of capacity

−7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5

2

4

6

8

10

12

14

16

log10

(gap)

log 10

(n)

β = 2β = 1.5‘‘balanced’’ gapsβ = 0.75β = 0.5


Asymptotic scaling of iterations and gap

AssumePe = gapβ and BSC channel:

If β ≥ 1, i ≥ 2 log 1gap

+ log log 1gap

+ cβ .

If β ≤ 1, i ≥ 2β log 1gap

+ log log 1gap

+ cβ .





+ log log 1gap

+ cβ .


+ log log 1gap

+ cβ .

Proved by takingg∗ = p + gapr and using careful Taylor expansions

aroundg = p.





+ log log 1gap

+ cβ .


+ log log 1gap

+ cβ .


aroundg = p.

Much better than semi-trivial bound ofΩ(log log 1gap

).





+ log log 1gap

+ cβ .


+ log log 1gap

+ cβ .


aroundg = p.

Much better than semi-trivial bound ofΩ(log log 1gap

).

Much more optimistic than Khandekar-McEliece conjecturedΩ( 1gap

).


Outline



Assume a fixed message size

Suppose message very large, but not urgent.



Suppose message very large, but not urgent.Why not increase rate?




Advantage: fewer samples to process




Advantage: fewer samples to process Disadvantage: need higher capacity and thus more power

Energy per message bit:1R

PTσ2

P+ max(1, 1

R)γi.




Advantage: fewer samples to process Disadvantage: need higher capacity and thus more power

Energy per message bit:1R

PTσ2

P+ max(1, 1

R)γi.

Optimize over the choice ofR asPe varies.


Results for BPSK signaling

0.88 0.9 0.92 0.94 0.96 0.98 1−100

−80

−60

−40

−20

Ropt

log 10

(⟨ P

e ⟩)

γ = 100γ = 0.4γ = 0.1


Results for BPSK signaling

2 4 6 8 10 12 14 16 18 20

−60

−50

−40

−30

−20

−10

Energy per bit

log 10

(⟨ P

e ⟩)

Limiting value of optimal transmit power

neglecting the proessing energy

γ = 0.4γ = 0.1

γ = 1

Uncoded transmission


Concluding remarks

More details:arXiv:0801.0352

www.eecs.berkeley.edu/∼sahai/

Bounds can probably be tightened.


Concluding remarks




Encoding power still needs to be better understood.


Concluding remarks





Is there a way around our model of decoding?

How to reconcile with “linear-time” codes like expanders?


Concluding remarks





Is there a way around our model of decoding?

How to reconcile with “linear-time” codes like expanders?

Expand scope to cover multiterminal problems and other components.


fundamental bounds for power consumption at the physical layer: 'waterslide curves...

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