Type-Based Distributed Estimation over Multiaccess Channels G - - PowerPoint PPT Presentation

Type-Based Distributed Estimation over Multiaccess Channels G¨

• khan Mergen

Joint work with Prof. Lang Tong School of Electrical and Computer Engineering Cornell University, Ithaca, NY

2

Motivation: Sensor Networks

Fusion centers

• Sense a physical phenomena, transmit it

to a fusion center.

• Fusion centers estimate the field param-

eters, and deliver the estimate.

3

Problem Formulation

Sensor 1 Sensor 2 Sensor n Sensor 1 Sensor 2 Sensor n Fusion center Fusion center

X1 X2 Xn θ

• Estimate θ ∈ R.
• Observation X1, · · · , Xn are i.i.d. con-

ditioned on θ. Assumptions:

• Xi takes values in {1, · · · , k}.
• Xi ∼ pθ, where pθ is a probability mass

function.

1 2 3

pθ pθ(1) pθ(2) pθ(3) x

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Problem Formulation Cont’d

Sensor 1 Sensor 2 Sensor n Fusion center

X1 X2 Xn h1 h2 hn

• Sensor i has a set of k channel wave-

forms si,1, · · · , si,k.

• Upon observing Xi, it transmits si,Xi.

t t t

si,1(t) si,2(t) si,3(t)

Received signal: z = n

i=1 hisi,Xi + w.

• Channel gains h1, · · · , hn ∈ R are i.i.d.
• The noise is white N(0, σ2).
• Energy constraint ||si,j||2 ≤ E.

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Problem Formulation Cont’d

Sensor 1 Sensor 2 Sensor n Fusion center

X1 X2 Xn h1 h2 hn

Estimator:

• The parameter θ is estimated based
• n the received signal z.
• ˆ

θ(z) is the estimate, ˆ θ is the estimator.

Objective: Design the channel waveforms and the estimator to minimize the Mean Square Error (MSE) E{(ˆ

θ − θ)2}.

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Classical Approach

time frequency

• Collision among users is “bad!”
• Solution:
• rthogonalize transmis-

sions.

• Can be done by time/frequency/code

division. Advantages:

• Allows us to use the standard layered approach.
• Well understood. Rather easy to implement.

Caveat:

• The bandwidth requirement is significant for large n.
• Neglects the dependency among sensor data.

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Proposed Approach: TBMA

Noise time

z=

√ E 2 √ E 3 √ E

• Nodes

transmit simultaneously with Pulse Position Modulation (TBMA).

• si,Xi =

√ EδXi, where δ1, · · · , δk are

• rthonormal pulses.
• When all hi = 1,

z = histogram+noise.

Advantages:

• Uses much less bandwidth/time than orthogonal approaches.
• The MSE with TBMA is asymptotically optimal as n→∞.

Remark:

• Any set of orthonormal δ1, · · · , δk can be used.

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Outline

• Introduction
• Performance analysis

Fundamental limits – TBMA with deterministic hi – TBMA with random hi – Orthogonal allocation

• Transmitter channel side information
• Conclusion

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Fundamental Limits

Cramer-Rao Bound: Let ˆ

θ be an unbiased estimator based on X1, · · · , Xn. Then, E{(ˆ θ − θ)2} ≥ 1 nI(θ), (1)

where I(θ) = k

i=1 (dpθ(i)/dθ)2 pθ(i)

is the Fisher information in Xi. Asymptotic Efficiency: There exists a class of estimators based on X1, · · · , Xn satisfying

ˆ θ . = N(θ, 1 nI(θ)),

for large n.

(2)

Notation “ .

=” means ˆ θ

p

→θ and √n(ˆ θ − θ)

d

→N(0,

1 I(θ)) as n→∞.

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Fundamental Limits Cont’d

Key observations:

• To achieve asymptotic efficiency, an estimator need not have access to all

data X1, · · · , Xn.

• Knowledge of a sufficient statistic is actually enough.

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Outline

• Introduction
• Performance analysis

– Fundamental limits TBMA with deterministic hi – TBMA with random hi – Orthogonal allocation

• Transmitter channel side information
• Conclusion

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TBMA with Deterministic hi

Noise time

z=

√ E 2 √ E 3 √ E

• A sufficient statistic is empirical

measure:

˜ p := histogram n .

• Scale the received signal:

y := z √ En = ˜ p + N(0, σ2 n2E)

• (∗)

.

Questions:

• How bad is (∗) ?
• What estimator should be used?

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TBMA with Deterministic hi Cont’d

Answers: (i)

˜ p . = N(pθ, 1

nΣ) for large n, where Σ = Diag(pθ) − pθpT θ .

y = ˜ p + N(0, σ2 n2E) ⇒ y . = N(pθ, 1 nΣ).

(ii) Maximum-likelihood estimator (MLE) based on y is prohibitive. Let y = N(pθ, 1

nΣ), then its pdf is

f(y1, · · · , yk | θ) = exp  − n k

i=1 (pθ(i)−yi)2 pθ(i)

+ log k

i=1 pθ(i)

2   g(y).

Given y, minimize

k

• i=1

(pθ(i) − yi)2 pθ(i)

for asymptotic MLE.

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TBMA with Deterministic hi Cont’d

Theorem 1: The proposed estimator ˆ

θ minimizing M(θ) :=

k

• i=1

(pθ(i) − yi)2 pθ(i) (3)

with respect to θ ∈ R satisfies ˆ

θ . = N(θ,

1 nI(θ)) for large n.

Remarks:

• The asymptotic performance of TBMA is as if the fusion center has

direct access to Xi’s.

• No unbiased ˆ

θ, even the ones with direct access to Xi’s, can do better

than this.

• The theorem holds independent of the noise power σ2.
• The σ2 determines the speed of convergence to the asymptotic MSE.

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Outline

• Introduction
• Performance analysis

– Fundamental limits – TBMA with deterministic hi TBMA with random hi – Orthogonal allocation

• Transmitter channel side information
• Conclusion

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TBMA with Random hi

Noise time

z=

√ E 2 √ E 3 √ E

• Assume hi has non-zero mean h :=

E(hi), and σ2

h := Var(hi).

• Define y =

z √ Ehn.

• Observe y .

= N(pθ, 1

nΣ), where

Σ = (1 +

σ2

h

h2)Diag(pθ) − pθpT θ .

f(y|θ) ∝ exp

• −n(y − pθ)TΣ−1(y − pθ) + log |Σ|

2

• .

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TBMA with Random hi

Theorem 2: The estimator ˆ

θ minimizing M(θ) = (y − pθ)TΣ−1(y − pθ), (4)

with respect to θ ∈ R satisfies

ˆ θ . = N(θ, 1 +

σ2

h

h2

nI(θ) ),

for large n. Remarks:

• The performance loss due to channel randomness is ∝ (1 +

σ2

h

h2).

• When h ≈ 0, the loss is significant.
• At the extreme case h = 0, the MSE does not go to zero even though

n→∞. This is true for all unbiased ˆ θ.

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Outline

• Introduction
• Performance analysis

– Fundamental limits – TBMA with deterministic hi – TBMA with random hi Orthogonal allocation

• Transmitter channel side information
• Conclusion

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Performance of Orthogonal Allocation

• Let s1, · · · , sk ∈ Cm, ||si||2 ≤ 1, be constellation points.
• Sensor i transmits the waveform corresponding to

√ EsXi.

• Signal received from the i’th sensor:

z(i) = hi √ EsXi + v(i), (5)

where v(i) ∼ N(0, σ2I).

• When hi = 1, the z(1), · · · , z(n) are i.i.d. with Gaussian mixture density:

z(i) ∼

k

• j=1

pjN( √ Esj, σ2I).

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Performance of Orthogonal Allocation Cont’d

Asymptotic Performance:

• For any unbiased ˆ

θ, E{(ˆ θ − θ)2} ≥ 1 nJ(θ), (6)

where J(θ) = Ez(i)

• d log f(z(i))

dθ

2

is the Fisher information in z(i).

• The MLE based on z(1), · · · , z(n) satisfies ˆ

θ . = N(θ,

1 nJ(θ)).

Remarks:

• For the best asymptotic performance J(θ) should be maximized with

respect to s1, · · · , sk.

• For hi = 1, the antipodal constellation maximizes J(θ) for k = 2.
• In general, the optimal constellation depends on the family {pθ : θ ∈ R}.

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Performance of Orthogonal Allocation Cont’d

−20 20 40 1 2 3 4 5 6 7 SNR (dB) Fisher information Fisher Information in z(i); Bernoulli(0.8)

I(θ) (Fisher information in Xi) J(θ)

−20 −10 10 20 30 40 0.2 0.4 0.6 0.8 1 1.2 1.4

SNR (dB) Fisher information

Simplex Orthogonal BPSK I(θ) (Fisher information in Xi) J(θ)

Fisher Information in z(i); Poisson(1)

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A Numerical Example

1 2 4 8 16 32 64 10

−3

10

−2

10

−1

10

• No. of nodes

Mean squared error Identical channels (hi=1); Bernoulli(0.8) TDMA(SNR=−10dB) TDMA(SNR=0dB) TBMA(SNR=−10dB) TBMA(SNR=0dB) Direct access + ML Asymptotic perf.

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Numerical Example - 2

1 2 4 8 16 32 64 10

−3

10

−2

10

−1

10

• No. of nodes

Mean squared error Rician fading; E/ σ2=0dB, Bernoulli(0.8) TBMA (K=0.01) TDMA (K=0.01) TDMA (K=1) TBMA (K=1) Directaccess + ML

• The channel is Rician distributed, i.e.,

hi =

• K

K + 1 +

• 1

K + 1CN(0, 1),

where K > 0 is a deterministic number (K = 0 ⇒ Rayleigh).

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Outline

• Introduction
• Performance analysis

– Fundamental limits – TBMA with deterministic hi – TBMA with random hi – Orthogonal allocation Transmitter channel side information

• Conclusion

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Channel Side Information (CSI) at the Transmitter

• In certain cases, the transmitter nodes may be able to learn their channel

states before the transmission.

• Transmitter CSI can be utilized to solve the problem of zero-mean hi.
• Let hi := riejρi.
• The i’th node transmits P(ri)e−jρi√

EδXi in TBMA, where P(·) is a power

control rule satisfying

Eri[P 2(ri)] ≤ 1.

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CSI at the Transmitter Cont’d

• One possibility is to invert the channel: P(r) = 1
• r. This effectively cancels
• ut the effect of fading.
• Complete inversion may require infinite energy Er(1/r2) as in the case of

Rayleigh channels.

• To circumvent this, consider the following generalization:

P(r) =

• α/r, r ∈ [β, ∞)

γ,

• th.

where α, β, γ ∈ R are constants independent of r.

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CSI at the Transmitter Cont’d

• The channel gain seen by the receiver is ˜

hi := riP(ri).

• The asymptotic MSE is given by (1 +

σ2

˜ h

˜ h2)/nI(θ0), where ˜

h = E(˜ hi), σ2

˜ h = Var(˜

hi).

Lemma: As α and β converge to zero,

1 + σ2

˜ h

˜ h2→1.

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Numerical Example - 3

1 2 4 8 16 32 64 10

−3

10

−2

10

−1

10

• No. of nodes

Mean squared error Rayleigh with Power Control; SNR0dB, Ber(0.8) TDMA (detect+MLE) TBMA

• Asy. perf. of TBMA

Direct Access

• Power controlled Rayleigh channel (α = 1, β = 0.89, γ = 1/β).

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Numerical Example - 4

1 2 4 8 16 32 64 10

−3

10

−2

10

−1

10

• No. of nodes

Mean squared error Rayleigh channel with pow. control; SNR0dB, Ber(0.8) TBMA (phase +−90) TBMA Direct access

30

Numerical Example - 5

1 2 4 8 16 32 64 10

−2

10

−1

10 10

1

• No. of nodes

Mean squared error Identical channels; SNR=0dB, Poisson(2) TDMA (detect) TBMA TBMA+new estimator Direct access Asymptotic perf.

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Conclusions

• We considered the problem of communicating sensor readings over a

Gaussian multiaccess channel.

• The TBMA scheme is proposed as an alternative to the conventional
• rthogonal allocation.
• We proposed an estimator and characterized its asymptotic MSE.
• The TBMA is bandwidth efficient and also has favorable MSE.
• The TBMA needs to be used with transmitter CSI in case the channel

has zero mean.

32

Related Publications

• G. Mergen and L. Tong ”Type Based Estimation Over Multiaccess

Channels,” submitted to IEEE Transactions on Signal Processing, July 2004.

• G. Mergen and L. Tong ”Estimation Over Deterministic Multiaccess

Channels ,” 42nd Annual Allerton Conference on Communications, Control and

Computing, 2004.

• K. Liu and A. Sayeed, ”Optimal Distributed Detection Strategies for

Wireless Sensor Networks,” 42nd Annual Allerton Conference on

Communications, Control and Computing, Monticello, IL, 2004.