The Scaling Law of SNR-Monitoring in Dynamic Wireless Networks

Soung Chang LiewHongyi Yao Xiaohang Li

Channel Gain or Single-Noise-Ratio (SNR)

The channel gain H of a wireless channel (S,R) is defined by: Y= H X + Z, where X is the signal sent by S, Y is the signal received by R and Z ~ N(0,1) is the noise term.

For the simplicity, both noise power and transmit power are normalized to be 1.

SH

ZChannel Model

R

1

Channel Gain Monitoring In a wireless network, the knowledge of channel gains

are needed to design high performance communication schemes.

Due to fading, node mobility and node power instability, channel gains vary with time.

Thus, tracking and estimating channel gains of wireless channels is fundamentally important

This work seeks the answer of the following question: What is the minimum communication overhead such

that all network channels can be tracked? 2

Toy Example

R

S1 S2 S3

H1 H2 H3

Prior Knowledge:

H1=1 and H2=1 and H3=1.

Update

There exists i in {1,2,3} such that Hi varied.

Monitoring Object: The receiver R wants to recover i and Hi.

3

Toy Example

Recovering i and x: Unit Probing

R

S1 S2 S3

1 1 1Time Slot

1:Time Slot 2:Time Slot 3:

Three time slots are required for probing.

4

Hi is unknown, Hj = 1 for .j i

Toy Example (Differential Group Probing)

R

S1 S2 S3

1 1 1

Time Slot 1:

Time Slot 2:

R

S1 S2 S3

1 2 3

Receive Y[1]=3+(Hi-1) Receive Y[2]=6+(Hi-1)i

Using the a priori knowledge of the channel gains, R computes [Y’[1],Y’[2]]=[3,6] and then the difference: [Y[1],Y[2]] - [Y’[1],Y’[2]]=(Hi-1)[1,i].

5Since [1,1], [1,2] and [1,3] are linear independent, R can decode i and then Hi. - One time slot saving !

Hi is unknown, Hj = 1 for .j i

Motivation Raised by the Toy Example

Unit Probing VS. Differential Group Probing. Unit Probing (Scheduling Interference): Since we

do not know which channel varied, all channels must be sampled one by one.

Differential Group Probing (Embracing Interference): All channels are sampled simultaneously to explore the a prior knowledge.

Question: Does differential group probing suffice to achieve the minimum communication overheads? Answer: YES!

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Outline of the Talk Fundamental setting: multiple transmitters and

one receiver. The scaling law of tracking all channel gains. Achieving the scaling law by ADMOT.

General setting: multiple transmitters, relay nodes and receivers.

The scaling law of above fundamental setting still holds. Achieving the scaling law by ADMOT-GENERAL.

Simulation results.

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Fundamental Setting

Multiple transmitters and one receiver:

R

S1 S2 Sn

H1 H2 Hn

… For Si, the probe in the s’th time slot is Xi[s].

R receives: 1

[ ] [ ] [ ].n

i ii

Y s H X s Z s

The term Z[s] is the noise i.i.

d. (of s) ~ N(0,1).

Definition (State): The state H is a length n vector, with the i’th component equaling Hi. The vector H’ is the a priori knowledge of H preserved by R.

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State Variation

The state variation H-H’ is said to be approx-k-sparse if there are at most k “significant” nonzero components in H-H’.

Practical interpretation: Approx-k-sparse state variation means there are at most k channels suffering significant variations, while the variations of other channels are negligible.

Details about “approx” can be found in paper [1].

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Main Theorem

Theorem: When the state variation H-H’ is approx-k-sparse, we have: Scaling Law: At least

time slots are required for reliably estimating all the n channels.

Achievability: There exists a monitoring scheme using time slots, such that R can estimate all the n channels in a reliable and computational efficient manner.

( log( / ))k n k

( log( / ))k n k

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Proof of the Scaling Law Assuming T time slots are used for allowing

R estimating H from the a priori knowledge H’.

For the clarity, we simplify the problem by assuming the noise term Z[s]=0 for each time slot s.

Thus, R receives for s={1,2,…T}. 1

[ ] [ ],n

i ii

Y s H X s

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Proof of the Scaling Law Using H’, R computes and

Note that recovering H is the same as recovering H-H’ by using the linear samples D[s] for s={1,2,…,T}.

Using the results in [2], at least linear samples are required for reliably recovering a approx-k-sparse vector H-H’ [1].

1

'[ ] ' [ ],n

i ii

Y s H X s

1

[ ] [ ] '[ ] ( ' ) [ ].n

i i ii

D s Y s Y s H H X s

Key Idea: Wireless interference only provides linear samples.

( log( / ))k n k

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Achieve the Scaling Law by ADMOT

Systematical View of ADMOT:

Core techniques in ADMOT: Differential Group Probing+ Compressive Sensing.

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The Training Data of ADMOT

The matrix of dimensions consists of the training data of ADMOT. Here, N is the maximum number of time slots allowed by ADMOT, and n is the number of transmitters.

Each component of is i.i.d. chosen from {-1,1} with equal probability.

The i’th column of is the training data of transmitter Si. To be concrete, in the s’th time slot, Si sends , as:

N n

( , )s i

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Construction of ADMOT ADMOT(m, H’)

Variables Initialization: H* is the estimation of H. Vector Y is of dimension m. Matrix consists of the 1,2,…,m’th rows of .

Step A (Probing): For s = 1, 2,…m, in the s’th time slot:

For each i in {1,2,…,n}, Si sends Receiver R sets Y[s] (i.e., the s’th componen

t of Y) to be the received sample. Thus, Then we have

m

( , ).m s i

[ ] ( , ) [ ].m ii

Y s s i H Z s mH Z.Y

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Construction of ADMOT ADMOT(m, H’) Continued from previous slide

Step B (Computing Differences): Receiver R computes

Step C (Norm-1 Sparse Recovering): Receiver R finds the solution E* of the following convex program:

Minimize , subject to

Step D (Estimating) : Receiver R estimates H as H*=H’+E*.

Step E: Terminate ADMOT.

' ( ') .m mD Y H H H Z

1|| E || 2|| E D || 2m.m

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Comments for ADMOT

The computational complexity of R is dominated by a norm-1 minimization convex program.

If H-H’ is approx-k-sparse, using the results of Compressive Sensing[3], E* is a reliable estimation of H-H’ provided that m=Cklog(n/k) for a constant C.

The receiver can adapt the system parameter m for future rounds of ADMOT by analyzing the square-root estimation error |H-H*|2. Details can be found in [1].

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Tightly Match the Scaling Law!

Outline of the Talk Fundamental setting: multiple transmitters and

one receiver. The scaling law of tracking all channel gains. Achieving the scaling law by ADMOT.

General setting: multiple transmitters, relay nodes and receivers.

The scaling law of above fundamental setting still holds. Achieving the scaling law by ADMOT-GENERAL.

Simulation results.

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General Communication Networks

There are multiple transmitters in S, multiple relay nodes in V and multiple receivers in R.

For each node , all its incoming channels (from S and V) require monitoring.

In the following toy network, the directed lines denote the channel requiring monitoring.

v V R

S R

V1

V2 19

Simplified Model

The challenging of general communication network rises from the existence of relay nodes in V.

For the simplicity, we consider a network with only relay nodes V={v1,v2,…,vn}.

Thus, for each node vi in V, it wants to track the channel (vj,vi) for each j=1,2,…,n.

20Complete Network!

The Scaling Law of General Setting

Assume for each node vi in V, the incoming channels of vi suffer approx-k-sparse variation.

Directly using the scaling law of the single receiver scenario, at least time slots are required.

Surprisingly, this scaling law is also tight for general communication networks.

( log( / ))k n k

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Achieving the Scaling Law Full-Duplex model: Any node in V can transmit and receive

in the same time slot. Due to the broadcast nature of wireless medium, each node in V

can probe under ADMOT, and in the mean time receive the probes of other nodes in V.

In the end, each node in V can estimate its incoming channels following ADMOT.

Thus, the overall overhead is

Half-Duplex Model: Any node in V can not transmit and receive in the same time slot.

The generalization is non-straightforward and shown in the following slides.

For both models, the achievability schemes are implemented in a distributed manner, i.e., no centralized controller is needed.

C log( / ).k n k

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Achieving the Scaling Law for Half-Duplex Model

We construct ADMOT-GENERAL to achieve overheads for a constant C’.

The matrix of dimensions consists of the training data.

Each component of is i.i.d. chosen from {0,-1,1} with probability {1/2,1/4,1/4}.

The i’th column of is the training data of vi.

3C' log(n / )k k

N n

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High-level Construction of ADMOT-GENERAL

ADMOT-GENERAL runs m time slots.

In the s’th time slot, if node vi receives in the time slot; Otherwise, vi sends in the time slot.

In the end, with large probability (Chernoff Bound), each node, say vi, received at least m/3 data.

Let the vector Yi consist of the received data of vi, and Hi be the vector consisting of all incoming channel gains of vi.

Each component of Yi is a linear sample (with noise) of Hi. That is, , where consists of at least m/3 rows of .

( , ) 0,s i ( , )s i

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i i i iY H Z i

High-level Construction of ADMOT-GENERAL

Node vi computes the difference

using the a priori knowledge Hi’ for its incoming channel gains.

Note each component of is i.i.d. sampled from {0,-1/2,1/2} with probability {0.5, 0.25, 0.25}, which are therefore sub-Gaussian ensembles.

Approx-k-sparse Hi-Hi’ can be recovered provided that RowNumber( ) for a constant C’ [4].

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' ( ')i i i i i i i iD Y H H H Z

i

m/3 C' log(n / )k k i

Tightly Match the Scaling Law!

Outline of the Talk Fundamental setting: multiple transmitters and

one receiver. The scaling law of tracking all channel gains. Achieving the scaling law by ADMOT.

General setting: multiple transmitters, relay nodes and receivers.

The scaling law of above fundamental setting still holds. Achieving the scaling law by ADMOT-GENERAL.

Simulation results.

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Simulations Setting:

n=500 transmitters. One receiver. Average SNR = 20 dB. Approx-k state variation. Define channel stability=1-

k/n. ADMOT is implemented as the consecutive manner:

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Simulations

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Future Works General Setting: Network Tomography +

Channel Gain Estimation?

Current ADMOT-GENERAL requires the internal nodes in V performing sophisticated protocol (ADMOT) for channel gain estimation.

Can we estimate internal channel gains as “tomography”, in which relay nodes do normal network transmission, only the transmitters and receivers perform sophisticated protocols?

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Thanks! & Questions?

[1]. H. Yao and X. Li and S. C. Liew, “Achieving the Scaling Law of SNR-Monitoring for Dynamic Wireless Networks”, arxiv 1008.0053.

[2]. K. D. Ba, P. Indyk, E. Price, and D. P. Woodruff, “Lower bounds for sparse recovery,” in Proc. of SODA, 2010.

[3]. E. Cand´es, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, 2006.

[4]. S. Mendelson, A. Pajor, and N. T. Jaegermann, “Uniform uncertainty principle for bernoulli and subgaussian ensembles,” Constructive Approximation, 2008.

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