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```					 The Scaling Law of SNR-Monitoring in
Dynamic Wireless Networks

Hongyi Yao   Xiaohang Li   Soung Chang Liew
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.
Channel Model   Z

H
S                                 R

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

Prior Knowledge:
S1            S2                 S3       H1=1 and H2=1 and H3=1.

H1      H2          H3                      Updat
e

R                      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
Hi is unknown, Hj = 1 for   j  i.

   Recovering i and x: Unit Probing

Time Slot 1:                S1             S2         S3
1       1         1
Time Slot 2:
Time Slot 3:
R

Three time slots are required for probing.

4
Toy Example (Differential Group Probing)
Hi is unknown, Hj = 1 for         j  i.
Time Slot 1:                                   Time Slot 2:
S1              S2               S3           S1              S2              S3
1          1             1                     1         2             3

R                                              R
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].

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

6
Outline of the Talk
   Fundamental setting: multiple transmitters and one
   The scaling law of tracking all channel gains.
   Achieving the scaling law by ADMOT.

   General setting: multiple transmitters, relay nodes and
   The scaling law of above fundamental setting still holds.
   Achieving the scaling law by ADMOT-GENERAL.

   Simulation results.

7
Fundamental Setting
    Multiple transmitters and one receiver:
For Si, the probe in the s’th
S1         S2        …    Sn    time slot is Xi[s].
n

H1    H2       Hn     R receives: Y [ s ]   H i X i [ s ]  Z [ s ].
i 1

The term Z[s] is the noise
R                 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.
8
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].

9
Main Theorem
   Theorem: When the state variation H-H’
is approx-k-sparse, we have:
 Scaling Law: At least (k log(n / k )) time
slots are required for reliably estimating all
the n channels.
   Achievability: There exists a monitoring
scheme using (k log(n / k )) time slots, such
that R can estimate all the n channels in a
reliable and computational efficient manner.

10
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.
n
   Thus, R receives Y [ s ]   H i X i [ s ], for
s={1,2,…T}.                i 1

11
Proof of the Scaling Law
n
   Using H’, R computes Y '[ s]   H 'i X i [ s], and
n             i 1
D[ s ]  Y [ s ]  Y '[ s ]   ( H i  H 'i ) X i [ s ].
i 1

   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 (k log(n / k ))
linear samples are required for reliably
recovering a approx-k-sparse vector H-H’ [1].
Key Idea: Wireless interference only provides linear samples.
12
Achieve the Scaling Law by ADMOT

   Core techniques in ADMOT: Differential Group
Probing+ Compressive Sensing.
13
   The matrix  of dimensions N  n 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 ( s, i ) , as:

14
   Variables Initialization: H* is the estimation
of H. Vector Y is of dimension m. Matrix  m
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 
m ( s , i ).
   Receiver R sets Y[s] (i.e., the s’th component
of Y) to be the received sample. Thus,
Y [ s ]    m ( s, i ) H i  Z [ s ].
i
   Then we have Y   m H  Z.
15
   ADMOT(m, H’)          Continued from previous slide
   Step B (Computing Differences): Receiver R
computes D  Y   m H '   m ( H  H ')  Z .
   Step C (Norm-1 Sparse Recovering): Receiver R
finds the solution E* of the following convex
program:
 Minimize || E ||
1 , subject to || m E  D ||2  2m.

   Step D (Estimating) : Receiver R estimates H as
H*=H’+E*.
16
   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.
Tightly Match the Scaling Law!
future rounds of ADMOT by analyzing the square-root
estimation error |H-H*|2. Details can be found in [1].

17
Outline of the Talk
   Fundamental setting: multiple transmitters and one
   The scaling law of tracking all channel gains.
   Achieving the scaling law by ADMOT.

   General setting: multiple transmitters, relay nodes and
   The scaling law of above fundamental setting still holds.
   Achieving the scaling law by ADMOT-GENERAL.

   Simulation results.

18
General Communication Networks
   There are multiple transmitters in S, multiple
relay nodes in V and multiple receivers in R.
   For each node v  V R , all its incoming
channels (from S and V) require monitoring.
   In the following toy network, the directed
lines denote the channel requiring monitoring.
S                    R

V       V
1       2
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.
Complete Network!
20
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 (k log(n / k )) time
slots are required.
   Surprisingly, this scaling law is also tight for
general communication networks.

21
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
   Thus, the overall overhead is Ck log(n / k ).

   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.

22
Achieving the Scaling Law for Half-Duplex Model

   We construct ADMOT-GENERAL to achieve
overheads 3C'k log(n / k ) for a constant C’.
   The matrix  of dimensions N  n 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.

23

   ADMOT-GENERAL runs m time slots.

   In the s’th time slot, if ( s, i)  0, node vi receives in the
time slot; Otherwise, vi sends ( s, i) 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, Yi   i H i  Zi , where  i consists of at least
m/3 rows of  .
24

   Node vi computes the difference
Di  Yi   i H i '   i ( H i  H i ')  Z i
using the a priori knowledge Hi’ for its incoming
channel gains.

   Note each component of  i 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(  i )  m/3  C'k log(n / k ) for a
constant C’ [4].
Tightly Match the Scaling Law!
25
Outline of the Talk
   Fundamental setting: multiple transmitters and one
   The scaling law of tracking all channel gains.
   Achieving the scaling law by ADMOT.

   General setting: multiple transmitters, relay nodes and
   The scaling law of above fundamental setting still holds.
   Achieving the scaling law by ADMOT-GENERAL.

   Simulation results.

26
Simulations
   Setting:
   n=500 transmitters.
   Average SNR = 20 dB.
   Approx-k state variation. Define channel stability=1-k/n.
   ADMOT is implemented as the consecutive manner:

27
Simulations

28
Future Works
   General Setting: Network Tomography +
Channel Gain Estimation?

   Current ADMOT-GENERAL requires the internal
nodes in V performing sophisticated protocol

   Can we estimate internal channel gains as
“tomography”, in which relay nodes do normal
network transmission, only the transmitters and

29
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.

30

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