The Scaling Law of SNR-Monitoring in
Dynamic Wireless Networks
Hongyi Yao
Xiaohang Li
Soung Chang Liew
Channel Gain or Single-NoiseRatio (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
S

H
Z
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
H1
H2
R
S3
H3
H1=1 and H2=1 and H3=1.
Updat
e
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:
Time Slot 2:
S1
S2
1
1
S3
1
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:
S1
S2
1
1
Time Slot 2:
S3
1
R
Receive Y[1]=3+(Hi-1)
S1
S2
1
2
S3
3
R
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].
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
achieve the minimum communication overheads?

Answer: YES!
6
Outline of the Talk

Fundamental setting: multiple transmitters and one
receiver.



General setting: multiple transmitters, relay nodes and
receivers.



The scaling law of tracking all channel gains.
Achieving the scaling law by ADMOT.
The scaling law of above fundamental setting still holds.
Achieving the scaling law by ADMOT-GENERAL.
Simulation results.
7
Fundamental Setting

S1
Multiple transmitters and one receiver:
…
S2
Sn
For Si, the probe in the s’th
time slot is Xi[s].
n
H1
H2
Hn
R receives: Y [ s]   H i X i [ s]  Z [ s].
i 1
R
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.
8
State Variation



The state variation H-H’ is said to be approx-ksparse 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
i 1
s={1,2,…T}.
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

Systematical View of ADMOT:

Core techniques in ADMOT: Differential Group
Probing+ Compressive Sensing.
13
The Training Data of ADMOT



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
Construction of ADMOT

ADMOT(m, H’)


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
Construction of ADMOT

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:
2m.
 Minimize || E ||
1 , subject to || m E  D||2 
Step D (Estimating) : Receiver R estimates H as
H*=H’+E*.
Step E: Terminate ADMOT.
16
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.
Tightly Match the Scaling Law!
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].
17
Outline of the Talk

Fundamental setting: multiple transmitters and one
receiver.



General setting: multiple transmitters, relay nodes and
receivers.



The scaling law of tracking all channel gains.
Achieving the scaling law by ADMOT.
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.




Half-Duplex Model: Any node in V can not 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 Ck log(n / k ).
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
High-level Construction of ADMOT-GENERAL





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 Hi  Zi , where i consists of at least
m/3 rows of  .
24
High-level Construction of ADMOT-GENERAL



Node vi computes the difference
Di  Yi  i Hi '  i ( Hi  Hi ')  Zi
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
/ ka)
RowNumber( ) i  m/3  C'k log(nfor
constant C’ [4].
Tightly Match the Scaling Law!
25
Outline of the Talk

Fundamental setting: multiple transmitters and one
receiver.



General setting: multiple transmitters, relay nodes and
receivers.



The scaling law of tracking all channel gains.
Achieving the scaling law by ADMOT.
The scaling law of above fundamental setting still holds.
Achieving the scaling law by ADMOT-GENERAL.
Simulation results.
26
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:
27
Simulations
28
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?
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