Distributed Beamforming in Sensor Networks

Distributed Beamforming in Sensor Networks

Daniel Tai

8/12/2013

Daniel Tai Distributed Beamforming in Sensor Networks 8/12/2013 1 / 21

References

1 R. Mudumbai et al., “Scalable feedback control for distributed beamforming in sensornetworks”, Proceedings. International Symposium on Information Theory, 2005.

2 Murali Tummala et al., “Distributed Beamforming in Wireless Sensor Networks”,Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems andComputers, 2005

3 John Litva and Titus Kwok-Yeung Lo, Digital Beamforming in Wireless Communications,Artech House, Norwood, MA, 1996.


Introduction

Distributed beamforming is using complex gains at distributed transceiver to beamformsignal toward desired receiver

Synchronization is one of the sources of error

In the 2 papers, 2 different techniques are used to generate beamforming vectors . Thefirst one explicitly takes phase error into consideration.

Both methods are based on gradient search(but with different objective).

In [2], effects of 3 sources of error are discussed:1 # of nodes used for beamforming2 position change after beamforming (modeled as amplitude + phase error)3 random node fail


Beamforming techniques

Two beamforming techniques are to be discussed.

Proposed in [1], the beamforming vector is computed by each node in a distributed anditerative manner by utilizing SNR information from receiver via perfect feedback.

In each iteration, total SNR information feedback from receiver through noiseless feedbackpathEach node then adjust its beamforming phase according to total SNR change at the receiver.The goal is to maximize SNR of receiver.The existence of optimum solution and convergence is proven.

In [2], beamforming is computed by head of sensor cluster trying to minimize the errorbetween received signal and reference signal from receiver using adaptive LSE algorithm

Receiver sends reference signal r(t) known to cluster headCluster head uses adaptive LSE algorithm (described in [3]) to get optimum beamformingvector iteratively. The goal is to minimize the squared error ||wT x − r(t)||2While this algorithm is already well-known, the contribution of this paper is to discuss theeffect of 3 factors listed in the last slide.


Beamforming techniquesDifferences

The 2 techniques differ in several ways

In [1], beamforming coefficients are computed by each sensor node in a distributedmanner. While in [2], sensors are assumed to form a cluster with a cluster head computescoefficients using data from nodes in the same cluster.

The method proposed in [1] needs feedback from receiver (UAV in military scenario)broadcasting its received power. In [2], the beamforming vector is computed from aknown reference signal r(t) from receiver.

Method in [2] assumes perfect synchronization between cluster head and receiver. (Iguess it can be solved by PLL or other techniques) while [1] take it into account.


System model [1]

Considering a system of N sensors sending a common message signal m(t) to a receiver.

Transmitting power for each sensor is constrained to P.

Assumptions:

Flat and slow fading channel is assumed. So channel for sensor i can be modeled by a singlecomplex scalar gain hi .The channels are time-slotted with slot length TS . Sensors only transmit at the beginning ofa slot. Hence

τmaxB � 1

, where τmax is upper bound of timing error and B is bandwidth. TS = 1/B.Sensors are synchronized to each other on carrier frequency fc so drift is small.Each sensor i has an unknown phase offset γi relative to receiver due to timing sync. errors.


Distributed transmission model [1] I

Each transmits: si (t) = A · gim(t − τi ), where A =√P is the amplitude subject to power

constraint P, gi is complex beamforming coef. (|gi | ≤ 1), τi ≤ τmax is timing error.The received signal at receiver can be modeled as

r(t) =N∑i=1

hi si (t)e jγi + n(t)

= AN∑i=1

aibiej(γi+θi+ψi )m(t − τi ) + n(t)

where hi = aiejψi , gi = bie

jθi


Distributed transmission model [1] II

In frequency domain,

R(f ) = AN∑i=1

aibiej(γi+θi+ψi )M(f )e−jf τi + N(f )

≈ A ·M(f )N∑i=1

aibiej(γi+θi+ψi )M(f )

using the assumption on Bτi � 1.It leads to beamforming gain

G =

∣∣∣∑Ni=1 aibie

j(γi+θi+ψi )∣∣∣2∑N

i=1 |bi |2


Distributed transmission model [1] III

From Cauchy-Schwartz Inequality, we can see that G is maximized when

γi + θopti + ψi = Const

The goal is to find θopti for each sensor i .

We use φi = γi + θi + ψi to denote received phase in following equations.


Feedback control protocol [1] I

The synchronization is processed in a iterative manner over timeslots. Let n be timeslot index.y [n] be the amplitude (y [n] ∝

∣∣∑i aibie

jφi [n]∣∣). θi is to be decided.

The protocol proceeds in each note distributively as follows:

1 Initialization: Let θi [0] = 0

2 At time slot n, y [n] is received through noiseless feedback. Based on that,

θi [n + 1] = θi [n] + ui [n] + δi [n]

where δi [n] = ±δ decided by random with equal probability (δ is a parameter controllingthe speed of convergence) and

ui [n] =

{δi [n − 1] y [n] > y [n − 1]

0 otherwise


Feedback control protocol [1] II

Simulation shows that φi [n] eventually converges to a constant.


Convergence of [1]

Convergence is proven in the paper. (omitted here)

[2] also gave a analytic approximation of y [n]. It is shown to be accurate in latersimulation results.


Results [1]Time vs. Received Amplitude

Number of sensor: N = 100

It takes about K = 5N timeslots toachieve signal amplitudes of about 0.75 ofmaximum


Results [1]Received Amplitude vs. Number of Sensors

More sensors lead to better receive gain


Results [1]Received Amplitude vs. Choice of δ

This shows the trade-off between initialconvergence rate and long-termperformance.


System model of [2] 1

[2] uses adaptive LSE technique to compute receiving beamforming coefficients. The receiversends a known reference signal r(t) to the sensor array. The squared error between referencesignal and beamformed received signal can be formalized:

ε2(t) = [r(t)−wHx(t)]2

where w is the beamforming vector and x(t) is received signal vector.Taking expectation,

E{ε2(t)} = E{r2(t)} − 2wHrrx + wHRxxw

where rrx = E{r(t)x(t)} and Rxx = E{xxH}.

∇w(E{ε2(t)}) = −2rrx + 2Rxxw

and it’s known thatwopt = R−1

xx rrx1Content of this and the next slide is from [3]: John Litva and Titus Kwok-Yeung Lo, Digital Beamforming

in Wireless Communications, Artech House, Norwood, MA, 1996.Daniel Tai Distributed Beamforming in Sensor Networks 8/12/2013 16 / 21

Adaptive LSE algorithm [2,3]

As in real scenario, prior knowledges of Rxx and rrx are required, which is not possible fordistributed sensors. Hence we use their instantaneous estimates,

R[n] = x[n]Hx[n] and r = r [n]x[n]

Estimatedbeamforming weights w is computed iteratively

w[n + 1] = w[n] +1

2µ[−∇(E{ε2[n]})]

= w[n] + µ(r− Rw[n]

)= w[n] + µx[n]

(r [n]− xH [n]w[n]

)= w[n] + µx[n]ε[n]

The gain constant µ controls convergence speed.Daniel Tai Distributed Beamforming in Sensor Networks 8/12/2013 17 / 21

Simulation configuration [2]

Reference signal r(t) is known and usedby cluster head

2 artificial interfering signal fromθ1 = 60◦, θ2 = −60◦ (refer to the figure)

UAV flying path is shown on the figure

There is perfect frequency, phase anddata synchronization among sensorsand between UAV and sensors

Cluster head has perfect knowledge ofsensors’ positions as well as AOA ofdesired signal and noise


Effects on Sensor Node Density [2]

Nodes in simulation are uniformlydistributed within a 1.2× 1.2m2 space

Control variable: number of nodes withinthe space

Nulls are generated on noise-injectedangles(±60◦)

The more sensors we have, the betterrejection level can we get


Effects on Position Error [2]

This simulation the effects on beamformingwithout considering position change

In [3], a analytical equation of its induced sidelobeincrease (∆s) in two-dimensional M × N array isgiven:

∆s =1

MN

[1

e−σ2∆p

(1 +

σ2∆a

I 2

)− 1

]where σ2

∆a is the variance of amplitude error involts2, σ2

∆p is the variance of phase error in rad2

(assumed to be Gaussian), I is amplitude ofbeamforming weights

also fractional loss Lp in mainlobe is given in [3]

Lp = e−σ2∆p


Effects on Sensor Node Failures [2]

In this simulation, sensors fail randomlyafter beamforming

In [3], a analytical equation of fractionalloss Lf in mainlobe is given

Lf =

(M ′N ′

MN

)2

where M,N are dimensions of originalcluster, M ′,N ′ are dimensions ofoperating clusters


Technology

Distributed Beamforming in Sensor Networks