performance of distributed beamforming for dense...

Performance of Distributed

Beamforming for Dense Cooperative

Networks with Limited Feedback

Turo Halinen, Alexis Dowhuszko, and Jyri Hämäläinen

Department of Communications and Networking

Aalto University, Espoo, Finland

IMANET+ Technical Seminar Program

30th August, 2013 · University of Oulu, Finland

2

Overview

Introduction

■ Massive Distributed Beamforming (DBF) scheme

System model and assumptions

Performance analysis

■ Probability Distribution Function (PDF)

approximation for large number of cooperative nodes

Obtained results (based on proposed approximation)

■ Outage probability analysis

■ Ergodic capacity analysis

■ Bit error probability analysis

Summary of conclusions

IMANET+ Technical Seminar

University of Oulu - 30.08.2013

3

Introduction (1)

The demand of mobile data has been growing at a

steady pace in the last few years

This tendency has been fueled by the introduction of new

types of mobile devices (smartphones and tablets)



Papal Conclave in Vatican City (2013) Papal Conclave in Vatican City (2005)

Main

Transmitter Main

Receiver

4

By 2020, it is expected that every one of us will be on

average surrounded by 10 wireless enabled devices

■ Prediction: 50 billon connected devices (10-fold increase)

Massive deployment of low-cost Relaying Stations (RSs)

Distributed Beamforming scheme

(boost received signal energy)



Introduction (2)

5

Cross-layer interference mitigation Co-layer interference mitigation

Proposal: Each User Equipment (UE) receives assitance

from a cooperative group of RSs placed in close proximity

References: MBS = Macro BS, MUE= Macro UE, FBS = Femto BS, FUE = Femto UE



Introduction (3)

System model: Illustration

6

Goal: Enhance quality

of Rx signal at the

main receiver side

Generalized system model

composed of a main

transmitter (Tx), a main

receiver (Rx), and a cluster of

”M” RSs. Feedback link between

main Rx and RS cluster enables

to implement DBF in the 2nd hop

Basic idea of proposed DBF scheme:

Share a common message in the 1st hop (i.e., similar to a distributed

antenna system, but without wired links)

Implement a Transmit Beamforming (TBF) scheme with limited

feedback in the 2nd hop (in a distributed way)

All devices are equipped with a single Tx/Rx antenna

Main Tx and RSs operate in half-duplex mode, in a

Decode-and-Forward (DF) fashion

7

Adopted assumptions

T1

1st hop (low

attenuation)

2nd hop (large

attenuation)

T2

Time frame

1st hop 2nd hop

Main Rx Main Tx

Cluster of RSs

Cluster of RSs shares same location with main Tx

Mean path loss in 1st hop is much lower than in 2nd hop

First link assumed ”almost” costless (i.e., T1«T2)



The signal that the main receiver experiences at

Transmission Time Interval (TTI) “i” attains the form

where

Row vector ” ” contains the transmitted

signals in the 2nd hop (one per element in the cluster of RSs),

Row vector ” ” is the aggregate vector

that contains the channel gains from each RS to the common Rx,

”n” is Additive White Gaussian Noise (AWGN) with power ”PN”

8

Signal model (1)

Transmit vector “x[i]” is related to common information

symbol “s” via linear beamforming. In this case,

Row vector ” ” is a TBF vector with the

co-phasing factor that each RS ”m” should apply in transmission



Uniform quantization set for ”N” bits

9

Adjust phase

of each antenna

Ideal co-phasing case for “2” antennas (full channel phase resolution):

Channel gains can be summed

with perfect co-phasing

Equivalent model for the 2nd hop Limited feedback scheme

Each array element adjusts its

transmission signal independently



Signal model (1)

Instant.

SNR at

main Rx

Mean received

SNR from mth

array element

Ergodic capacity. Represents the long-term average transmission data

rate. Valid for applications with no strict delay constraints (code words

span over several coherence time intervals of a fading channel)

Outage probability. Represents the probability that an outage occurs in

an specified TTI because target rate cannot be supported with actual

channel state. Appropriate for contant-rate delay-limited transmissions

Bit error probability (BEP). Probability of making a wrong estimation of

the information bit that is being transmitted

10

Performance analysis (1)

To compute the previous performance measures, we need to

determine a suitable expression for the Probability Density Function

(PDF) of the instantaneous received SNR at the main receiver

”Lm[i]” is the total path loss attenuation for the mth signal path

Performance measures

Based on the previous assumptions, it is possible to show

that the expression for SNR attains the following form:

where individually received SNRs ” ”

are assumed to be known, and remain constant due to

static location of transmitters (i.e., RSs) and main receiver

11

Performance analysis (2)

Suitable expressions for conditional distributions required

Unfortunable, treatable closed-form expressions can only

be derived in specific cases (i.e., not for all ”M” and ”N”)

”M” = number of elements in distributed antenna array

”N” = number of feedback bits per antenna element

Approximated as non-central

chi-squared distributed

Assumption: The number of elements in the distributed

antenna array is high (e.g., “M >=10”)

Proposal: Use central limit theorem to obtain a suitable

distribution approximation for the PDF of Random Variable

where

12

PDF approximation (large virtual array) (1)

Approximated as central

chi-squared distributed



Based on the fact that “M” is large, we use the central limit

theorem to claim that both real and imaginary parts of RV

“H[i]” are Gaussian with means “μR” and “μI”, respectively

When a quantized co-phasing scheme is used, i.e.,

The co-phasing quantization errors, in the form of RV ”θm”,

are uniformly i.i.d. in the interval ”(-π/2N, π/2N)”. Therefore,

13


”M” = number of elements in distributed antenna array

”N” = number of feedback bits per antenna element

Main result: The PDF for the received SNR of the DBF

scheme, i.e., “Z = |XR|2+ |XI|2”, can be expressed as a

weighted sum of non-central chi-squared PDFs

where

are the standard deviations of the real and imaginary

parts of RV ”H[i]”, and

is the corresponding weighting factor

14




In the previous equation,

is a non-central chi-squared PDF with ”2(k+1)” Degrees

of Freedom (DoF), non-centrality parameter ”s12 = μR

2”.

”Ik” is the kth order modified Bessel function of the first kind

15


Second raw moment for the

real part of RV ”H[i]”

Second raw moment for the

imaginary part of RV ”H[i]”

Amplitude of different weighting factors

PDF for received SNR of DBF scheme

References: ”N = 1” (green), ”N = 2” (blue), ”N = 3” (magenta)

”M = 10” array elements

16 IMANET+ Technical Seminar


Note: Only few terms in

the sum are required

Outage probability (1)

17

To evaluate the performance of a mobile communications

system, it is usually assumed that transmission is

successful if

where ”γ0” is selected to guarantee a certain QoS, i.e.,

The closed-form formula for the received SNR

Cumulative Distribution Function (CDF) is required, and

can be obtained from the previously derived PDF formula

approximation.

”QM(a,b)” is the generalized Marcum Q-function of order ”M”

Outage Probability (2)

PDF for received SNR of DBF scheme

References: ”N = 0” (red), ”N = 1” (green), ”N = 2” (blue), ”N = 3” (magenta)




Larger number

of feedback bits Point values ’*’

obtained with

numerical

simulations

Outage Probability (3)

Power

imbalance

situation




Array

elements

grouped in

two cluster

References: ”N = 0” (red), ”N = 1” (green), ”N = 2” (blue), ”N = 3” (magenta). Solid lines:

Perfect channel power balance (”δ = 0 dB”). Dashed lines: Medium channel power

imbalance (”δ = 3 dB”). Dashed-dotted lines: large channel power imbalance (”δ = 6 dB”).

Larger number

of feedback bits

Point values ’*’

obtained with

numerical

simulations

Ergodic capacity (Mean data rate) (1)

20

The ergodic capacity is defined as follows:

where

As previously mentioned, instantaneous SNR gain ”Z”

can be approximated as the sum of 2 chi-squared RVs

The ergodic capacity is analyzed in two parts:

No channel phase signaling case (i.e., ”N = 0”)

Limited channel phase signaling case (i.e., ”N >= 1”)

Average SNR for

the 2nd hop

Instantaneous

SNR gain




21

No channel signaling (i.e., “N = 0”)

In this case, the real and imaginary parts of RV “H[i]”

follow a Gaussian distribution with zero mean and

identical variance (i.e., “σ12 = σ2

2 = σ2”)

Then, RV ”Z” is exponentially distributed

After few derivations,

results, where ”E1(z)” is the exponential integral

function of the first order




22

Limited channel phase signaling (i.e., “N >=1”)

In this situation, aim is to calculate

where ”Wk(σ1, σ2)” is the weighting factor and ”fk(z)” is a

non-central chi-squared PDF

At this point of the analysis, Jensen’s inequality

is used (valid when ”g(x)” is a concave function)

Jensen’s inequality becomes particularly accurate when

the values of RV ”Z” become concentrated near the mean




23

Limited channel phase signaling (cont'd)

The fading figure is used to analyze the degree of

variability of RV ”Z”

Finally, the following closed-form approximation is derived

Note: This approximation provides an strict upper bound for the ergodic capacity


References: ”N = 0” (red), ”N = 1” (green), ”N = 2” (blue), ”N = 3” (magenta)



Mean average SNR = 10 dB

Upper bound (full CSI at RSs)

Larger number

of feedback bits


obtained with

numerical

simulations


References: ”M = 10” (green), ”M = 20” (blue), ”M = 30” (magenta)



Mean average SNR = 10 dB

Larger number

of array elements

(cooperative nodes)


obtained with

numerical

simulations

Bit error probability (1)

26

The average BEP can be expressed using

where “fZ(z)” is the PDF of the instantaneous SNR, and

is the error rate when the modulation scheme is BPSK

It is possible to show that the BEP of the distributed

antenna system attains the form

where ”Pe(k+1)” admits some closed-form expression



Ratio between energy

deterministic and

random components

Bit error probability (2) BEP for a non-central chi-squared distributed RV with ”n” degrees of freedom

Multichannel

order

Confluent

hypergeometric

funtion

Pochhammer’s

symbol

BEP formula when ”M = 1”

Bit error probability (3)

References: ”N = 1” (green), ”N = 2” (blue), ”N = 3” (magenta)



Active array elements ”M = 20”

Larger number

of feedback bits


obtained with

numerical

simulations

29

Summary of results The performance of a DBF scheme was studied in case of

different channel phase feedback resolutions

Study was done in the context of a massive DBF architecture,

composed by a main Tx, a main Rx, and a cluster of RSs

■ First hop is considered practically costless

■ Bottleneck of the cooperative architecture assumed in 2nd hop

Closed-form approximations for three different performance

measures were derived (PDF/CDF approximations for Rx SNR)

■ Outage probability (Outage capacity)

■ Ergodic capacity (Mean data rate)

■ Bit error probability

It was observed that the achievable end-to-end performance,

when using small amounts of channel phase information (per

RS) is close to the full channel phase information upper bound

IMANET+ Technical Seminar University

of Oulu - 30.08.2013

Performance of Distributed

Beamforming for Dense Cooperative

Networks with Limited Feedback

Questions and/or Comments?

IMANET+ Technical Seminar Program

30th August, 2013 · University of Oulu, Oulu

30

performance of distributed beamforming for dense...

Documents