1

Presenter: Chun-Hsien Peng ( 彭俊賢 )Advisor: Prof. Chong-Yung Chi ( 祁忠勇 教授 )

Institute of Communications Engineering &Department of Electrical Engineering

National Tsing Hua UniversityHsinchu, Taiwan 30013, R.O.C.

E-mail: d905610@oz.nthu.edu.tw

Blind Beamforming for Multiuser OFDM Systems by Kurtosis Maximization Based on Subcarrier Averaging

2

2. MIMO Models for Beamforming of Multiuser OFDM Systems

1. Introduction

3. Post-FFT Fourier Beamformer by Subcarrier Averaging

OUTLINE

4. Blind Post-FFT KMBFA by Subcarrier Averaging

5. Simulation Results

6. Conclusions and Future Researches

KMBFA: Kurtosis Maximization Beamforming Algorithm

3

f1

f3

f5

f4f6

f2

f1

f7

CCICCI

CCI:CCI: Co-channel Interference

ISI:ISI: Intersymbol Interference (due to Multipath)

1. Introduction

][nh][nu

][nx

(Nois(Noise)e)

(Multipath (Multipath channel)channel)

][nw

Wireless communication problems: ISI, MAI,ISI, MAI, and CCICCI suppression in cellular wireless communication systems

MAI:MAI: Multiple Access Interference (Caused by Multiple Users) in a Cell

MAIMAI

ISIISI

A multiuser OFDM system with antenna arrays such as the pre-FFT pre-FFT and pospost-FFT beamfoming receiverst-FFT beamfoming receivers have been considered for combating CCI, MAI combating CCI, MAI and ISIand ISI in the receiver design.

4)2 ( /N

2. MIMO Models for Beamforming of Multiuser OFDM Systems

Transmitter for “Quasi-synchronous” Multiuser OFDM Systems

][kup : data sequence of userp

: number of subcarriers

NgN : length of guard

interval (GI)

][kup ][ns p

S/ PN-pointI FFT

P/ SGI

I nsertion D/ A

User p

approximately Gaussian (by Central Limit Theorem)

1

0

/2 ,][1

][N

k

Nknjpp eku

Nns 1 , ,1 , ... NNNn gg

1 , ,1 ,0 Nk

5

(Received signals)

(Transmitted signals)

(Noise vector)

1

2

),,( 1,11,11,1

),,( 2,12,12,1

),,( 1,1,1, PPP

),,( 2,2,2, PPP

][1 ns

][nsP

][nx

][nw

Qtime delay

DOApath gain

,][ ] [)( ][1 1

,,,

P

p

pL

llpplplp nnsn wax

T) ,sin()1() ,sin( , )..., ,,1( )( lpjlpjlp ee

Qawhere

( steering vector)Q 1

1 ..., ,1 , NNNn gg

Baseband discrete-time received signal:

: total number of paths (or DOAs) associated with user pL p

: ( ) total number of paths (or DOAs) of all the usersL PLLL ... 21

Q : number of receive antennas

DOA: Direction of Arrival

6

Some general assumptions : (A1) are i.i.d. QPSK symbol sequences (i.e., for each k is a random variable with uniform probability mass function over the sample space ), and is statistically independent of for .

][, ],...[ ],[ 21 kukuku P ][kup

][kup ][kuq

Non-Gaussianprocess

QPSK: Quadriphase-shift Keying

QI : identity matrixQQ

pq

(A4) is zero-mean white Gaussian with and statistically independent of 's.

QIww 2H ]}[][{ wnnE ][nw][kup

} , { 4/4/ jj ee

i.i.d.: Independent Identically Distributed

(A3) , ... ,2,1, gpLppp N0 . p Quasi-synchronous OFDM Systems

(A2) , for all ;lpmq ,, ),( ),( lpmq 21 PLLLL Q , and L is known.

number of receive antennas

total number of paths ofall the users

7

Pre-FFT Beamforming Structure (Pre-FFT BFS) [18,19]

A/DA/D

A/DA/D GI Removal

N-point FFT

S/P

A/D

P/S

][ , ku lpSpatial Processor

(Beamformer)

Spatial Processor

(Beamformer)

][nx

[18] M. Okada and S. Komaki, “Pre-DFT combining space diversity assisted COFDM,”

IEEE Trans. Vehicular Technology, vol. 50, pp. 487-496, Mar. 2001.[19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43.

Nlpkjplp ekuku

/ , 2 , ][ ][

][ , ns lp

][][ , , lplp nsns

8

MIMO Model for Pre-FFT BFS

],[ ][

][ ] [)( ][1 1

,,,

nn

nnsnP

p

pL

llpplplp

wAs

wax

1 ..., ,1 , NNNn gg

( DOA matrix)Q L

full column rank with by Assumption

(A2)Q L

))(, ... ),(),(( ,1,2,1,1,1, pLppppppp aaaA

)..., ,,( ), ... ,,( )((2)(1)21

LP aaaAAAA

approximately Gaussian (by Central Limit Theorem)

T)()2()1(TTT2

T1 ])[],...,[],[( ])[, ... ],[],[( ][ nsnsnsnnnn L

Pssss

T,2,1, ])[, ..],.[ ],[( ][ nsnsnsn

pLpppps

] [ ][ ,, lplp nsns

(A2) , for all ;lpmq ,, ),( ),( lpmq . 21 PLLLL Q

9

Remarks:

MIMO Model:

By Assumptions (A1) and (A3), one can observe that for each fixed n is a zero-mean L ×1 random vector with all the L random components being mutually statistically independent with .

LNnnE Iss

1 ]}[][{ H

][ns

SOS based blind beamforming algorithms can be applied, but HOS based blind beamforming algorithms are not applicable because is approximately a Gaussian vector process.][ns

Each column of the mixing matrix A only comprise the energy from a single path. Though beamforming algorithms using SOS can be applied to extract each source , their performance is limited due to lack of path diversity.

)(ia

][)( ns i

],[ ][ ][ nnn wAsx 1 ..., ,1 , NNNn gg

(A1) are i.i.d. QPSK symbol sequences (i.e., for each k is a random variable with uniform probability mass function over the sample space ), and is statistically independent of for .

][, ],...[ ],[ 21 kukuku P ][kup

][kup ][kuq pq

} , { 4/4/ jj ee

(A3) , ... ,2,1, gpLppp N0 . p

10

……

…

……

…

A/D GI Removal

S/PN-point

FFT

GI Removal

S/PN-point

FFT

GI Removal

S/PN-point

FFT

A/D

A/D

P/S

P/S

P/S

][nx ][kx

][kuProposed Blind Post- FTT

KMBFA

Post-FFT BFS [19,20]

[19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43.

[20] D. Bartolome and A. I. Perez-Neira, “MMSE techniques for space diversity receivers in OFDM-based wireless LANs,” IEEE J. Sel. Areas Commun., vol. 21, pp. 151-160, Feb. 2003.

KMBFA: Kurtosis Maximization Beamforming Algorithm

11

MIMO Model for Post-FFT BFS After the processes of the removal of GI, S/P conversion, N-point

FFT operation, and P/S conversion at each receive antenna, the MIMO model for each subcarrier k of the post-FFT BFS can be established as follows:

][

][ ][)( ][

)()(

1 1

/,2,,

k

kkuek

kk

P

p

pL

lp

Nlpkjlplp

wu

w

A

a

x

T21

)( ])[ , ,... ][ ],[( kukuku Pku

pL

l

Nlpkjlplp

kp e

1

/,2,,

)( )( aa

( vector)P 1

), ... ,,( )()(2

)(1

)( kP

kkk aaaA ( matrix)Q P

full column rank and jkjk )()( AA

)(kpa ] [ ,lpp ns

12

Remarks:

Each column of the mixing matrix comprises multipath energy implying a path diversity gain in the estimation of each source can be foreseen.

)(kpa )(kA

][kup

MIMO Model:

][ ][ )()( kk kk wuAx

All the components 's (QPSK signals) of the P × 1 random input vector are zero-mean non-Gaussian mutually statistically independent with .

)(ku][kup

PkkE I })({ H)()( uu

However, a set of N estimators is needed each for one subcarrier k because of for all for all k j. This leads to high computational complexity.

)()( jk AA

T21

)( ])[ , ,... ][ ],[( kukuku Pku ( vector)P 1

pL

l

Nlpkjlplp

kp e

1

/,2,,

)( )( aa

13

Theoretically, the nonblind MMSE beamformer associated with post-FFT BFS is optimum and performs much better than that associated with pre-FFT BFS owing to lack of path diversity for the latter. However, the latter only needs one OFDM pilot block for the estimation of channel matrix, but the former may need many pilots to accurately estimate the channel matrix (and thus needs many OFDM blocks provided over which the channel is static) [19,20,21].

Remarks:

)(kA

[19] Z. Lei and F.P.S. Chin, “Post and pre-FFT beamforming in an OFDM system,” IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May 17-19, 2004, pp. 39-43.

[20] D. Bartolome and A. I. Perez-Neira, “MMSE techniques for space diversity receivers in OFDM-based wireless LANs,” IEEE J. Sel. Areas Commun., vol. 21, pp. 151-160, Feb. 2003.

[21] M. Budsabathon, Y. Hara, and S. Hara, “Optimum beamforming for pre-FFT OFDM adaptive antenna array,” IEEE Trans. Vehicular Technology, vol. 53, pp. 945-955, Jul. 2004.

][kup

Blind algorithms associated with post-FFT BFS using HOS (such as FKMA) are applicable to the estimation of , but in general, they also require many OFDM data blocks with the assumption that the channel is static over these OFDM data blocks, and, again, a set of N estimators is needed.

14

GOALGOAL To design a block-by-block blind beamforming algorithm which is exactly the same for all the subcarriers, and attains “maximum multipath diversity gain” in the meantime.

15

Time Delays

3. Post-FFT Fourier Beamformer by Subcarrier Averaging

,),...,,( T,2,1, pLpppp τ

L × 1 vector ,),...,,( ),...,,( T21TTT2

T1

LP τττττττ

,),...,,( ),...,,( T21TTT2

T1

LP θθθθθθθ

,),...,,( ),...,,( T21TTT2

T1

LP αααα

L × 1 vector

L × 1 vector

where

,),...,,( T,2,1, pLpppp θ

,),...,,( T,2,1, pLpppp αααα

Lp × 1 vector

Lp × 1 vector

Lp × 1 vector

DOAs

Path gains

Notations:

16

Nlpkjplp ekuku

/ , 2 , ][ ][

1 , ,1,0 ],[ ][

][ ][)( ][1 1

/,2,,

Nkkk

kkuekP

p

pL

lp

Nlpkjlplp

wu

w

A

a

x

Alternative form for :][kx

(correlated sources)

T,2,1, ])[, ...],[ ],[( ][ kukukuk

pLppppu

T)()2()1(

TTT2

T1

])[, ... ],[],[(

])[, ... ],[],[( ][

kukuku

kkkkL

Puuuuwhere

zero-mean wide-sense non-Gaussian stationary process by treating k as a time index

ij

ekukuE Nkjipjp

ipjp

allfor ,0

]}[][{/) (2 *

,,,,

][ , ku lp

17

In spite of , , which implies that all the components of become “uncorrelated” by subcarrier averaging.

1

0

)()( ][1

][N

k

ii kuN

ku

Lemma 1. Under the assumptions (A1) and (A3), it can be shown that,

, ,0)][( 2)( iku i P

, ,0])[]([ *)()( ijkuku ji P

where denotes “convergence in probability” as .

P N

NOTENOTE

0 ]}[][{ ,, kukuE ipjp ,0 ][][ P

,, kuku ipjp ij ][ku

ij

Subcarrier average of ][)( ku iT)()2()1(

TTT2

T1

])[, ... ],[],[(

])[, ... ],[],[( ][

kukuku

kkkkL

Puuuu

18

Let be a beamformer (a spatial filter) with the input being . Its output is then

][ ][ ][ ][ TT kwkkke uv gx

where

).,,,( )()2()1(TT Lggg Ag v

][kxv Q 1

2

H

2/ 2/][)(

1maxarg

~kx

a

Qwhere

Post-FFT <Fourier> beamformer:

)~

(1

*FB a

Qv

)(P ~ r

for sufficiently large where .},,2,1{ Lr Q

19

For finite Q, the output of spatial filter is thenFBv

][ ][ ][

][ ][ ][ ][

,1

)()(FB

)()(FB

TFB

TFBFB

kwkugkug

kwkkkeL

rii

iirr

uv gx

)()~

( )( )(H)(

)(TFB

)()(FB

ii

iiig

aaaQ

vwhere

Under the noise-free assumption, and the assumptions (A1) through (A3), the rth column of A can be estimated by input- output cross-correlation (IOCC) as follows:

})()(

)(){(1

][

][][

,1

)()()(FB

)()()(FB

1

2)(FB

P

2FB

FBFB

L

rii

iii

rrrL

i

i

g

ggke

kek

a

ax

a

Channel of interest

Bias

. |,| || )(FB

)(FB rigg ri and

)( ~ r

20

FB

FBH

FB)

~(

)~

( )),

~((

a

aa

a

aa

(magnitude of the normalized cross correlation between and and ).

A “blind performance index” for post-FFT <Fourier> beamformer:

)~

(aFBa 1 )),

~(( 0 FBa a

is an estimate of and is an estimate of implying that the better the estimation accuracy of both and , the larger the value of .

NOTENOTE

~ )(r FBa )( )()( rr a

FBa )),~

(( FBa a)(r

21

][ kup ][k

4. Blind Post-FFT KMBFA by Subcarrier Averaging

Proposed Blind Post-FFT KMBFA

TDEC: Time Delay Estimation and Compensation

KMBFA: Kurtosis Maximization Beamforming Algorithm

at the th stage

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

TDEC

Classification

Deflation

Update through BMRC

][ k u1 ][ k u2 ][ ku },, ...,{ P

No

Yes

][ kup ][ kup

][ k u1 ][ k u2 ][ ku },, ...,{ P

Update by?

p

p][1 kx

P

][(opt) ke][kx

][kx][k

][k

a (opt)

1 P1 p P

22

2

2)(2

2)(4)()( ][ ][2 ][ ]}[{ kukukuku iiii

Kurtosis Maximization Based on Subcarrier Averaging

Let us define the kurtosis of by subcarrier averaging as follows:

][)( ku i

iku ii ,1 ]}[{ )()( P

By Lemma 1, it can be easily shown that

Lemma 1. , ,0)][( 2)( iku i P

=1 =1 0 P

(may depend on i for other modulations such as BPSK signals)

23

The objective function to be maximized for the design of the beamformer : “magnitude of normalized kurtosis” of

2 2][

]}[{ ])[( )(

ke

kekeJJ

v

v ][ke

Maximization

?

][][ T kke xv

24

(A5) if , where are distinct integers.

(A6) if , where are distinct

integers. (A7) and if and

Theorem 1: Under the assumptions (A1) through (A3), (A5) through (A7), and the noise-free assumption, attains maximum, and

where is an unknown constant, and is anunknown integer.

0 )(r } ..., ,2 ,1{ Lr

1 ])[( )( )(optopt

rkeJJ pv

][][ )()(opt kuke rr p

Assumptions:

])[( )( optopt keJJ v

lji , ,

mlji , , ,

3 pL, ) ( ,,, plpipjp ,2

, , ,,,, pmpjplpip 4 pL

, , |, | | | ,,,, ijqplqmqipjp , lm 2 pL .2 qL

25

][)( ke i

][)( ke i

][)1( ke i

No

)1(*1

)1(*1)(

)(

)(

i

ii

dR

dR

x

xv

Compute

at the th iterationi

][kxYesSuper-expo

Algorithm(SEA)

nential

)( )( )1()( ii JJ vv

?

To the thiteration

)1( i

Update through a gradient type optimization algorithm such that

)(iv

)( )( )1()( ii JJ vv

( matrix) Q Q

kkekekkeke

kkeke

iiii

iii

][ ])[ (])[( ][ ][][2

][ ][][

*)1(2)1()1(2)1(

)1(2)1()1(

d

][ ][ H kkxR x

Post-FFT <FKMA> by Subcarrier Averaging

x

x

x x

26

Remarks:

Empirically, we find that the proposed iterative post-FFT <FKMA> also shares the fast convergence and guaranteed convergence advantages of the conventional FKMA.

An initial condition is needed to initialize the proposed post-FFT <FKMA>. For finite N and finite SNR,

)0(v

].[][ ][ )()()( kukuke rrr

The proposed post-FFT <FKMA> may fail to extract the sources when any of the assumptions (A5), (A6) and (A7) are not satisfied, while the probability of the event that violation of any of the three assumptions occurs depends on values of (length of GI) and (number of paths of each user).

gN pL

27

The proposed post-FFT <FKMA> is also applicable to thecase of BPSK symbol sequence.

Three ExtraAssumptions

;2 , ) ( ,,, pjpipmq Ljipq ifand (A8) (A9)

, , ,, pqlpiq (A10)

, , , , ,,,, mjlipqmpjplqiq if 2 pL and ;2 qL

if and

,2

,ip ;0 ,lq0

Therefore, the proposed post-FFT <FKMA> may fail to extract a source with higher probability for the BPSK case than for the QPSK case due to the above three extra assumptions required.

.0 as ,1

0 as ,2

)(

)()(

i

ii

ii ,1 )( (for the QPSK case)×

28

][(opt) ke

At th stage:

Multistage Source Extraction

Assume that and , are the source estimate and the associated channel estimate obtained at stage .

][1)opt( ke

)opt(

1a

1

][11 ][1 ][ )opt()opt( kkk e axx

which basically corresponds to the MIMO signal withall the contributions from removed.

][1 , ],[ ],[ )opt()opt(2

)opt(1 kkk eee

][kx

Cancellation (or Deflation Processing)

...

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

Deflation

][1 kx

][kx

][kx

a (opt)

29

Initial Condition: Post-FFT <Fourier> beamformer

)~

(1

*)0(

aQ

v

where

)(

2H

|][)(1

| 2/ 2/

maxarg ~

r

k

xaQ

][)( ][ T)0()0( kke x v

][

])[]([

2)0(

)0()0(

ke

kek

xa

(Output)

(Channel estimate)

][(opt) ke

...

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

Deflation

][1 kx

][kx

][kx

a (opt)

30

Post-FFT <FKMA>:initialized by

(Output of <FKMA>)

(Channel estimate)

)0(

v

][ ][ ][ )(T kukke rx v

)(

)()(

)(

)(

2*

*)(

)(

][

][][

r

rr

r

r

r

ke

kek

aa

aa

x

][(opt) ke

...

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

Deflation

][1 kx

][kx

][kx

a (opt)

otherwise ), ],[(

)),~

(()),~

(( if ),],[(])[ ],[(

)0()0(

)0(

)opt()opt(

a

aaaa

ke

kekke

aa <FKMA><Fourier>

Hybrid-<Fourier>-<FKMA>:

31

Nrkjrp

r ekukuke /)(2)(

)()opt( ][ ][ ][

where is an unknown constant, ,

Remark:

)(rp

1 1 Lr

2 1 1 1 LLrL

LrLLL P ... 1 2 1

,1

,2

,P

unknown time delay

} ..., ,2 ,1{ Lr

The proposed blind Hybrid-<Fourier>-<FKMA> performs well only with Assumptions (A1) through (A4) required, and it is applicable to both the BPSK case and the QPSK case.

32

][(opt) ke

1 p P1 P

][ kup ][k

Time Delay Estimation and Compensation (TDEC)

Nrkjrp

r ekukuke /)(2)(

)()opt( ][ ][ ][

at the th stage

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

TDEC

Classification

Deflation

Update through BMRC

][ k u1 ][ k u2 ][ ku },, ...,{ P

No

Yes

][ kup ][ kup

][ k u1 ][ k u2 ][ ku },, ...,{ P

Update by?

p

p][1 kx

P

][kx

][kx][k

][k

unknown time delay

a (opt)

33

][ ][ ][ ][ )(/)(2)opt( kkuekek rpNrkj

)}],[({max arg / 2)opt(

0

)( Nkj

gN

r eke

4

)opt(

/ 2)opt(

/2)opt(

][

][ )],[(

ke

ekeeke

NkjNkj

)(

)(

,0

,1r

r

where

The unknown time delay in the extracted source can be estimated also by subcarrier averaging

][)opt( ke)(r

34

1 p P1 P

][ kup ][k

Update byP

Classification and BMRC

at the th stage

Source Extraction

Using Hybrid-

<Fourier>-<FKMA>

TDEC

Classification

Deflation

Update through BMRC

][ k u1 ][ k u2 ][ ku },, ...,{ P

No

Yes

][ kup ][ kup

][ k u1 ][ k u2 ][ ku },, ...,{ P

?

p

p][1 kx

][kx

][kx][k

][k

BMRC: Blind Maximum Ratio Combining

a (opt)

][(opt) ke

]}[1..., ],[],[{ 21 kkk

22][][

][][

kuk

kuk

q

q

q

<FKMA>

35

Correlated Sources

)1(1,p

......

Clus

ter 1

Clus

ter 2

Clus

ter 3

)2(1,p

)3(1,p

Assume that is a cluster of correlated sources impinging on the receiver antenna array where

},1 ], [)({ ,,)(

,)(

, lplppmlp

mlp Lmns a

lpL ,

)( ,

mlp : path gain of each correlated

signal in the cluster: DOA of each correlated signal in the cluster

)( ,

mlp

)( )( ,

,

1

)( , ,

mlp

lpL

m

mlplp

aa

1 , ,1 , ,][ ][)( ][1 1

,,, NNNnnnsn gg

P

p

pL

llpplplp

wax MIMO

signal

models replaced by

,][ ][)( ][1 1

/,2,,

P

p

pL

lp

Nlpjlplp kkuek w

ax 1 , ,1 ,0 Nk

][1 nx

][2 nx

][nxQ

][nsp

),,( )1(1,1,

)1(1, ppp

),,( )2(1,1,

)2(1, ppp

),,( )3(1,1,

)3(1, ppp

)( )(1 ,

3

1

)(1 ,1 ,

mp

m

mpp

aa

36

Remarks:

The proposed post-FFT KMBFA is able to accurately estimate the associated source signal as long as is sufficiently large, implying its robustness to correlated signals. On the other hand, the Capon's MV beamformer is incapable of extracting the associated source regardless of the value of because is no longer a steering vector of a certain DOA required by the Capon's MV beamformer.

Nlpjplp ekuku

/,2, ][ ][

lp ,a

MVv][ ,lpp ns

lp ,a lp ,a

37

5. Simulation Results

Parameters Used:

Consider a four-user ( ) OFDM system with , and .

's: zero-mean, i.i.d. BPSK (or QPSK) signals used with ][kup

: i.i.d. zero-mean Gaussian with . ][nw

1024 N 20 gN 20 Q

QIww 2H ]}[][{ wnnE

.1 }|][{| 2kuE p

}][{

} ][][ {2

2

nEP

nnE

w

wx

Input SNR:

4 P

performance index: average symbol error rate (SER)

38

MultipathChannel

Parameters

)2 ,2 ,2 ,4( ) , , ,( 4321 LLLL

)0,20,30 ,45( ), , ,( οοοο4,13,12,11,1

)35 ,45( ), ( οο2,21,2

)80,60( ), ( οο2,21,2

)10,60( ), ( οο2,41,4

Example 1 (Environment without Correlated Sources):

Fifty sets of time delay parameters were generated randomly. For each set of time delay parameters, fifty sets of data were generated.

pL

l lp12

, 1 ||

} , 2, ,1 , , 2, ,1 , { , pglp LlPpN

),0.4535 0.4837 , 5140.0 , 5442.0( ), , ,( 4,13,12,11,1

)0.6459 ,7634.0( ),( 2,21,2

)0.4961 , 8682.0( ),( 2,31,3

)0.4472 ,8944.0( ),( 2,41,4

39

(a)(a)

QPSK signals

( ) post-FFT KMBFA if post-FFT <FKMA> is used

( ) post-FFT KMBFA if post-FFT <Fourier> Beamformer is used

( ) post-FFT KMBFA if Hybrid-<Fourier>-<FKMA> is used

(b)(b)

BPSK signals

INPUT SNR (dB)

INPUT SNR (dB)

40

(b)(b)

(a)(a)

BPSK signals

QPSK signals

Nonblind MMSE beamformerNonblind MRC beamformerProposed Blind Post-FFT KMBFA Theoretic Capon's MV beamformerActual Capon's MV beamformer

INPUT SNR (dB)

INPUT SNR (dB)

41

, 1 1 4 plp LlPL andfor

)(,mlp 's are all distinct DOAs

lp , 's ( ) were generated randomlygN

Example 2 (Environment with Correlated Sources):

MultipathChannel

Parameters

p

)2 ,2 ,2 ,4( ) , , ,( 4321 LLLL

total number of correlated sources of a cluster

42

(b)(b)

(a)(a)

BPSK signals

QPSK signals

Nonblind MMSE beamformerNonblind MRC beamformerProposed Blind Post-FFT KMBFA Theoretic Capon's MV beamformerActual Capon's MV beamformer

INPUT SNR (dB)

INPUT SNR (dB)

43

6. Conclusions and Future Researches

Under Assumptions (A1) through (A4), we have presented a block-by-block processing algorithm based on subcarrier averaging, namely the post-FFT KMBFA, for the estimation of symbol sequences of all the active users of an OFDM system. It is also a multistage blind beamforming algorithm consisting of source extraction using the proposed blind Hybrid-<Fourier>-<FKMA>, TDEC processing, classification and BMRC processing at each stage.

The proposed blind Hybrid-<Fourier>-<FKMA>, which is the kernel of the proposed post-FFT KMBFA, is also a selection scheme by the performance of two blind beamformers, the post-FFT <FKMA> and the post-FFT <Fourier> beamformer, using subcarrier averages of one OFDM block. Moreover, as the conventional FKMA, the post-FFT <FKMA> supported by Theorem 1 is also a computationally fast source extraction algorithm.

Conclusions

44

Some simulation results were provided to support that the proposed post-FFT KMBFA performs well no matter whether correlated sources are present or not, and its performance is close to the “optimal” nonblind MMSE beamformer associated with the post-FFT BFS, in addition to a performance comparison of some existing beamformers

The results of this chapter have been partly presented at

IEEE ISPACS'05 (Hong Kong, Dec. 13-16, 2005), co-authored by Chun-Hsien Peng, C.-C. Lin, Y.-H. Lin, and C.-Y. Chi,

and have been submitted to IEEE Trans. Signal Processing for publication, co-authored by Chun-Hsien Peng, K.-C. Huang, C.-Y. Chi, and W.-K. Ma.

ISPACS: Intelligent Signal Processing and Communication Systems

45

We considered the uplink of a multiuser OFDM system under the scenario of multiple distinct DOAs for each user, and a flat fading channel for each DOA.

The extension of the proposed blind beamforming algorithm

to more general scenarios is a worthwhile research.

The feasibility of other digital symbols such as 16-QAM isa future research in addition to the estimation of L.

Subcarrier averaging may open a door for efficient post-FFTblind beamforming algorithms of multiuser OFDM systemsusing one OFDM block.

Future Researches

46

Thank you very muchThank you very much

47

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