channel estimation - 140.117.160.140140.117.160.140/commeduimp/pdfdownload/9222/channel...

Channel Estimation

Cheng-Che Chan（詹政哲）

2004/4/29 WLAN Group2

OutlineIntroduction to IEEE 802.11a WLAN Packet Structure

Short Training SymbolLong Training Symbol

Introduction to IEEE 802.11a WLAN Channel Estimation

Channel Estimation TechniquesPilot ArrangementChannel Estimation @ Block-TypeChannel Estimation @ Comb-Type PilotInterpolation @ Comb-Type

IEEE 802.11a channel estimation methodsFrequency-Domain Adaptive FilterAverage in Time DomainDecision-Aided AIT


Introduction to IEEE 802.11a WLAN Packet StructureShort Training SymbolLong Training Symbol





Packet for WLAN 802.11a


Packet for WLAN 802.11aThe PLCP ( Physical Layer Convergence Protocol ) preamble field is used for synchronization. It consists of 10 short training symbols ( t1 to t10 ) and 2 long training symbols ( T1 & T2 ).The PLCP preamble is followed by the SIGNAL field and DATA. The total training length is 16us.Composed of 10 repetitions of a “shorting training sequence”.

used for AGC (Automatic Gain Control) convergence, diversity selection, timing acquisition,& coarse frequency acquisition in the receiver.

Two repetitions of a “long training sequence”.used for channel estimation & fine frequency acquisition in the receiver.


Short training symbolA short OFDM training symbol consists of 12 subcarriers, which are modulated by the elements of the sequence S, given by

The multiplication by a factor of is in order to normalizethe average power of the resulting OFDM symbol, which utilizes 12 out of 52 subcarriers.The fact that only spectral lines of S-26:26 with indices that are a multiple of 4 have nonzero amplitude results in a periodicity of 0.8 us.

26,26 13 6 {0, 0, 1 , 0, 0, 0, 1 , 0, 0, 0, 1 , 0, 0, 0, 1 , 0, 0, 0,

1 , 0, 0, 0, 1 , 0, 0, 0, 0, 0, 0, 0, 1 , 0, 0, 0, 1 ,

0, 0, 0, 1 , 0, 0, 0, 1 , 0, 0, 0, 1 , 0, 0, 0, 1 , 0, 0}

S j j j j

j j j j

j j j j

− = × + − − + − −

− + + − − − −

+ + + +

-24 -20 -16

-8 -4 0 4 8

12 16 20 24

613

-12


Short training symbol The signal shall be generated according to the following equation:

∑−=

∆=2

2)2exp()()(

ST

ST

N

NkFkTSHORTSHORT tkjStwtr π


Long training symbolA long training symbol consists of 53 subcarriers (including a zero value at dc), which are modulated by the elements of the sequence L, given by

A long OFDM training symbol shall be generated according to the following equation:

where

2

22

( ) ( ) exp( 2 ( ))ST

ST

N

LONG TLONG k F GIk N

r t w t L j k t Tπ=−

= ∆ −∑

2 1.6GIT sµ=

}1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1{26,26

−−−−−−−−−−−−−

−−−−−−−−=−L


Long training symbolTwo period of the long sequence are transmitted for improved channel estimation accuracy, yielding

The sections of short repetitions and long repetitions shall be concatenated to form the preamble

sTLONG µ82.326.1 =×+=

)()()( SHORTLONGSHORTPREAMBLE Ttrtrtr −+=


Pilot subcarriersIn each OFDM symbol, four of the subcarriers are dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase noise.

These pilot signals shall be put in subcarriers –21, –7, 7 and 21. The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines.

d5 P7d4 P-21 d17 P-7d18 d23 DC d24d29 d30 d42 P21 d43 d47d0

-26 -21 -7 0 7 21 26


Channel Estimation for WLANChannel estimation can be performed with frequency domain and time domain.

The long training symbols in the WLAN preamble facilitate an easy and efficient estimate of the channel frequency and impulseresponse for all the subcarriers.

The contents of the two long training symbols are identical, so averaging them can be used to improve to quality of the channel estimate.

DFT is a linear operation, hence the average can be calculated before the DFT (time domain), or after the DFT (frequency domain).


Frequency Domain Approach for Channel Estimation

After the DFT processing, the received training symbols Y1, j and Y2, j are a product of the training symbols Xj and the channel Hjplus additive noise Ni, j.

Thus the channel estimate can be calculated as, ,i j j j i jY H X N= +

( )( )

( )( )

1, 2,

1, 2,

2

1, 2,

1, 2,

1ˆ21 2

1 2

1 2

j j j j

j j j j j j j

j j j j j

j j j j

H Y Y X

H X N H X N X

H X N N X

H N N X

∗

∗

∗

∗

= +

= + + +

= + +

= + +


Frequency Domain Approach for Channel Estimation

The noise samples N1, j and N2, j are statistically independent, thus the variance of their sum divided by two is a half of the variance of the individual noise samples.


Time Domain Approach for Channel Estimation

The channel estimation can also be performed using the time domain approach, before DFT processing of the training symbols.In this case, the channel impulse response, instead of the channel frequency response, is estimated.The received time domain signal during the two long training symbols is

1, 1,j j jy h x n= ∗ +



The time domain convolution can be expressed as a matrix vector multiplication. The circular convolution matrix is formed from the training data as

The parameter M defines the maximum length of the impulse response that can be estimated, and X is in general a rectangular matrix.

1 64 64 2

2 1 64 3

64 63 64 1

M

M

M

x x xx x x

x x x

− +

− +

− +

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

L

L

M M M

L



The channel impulse response vector is

Then the convolution is expressed as

The channel impulse response estimate can then be formed by

where X+ denotes Moore-Penrose generalized inverse of X

[ ]1 1T

M Mh h h−=h L

( )( )

( )

( )

1, 2,

1 2

1 2

1 2

1h X21 X Xh Xh2

1 X Xh X2

1 h X2

j jy y

n n

n n

n n

+

+

+ +

+

= +

= + + +

= + +

= + +

ijijji nnxhy +=+∗= Xh,,



The channel frequency response estimate is then formed by calculating the DFT of the impulse response estimate

{ }ˆˆ hH DFT=


Analysis of the Time Domain and Frequency Domain Approaches for Channel Estimation

The advantage of the time domain approach is improved performance, when the maximum length of the impulse response can be limited to a number significantly less than the number ofCP.The rationale is that the frequency domain estimator has to simultaneously estimate all the subcarriers, whereas the time domain estimator needs to estimate only the taps of the impulse response.For example, in the IEEE 802.11a system the number of subcarriers is 52, and the maximum length of the channel can be assumed to be less than the cyclic prefix length of 16 samples.The drawback of the time domain method is that additional computations are required (requires 64．M multiplications).


Pilot ArrangementComb Type

Some sub-carriers are reserved for pilots for each symbol

Block TypeAll sub-carriers reserved for pilots with a specific period


Channel Estimation @Block-TypeIf ISI is eliminated by the guard interval, we can write

whereWXHWXFhY +=+=

kNnjnk

N

NNN

NN

NNN

TN

NT

N

TN

N

eN

W

WW

WW

F

GaussianmeanzeroiidWWWW

hDFTHHHH

YYYY

XXXdiagX

π2

)1)(1(0)1(

)1(000

110

110

110

110

1.

..

.

.

.

.

,,,],...,,[

)(],...,,[

],...,,[

},...,,{

...

.

.

...

−

−−−

−

−

−

−

−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

=

==

=

=


Channel Estimation @Block-TypeIf the time domain channel vector h is Gaussian and uncorrelated with the channel noise W, the frequency domain MMSE estimate of h is given by:

where

The LS estimate is represented by :

YRFRFRH YYhYMMSEMMSE1−==

NHH

hhyy

HHhhhY

IXFXFRYYER

XFRhYER2][

][

σ+==

==

YXH LS1−=


Channel Estimation @Block-TypeWhen the channel is slow fading, the channel estimation inside the block can be updated using the decision feedback equalizer at each sub-carrier.

For fast fading, the comb-type estimation performs much better.

Decision Feedback Equalizer

1,...,1,0)(

)()( −== NkkH

kYkXe

e

)(mapper signaldemapper signal)( kXkX puree →→→

1,...,1,0)(

)()( −== NkkX

kYkHpure

e


Channel Estimation @ Comb-Type Pilot

The Np pilot signals uniformly inserted in X(k) according to the following equation:

where L= # of Carriers / Np and xp(m) is the mth pilot carrier value.We define {Hp(k) k=0,1,…,Np} as the frequency response of the channel at pilot sub-carriersYp(k) and Xp(k) are output and input at the kth pilot sub-carrier respectively.

⎩⎨⎧

−=

==

+=

1,...,1 . inf

0),()()(

Lldata

lmxlmLXkX

p


Channel Estimation @ Comb-Type Pilot

The estimate of the channel at pilot sub-carriers based on LS estimation is given by:

Yp(k) and Xp(k) are output and input at the kth pilot sub-carrier respectively.Since LS estimate is susceptible to noise and ICI, MMSE is proposed while compromising complexity.

1,...,1,0 −== pp

pe Nk

XY

H


Interpolation @ Comb-TypeLinear Interpolation

Second Order Interpolation

( ) ( )( ) ( )( ) ( )

LlmH

LlmHmH

lmLHkH

ppp

ee

<≤

+−+=

+=

01

( ) ( )( ) ( ) ( )( )

( )( )( )

⎪⎪

⎩

⎪⎪

⎨

⎧=

+=

=

+++−

+=

−

+−−=

−=

−

Nl

mHcmHcmH

c

c

c

where

clmLHkH

ppp

ee

/

11

,2

11

,110

,2

11

101

ααα

αα

αα


The Frequency-Domain Adaptive Filter (FDAF)

Frequency-domain adaptive filter is a frequency domain implementation of the LMS algorithm.The block diagram is



The processing of the frequency-domain adaptive filter for one subcarrier is illustrated as

f(k,l) Decision

+

x(k,l)dy(k,l) x(k,l)e

-

e(k,l) +



The mathematical representation is as follower：

where is the received kth subcarrier signal in the lth

symbol, is the inverse of the channel frequency response of the kth subcarrier signal in the lth symbol and is a one-tap filter coefficient. is the signal point after decision.

is the step size, and is assumed the same for each subcarrierin all symbols.

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

),(),(),(),(*),(),(

lkylkylkelkflkf

lkxlkxlkelkflkylkx

ed

e

⋅⋅+=+

−==

α

),( lky

),(1),(

lkHlkf =

dlkx ),(

α



Since the initial value of the filter coefficient uses the estimated value in the pre-symbol, the method using frequency-domain adaptive is suitable for slowly time-variant channel.


References[1]John Terry, and Juha Heiskala, “OFDM Wireless LANs: Practical Guide,” Sams Publishing, 2002.[2]Steven M. Kay, “Fundamentals of Statistical Signal Processing Estimation Theory,” Prentice Hall, 1993.[3]Sinem Coleri, Mustafa Ergen, Anuj Puri, and Ahmad Bahai, “Channel Estimation Techniques Based on Pilot Arrangement in OFDM Systems,” IEEE transactions on Broadcasting, Vol. 48, No. 3, September 2002.[4]Yun-Yi Shih, “Channel Estimation Techniques for the IEEE 802.11a Wireless Local Area Network,”Institute of Communications Engineering National TsingHua University, July 2002.


Appendix

The Approach of Average in Time Domain (AIT)



For the 802.11a, there are two long training sequences in the preamble, which are the same, and assume the channel does not change during the interval, the received signal of two long training sequences can be represented as:

and is the ith sample of the complex Gaussian noise with zero mean and

1,0=+⊗= lnhxy lll

),,,( ,1,1,0 lNlll nnnn −= L lin ,

2,, })(Im{})(Re{ nlili nVarnVar σ==



Since there is no ISI in the OFDM symbol, the received signal after FFT can be expressed as:

is complex Gaussian random variables.and are Gaussian random variables with:

51,,01,0,, K==+⋅= klZHXY lkkklk

)(}{ , llk nFFTZ =}Re{ ,lkZ }Im{ ,lkZ

NZVarZVar

ZEZE

nlklk

lklk

2,,

,,

}][Im{}][Re{

0}][Im{}][Re{

σ==

==



It can be revealed that and are uncorrelated, which are both Gaussian random variables, we can conclude that they are independent.

So,

Thus, Zk,l’s are uncorrelated Gaussian random variables and they are indendent.

0}]Im{}[Re{})Im{},(Re{ ,,,, == lklklklk ZZEZZCov

}Re{ ,lkZ }Im{ ,lkZ

⎩⎨⎧

≠=

=21

212

*,, 0

2][

21 iiforiifor

nnE nkiki

σ

)(][21),( 122

2*

,,,, 2121kk

nZZEZZCov n

lklklklk −== δσ



The two long train sequences can be represented as:

where

And

Therefore, Yk,0 and Yk,1 can be treated as they are independent.

1,1,

0,0,

kkkk

kkkk

ZHXYZHXY

+⋅=

+⋅=

kkkk HXYEYE ⋅== ][][ 1,0,

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

02121),(

*1,0,

*1,1,0,0,1,0,

=

=

−−=

kk

kkkkkk

ZZE

YEYYEYEYYCov



According to independent characteristic, the maximum likelihood function of Hk can be represented as:

( )⎥⎦

⎤⎢⎣

⎡−+−−=

=

=

21,

20,22

1,1,

0,0,

1,0,

21exp

21

})Im{};(Im{})Re{};(Re{})Im{};(Im{})Re{};(Re{

);,()(

kkkkkknn

kkkk

kkkk

kkkk

XHYXHYNN

HYfHYfHYfHYf

HYYfHL

σσπ



The estimation of Hk is

k

kk

kkkkkkH

kHk

XYY

XHYXHY

HLH

k

k

2

minarg

)(maxarg

1,0,

21,

20,

+=

−+−=

=∧


The Approach of Decision-Aided AIT (DA-AIT)

DA-AIT estimator follows the concept of using average techniques to reduce the interference of AWGN. First, the estimation result of AIT estimator is taken as the initial channel response, and the received signals are compensated by the initial channel response.After the compensated received signals are detected, they could be used as the pilots. Using these “virtual” pilots, the channel response of the next symbol is obtained.Then, the more channel responses can be used to do average, and the estimated result will be closer to the real channel response.



The estimated channel response form symbol i is denoted as .

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−=

⎯⎯⎯ →⎯

⎥⎦

⎤⎢⎣

⎡−

−=

∧∧∧

∧

∧

)1(

)1(

)1(

)1(

)0(

)0(

)1()1(

)1()1(

)0()0(

NX

NY

X

Y

X

YH

XX

NHNY

HY

HYX

i

i

i

i

i

ii

iDecision

i

AIT

i

AIT

i

AIT

ii

L

L

∧

iH



Assuming that the channel is fixed for one packet and there are M symbols chosen to estimate channel response, the decision-aided AIT estimation result could be presented as:

121

+++++

=

∧∧∧∧

−

∧

MHHHHH AITM

AITDAL



AITH∧

AITH∧

AITH∧

Y1 Decision Y1

Y2 Decision Y2

YMDecisionYM

+…AITH

∧

1x

2x

Mx

1

∧

x

2

∧

x

Mx∧

1

∧

H

2

∧

H

MH∧

11+M

AITDAH −

∧



The DA-AIT estimator averages several channel responses in the time domain.

According to the large number law, the noise variance will be reduced as the increasing of averaging number, and the more precise channel estimation result will be obtained.

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