CE in OFDM Sytems 2012

8/10/2019 CE in OFDM Sytems 2012

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Wireless Information Transmission System Lab.

National Sun Yat-sen UniversityInstitute of Communications Engineering

Channel Estimation

2012/08/13


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2

Outline

IntroductionChannel Estimation Techniques in OFDM Systems

LS Channel Estimation

Linear Interpolation Channel Estimation

MMSE Channel EstimationMLS Channel Estimation

Pilot Arrangement in OFDM Systems and Decision-

Feedback Channel Estimation

Channel Equalization in Timing Varying Channel


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Introduction


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4

Small Scale Fading

Multi-path channel


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Channel Impulse Response

A mobile radio channel may be modeled as a linear

filter with a time varying impulse response, where the

time variation is due to receiver motion in space.

The filtering nature of the channel is caused by the

summation of amplitudes and delays of the multiple

arriving waves at any instant of time.

5


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6

Channel Impulse Response

In the absence of noise, the received signal can be expressed as

( ) ( ) ( ) ( ) ( )

where

( ) is the channel impulse resonse.

( ) is the transmitted signal.

( ) i

y t x t h t x h t d

h t

x t

y t

1

0

s the received signal.

After sampling, the discrete received signal is given by

[ ] [ ] [ ]L

k

y n x k h n k


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The Multi-Path Channel Effect

The multi-path channel effect

1 2 3 4

1 2 3 4

EX:

Data = [1 2 3 4]

Channel = [1 1]

y = 1 2 3 41 2 3 4

= 1 3 5 7 4

path1path2


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The Matrix Form of Channel

The wireless stationary channel impulse response is givenby , where Lis the total number of

resolvable paths.

We assume that each tap of the channel impulse responses ,, are independently distributed complex Gaussian

random variables with zero-mean and variance .

[ 0 , 1 ,..., 1 ]Th h h L h

h l0 1l L

2

h l

1 2 3 4 5 6 7 80

0.5

1

the lth tap of the channel

themagnitude


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The matrix Gis constructed as follows:

0 0 0

0

1 0

0 1 0

0

0 0 1 0

h

h

h L h

h L h

h L h

G

EX:

Data = [1 2 3 4]

Channel = [1 1]

1 0 0 0 1 1

1 1 0 0 2 3

0 1 1 0 3 5

0 0 1 1 4 7

Gx


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The matrix is constructed as follows:tailG

tail

0 0 1 1

0

1

0

0 0

h L h

h L

G

CP


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Furthermore, a circular convolution matrix can beobtained:

circularG

circular tail=

0 0 0 1 1

1 0 0

1

1 0 0

0 1 00

0 0 1 0

h h L h

h h

h L

h L h

h L h

h L h

G G + G


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The Transmitted OFDM Signal

After the inverse discrete Fourier transform (IDFT) operation, theith transmitted OFDM symbol in time domain can be expressed

by:

where and are an vector and an matrix standing

for modulated symbols and an IDFT matrix.

1NiX

H

i ix F X

HF N N


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The Received CP-OFDM Symbol

Assuming the synchronization is perfect and CP is adopted, thereceived ith OFDM symbol can be expressed as:

where denotes the circular convolution and denotes the

additive white Gaussian noise (AWGN) vector in the time domain

with zero mean and variance . We note that there is no ICI and

ISI in each OFDM symbol.

tail

circular

i i i i

i i

i N i

r Gx G x w

G x w

x h w

N iw

2

w


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After DFT operation, the ith received OFDM symbol in thefrequency domaincan be expressed as:

where is the AWGN in the frequency domain and H is aNN

diagonal matrix denoting the channel response in frequency

domain.

i i

i N i

i i

R Fr

F x h w

HX W

iW


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Any circular matrix can be diagonalized by DFT matrix:

The received signal can be expressed as:

circular

0 0 0

0 10

0 0 1

H

H

H

H N

FG F H

i circular circular

circular

i i i i i

i i i i

R Fr F G x w F G FX w

FG FX Fw HX W


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Channel Estimation Techniques in OFDM

Systems


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Introduction

In general, channel estimation can be cataloged intothree kinds of estimation schemes:

1. Blind 2. Superimposed 3. Pilot-based

The first two structures can obtain some bandwidth

merit, but the computational complexity is usually not

acceptable in practical realization.

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Introduction

The pilot-based estimation can be cataloged into twokinds of approaches:

1. The parameters are deterministic but unknown

constant, such as maximum likelihood (ML) estimator

and least square (LS) estimator.

2. The parameters are random variables, such as

minimum mean square error (MMSE) estimator and

maximum a posteriori (MAP) estimator.

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System architecture


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System architecture

1,...,2,1,0

Nn

kXIDFTnx

1,...,1,0,

1,...,1,,

Nnnx

NNnnNx

nx gg

f

nwnhnxy ff

1,...,1,0 Nnnyny f

1,...,2,1,0

Nk

nyDFTkY

1,...,1,0

Nk

kWkIkHkXkY

1,...,1,0 NkkH

kYkX

e

e

1

76

54

32

Input to Time Domain

Guard Interval Channel

Guard Removal Output to Frequency Domain

Output Channel EstimationICI AWGNChannel EstimatedChannel


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System Model

Generally, the pilot symbols are multiplexed into an OFDMsymbol in frequency domain:

In addition, the power allocation of data and pilot symbols aregiven by:

, , 0,1,..., 1.

, others.

k

k

k

P k qT q QX

S

2

2

2

, , 0,1,..., 1.

1 , others.

k

k

k

P k qT q QQ

X

SN Q

: Total Power : Power Allocation Factor

N: Number of Subcarriers


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22

System Model

If system is perfectly synchronized, and the CP is added andremoved appropriately, there is no ISI and inter-carrier

interference (ICI). As a result, the ith received OFDM symbol

after DFT can be expressed as:

where is a diagonal channel matrix with the kth element

standing for the channel frequency response of the kth sub-carrier

and Wis a complex white Gaussian noise vector with covariance

matrix .

i i i R X WH

H N N

2

W W NC I


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The LS Channel Estimator

Define , where denotes the received pilot signal.

The channel estimator based on the LS method is given by:

where denotes a diagonal matrix whose diagonal

elements are given by

11

LS

1

M M

RH H W

1

M

ii R R

Q Q

, , , 0,1,..., 1.q q kP k qT q Q

iR


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24

Linear and Second Order Interpolation

Linear Interpolation

Second Order Interpolation [1-2]

Ll

mHL

lmHmH

lmLHkH

ppp

ee

0

1

1 0 1

1

0

1

1,

2

1 1 ,

1,

2

1 1

/

e e

p p pH m c H m c H m

l N

H k H mL l

c

c

where c

c


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Linear Interpolation

25

Real Channel

Frequency Domain

Estimated Channel

(Channel + Noise)s Lower Bound

(Channel + Noise)s Upper Bound

Pilot Subcarrier

Data Subcarrier


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26

The MMSE Channel Estimator

The MMSE channel estimator is given by [3, 4]

where represents the cross-correlation between all the

subcarriers and the pilot subcarriers, and represents theautocorrelation matrix between the pilot subcarriers.

1

2 1MMSE LSW ( )p p p

H

HH H H

H R R H

pHHR

p pH HR


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The Low-Rank MMSE Channel Estimator

A low-rank MMSE channel estimator is given by [1]:

where is a diagonal matrix with entries

Note that can be viewed as the attenuation of the lth tap of

the channel impulse response:

and ccan be expressed as [3]:

2ll h

l

2 21k kc E x E x

MMSE LSH

L H U U H

SNR , 0,1,2,..., 1,0, ,..., 1,

l

l l c l L

l L N

L

Th R li ti f MMSE Ch l


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The Realization of MMSE ChannelEstimator

In practice, the channel power of the lth transform coefficientcan be obtained from the results of the LS channel estimation.

First, the estimate of the channel impulse response can be

acquired by taking the IDFT of the channel frequency responseobtained from the LS channel estimate:

And then theis obtained.

l

h

LSHh F H

2 2 2 2

0 1 1 0 1 1

, ,..., , ,...,L Lh h h

h

Th M difi d LS (MLS) Ch l


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The Modified LS (MLS) ChannelEstimator

The modified LS (MLS) channel estimator is given by

where is a diagonal matrix. The entries of are:

The MLS channel estimator can be considered as a low-pass filter,

which is also termed as DFT-based scheme.

MLS LSH

L H F F H

L l L

1, 0,1,..., 1.

0, ,..., 1.

l L

l l L N

Wir l I f r ti Tr i i S t L b


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Pilot Arrangement in OFDM Systems [2]

and Decision-Feedback Channel Estimation


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Introduction

The channel estimation can be performed by eitherinserting pilot tones into all of the subcarriers of OFDM

symbols with a specific period (block type) or inserting

pilot tones into each OFDM symbol (comb type).

Time

Freq. Comb Type Block Type


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LTE Reference Symbol Arrangement

LTE pilot symbol arrangement

32

1 2


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Band Edge

33

f

Virtual

subcarriers

Active Band

Virtual

subcarriersDC

Pilot Data


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34

Decision-Feedback Channel Estimation

When the channel is slow fading, the channel estimation insidethe block can be updated using the decision feedback equalizer

at each sub-carrier.

Decision Directed Channel Estimation (DDCE) is one of theearliest methods studied for OFDM, mainly because of its

popularity in legacy systems. In the earlier studies, DDCE was

applied mostly in training based systems.


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Decision-Feedback Channel Estimation

The main idea behind DDCE is to use the channel estimation of aprevious OFDM symbol for the data detection of the current

estimation, and thereafter using the newly detected data for the

estimation of the current channel.

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

35

Wireless Information Transmission System Lab


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OFDM Systems in Time-Variant Multipath

Channels


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Introduction

Orthogonal frequency-division multiplexing (OFDM) is generallyknown as an effective technique for high bit rate applications such

as DAB, DVB and WiMAX, since it can prevent intersymbol

interference (ISI)by inserting a guard intervaland can mitigate

frequency selectivity of a multipath channel using a simple one-

tap equalizer.


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Introduction

In an OFDM system, although the degree of channel variationover the sampling period becomes smaller as data rates increase,

the time variation of a fading channel over an OFDM block

period causes a loss of subchannel orthogonality, resulting in an

error floor that increases with the Doppler frequency.

The performance degradation due to the interchannel

interference (ICI)becomes significant as the carrier frequency,

block size, and vehicle velocityincrease.


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39

Block Diagram for An OFDM System

The time-domain transmitted signal is given by

The time-domain received signal is then given by

12

0

, 0 -1N

j nm N

n m

m

x X e n N

DFT

Data

Symbol

ModulatorChannel

IDFT Add CPBinary Data

Data

Symbol

Demodulator

Binary Data Remove

CP

+Channel

Estimator

Equalizer mY ny

nxmX

nwmH

mX

1

0

L

n n n l n

l

y h x w


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ICI Analysis

The frequency-domain received signal is then given by

where denotes the frequency-domain noise and

represents the FFT of timing-variant channel, i.e.,

1 1( ) 2

0 0

1 1 10 2 ( ) 2

0 0

, 0 1 *

N Lm k j lk N

m k l m

k l

L N Lj lm N m k j lk N

l m k l ml k m l

m m m m

Y X H e W

H e X X H e W

X W m N

mW

( )m k

lH

1( ) ( )

,

0

1 Nm k j m k N l n l

n

H h eN


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41

ICI Analysis

In the general case where the multipath channel cannot beregarded as time-invariant during a block period, the received

signal can be expressed in vector form as

where , ,and

with

0 1 1, ,...,

T

NX X X

X

Y = HX + W

0 1 1, ,...,

T

N

Y Y Y

Y =

0 1 1, ,...,

T

N

W W W

W

0,0 0,1 0, 1

1,0 1,1 1, 1

1,0 1,1 1, 1

N

N

N N N N

a a a

a a a

a a a

H

( ) ( ) ( ) ( 1)

, 0 1 1... ,

0 , 1

m k m k j k N m k j k L N

m k La H H e H e

m k N


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Channel Equalization

After performing channel equalization, the equalized signal can beexpressed as

where

The inverse operation increases the system computation

complexity.

1

1

X H Y

X + H W

1

0,0 0,1 0, 1

1,0 1,1 1, 11

1,0 1,1 1, 1

N

N

N N N N

a a a

a a a

a a a

H


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Conclusions

Assuming that the channel is stationary over the period of an

OFDM symbol, the conventional frequency-domain equalizer

with one-tap in an OFDM system compensates the frequency-

selectivity of a multipath fading channel.

The one-tap frequency-domain equalizer cannot eliminate ICI for

the case of a time-varying channel.

In time-varying channel, the computation complexity of thefrequency-domain equalizer is increased.


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References

[1] Sinem Coleri, Mustafa Ergen, Anuj Puri, and Ahmad Bahai, Channel Estimation

Techniques Based on Pilot Arrangement in OFDM Systems,IEEE transactions onBroadcasting, Vol. 48, No. 3, September 2002.

[2] G.-S. Liu and C.-H. Wei, A new variable fractional sample delay filter with

nonlinear interpolation,IEEE Trans. Circuits and Systems-11: Anulog andDigiral

Signal Processing, vol. 39, no. 2, Feb. 1992.

[3] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Borjesson,OFDM Channel Estimation by Singular Value Decomposition, IEEE

Transactions on Communications, vol. 46, no. 7, pp. 931-939, Jul. 1998.

[4] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory,

New Jersey: Prentice Hall, 1993, pp. 380-382.

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Appendix A. LMMSE estimator

45

1

2

2

LMMSE HY YY

H H H

HY HH

H H HYY N

H

HH N

Y XH W

H R R Y

R E HH X R X

R E XH W XH W E XHH X I

XR X I


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Appendix A. LMMSE estimator

1

2

1 11 1 1 1 12

11 11 1 12

11 12

H HLMMSE HH HH N

H H H H

HH HH N

H H H

HH HH N

H

HH HH

H R X XR X I Y

R X X X XR X I X X Y

R X XR X X X I X X Y

R R X X X Y

CE in OFDM Sytems 2012

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