ce in ofdm sytems 2012
TRANSCRIPT
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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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Wireless Information Transmission System Lab.
National Sun Yat-sen UniversityInstitute of Communications Engineering
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.
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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 Form of Channel
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 Form of Channel
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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The Matrix Form of Channel
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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The Received CP-OFDM Symbol
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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The Received CP-OFDM Symbol
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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Wireless Information Transmission System Lab.
National Sun Yat-sen UniversityInstitute of Communications Engineering
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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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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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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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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Wireless Information Transmission System Lab.
National Sun Yat-sen UniversityInstitute of Communications Engineering
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
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1 2
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Band Edge
33
f
Virtual
subcarriers
Active Band
Virtual
subcarriersDC
Pilot Data
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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.
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Wireless Information Transmission System Lab
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Wireless Information Transmission System Lab.
National Sun Yat-sen UniversityInstitute of Communications Engineering
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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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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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