Wireless Information Transmission System Lab.
Channel Estimation
2012/08/13 蒲俊瑋
Institute of Communications Engineering
National Sun Yat-sen University
Outline
Introduction
Channel Estimation Techniques in OFDM Systems
LS Channel Estimation
Linear Interpolation Channel Estimation
MMSE Channel Estimation
MLS Channel Estimation
Pilot Arrangement in OFDM Systems and DecisionFeedback Channel Estimation
Channel Equalization in Timing Varying Channel
2
Wireless Information Transmission System Lab.
Introduction
Institute of Communications Engineering
National Sun Yat-sen University
Small Scale Fading
Multi-path channel
4
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
Channel Impulse Response
In the absence of noise, the received signal can be expressed as

y (t )  x(t )  h(t ) 
 x( )h(t   )d

where
h(t ) is the channel impulse resonse.
x(t ) is the transmitted signal.
y (t ) is the received signal.
After sampling, the discrete received signal is given by
L 1
y[n]   x[k ]h[n  k ]
k 0
6
The Multi-Path Channel Effect
The multi-path channel effect
EX:
Data = [1 2 3 4]
Channel = [1 1]
y= 1234
1234
= 13574
1 2 3 4
1 2 3
4
7
path1
path2
The Matrix Form of Channel
The wireless stationary channel impulse response is given
by h  [h  0 , h 1 ,..., h  L 1]T , where L is the total number of
resolvable paths.
We assume that each tap of the channel impulse responses h  l  ,
0  l  L  1 , are independently distributed complex Gaussian
2

random variables with zero-mean and variance h l  .
the magnitude
1
0.5
0
1
2
3
4
5
the lth tap of the channel
8
6
7
8
The Matrix Form of Channel
The matrix G is constructed as follows:
 h  0
0

h  0

 h  L  1
h  0
G
h  L  1
h 0
 0


0
h  L  1
 0
1
1
Gx  
0

0
EX:
Data = [1 2 3 4]
Channel = [1 1]
9
0
1
1
0
0 





0 

h  0  
0
0
1
1
0  1  1 
0   2   3 

0 3 5 
   
1   4  7 
The Matrix Form of Channel
The matrix G tail is constructed as follows:
G tail
0







0
0
h  L  1
0
CP
10
h 1 


h  L  1 

0 


0 
The Matrix Form of Channel
Furthermore, a circular convolution matrix G circular can be
obtained:
G circular =G + G tail
 h  0
0

h 0
 h 1


  h  L  1
 0
h  L  1


 0

0
h  L  1
0
h  0
0
h  L  1
11
h 0
h 1 


h  L  1

0 


0 
h  0  
The Transmitted OFDM Signal
After the inverse discrete Fourier transform (IDFT) operation, the
ith transmitted OFDM symbol in time domain can be expressed
by:
xi  F H Xi
where Xi and F H are an N  1 vector and an N  N matrix standing
for modulated symbols and an IDFT matrix.
12
The Received CP-OFDM Symbol
Assuming the synchronization is perfect and CP is adopted, the
received ith OFDM symbol can be expressed as:
ri  Gxi  G tail xi  wi
 G circular xi  wi
 xi  N h  w i
where  N denotes the circular convolution and wi denotes the
additive white Gaussian noise (AWGN) vector in the time domain
2

with zero mean and variance w . We note that there is no ICI and
ISI in each OFDM symbol.
13
The Received CP-OFDM Symbol
After DFT operation, the ith received OFDM symbol in the
frequency domain can be expressed as:
R i  Fri
 F  xi  N h  w i 
 HXi  Wi
where Wi is the AWGN in the frequency domain and H is a N × N
diagonal matrix denoting the channel response in frequency
domain.
14
The Received CP-OFDM Symbol
Any circular matrix can be diagonalized by DFT matrix:
FG circular F H  H
 H  0
0

0
H 1




 0




0

H  N  1 
0
0
The received signal can be expressed as:
Ri  Fri  F  G circular xi  w i   F  G circular FXi  w i 
 FG circular FXi  Fwi  HXi  Wi
15
Wireless Information Transmission System Lab.
Channel Estimation Techniques in OFDM
Systems
Institute of Communications Engineering
National Sun Yat-sen University
Introduction
In general, channel estimation can be cataloged into
three 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.
17
Introduction
The pilot-based estimation can be cataloged into two
kinds 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.
18
System architecture
19
System architecture
1
Input to Time Domain
xn   IDFTX k 
n  0,1,2,..., N  1
2
4
Guard Interval
 xN  n, n   N g , N g  1,...,1
x f n  
n  0,1,..., N  1
 xn,
Guard Removal
3
y f  x f n  hn  wn
5
yn  y f n n  0,1,...,N 1
6
Output
Channel
ICI
Channel
Output to Frequency Domain
Y k   DFTyn 
k  0,1,2,..., N  1
AWGN
Y k   X k H k   I k   W k 
k  0,1,..., N  1
20
7
Channel Estimation Estimated
Channel
Y k 
X e k  
k  0,1,...,N  1
H e k 
System Model
Generally, the pilot symbols are multiplexed into an OFDM
symbol in frequency domain:
 Pk , k   qT , q  0,1,..., Q  1.
Xk  
 Sk , others.
In addition, the power allocation of data and pilot symbols are
given by:

2

k   qT , q  0,1,..., Q  1.
 Pk  Q ,
2

Xk  
 S 2  1     , others.
 k
N Q
ρ: Total Power β: Power Allocation Factor
N: Number of Subcarriers
21
System Model
If system is perfectly synchronized, and the CP is added and
removed appropriately, there is no ISI and inter-carrier
interference (ICI). As a result, the ith received OFDM symbol
after DFT can be expressed as:
Ri  HXi  Wi
where H is a N  N diagonal channel matrix with the kth element
standing for the channel frequency response of the kth sub-carrier
and W is a complex white Gaussian noise vector with covariance
matrix CW  W2 I N .
22
The LS Channel Estimator
M
Define R   i1 R i , where Ri denotes the received pilot signal.
The channel estimator based on the LS method is given by:
Γ 1R
1 1
H LS 
H Γ W
M
M
where Γ denotes a Q  Q diagonal matrix whose diagonal
elements are given by
q,q  Pk , k   qT , q  0,1,..., Q 1.
23
Linear and Second Order Interpolation
Linear Interpolation H e k   H e m L  l 
l
 H p m  1  H p m   H p m 
L
0l  L
Second Order Interpolation [1-2]
H e  k   H e  mL  l 
 c1 H p  m  1  c0 H p  m   c1H p  m  1
   1

,
c1 
2

where c0     1  1 ,

   1
,
c1 

2
24
 l/N
Linear Interpolation
Estimated Channel
(Channel + Noise)’s Upper Bound
Real Channel
Pilot Subcarrier
(Channel + Noise)’s Lower Bound
Data Subcarrier
Frequency Domain
25
The MMSE Channel Estimator
The MMSE channel estimator is given by [3, 4]

HMMSE  RHH p RH p H p   (ΓΓ )
2
W
H 1

1
 HLS
where R HH p represents the cross-correlation between all the
subcarriers and the pilot subcarriers, and RH p H p represents the
autocorrelation matrix between the pilot subcarriers.
26
The Low-Rank MMSE Channel Estimator
A low-rank MMSE channel estimator is given by [1]:
HMMSE  UΔL U H  HLS
where Δ L is a diagonal matrix with entries

  l     l   c SNR  , l  0,1, 2,..., L  1,
l  
0,
l  L,..., N  1,


Note that   l  can be viewed as the attenuation of the lth tap of
the channel impulse response:
2
  l   hl
and c can be expressed as [3]:
2
2
c  E  xk  E 1 xk 

 

27
The Realization of MMSE Channel
Estimator
In practice, the channel power of the lth transform coefficient   l 
can be obtained from the results of the LS channel estimation.
First, the estimate of the channel impulse response h can be
acquired by taking the IDFT of the channel frequency response
obtained from the LS channel estimate:
h  FH HLS
2
2
2
2

ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
And then the λ  0 , 1 ,..., L 1   h  h0 , h1 ,..., hL 1 


is obtained.
28
The Modified LS (MLS) Channel
Estimator
The modified LS (MLS) channel estimator is given by
HMLS  FΔLF H  HLS
where ΔL is a diagonal matrix. The entries   l  of ΔL are:
1, l  0,1,..., L  1.
 l   
0, l  L,..., N  1.
The MLS channel estimator can be considered as a low-pass filter,
which is also termed as DFT-based scheme.
29
Wireless Information Transmission System Lab.
Pilot Arrangement in OFDM Systems [2]
and Decision-Feedback Channel Estimation
Institute of Communications Engineering
National Sun Yat-sen University
Introduction
The channel estimation can be performed by either
inserting 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).
Freq.
Comb Type
Block Type
Time
31
LTE Reference Symbol Arrangement
LTE pilot symbol arrangement
1
2
32
Band Edge
Virtual
subcarriers
DC
Virtual
subcarriers
f
Active Band
…
…
…
…
Pilot
Data
33
Decision-Feedback Channel Estimation
When the channel is slow fading, the channel estimation inside
the block can be updated using the decision feedback equalizer
at each sub-carrier.
Decision Directed Channel Estimation (DDCE) is one of the
earliest methods studied for OFDM, mainly because of its
popularity in legacy systems. In the earlier studies, DDCE was
applied mostly in training based systems.
34
Decision-Feedback Channel Estimation
The main idea behind DDCE is to use the channel estimation of a
previous 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.
OFDM Systems in Time-Variant Multipath
Channels
Institute of Communications Engineering
National Sun Yat-sen University
Introduction
Orthogonal frequency-division multiplexing (OFDM) is generally
known 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 interval and can mitigate
frequency selectivity of a multipath channel using a simple onetap equalizer.
37
Introduction
In an OFDM system, although the degree of channel variation
over 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 velocity increase.
38
Block Diagram for An OFDM System
The time-domain transmitted signal is given by
N 1
xn   X me j 2 nm N , 0  n  N -1
m0
The time-domain received signal is then given by
L 1
yn   hn xnl  wn
l 0
Binary Data
Data
Symbol
Modulator
Xm
IDFT
Data
Symbol
Demodulator
Add CP
Channel
Hˆ m
Binary Data
xn
Xm
Channel
Estimator
Equalizer
39
Ym
wn
DFT
+
yn Remove
CP
ICI Analysis
The frequency-domain received signal is then given by
N 1 L 1
Ym   X k H l( m  k ) e  j 2 lk
k 0 l 0
N
 Wm
N 1 L 1
 L 1 0  j 2 lm N 
( m  k )  j 2 lk
  H l e
X

X
H
e
 m  k l
k  m l 0
 l 0

  m X m   m  Wm , 0  m  N  1
N
 Wm
where Wm denotes the frequency-domain noise and Hl( mk )
represents the FFT of timing-variant channel, i.e.,
N 1
1
H l( mk )   hn,l e j ( mk ) N
N n 0
40
 *
ICI Analysis
In the general case where the multipath channel cannot be
regarded as time-invariant during a block period, the received
signal can be expressed in vector form as
Y = HX + W
T
T
T
where Y = Y0 , Y1 ,..., YN 1  , X   X 0 , X 1 ,..., X N 1  , W  W0 ,W1 ,...,WN 1 
and
 a0,0
 a
1,0
H


 aN 1,0
a0,1
a1,1
aN 1,1
( mk )
( m  k )  j k
a

H

H
e
with m,k
0
1
N
a0, N 1 
a1, N 1 


aN 1, N 1 
 ...  H L( m1k ) e j k ( L1) N ,
0  m, k  N  1
41
Channel Equalization
After performing channel equalization, the equalized signal can be
expressed as
X  H1Y
 X + H 1W
where
 a0,0
 a
1,0
H 1  


 aN 1,0
a0,1
a1,1
aN 1,1
a0, N 1 
a1, N 1 


aN 1, N 1 
1
The inverse operation increases the system computation
complexity.
42
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 frequencyselectivity 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 the
frequency-domain equalizer is increased.
43
References
[1] 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.
[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.
44
Appendix A. LMMSE estimator
Y  XH  W
1
Hˆ LMMSE  RHY  RYY  Y
RHY  E  HH H X H   RHH X H
H

RYY  E  XH  W  XH  W    E  XHH H X H    2 I N


 XRHH X H   2 I N
45
Appendix A. LMMSE estimator
1
H
H
2
ˆ
H LMMSE  RHH X  XRHH X   I N  Y
 RHH X
 RHH
 RHH
H
 X   X 
H
1
H

1 1
 XR
X  IN 
H
HH
 X  XR X  X    X 
R   X X    X  Y
1
H
H
1
HH
2
H
1 1
1
HH
46
1
2
 IN  X
2
1
H
 X  

1 1

1 1
X 
X 
1
Y
1
Y