Optimum Pilot Pattern for Channel Estimation in OFDM Systems

高永安老師

學生：吳林達報告日期： 2006/8/4

Ji-Woong Choi, Member, IEEE, and Yong-Hwan Lee, Member, IEEE

,” Optimum Pilot Pattern for Channel Estimation in OFDM Systems,

” IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005

Outline

• Introduction

• System Model

• Optimal Pilot Pattern

• Simulation Condition

• Simulation Result

• Conclusion

Introduction

• The performance of channel estimation in orthogonal frequency division multiplexing (OFDM) systems significantly depends on the pilot signal, which is usually scattered in time and frequency domains.

• The authors optimally design the pilot pattern so as to minimize the mean squared error (MSE) of the channel estimate with the use of a general interpolator.

• The Channel Impulse Response(CIR) can be estimated using predetermined pilot symbols in real time.

• In the OFDM system, pilot symbols are scattered in the time and frequency domain to track time-variant and frequency-selective channel characteristics.

• This paper considers the design of the optimal 2-D pilot pattern so as to minimize the MSE of the estimated CIR assuming the use of a conventional interpolator.

• The authors first analytically verify the optimum pilot shape that was suggested in [5] by computer simulation.

• For a given pilot density, the authors derive the optimal pilot spacing in terms of the moment of the Doppler spectrum and power delay profile representing the channel characteristics in the time and frequency domain, respectively.

• 本篇主要是以計算 (估測的 CIR-實際的 CIR)的MSE,在給定的 pilot density下 ,決定最佳的 pilot pattern

System Model

• The authors consider a wireless channel whose impulse response is represented as:

1

0

,

:

:

:

L

l ll

l

l

h t h t

L number of paths

delay at time t of the lth path

h t complex valued CIR at time t of the lth path

• The authors assume that is zero-mean Gaussian, statistically independent of each other paths and has the same normalized correlation function for all l.

• Then, the time-domain correlation function of the lth path CIR can be represented as:

lh t

tr t

2

2 :

l l l l t

l

r t E h t t h t r t

the average power of the lth path

• The frequency response of the CIR at time t can be represented as:

• Assuming a normalized average path power (i.e. ), the correlation function of the frequency response can be represented as:

2, , j fH t f h t e d

12

0

1L

ll

1

22

0

, , ,

l

H

t f

Lj f

f ll

r t f E H t t f f H t f

r t r f

where r f e

• Denoting the sub-carrier spacing by Δfc, the correlation function can be represented as:

• The corresponding spectrum of the channel correlation can be represented as:

,

;

H t f

t t s f f c

r n k r n r k

r n r nT r k r k f

1 2

1 2

1 2

1 2

1 2

, ,

j n kH H

n k

j n j kt f

n k

H H

S r n k e

r n e r k e

S S

• Assuming ideal synchronization at the receiver, the received symbol of the kth sub-carrier at the nth symbol time can be represented by:

2

, , , ,

, :

in

, : z

Y n k X n k H n k Z n k

H n k frequency response of the kth sub carrier

the kth OFDM symbol

Z n k background noise approximated as AWGN with

Optimal Pilot Pattern

• A regular pilot arrangement can be represented using two basis vectors as shown in Fig. 2, where the vectors are represented in Cartesian coordinates: the time axis by the abscissa and the frequency axis by the ordinate.

1 1 1 2 2 2, ; ,T T

v x y v x y

• Since the pilot density D is inversely proportional to the pilot spacing, it can be defined as the inverse of the area of a parallelogram formed by and , i.e.,1v 2v

1 1

1 2 2 1

1 21 2

1 2

det

D V x y x y

x xV v v

y y

• The CIR is first estimated from the pilot signal as:

• CIR can be estimated by interpolating the received pilot symbols as:

• where is equal to for the pilot symbol and zero otherwise, and denotes the coefficient of the interpolator.

'

,,

,

, ,

p p

p p

p p

p p p p

Y n kH n k

X n k

H n k Z n k

, , , (1)sp q

H n k H n p k q p q

,sH n k

,H n k

,p q

• When there is no interference, the CIR can be perfectly estimated using an ideal interpolator:

• is the coefficient of an ideal 2-D brick-wall-type non-causal filter represented as [14]:

, , , (2)s idp q

H n k H n p k q p q

,id p q

1 2

1 2

1 2

sin sin

, sin sinid

p qx y p q

p q c cx yp q

x y

• Using(1),(2):

, , , ,

, , ,

, , , ,

, ,

s sp q

s idp q

s idp q

sp q

H n k H n p k q Z n p k q p q

H n k H n p k q p q

H n k H n p k q p q p q

Z n p k q p q

desired CIR self-distortion noise due to the useof a non-ideal interpolator

the interference due to the background noise plus interference

• The MSE of the CIR estimate can be represented as:

22

2

1 2 1 2 1 22

22

1 2 1 22

2 2

2

2

, ,

1 , ,

2

,2

:

: int

s

e

H e

Z

S I

S

I

E H n k H n k

S W d d

DW d d

MSE due to self distortion

MSE due to erference

1 2

, , ,

, 2 . ,

e idp q p q p q

p q D F T W

• is a sampled version of [15]:

• the authors can represent the 2-D FT of as

1 2,sHS 1 2,HS

1 2

2

1 2 1 2

1 12

1 20 0 1 2 1 2

,

2 2 2 ,

sH

x y

Hn k

S x y

xS n k n

x y x y

e ,p q

1 2 1 2 1 21 21 2

1 2

, , ,

, , e

W x y andx yW

W otherwise

1 2

, , ,

, 2 . ,

e idp q p q p q

p q D F T W

• of a linear interpolator can be represented as [14]:

• For example, Fig. 3 depicts and when a linear interpolator is employed, where denote the maximum Doppler frequency and maximum delay of the channel, respectively.

1 2,W

2

1 1 2 2

1 21 21 2

sin sin1 2 2

,sin sin

2 2

x y

Wx y

1 2,sHS 2

1 2,eW

max df and

• Using the symmetrical property, it can easily be shown that is minimized when This means that the optimum pilot pattern has a diamond shape that was suggested by the simulation results in [5].

2S 1

2 2

xx

• Neglecting the MSE in the out-of-pass-band spectrum and using a Taylor series approximation of

then:

2

1 2,eW

2 1

2 1

1 2

2 2 2 4 41 2 ,0 1 2 ,1 1 ,2 2 1 22

2 2 4 41 2 ,0 1 2 ,1 1 ,2 2 1 22 2

1 2

2 2 4 4_ _ _ _

,0 1 2 ,1 1 ,2 22

1 2

_

1

1,

2

1

2

1

:

S

y x

S H IF IF IF

y x

H H IF IF IF

IF IF IF

n

th

S c c c d d

S S c c c d dx y

c c cx y

n o

_

2

2

,0 ,1 ,2 1 2

:

, , : ,

n

th

IF IF IF e

rder moment of the Doppler spectrum

n order moment of the power delay profile

c c c coefficients of approximated polynomial of W

• Assume that are continuous parameters for ease of analytical design. The optimal spacing

can then be uniquely determined by solving:

1 2 x and y

1 2x and y

1 1 2 2

2 2

1 2

| 0 | 0S S

x x y yand

x y

• Assuming , the optimum spacing is determined as:

2 21 2, 1x y 1 2x and y

11

821

11

822

11 2

4_

24_

1

H

H

H

x D

y D

where D x y

Little Conclusion

• mobility decreases ( decreases):

increase and decrease

• goes to zero

goes to infinity and goes to zero• decrease

increase and decrease

4_

1

1x

2y

4_

1

4_

2

2y

1x

2y

1x

Simulation Condition

Simulation Result

Conclusion

• This paper analytically determined the optimum pilot pattern for channel estimation in an OFDM system with the use of conventional interpolators.

• The authors have verified that the optimum pilot pattern has a diamond shape.

• The spacing of the pilot symbols is optimally determined in terms of the pilot density, Doppler spectrum, and power delay profile.