高永安 老師 學生：吳林達 報告日期： 2006/7/28

ON THE NUMBER OF PILOTS FOR OFDM SYSTEM IN MULTIPATH F

ADING CHANNELSWei Zhang, Xiang-Gen Xia, P. C. Ching and Wing-Kin Ma,

ICASSP2004

高永安老師

學生：吳林達報告日期： 2006/7/28

Outline

• Introduction

• Channel estimation for OFDM

• Analytical BER of OFDM System

• Process of searching the optimal number of pilots

• Simulation environment

• Simulation result

Introduction

• Orthogonal frequency division multiplexing (OFDM) system has been demonstrated to be effective in a frequency-selective Multi-path fading environment.

• To compensate the deleterious effects of the fading channels, pilot symbol assisted modulation (PSAM) technique has been used to obtain the channel estimate.

• The number of pilot symbols is a tradeoff between channel estimation accuracy and bandwidth efficiency.

• We propose a novel method which is based on the derivation and analysis of BER for a given channel SNR.

• Through examination of the analytical BER curves with different pilot spacing, the smallest number of pilots can be easily found.

在給定的 SNR下與期望的 BER下 ,求所需之最少 pilot數

Channel estimation for OFDM• Pilot symbol assisted OFDM channel estimation metho

d is viewed as follows• Through multi-path fading channel, the received OFD

M signal can be written as:

1

: impulse

:

:

: -

:

:

pN

k k kk

k k k k

p

k

k

k

y t h t x t n t

h t c r t t channel response of the kth path

x t transmitted signal

N number of transmission paths

n t zero mean AWGN

c normalize amplitude of kth path

propagation delay

:

var k

of kth path

r t independent stationary complex zero mean Gaussian process with

unit iance for the kth path

• After removing the guard interval and performing DFT on y(t), we obtain:

• The pilot sub-channel can be estimated using the least square method:

2

1

, , , ,

, :

, :

, :

, : int

:

kp

s

j m nNT

kk

s

Y m n H m n X m n N m n

H m n h n e frequency response of the channel

X m n transmitted symbol

Y m n received symbol

N m n AWGN at the mth subchannel and the nth OFDM symbol erval

T duration of o

ne OFDM symbol

, , / ,

:

:

:

p p pH j n Y j n X j n

p pilot position

j index of pilot sub channel

n nth OFDM symbol

• The data sub-channel information can be estimated by taking interpolation between the pilot sub-channel estimates.

• In linear interpolation, the data sub-channel estimate is given by:

• Thus, the transmitted data symbol at the ith data sub-channel and the nth OFDM symbol can be estimated by

, 1 , 1,

:

:

:

:

: tan

d p pl l

H i n H j n H j nL L

d data position

i index of data sub channel

n nth OFDM symbol

L pilot spacing

l dis ce between jth pilot sub channel and the ith data sub channel

, , / ,d d dX i n Y i n H i n

Analytical BER of OFDM System

• We assume the channel is a wide sense stationary uncorrelated scattering (WSSUS) frequency-selective Rayleigh fading channel with a Jakes power spectrum density (PSD).

11 2 1

' 23 4 2

:

:

i

i

Y HX N

H x ix re

H H W x ix r e

N the noise component at a data symbol location

W function of noise samples at pilot locations

Because W is a function of noise samples at pilot locations while

1 2 3 4

1 2

22 2 21 2 1 1 2

,

, , , - -

R

, . .

1,

2

N is the noise component at a data symbol location W is independent of N

x x x x all gaussian

Invoking general ayleigh fading process assumption

x x i i d

E x E x E H E x x

22 2 2

3 4 2 3 4

1 3 2 4 1

1 4 2 3 2

0

1, 0

2

1Re

2

1Im

2

E x E x E H E x x

E x x E x x E H H

E x x E x x E H H

Frequency selective fading case

• The average bit error rate for an OFDM-QPSK system is given by

1 2 1 2

2 2

1 2 1 2

21

2

1 2 1 2

1 1 12 212 2 21 1

1 12 22 2

: /

, , , ,

b

b b

bn

n b

P E

average SNR bit

P E

必須知道 才可算

1 21 2

1 2 1 2

= , =

• To evaluate (1) and (2), we need to know the channel’s frequency correlation function (FCF). Assuming a 6-tap typical urban (TU) channel model ,the resulting channel’s FCF in one OFDM symbol is given by

2

21 2 2+ = (1), (2)

2 2

dd dE HE H H

j

62

1

2(3)

:

:

kk

k s

j m nE H m H n c e

T

m index of the mth frequency sub channel

n index of the nth frequency sub channel

Frequency-selective and time-selective fading case

• Considering the Doppler spread induced by the mobile receiver, the resultant time-varying channel is approximated as a time-invariant channel and the approximation error is considered as Gaussian noise

• The variance of the approximation error can be denoted by

• the total mean noise power equals to

20 0

1

0

1= 0 2 1

: 0

: max

: int

N

mk

m

J J f T kN

J x th order first class Bessel function

f Doppler shift

T sampling erval

2 2 2n = +

:AWGN

Multi-path correlated fading case

• In wide-band radio-wave propagation, the correlation coefficients between transmission paths may have values ranging from 0 to larger than 0.8

• Thus the FCF depends on the actual frequencies under consideration, rather than just the frequency spacing.

• Here, the channel’s FCF can be expressed as

• If Φ is a diagonal matrix that indicates the multiple paths are uncorrelated, (4) will lead to (3).

(4)

:

:

:

:

m n

m

n

k

E H m H n F F

F mth row of fourier coefficient matrix

F nth row of fourier coefficient matrix

diagonal matrix with the element c

correlation matrix of the six paths

Process of searching the optimal number of pilots

• Substitute an initial pilot spacing, L0, and other OFDM system and channel parameters into the analytical BER formula. So the BER in terms of SNR at pilot spacing L0 can be obtained and represented by Pb(L0).

• Increase the pilot spacing by one, i.e. L1 = L0 + 1, and obtain Pb(L1) similarly. Repeat the process iteratively and a set of Pb(L) (L = L0, L1, · · · , LP ) can be obtained.

• Plot these BER curves in one figure, and compare them with the desired BER at the given SNR.

• The curve having the maximal pilot spacing yet satisfying the requirement is identified and from which the minimal number of pilot symbols can be found.

Simulation environment

• An un-coded PSAM-OFDM system• OFDM symbol duration=224μs• Sampling interval = 7/64μs.• Bandwidth = 8MHz.• Center carrier frequency =470MHz.• Length of cyclic prefix = 1/16• Number of carriers=2048.• 6-tap TU channel model is considered• the uncorrelated multi-path fading are simulated with Jakes PS

D.• Modulation of the sub-carrier is assumed to be QPSK.

Simulation result

Example

• A mobile receiver is moving with 100 km/h in an uncorrelated multi-path fading channel

• how many pilots should at least be used in a PSAM-OFDM system in order to achieve a BER lower than at 30 dB SNR?

310

Example

Example