residual frequency offset tracking - welcome. wits lab

Residual frequency offset tracking

Po-Lin Chen (陳柏霖)

2004/4/29 WLAN Group2

OutlinePosition in receiverAfter pilot estimationCarrier phase tracking [1] [2]

Data-aided carrier phase trackingAveraging processL-extension Method

Nondata-aided carrier phase tracking

ImplementationReferences


Position in receiver

Timing,frequencyand phaseestimation

Frequencycompensation FFT

Phaseestimation

Phasecompensation

De-mapping

Removecyclic

extension

Channelcompensation

Channelestimation

De-interleave Decoding Received

data

Time &

phase

tracking

Time &phase

compensation

X(n) after A/D,AGC


After pilot estimation

Pilot estimated

Data symbols Pilot subcarriers.*

phase tracking

Corrected symbols

TrackingChannel compensation

y(n)


Carrier phase trackingFrequency estimation is not a perfect process, so there is always some residual frequency error. ICI here shouldn’t be a problem, but constellation rotation may be a main one.


Data-aided carrier phase tracking802.11a include four predefined subcarrier, [1 1 1 -1] or [-1 -1 -1 1] , among the transmitted data.

These subcarriers are referred to as pilot subcarriers.

The received pilot subcarrier

where n is the OFDM symbol index, k is subcarrier index, Hk is channel frequency response, Pn,k is the predefined subcarrier and is the residual frequency error.

∆= nfjknkkn ePHR π2

,,

∆f


Averaging processAssuming the estimate Hk for channel is perfectly accurate, we can get the estimator

If not,

⎥⎦

⎤⎢⎣

⎡∠=

∠=Φ

∑

∑

=

=

∧

∆

pilotskk

nfj

pilotskknkknn

He

PHR

22

*,,

])([

π

])([

])([

*,

2,

*,,

∑

∑

=

∧

=

∧∧

∆∠=

∠=Φ

pilotskknk

nfjknk

pilotskknkknn

PHePH

PHR

π


Averaging processFor one OFDM symbol in AWGN, Sk denotes the k-thpilot signal, and W denotes the AWGN noise whose distribution is ~N (0, ). Then, the average of the four pilot signals is

(S1+W + S2+W + S3+W + S4+W) / 4 = (S1 + S2 + S3 + S4 + W’) / 4= Save+Wave

where W’ and Wave are the new AWGN, whose distributions are ~N (0, 4 ) and ~N (0, ), respectively.

2nσ

4/2nσ2

nσ


Averaging processWe use the equation

So, we get )var( )var( 2 xbbxa =+

n

n

ave

W

W

W

2

2

41

4161

)'var(161

)'41 var(

)var(

σ

σ

=

⋅=

=

=

Average makes variance smaller


L-extension Method

ADPLL (All Digital Phase-Locked Loop)


L-extension MethodConsider L = 3, pilots are extracted from symbol 1 and held in the register. When symbol 3 arrives, error phase detector will calculate the error phases between the same pilots of the symbols and take the average of them.

The result will be fed into the ADPLL to generate an proper frequency to compensate the frequency offset of symbols 4, 5 and 6.

Now, symbols 4 and 6 replace the position of symbols 1 and 3, respectively.


L-extension MethodThis time the frequency offset estimation will be added to original frequency offset, i.e. accumulation.

The accumulated offset is used to compensate symbols 7, 8 and 9.

This process continues until the end of a packet.


L-extension MethodFirst, let us discuss a portion of one OFDM symbol produced after IFFT in the baseband of transmitter

where Xk is a complex value in the k-th carrier, N is the number of OFDM subcarriers, s(n) is a complex value in the n-th output carrier.

1,......,1,0 1)(1

0

/2 −== ∑−

=

NneXN

nsN

k

Nnkjk

π


L-extension MethodAssume that the wireless environment brings in a phase offset and frequency offset (after coarse and fine frequency compensation) and assume that we have the perfect symbol boundary with sampling clock T = Ts/N then

where Ts is symbol duration and is a normalized frequency offset of the carrier spacing.

θ

∑

∑−

=

∆+

−

=

+∆

+∆

=

=

=

1

0

/)(2

1

0

/2)2(

)2(

1

1

)(

N

k

Nkknjk

j

N

k

Nnkjk

fnTj

fnTjn

eXN

e

eXN

e

nser

πθ

πθπ

θπ

sfffNTk /∆=∆=∆

f∆


L-extension MethodAfter FFT in the baseband of receiver,

)(1 1

0

)(21

,0

1

0

/2

/21

0

^

∑ ∑∑

∑−

=

∆+−−

≠=

−

=

∆

−−

=

+=

=

N

n

NkpkpjN

pnnk

N

n

Nknjk

j

NnpjN

nnp

eXeXeN

erX

ππθ

π

ICI (intercarrier interference)


L-extension MethodIf the offset is small enough, the second term can be ignored. It means

and the (L-1)th symbol before it is

Hence, the estimated frequency offset is given by

∑−

=

∆=1

0

/2^ 1 N

n

Nknjk

jp eXe

NX πθ

∑−

=

∆+∆++−− =

1

0

/2]))(1(2[,

^ 1 N

n

Nknjk

fTNNLjpL eXe

NX g πθπ

))(())(1(2

1,

^^^

∑=

−∠−∠+−

=∆pilotp

pLp

g

XXNNLT

fπ


L-extension MethodThe range of

then, the observation range is

)1()(21

)1()(21

~ )(

^

,

^^is

−+≤∆≤

−+−

−∠−∠∑=

−

LTNNf

LTNN

XX

gg

pilotppLp ππ


L-extension MethodIf L is too long, the first uncompensated symbols may suffer large phase rotation and degrade the performance.


L-extension MethodWhere t1 and t2 denote the durations of different L.

We can observe that large t2 cause a large phase rotation such that the constellation points exceed the demodulation range and errors take place.

Therefore, it is of significance to discover an optimal L.


Simulation in AWGN


Simulation in AWGN

2020


Simulation in AWGN


Simulation in AWGNFrom above graphs, we know that the higher the mapping scheme is used, the smaller the optimal L is.

At the initial state, the system isn’t able to acquire the best performance by optimizing L, hence a trade-off should be made between L and performance according to different modulation schemes.



The phase error can be estimated without pilot symbols.

The phase error resulting from frequency offset is identical for subcarriers.

The angle can be estimated without any knowledge ofthe data by performing hard decision on the received data symbol after they are corrected for the channel effect.



Here Xn,k doesn’t necessarily 1 or -1.

⎥⎦

⎤⎢⎣

⎡∠=

∠=Φ

∑

∑−

=

−

=

∧

∆

1

0

2,

22

1

0

*^

,

^

,

||||

])([

N

kknk

nfj

N

kknkknn

XHe

XHR

π


Implementation


Table


IdeaReplace arctan by tangent to simplify the challenge, i.e.,error_pilot_sum_Q[input_length:0] >= error_pilot_sum_Ierror_pilot_sum_Q[input_length:0] >= (error_pilot_sum_I>>1)error_pilot_sum_Q[input_length:0] >= (error_pilot_sum_I>>2)error_pilot_sum_Q[input_length:0] >= (error_pilot_sum_I>>3)

……

……

……

….


References[1] John Terry and Juha Heiskala “OFDM Wireless LAN_ A Theoretical and Practical Guide.”

[2]賴祐徵 “Research on Residual Carrier Frequency Offset Tracking in OFDM Wireless LAN Systems”

residual frequency offset tracking - welcome. wits lab

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