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
2004/4/29 WLAN Group3
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
2004/4/29 WLAN Group4
After pilot estimation
Pilot estimated
Data symbols Pilot subcarriers.*
phase tracking
Corrected symbols
TrackingChannel compensation
y(n)
2004/4/29 WLAN Group5
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.
2004/4/29 WLAN Group6
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
2004/4/29 WLAN Group7
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
2004/4/29 WLAN Group8
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
2004/4/29 WLAN Group9
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
2004/4/29 WLAN Group10
L-extension Method
ADPLL (All Digital Phase-Locked Loop)
2004/4/29 WLAN Group11
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.
2004/4/29 WLAN Group12
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.
2004/4/29 WLAN Group13
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
2004/4/29 WLAN Group14
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∆
2004/4/29 WLAN Group15
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)
2004/4/29 WLAN Group16
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
2004/4/29 WLAN Group17
L-extension MethodThe range of
then, the observation range is
)1()(21
)1()(21
~ )(
^
,
^^is
−+≤∆≤
−+−
−∠−∠∑=
−
LTNNf
LTNN
XX
gg
pilotppLp ππ
2004/4/29 WLAN Group18
L-extension MethodIf L is too long, the first uncompensated symbols may suffer large phase rotation and degrade the performance.
2004/4/29 WLAN Group19
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.
2004/4/29 WLAN Group20
Simulation in AWGN
2004/4/29 WLAN Group21
Simulation in AWGN
2020
2004/4/29 WLAN Group22
Simulation in AWGN
2004/4/29 WLAN Group23
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.
2004/4/29 WLAN Group24
Nondata-aided carrier phase tracking
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.
2004/4/29 WLAN Group25
Nondata-aided carrier phase tracking
Here Xn,k doesn’t necessarily 1 or -1.
⎥⎦
⎤⎢⎣
⎡∠=
∠=Φ
∑
∑−
=
−
=
∧
∆
1
0
2,
22
1
0
*^
,
^
,
||||
])([
N
kknk
nfj
N
kknkknn
XHe
XHR
π
2004/4/29 WLAN Group26
Implementation
2004/4/29 WLAN Group27
Table
2004/4/29 WLAN Group28
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)
……
……
……
….
2004/4/29 WLAN Group29
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”