Residual frequency offset tracking

Residual frequency offset tracking
Po-Lin Chen ( 陳 柏 霖 )

Position in receiver
After pilot estimation
Carrier phase tracking [1] [2]
Outline
Data-aided carrier phase tracking
Averaging process
L-extension Method
Nondata-aided carrier phase tracking
Implementation
References
2004/4/29 2
WLAN Group

Position in receiver
Timing,
frequency
and phase
estimation
Channel
estimation
Time &
Phase
phase
estimation
tracking
Frequency
compensation
Remove
cyclic
extension
FFT
Channel
compensation
Time &
Phase
compensation
phase
compensation
Demapping
Deinterleave
Decoding
Received
data
X(n) after A/D,AGC
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WLAN Group

After pilot estimation
Channel compensation
Pilot estimated
Tracking
Data symbols
.*
Pilot subcarriers
y(n)
phase
tracking
Corrected
symbols
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WLAN Group

Carrier phase tracking
Frequency 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.
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WLAN Group

Data-aided carrier phase tracking
802.11a 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
R
n , k
= H
k
P
n , k
e
j 2πnf
∆
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.
f ∆
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Averaging process
Assuming the estimate Hk for channel is perfectly
accurate, we can get the estimator
If not,
∧
Φ
∧
Φ
n
n
= ∠[
⎡
= ∠⎢e
⎣
= ∠[
∑
R
k = pilots
j2πnf
∑
R
k = pilots
∆
*
n, k
( H
k
Pn
, k
) ]
n,
k
∑
k = pilots
∧
( H
k
H
P
k
n,
k
2
)
⎤
⎥
⎦
*
]
= ∠[
∑
H
k = pilots
k
P
n,
k
e
j2πnf
∆
∧
( H
k
P
n,
k
)
*
]
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WLAN Group

Averaging process
For one OFDM symbol in AWGN, Sk denotes the k-th
pilot signal, and W denotes the AWGN noise whose
2
distribution is ~N (0, σ n ). 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
2
2
distributions are ~N (0, 4 σ n ) and ~N (0, σ n
/ 4),
respectively.
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We use the equation
So, we get
Averaging process
var( a + bx)
= b
=
var( W
1
var(
4
var( x)
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2
ave
1
=
16
1
= ⋅4σ
16
1 2
= σ n
4
)
W ')
var( W ')
2
n
Average makes
variance smaller

L-extension Method
ADPLL (All Digital Phase-Locked Loop)
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L-extension Method
Consider 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.
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L-extension Method
This 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.
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WLAN Group

L-extension Method
First, let us discuss a portion of one OFDM symbol
produced after IFFT in the baseband of transmitter
s ( n )
N
1
−
= ∑
N
1
k = 0
X
k
e
j 2 πnk
/
N
n
=
0,1,......,
N
−
1
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.
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L-extension Method
Assume that the wireless environment brings in a phase
offset θ and frequency offset ∆f (after coarse and fine
frequency compensation) and assume that we have the
perfect symbol boundary with sampling clock T = Ts/N
then
j(2π∆fnT
+ θ )
r
n
=
=
=
e
e
e
j(2π∆fnT
+ θ )
jθ
1
N
N −1
∑
k = 0
s(
n)
where Ts is symbol duration and ∆k = NT∆f
= ∆f
/ f s
is a
normalized frequency offset of the carrier spacing.
N −1
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X
1
N
k
e
∑
k = 0
X
k
e
j2πnk
/ N
j2πn(
k +∆k
)/ N

L-extension Method
After FFT in the baseband of receiver,
^
X
p
=
=
N −1
∑
n=
0
n
− j2πnp
/ N
N −1
N −1
N −1
1 jθ
2 /
( ∑
j πn∆k
N
e X + ∑∑
ke
n=
0
n=
0 n=
0, n≠
p
N
r
e
X
k
e
2πp(
k − p+∆k
)
j
N
ICI (intercarrier interference)
)
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WLAN Group

L-extension Method
If the offset is small enough, the second term can be
ignored. It means
X
^ N − 1
1 jθ
= ∑ p
e
N n=
0
X
k
e
j2πn∆k
/ N
and the (L-1)th symbol before it is
^ N
− + + ∆ +
− 1
1 j[2π
( L 1)( N N g ) T f θ ]
∑ − L,
p = e
N
n=
0
X
j2πn∆k
/ N
Hence, the estimated frequency offset is given by
∆
^
f
1
=
2πT
( L −1)(
N + N
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WLAN Group
g
)
(
∑
p=
pilot
^
( ∠ X
X
p
k
e
− ∠
^
X
−L,
p
))

L-extension Method
The range of
−L,
p
then, the observation range is
^
∑(
∠ X
p=
pilot
p
− ∠
^
X
)
is
−π
~ π
2( N
+
−1
N ) T ( L
g
−1)
≤ ∆
^
f
≤
2( N
+
1
N ) T ( L
g
−1)
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WLAN Group

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

L-extension Method
Where 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.
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WLAN Group

Simulation in AWGN
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Simulation in AWGN
20
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Simulation in AWGN
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Simulation in AWGN
From 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.
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WLAN Group

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 of
the data by performing hard decision on the received data
symbol after they are corrected for the channel effect.
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WLAN Group

Nondata-aided carrier phase tracking
Here Xn,k doesn’t necessarily 1 or -1.
∧
Φ
n
= ∠[
N −1
∑
k = 0
⎡
= ∠⎢e
⎣
R
j2πnf
n,
k
∆
N −1
∑
k = 0
^
( H
|
k
H
X
k
^
n,
k
|
2
|
)
*
]
X
n,
k
|
2
⎤
⎥
⎦
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WLAN Group

Implementation
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WLAN Group

Table
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WLAN Group

Idea
Replace arctan by tangent to simplify the challenge, i.e.,
error_pilot_sum_Q[input_length:0] >= error_pilot_sum_I
error_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)
………………….
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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”
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