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OFDM Synchronization
Speaker:
1
Wireless Access Tech. Lab.
Outline
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
2
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Block
Xi
P/S
S/P IFFT x i
channel
Yi
' S/P
P/S FFT x i
3
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
An OFDM symbol structure
Cyclic prefix
interval
Symbol interval
CP xi,0,xi,1,……………………………,xi,N-1
copy
[
x i ≡ xi , 0 , xi ,1 , , xi ,( N −1) ] 4
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Signal Model
2π
1 − j N nk
W kn
N = e k = 0 ,1,2 , ,N-1
N
⎡ WN ⋅0
0
WN ⋅1
0
WN ⋅ 2
0
WN ⋅( N −1) ⎤
0
⎢ ⎥
WN⋅0
1
WN⋅1
1
WN⋅2
1
WN⋅( N −1) ⎥
1
W=⎢
⎢ ⎥
⎢ ( N −1)⋅0 ( N −1)⋅( N −1) ⎥
⎢WN
⎣ WN N −1)⋅1 WN N −1)⋅2
( (
WN ⎥ N ×N
⎦
N −1 2π
1 −j nk
X i ,k =
N
∑x i ,n e N
- - > X i = Wx i . X i is frequency domain signal.
n =0 FFT
N −1 2π
1 j nk
xi ,n =
N
∑X i ,k e
N
- - > x i = W * X i . x i is time domain signal.
k =0
IFFT
5
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
6
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
What is Synchronization?
Symbol synchronization
Symbol Synchronization refers to the task of finding the
precise moment of when individual OFDM symbols start
and end.
Frequency synchronization
Let the transmitted signal be sn, then the complex
baseband model of the passband signal yn is
yn = sn e j2 πftx nT s
Then the received complex baseband signal rn is
rn = sn e j2πftx nTs e -j2πfrx nTs
= sn e j2πf∆ nTs
where f and f is the transmitter and receiver carrier frequency
tx rx
7
respectively, and f∆ is frequency offset.
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Received Signal
Transmitted signal
Cyclic prefix
interval Symbol interval
xi ,( N −CP ) , , xi ,( N −1) , xi , 0 , xi ,1 , , xi ,( N −1)
Received signal
2π
jφ + j εn
xi' ,k = xi ,k ⋅ e N
+ wi ,k
where φ is initial phase,
ε is frequency offset,
xi ,k represents kth sample in ith OFDM symbol,
8
wi ,k is AWGN term.
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
CP ith OFDM symbol CP
Corrert sampling, φ = 0, ε = 0, and wi ,k = 0 :
⎡ WN ⋅0
0
WN ⋅1
0
WN ⋅ 2
0
WN ⋅( N −1) ⎤ ⎡ xi , 0 ⎤
0
⎡ X i ,0 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ X ⎥
⎢ WN⋅01
WN⋅11
WN⋅21
WN⋅( N −1) ⎥ ⎢ xi ,1 ⎥
1
⎢ i ,1 ⎥
Wx = ⎢ WN ⋅0
' 2
WN ⋅1
2
WN ⋅2
2
WN ⋅( N −1) ⎥ ⋅ ⎢ xi , 2 ⎥ = X i = ⎢ X i , 2 ⎥
2
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢W ( N −1)⋅0 W ( N −1)⋅1 W ( N −1)⋅2 WN N −1)⋅( N −1) ⎥ ⎢ xi ,( N −1) ⎥
( ⎢ X i ,( N −1) ⎥
⎣ N N N ⎦ ⎣ ⎦ ⎣ ⎦
9
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
CP ith OFDM symbol CP
Error sampling, φ = 0, ε = 0, wi ,k = 0, a is an integer number :
⎡ WN ⋅0
0
WN ⋅1
0
WN ⋅ 2
0
WN ⋅( N −1) ⎤ ⎡ xi ,( N − a ) ⎤
0
⎢ 1⋅0 ⎥ ⎢ ⎥
⎢ WN WN⋅11
WN⋅21
WN⋅( N −1) ⎥ ⎢
1
⎥
Wx ' = ⎢ WN ⋅0
2
WN ⋅1
2
WN ⋅ 2
2
WN 2⋅( N −1) ⎥ ⎢
⋅ xi , 0 ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢W ( N −1)⋅0 W ( N −1)⋅1 W ( N −1)⋅2 ( N −1)⋅( N −1) ⎥ ⎢
WN xi ,( N − a −1) ⎥
⎣ N N N ⎦ ⎣ ⎦
⎡ xi ,( N − a ) ⎤
⎡WN ( N − a )⋅0
−
0 0 ⎤ ⎡ WN ⋅( N − a )
0
WN ⋅( N − a +1)
0
WN ⋅( N −1)
0
WN ⋅0
0
WN ⋅( N − a −1) ⎤ ⎢
0
⎥
⎢ − ( N − a )⋅1 ⎥ ⎢ 1⋅( N − a ) 1⋅( N − a −1) ⎥ ⎢ ⎥
0 WN 0 ⎥ ⋅ ⎢ WN WN⋅( N − a +1)
1
WN⋅( N −1)
1
WN⋅0
1
WN
=⎢ ⎥⋅⎢ x ⎥
⎢ ⎥ ⎢ ⎥ ⎢ i ,0 ⎥
⎢ ⎥ ⎢ ( ⎥ ⎢ ⎥
⎢ 0
⎣ 0 WN ( N − a )⋅( N −1) ⎥ ⎢WN N −1)⋅( N − a ) WN N −1)⋅( N − a +1)
−
⎦ ⎣
(
WN N −1)⋅( N −1) WN N −1)⋅0
( (
WN N −1)⋅( N − a −1) ⎥ ⎢
(
⎦ x ⎥
⎣ i ,( N − a −1) ⎦
⎡ X i ,0 ⎤
( N −a ) ⎢ ⎥ ( N −a)
=P ⋅⎢ ⎥=P ⋅ Xi
⎢ X i ,( N −1) ⎥
⎣ ⎦
10
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
CP ith OFDM symbol CP
Error sampling, φ = 0, ε = 0, wi ,k = 0, a is an integer number :
⎡ xi ,a ⎤
⎡ WN ⋅0
0
WN ⋅1
0
WN ⋅ 2
0
W 0⋅( N −1)
N
⎤ ⎢ ⎥
⎢ 1⋅0 ⎥ ⎢ ⎥
⎢ WN WN⋅11
WN⋅21
W 1⋅( N −1)
⎥ ⎢ x
N
i ,( N −1)
⎥
Wx ' = ⎢ WN ⋅0
2
WN ⋅1
2
WN ⋅2
2
W 2⋅( N −1)
N
⎥⋅⎢ ⎥
⎢ ⎥ ⎢ xi +1,( N −CP ) ⎥
⎢ ⎥ ⎢ ⎥
⎢W ( N −1)⋅0 W ( N −1)⋅1 W ( N −1)⋅2 ( N −1)⋅( N −1) ⎥
⎣ N N N WN ⎦ ⎢x ⎥
⎢ i +1,( N −CP + a ) ⎥
⎣ ⎦
11
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
CP ith OFDM symbol CP
Corrert sampling, wi ,k = 0 :
⎡ x[0] ⋅ e jφ ⎤
⎡ WN ⋅0
0
WN ⋅1
0
WN ⋅2
0
W 0⋅( N −1)
N
⎤ ⎢ 2π ⎥
⎢ 1⋅0 ⎥ ⎢ x[1] ⋅ e
jφ + j ε
⎥
⎢ WN WN⋅11
WN⋅21
W 1⋅( N −1)
N ⎥ ⎢
N
2π ⎥
Wx' = ⎢ WN ⋅0
2
WN ⋅1
2
WN ⋅ 2
2
W 2⋅( N −1) ⎥⋅⎢ jφ + j 2ε
⎥
⎥ ⎢ x[2] ⋅ e
N N
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢W ( N −1)⋅0 W ( N −1)⋅1 W ( N −1)⋅2 WN N −1)⋅( N −1) ⎥ ⎢
( 2π
jφ + j ε ( N −1) ⎥
⎣ N N N ⎦ x[ N − 1] ⋅ e N
⎣ ⎦
⎡ WN ⋅0 ⎡ x[0] ⎤
0
WN ⋅1
0
WN ⋅2
0
WN ⋅( N −1) ⎤ ⎢
0
2π ⎥
⎢ 1⋅( N −1) ⎥ x[1] ⋅ e N
j ε
⎥ ⎢ ⎥
1⋅0
⎢ WN WN⋅11
WN⋅21
WN
⎢ 2π ⎥
= e j φ ⋅ ⎢ W N ⋅0
2
WN ⋅1
2
WN ⋅2
2
WN ⋅( N −1) ⎥ ⋅ ⎢ x[2] ⋅ e N
2 j 2ε
⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢W ( N −1)⋅0 W ( N −1)⋅1 W ( N −1)⋅2 ( N −1)⋅( N −1) ⎥ 2π 12
⎣ N N N WN ⎦ ⎢ x[ N − 1] ⋅ e N
j ε ( N −1)
⎥
⎣ ⎦
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Synchronization task
Data-aided method
Non-data-aided method
13
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
14
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Maximum likelihood)
To consider the received signal
2π
j ε ( k −τ )
r (k ) = st (k − τ )e N
+ w(k )
⎧s (k + N − CP ) ,0 ≤ k ≤ CP − 1
where st (k ) = ⎨ ,
⎩s (k − CP ) , CP ≤ k ≤ CP + N − 1
CP is the length of cyclic prefix,
N is the number of subcarriers,
s (k ) is transmitted signal,
τ and ε are timing shift and carrier frequency offset respectively,
and w(k ) is AWGN term.
15
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Maximum likelihood)
⎧ σ s2 + σ n , m = 0
2
{ }⎪
∀k ∈ Γ, E r (k )r * (k + m) = ⎨σ s2 e − j 2πε , m = N
⎪ 0, otherwise
⎩
Observation interval
Symbol i-1 Symbol i Symbol i+1
Γ Γ'
τ
1 2N+CP
16
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Maximum likelihood)
The log - likelihood function is
Λ (τ , ε ) = log f (r | τ , ε )
⎛ ⎞
= log⎜ ∏ f (r (k ), r (k + N ) ) ∏ f (r (k ) )⎟
⎜ ⎟
⎝ k∈Γ k∉Γ ∪Γ ' ⎠
⎛ f (r (k ), r (k + N ) ) ⎞
= log⎜ ∏
⎜ ∏ f (r (k ) )⎟
⎟
⎝ k∈Γ f (r ( k ) ) f (r ( k + N ) ) k ⎠
where f (⋅) denotes the probability desity function of the variables,
⎛ r (k ) ⎞
2
exp⎜ − 2 ⎟
⎜ (σ s + σ n ) ⎟2
f (r (k ) ) = ⎝ ⎠ , and
π (σ s + σ n )
2 2
{ }
⎛ r (k ) 2 − 2 ρ Re e j 2πε r (k )r * (k + N ) + r (k + N ) 2 ⎞
exp⎜ − ⎟
⎜ (σ s + σ n )(1 − ρ )
2 2 2
⎟
f (r (k ), r (k + N ) ) = ⎝ ⎠
17
π (σ s + σ n )(1 − ρ )
2 2 2 2
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Maximum likelihood)
Thus,
Λ (τ , ε ) = γ (τ ) cos(2πε + ∠γ (τ )) − ρΦ (τ ),
where ∠ deontes the argument of a complex number,
∆ m + L −1
γ ( m) = ∑ r (k )r * (k + N ),
k =m
∆1 m + L −1
Φ ( m) = ∑ r ( k ) + r ( k + N ) ,
2 2
2 k =m
and ρ =
{ }
E r (k )r * (k + N )
= 2
σ s2
=
SNR
{ }{
E r (k ) E r (k + N )
2 2
} σ s + σ n SNR + 1
2
is the magnitude of the correlation coefficien t. 18
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Maximum likelihood)
Because
max Λ (τ , ε ) = max max Λ (τ , ε ) = max Λ (τ , ε ML (τ )),
ˆ
(τ ,ε ) τ ε τ
Λ (τ , ε ) = γ (τ ) cos(2πε + ∠γ (τ )) − ρΦ (τ )
Thue, the estimated frequency offset ε is
1
ε ML = −
ˆ ∠γ (τ ) + n, n is an integer,
2π
and the estimated symbol timing τ is
τˆML = arg max{γ (τ ) − ρΦ (τ )}
τ
19
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
20
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
Signal model in BPSK-OFDM system
The transmitted signal is
2π
1 N −1 j nk
si (k ) = ∑ xi,ne
N n =0
N
, 0 ≤ k ≤ N −1
where N denotes the IFFT window size,
si (k ) presents kth sample of ith OFDM symbol,
xi ,n presents the data of nth subcarrier in ith symbol interval.
Because BPSK mapping is used, xi ,n is real value.
21
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
2π
⎛ 1 N −1 ⎞ N −1
2π
real (si (k ) )
j nk 1
= real ⎜
⎜ N ∑ xi ,n e N ⎟ =
⎟ N
∑ xi ,n cos( N
nk )
⎝ n =0 ⎠ n =0
2π
⎛ 1 N −1 nk ⎞ 1 N −1 2π
imag (si (k ) )
j
= imag ⎜ ⎜ N ∑ xi ,ne ⎟ = N ∑ xi,n sin( N nk )
N
⎟
⎝ n =0 ⎠ n=0
2π 2π
⎛ 1 N −1 n( N −k ) ⎞ ⎛ 1 N −1 −j ⎞
real (si ( N − k ) ) = real ⎜
j nk
⎜ ∑
N n =0
xi ,n e N ⎟ = real ⎜
⎟ ⎜ N∑ xi ,n e N
e j 2πn ⎟⎟
⎝ ⎠ ⎝ n =0 ⎠
( )
2π
⎛ 1 N −1 nk ⎞ 1 N −1 2π
∑ xi,n cos( N nk ) = real si (k )
−j
= real ⎜
⎜ N ∑ xi ,ne N ⎟=
⎟ N n=0
⎝ n =0 ⎠
2π 2π
⎛ 1 N −1 n( N −k ) ⎞ ⎛ 1 N −1 −j ⎞
imag (si ( N − k ) ) = imag ⎜
j nk
⎜ ∑ xi ,ne N ⎟ = imag ⎜
⎟ ⎜ N ∑ xi ,n e
N
e ⎟j 2πn
⎟
⎝ N n =0 ⎠ ⎝ n =0 ⎠
2π
⎛ 1 N −1 ⎞ −1 N −1
2π
nk ) = − imag (s i (k ) )
−j nk
= imag ⎜
⎜ N ∑ xi ,ne N ⎟=
⎟ N
∑ xi ,n sin( N
⎝ n =0 ⎠ n =0
22
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
Thus, the symmetry characteristic in BPSK - OFDM
symbol can be written as
N
si (k ) = s ( N − k ) , k ≠ 0 or .
*
i
2
23
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
N
Because si (k ) = si* ( N − k ) when k ≠ 0 or , si (k ) can be expressed as
2
jθ i , 0
si (0) ≡ Ai , 0 e
jθ i , k − jθ i ,k N
si (k ) ≡ Ai ,k e ⇒ si ( N − k ) = Ai ,k e , k ≠ 0 or
2
jθ i , N
si ( N 2) ≡ Ai , N 2 e 2
where Ai ,k and θ i ,k denote the magnitude and the phase of si (k ) respectively.
24
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
The received signal is
ri (k ) = vi (k − τ ) + wi (k )
jφ + j
2π
εk ri (τ ) = vi (0) + wi (τ )
vi (k ) = si (k )e N
τ
where τ is timing shift,
CP
φ is initial phase,
ε is carrier frequency offset,
si (k ) presents kth sample of ith OFDM symbol,
and wi (k ) is AWGN term.
25
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
First group of symmetry relationship :
When vi (0) is taken as the central point for the two opposite samples
vi (k ) and vi (− k ), it has the character
N
τ τ +
vi (k ) ⋅ vi (−k ) = A e
2 j 2φ
, 1 ≤ k ≤ CP 2
CP
i ,k
Second group of symmetry relationship :
When vi ( N 2) is taken as the central point for the two opposite samples
vi (k ) and vi ( N − k ), it has the character
vi (k ) ⋅ vi ( N − k ) = Ai2,k e j 2φ + j 2πε , 1 ≤ k ≤ N 2 − 1
N
τ τ +
2
CP
26
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
There are three kinds of useful property can be
employed as follows:
The first group of symmetry relationship.
The second group of symmetry relationship.
The cyclic prefix copied from the tail of OFDM symbol.
27
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
- Using angle
In second group of symmetry relationship,
vi (k ) ⋅ vi ( N − k ) = Ai2,k e j 2φ + j 2πε where 1 ≤ k ≤ N 2 − 1.
C k = {Ck ,1 , Ck , 2 , , Ck , N 2−1 } where
N
τ τ +
2
Ck ,1 = angle(ri (k + 1) ⋅ ri (k + N − 1) ) CP CP
Ck , N 2−1 = angle(ri (k + N 2 − 1) ⋅ ri (k + N − N 2 + 1) )
The mean of Cτ is
1 N 2−1
E{C k } = ∑1 Ck ,m When the central point of opposite samples is ri ( N 2 + τ ),
N 2 − 1 m= the mean of Cτ = {Cτ ,1 , Cτ , 2 , , Cτ , N 2−1 }is the twice initial
The cost function f {Cτ } is phase of ri ( N 2 + τ ), and the cost function f {Cτ } is zero
N 2 −1
for noise - free case.
f {C k } = ∑C k ,m − E{C k }
m =1 28
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
- Using angle
First useful property :
Bk ,m = angle(ri (k + m) ⋅ ri (k − m) )
1 CP
E{B k } = ∑ Bk ,m N
CP m =1 τ τ +
2
CP
f {B k } = ∑ Bk ,m − E{B k }
CP
m =1
Second useful property :
Ck ,m = angle(ri (k + m) ⋅ ri (k + N − m) )
Fk ≡ f {B k }+ f {C k }+ f {D k }
1 N 2−1
E{C k } = ∑ Ck , m N
N 2 − 1 m =1 τ τ +
2
N 2 −1
CP
f {C k } = ∑C k ,m − E{C k }
m =1
Third useful property :
(
Dk ,m = angle ri (k − m) ⋅ ri* (k + N − m) )
1 CP
E{D k } =
N
∑ Dk ,m
CP m =1
τ τ +
2
CP CP 29
f {D k } = ∑ Dk ,m − E{D k }
m =1
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
- Using correlation
In second group of symmetry relationship,
vi (k ) ⋅ vi ( N − k ) = Ai2,k e j 2φ + j 2πε where 1 ≤ k ≤ N 2 − 1.
N
τ τ +
2
N 2 −1
CP CP
C (k ) = ∑ r ( k + m) ⋅ r ( k + N − m)
m =1
i i
When the central point of opposite samples
is ri (τ + N 2), k = τ , C (τ ) has the maximum
value in observation region.
30
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (BPSK-OFDM case)
- Using correlation
First useful property :
CP
B(k ) = ∑ r ( k + m) ⋅ r ( k − m)
i i N
m =1 τ τ +
2
CP
Second useful property :
N 2 −1
C (k ) = ∑ r ( k + m) ⋅ r ( k + N − m)
i i
Fk ≡ B(k ) + C (k ) + D(k )
m =1
N
τ τ +
2
CP
Third useful property : τˆ = arg max Fk
CP
τ
D(k ) = ∑ r ( k − m) ⋅ r ( k + N − m )
i i
*
N
m =1
τ τ +
2
CP
31
CCU
Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Performance Analysis
AWGN channel
0
AWGN channel 3
10
10
2
-1 10
10
1
-2 10
Lost symbol timing rate
10
Mean-squared error
0
-3
10 10
-1
-4 10
10
-2
-5 Using angle 10 Using angle
10
Using correlation Using correlation
ML ML
-6 -3
10 10
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
SNR(dB) SNR(dB)
32
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Wireless Access Tech. Lab.
4 4 Using angle (Using two groups of symmetry relationship)
x 10 Maximum likelihood algorithm x 10
8 10
9
7
8
6
7
5 6
Statistics
Statistics
4 5
4
3
3
2
2
1 1
0 0
0 50 100 150 20 40 60 80 100 120 140
Time index, n Time index, n
CP CP CP
33
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Wireless Access Tech. Lab.
Solution
Image part
Real part
: A pair of sample which has symmetry
characteristic in noise free case 34
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Solution
Image part
Real part
: A pair of sample which has symmetry
characteristic in noise free case 35
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
36
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Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
Using the DFT based OFDM modulation with N subcarrier,
the continuous time complex baseband OFDM symbol
N −1 2πk
1 j ( t −t s )
s (t ) =
N
∑d e
k =0
k
T
where d k is used to modulate the subcarrier e j 2πk T ,
t s is the starting time of the current OFDM symbol
(excluding the guard time),
T is the OFDM symbol duration (DFT/IDFT interval).
37
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Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
We define Ts = T N as the symbol interval.
Assume that the sampling instances for s (t ) are at t = t s + nTs + τ
where τ is the initial time shift and n = 0, , N − 1, we have two
sets of discrete - time observatio ns depending on τ .
• τ = 0. This is the usual IDFT OFDM symbol model
2πnk
1 N −1 j
s1 ( n) = ∑ dk e N
N k =0 s1(n) s2(n)
Ts T
•τ = = . In this case, we get the signal model
2 2N
2πk
1 N −1
1
j (n+ )
s2 ( n) = ∑ dke N 2
N k =0 0 Ts 2Ts ( N −1)Ts T
38
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
Written in matrix form, we have
s1 = Wd
s2 = WEd
where W is the IDFT matrix; d is the symbol vector
[d 0 , , d N −1 ] ; and
T
π ( N −1)π
⎛ j j ⎞
E = diag ⎜1 e N
⎜ e N ⎟
⎟
⎝ ⎠
39
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
In the presence of time dispersive channel, additive
noise, and carrier frequency offset, the continuous time
baseband received signal is then,
2πk
1 N −1 j( + ∆ω )( t −t s )
x(t ) = ∑ H (k )d k e
N k =0
T
+ z (t )
where H (k ) is the channel frequency response
corresponding to subcarrier k and ∆ω is the frequency
offset.
40
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
Under the sampling time at t = t s + nTs + τ , we have two
sets of discrete - time observations depending on τ .
• τ = 0.
2πk
1 N −1 j( + ∆ω ⋅Ts ) n
x1 (n) = ∑ H (k )d k e
N k =0
N
+ z1 (n)
T T
•τ = s =
2 2N
2πk
1 N −1
1
j( + ∆ω ⋅Ts )( n + )
x2 ( n ) = ∑ H (k )d k e N
N k =0
2
+ z 2 ( n)
where z1 (n) and z 2 (n) are additive complex Gaussian
noise and are usually assumed to be uncorrelated to each
∆
other. We defined as φ = ∆ω ⋅ Ts . 41
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
The above signal model can be written in a more compact matrix form
as follows :
~
x1 = PW d + z1 (1)
~
x 2 = e jφ 2 PWE d + z 2 (2)
~
where d = Hd with H being a diagonal matrix with diagonal element H (k ).
The matrix E reflects the phase shift between x1 and x 2 due to the
difference in subcarrier frequencies, while the phase shift due to the
frequency offset differs by a scalar constant e jφ 2 .
Where P = diag (1, e jφ , , e j ( N −1)φ )
42
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
Perfect recovery of carrier frequency offset in the
absence of noise
~
From (1) and (2), we notice that d is unknown, and we
want to blind estimation of the frequency offset φ . Define
y 1 = W H P H x1 (3)
φ
−j
y2 = e 2
EH W H P H x2 (4)
It is straightfo rward to verify that in the absence of noise
~
y1 = y 2 = d (5)
Thus it is intuitive to find φ such the resulting y 1 and y 2
are equal. 43
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method (Via oversampling)
Frequency offset estimation in the present of noise
In presence of noise, it is unlikely that (5) will be true
for any value of φ . However, we propose, intuitively,
to minimize the distance between y 1 and y 2 .
min(y1 − y 2 )H (y1 − y 2 )
φ
Above minimization criterion using maximumlikelihood
principles is provided in the next section.
44
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
45
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
At the transmitter, the training-symbol-block
contains two equal-length training symbols in time
domain, and the second training symbol is the inverse
repeat of the first one.
The training - symbol- block :
S = [s(0) ,s(1) , ,s( N − 1) ,s( N − 1) , ,s(1) ,s(0)]
where N denotes the DFT length of training symbols.
46
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
At receiver, without considering of the channel
attenuation and additive AWGN, the relationship
between corresponding samples in a received training-
symbol-block is :
r ( 2 N − 1 − k ) = r ( k ) e j 2 πε ( 2 N −1− 2 k ) / N k ∈ [ 0 , N − 1]
where r ( k ) denotes the k th sample and ε denotes
the carrier frequency offset normalized to a
subcarrier spacing of training symbols.
47
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
A timing metric is defined:
N −1
2 ∑ r (2 N + d − 1 − k )r * (k + d )
k =0
M (d ) = d + 2 N −1
∑
2
r (k )
k =d
N -1
2 ∑ r(k - d)r * (k + d )e j 2πε ( 2 N + 2 d −1− 2 k ) / N
k =0
= d + 2 N −1
∑
2
r (k )
k =d
where d is a time index corresponding to the first sample
in a window of 2 N samples. 48
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
49
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
Shown in Fig. 1, that the expectation of M (0) is a function
of ε with period of N / 2, and within each period, a main lobe
appears.
The expectation of M (d <> 0) is a constant independen t of ε .
Within one period (without loss of generality, ε ∈ [− N 4 , N 4]
is assumed), M (0) > M (d <> 0) is satisfied only in a small
vicinity which stands in the center of the main lobe.
50
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
51
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
At the transmitter, 2 identical training - symbol - blocks are
transmitted. At the receiver, 3 N samples of the training
sequence are buffered. The length of 3 N guarantees one
integral training - symbol - block being buffered.
52
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
Because the carrier frequency offset is fully unknown at the
start of acquisition, a method of lookup is used :
(2k + 1) presupposed values :
( ε− k = -k ⋅ ∆ε , ε −( k −1) = −(k − 1) ⋅ ∆ε ,
ˆ ˆ , ε −1 = − ∆ε , ε 0 = 0,
ˆ ˆ
ε 1 = ∆ε ,
ˆ , ε k −1 = (k − 1) ⋅ ∆ε , ε k = k ⋅ ∆ε )
ˆ ˆ
where k ⋅ ∆ε < N 4 and 0 < ∆ε < N (2(2 N − 1))
∆ε denotes the lookup interval.
53
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
For jth ( −k ≤ j ≤ k ) compensated training sequence, the
maximum M (d ) (0 ≤ d ≤ N − 1) is represented as Γ j which
implies the appearance of an integral training - symbol -
block ; if Γq > Γp ≠ q for each − k ≤ p ≤ k , then q ⋅ ∆ε denotes
the carrier frequency offst estimated by acquisition
algorithm. The estimated start of that integral training -
symbol - block d is also the output of acquisition.
ˆ
0
54
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
After acquisition, if the remaining carrier frequency
offset exceeds the tracking range, tracking algorithm
will not work correctly, and this is called Missing Lock.
For usable SNR, if N is large enough, the probability
of Missing Lock is negligible.
55
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
After acquisition, the remaining frequency offset
needs to be further corrected.
For a receivedtraining- symbol- block ℜ under certain SNR
condition, the log - likelihood function for the carrier frequency
offset ε , Λ(ε ) is the logarithmof the probability density
function f (ℜ | ε ) .
Λ(ε ) = log f (ℜ | ε )
⎛ N −1 ⎞
= log⎜ ∏ f (r (k ), r (2 N − 1 − k ))⎟
⎜ ⎟
⎝ k =0 ⎠
N −1
= ∑ log( f (r (k ), r (2 N − 1 − k )))
56
k =0
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
The 2 − D complex - valued Guassian distribution
f (r (k ), r (2 N − 1 − k ) ) =
{ }
⎛ r (k ) 2 − 2 ρ Re e j 2πε ( 2 N −1− 2 k ) / N r (k )r * (2 N − 1 − k ) + r (2 N − 1 − k ) 2 ⎞
exp⎜ − ⎟
⎜
⎝ ( )(
σ s + σ n 1− ρ
2 2
) 2
⎟
⎠
(
π 2 σ s2 + σ n 1 − ρ 2)(
2 2
)
where
ρ=
{
E r (k )r * (2 N − 1 − k ) }
{ 2
}{
E r (k ) E r (2 N − 1 − k )
2
}
σ s2 SNR
= 2 =
σ s + σ n SNR + 1
2
σ s2
SNR = 2 57
σn
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
{ }
N −1
N ∑ r (k )r * (2 N − 1 − k ) ⋅ angle r (k )r * (2 N − 1 − k ) ⋅ (2 N − 1 − 2k )
ε =−
ˆ k =0
N −1
2π ∑ r (k )r * (2 N − 1 − k ) ⋅ (2 N − 1 − 2k )
2
k =0
In order to make the proposed tracking algorithm work correctly,
{ }
angle r (k )r * (2 N − 1 − k ) < π for each k ∈ [0, N − 1] should satisfied,
i.e., the tracking range is ± ( N /(2(2 N − 1))) subcarrier spacing.
58
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Wireless Access Tech. Lab.
Data-aided Method (Special training-symbol-block)
The Cramer - Rao lower bound is
⎧ N ⎫ 3N
Var ⎨e | ε < ⎬≥
⎩ 2(2 N − 1) ⎭ 4π 2 (4 N 2 − 1) SNR
59
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
OFDM System Description
Synchronization
What is Synchronization?
Symbol Synchronization
Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)
Frequency Synchronization
Non-data-aided Method (Via oversampling)
Data-aided Method (Special training-symbol-block)
Joint Symbol and Frequency Synchronization
Non-data-aided Method (Exploiting second-order cyclostationarity)
60
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
The discrete - time baseband equivalent OFDM signal
transmitted through a frequency - selective fading
channel is given by
L −1
y[n] = ∑ h[m]x[n − m]
m =0
where h[n] is the impulse response of channel,
L is channel order.
61
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
The transmitted signal x[n] is
N −1 ∞ 2π
j kn
x[n] = ∑ ∑ ck ,l g[n − lP ]e N
k = 0 l = −∞
where g[n] is the transmitter pulse shaping filter,
N is the number of sub - carrier,
ck ,l are the complex informatio n symbols.
62
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
At the receiver
y[n] = y[n − nε ]e j ( 2πθε n +φ ) + w[n]
ˆ
where nε is the integer - valued unknown arrival time of a symbol,
θ ε is the frequency offset,
φ is the initial phase.
63
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
The following assumptions are hold in this paper :
1. ck ,l is a zero - mean independen tly identicall y distributed(i.i.d.)
sequence with values drawn from a finite - alphabet complex
constellation, with variance σ c2 .
2. For 0 ≤ n ≤ L − 1, each h[n] is zero - mean independen t gaussian
random variable with variance σ h[ n ] .
2
3. x[n] is uncorrelated with h[n] and w[n].
64
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
The goal of synchronization is to estimate nε and θ ε .
The proposed algorithm is based on cyclostationarity
of the receiver OFDM signal caused by pulse shaping
filter.
65
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
ˆ {
ry [n;τ ] = E y[n] y *[n − τ ]
ˆ ˆ }
L −1 N −1 2π
j kτ
= ∑σ 2
h[ m ] e j 2πθε τ
σ c2 ∑ e N
m =0 k =0
∞
⋅ ∑ g ′[n − lP ]g ′*[n − lP − τ ] + rw [τ ]
l = −∞
= ry [n + kP;τ ]
ˆ
where g ′[n] = g[n − nε ]
66
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
For a fixed τ , ry [n;τ ] is periodic in n with period P.
ˆ
Thus, the cyclic correlation turns out to be
1 L-1 2 j 2πθετ
R y [ k ;τ ] = ∑ σ h[ m ] e
ˆ ⋅ Γ[τ ]
P m =0
∞ 2π
−j kn
⋅ ∑ g ′[n]g ′*[n + τ ]e
n = −∞
P
+ rw [τ ]δ [k ]
N −1 2π
j kτ
where Γ[τ ] = ∑ e N
k =0
67
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
Using Parseval's theorem
∞ 2π
−j kn
∑ g ′[n]g ′*[n + τ ]e
n = −∞
P
k 12
− j 2π ⎛ k⎞
G * ⎜ β − ⎟G (β )e − j 2πβτ dβ
nε
=e P
∫2 ⎝ P ⎠
−1
where G (β ) denotes Fourier transform of g[n].
68
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
Let G2 (k ;τ ) = ∫ G * (β − k P )G (β )e − j 2πβτ dβ .
12
−1 2
Then cyclic correlation is expressed as
2π
1 L −1 2 j 2πθετ 2 −j knε
R y [ k ;τ ] = ∑ σ h[ m ] e
ˆ σ c ⋅ Γ[τ ] ⋅ G2 (k ;τ )e P
+ rw [τ ]δ [k ]
P m =0
where G ( β ) denotes Fourier transform of g[n].
Upon defining
M [k ;τ ] = G2 1 (k ;τ ) R y [k ;τ ]
−
ˆ
2π
1 L −1 2 j 2πθετ 2 −j knε
= ∑ σ h[ m ] e σ c ⋅ Γ[τ ]e P
+ G2 1 (k ;τ )rw [τ ]δ [k ]
−
P m =0
69
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Wireless Access Tech. Lab.
Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
Consider the case only k ≠ 0.
For τ = 0, we can retrieve nε from the phase of cyclic correlation
arg[M [k ;0]] for k > 0
1
nε = −
ˆ
2π k P
Given timing offset nε , the frequency offset θ ε can be derived as
ˆ
⎡ j 2π nε ⎤
k
arg ⎢ M [k ; N ]⋅ e
1 ˆ
θε
ˆ = P
⎥
2πN ⎣ ⎦
Timing offset and frequency offset appear as the phase of cyclic
correlation, and the impluse response of channel h[n] does not affect
the phase of cyclic correlation.
70
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
Since we can not have access to ensemble cyclic quantity, we should
estimate it from finite data samples. We obtain R ˆ [k ;τ ] from data set
ˆ
y
{y[n]}n−10 . If number of data I is large enough, Ryˆ [k ;τ ] would be asymptotically
ˆ I
=
ˆ
unbiased and consistent in mean square sense, i.e.
lim R ˆ [k ;τ ] ≈ R ˆ [k ;τ ].
I →∞
ˆ
y y
71
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Wireless Access Tech. Lab.
Non-data-aided Method
(Exploiting second order cyclostationarity)
The effect of imperfection of cyclic correlatio n can be potentially
reduced average. The estimator is obtained by
nε = −
ˆ
1 1 Q −1
2π k P Q q =0
ˆ [
⋅ ∑ arg M [k ;0] ] for k > 0
1 1 Q −1 ⎡ ˆ j 2π nε ⎤
k
⋅ ∑ arg ⎢ M [k ; N ]⋅ e
ˆ
θε
ˆ = P
⎥
2πN Q q =0 ⎣ ⎦
Thus, we can observe that the number of average, Q, increases the
complexity of the estimator.
Large Q makes more reliable estimation.
72
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Wireless Access Tech. Lab.
Reference
1) J.J. van de Beek, M. Sandell and P. O. Börjesson, “ML
estimation of time and frequency offset in OFDM systems,”
IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800-1805,
July 1997.
2) B. Chen, H. Wang, “Blind OFDM carrier frequency offset
estimation via oversampling”, 2001 IEEE, vol. 2, pp. 1465 –
1469, Nov. 2001.
3) Z. Zhang, M. Zhao, H. Zhou, Y. Liu, J. Gao, “Frequency offset
estimation with fast acquisition in OFDM system”, IEEE
Communications Letters, vol. 8, no. 3, pp. 171 – 173, Mar.
2004.
4) B. Park, H. Cheon, E. Ko, C. Kang, D. Hong, “A blind OFDM
synchronization algorithm based on cyclic correlation”, IEEE
Signal Processing Letters, vol. 11, no. 2, pp. 83 – 85, Feb. 73
2004.
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