Timing misalignment estimation
View Patent ↗Orthogonal frequency division multiplexing (OFDM) receiver embodiments of the invention provide timing misalignment estimation by calculating the intra-baud timing differential. The preferred method exploits the spectral structure of the short-preamble in that a time delay in the time-domain is manifest as a phase rotation in the frequency-domain. The timing misalignment is determined by special processing the spectral peaks of the modified short-preamble. An alternative embodiment linearly combines the two long-preamble symbols to construct a single “best estimate” long-preamble symbol. A normalized dot product of the “best estimate” long-preamble symbol and the ideal on-baud symbol is computed to obtain the magnitude of the timing misalignment. A dot product between the “best estimate” long-preamble symbol and the time derivative of the ideal on-baud sampled sequence is computed to obtain the sign of the timing misalignment.
1. A method for timing misalignment determination in a radio receiver, comprising the steps of:
correcting a frequency offset and determining a boundary between a short and long preamble in a received radio signal;
constructing a real signal from a complex time-domain signal associated with said short preamble;
extracting an n-sample sequence from a portion of said short preamble to obtain a plurality of equidistant equal amplitude frequency peaks wherein n is an integer greater than or equal to zero; and
determining a timing offset estimate based on the relative phases of said plurality of equidistant equal amplitude frequency peaks.
2. The method of claim 1 , wherein:
the step of constructing includes in-phase and quadrature-phase sampling of said received radio signal to obtain a real part and an imaginary part;
wherein said real and imaginary parts are similar to one another except for a fixed time-skew between them.
3. The method of claim 2 , wherein:
the step of constructing includes a simple addition of said real and imaginary parts to obtain said real signal.
4. The method of claim 1 , wherein:
the step of determining is such that the phase of a set of three frequency peaks Φ 1 , Φ 2 and Φ 3 is assumed to vary with timing misalignment between Δ(t) and δ(t), and an intra-baud timing offset τ is derived from Φ 1 , Φ 2 and Φ 3 wherein, a received signal is represented by,
Ψ
(
t
)
=
Φ
1
(
t
)
+
2
Φ
2
(
t
)
+
Φ
3
(
t
)
=
π
4
(
1
+
t
T
s
)
+
2
π
4
(
1
+
2
t
T
s
)
+
3
π
4
(
-
1
+
2
t
T
s
)
=
2
π
t
T
s
and, Ψ=(X 8 P 1 )(X 16 P 2 ) 2 (X 24 P 3 ), where X k and P n respectively designate 64-point fast Fourier transform frequency components and phase correcting coefficients needed to compensate for phase offset errors caused by said timing misalignment between Δ(t) and δ(t), and the timing misalignment τ is expressed as a fraction of a sampling period, T s , and is
τ
=
Ψ
π
.
5. The method of claim 1 , wherein:
the step of determining computes a 64-point fast Fourier transform rather than a three-point discrete Fourier transform.
6. A method for timing misalignment determination in a radio receiver, comprising the steps of:
determining a boundary between a short preamble and a long preamble in a received radio signal;
linearly combining samples of two long sequences from said long preamble to obtain an idealized sequence of samples that best approaches under a certain criterion an ideal sequence of samples;
computing a normalized dot product of said idealized sequence of samples and an ideal on-baud sampled sequence to obtain a magnitude estimate of any timing misalignment; and
computing a dot product of said idealized sequence of samples and the time derivative of the ideal on-baud sampled sequence mentioned above to obtain a sign of any timing misalignment.
7. The method of claim 6 , wherein the steps of linearly combining and computing use a cost function mathematically described by,
C
(
α
1
,
α
2
)
=
R
→
on
-
[
X
→
1
X
→
2
]
[
α
1
α
2
]
2
=
(
R
→
on
-
[
X
→
1
X
→
2
]
[
α
1
α
2
]
)
H
(
R
→
on
-
[
X
→
1
X
→
2
]
[
α
1
α
2
]
)
=
(
R
→
on
H
-
α
1
*
X
→
1
H
-
α
2
*
X
→
2
H
)
(
R
→
on
-
α
1
X
→
1
-
α
2
X
→
2
)
=
R
→
on
2
-
2
Re
(
α
1
*
X
→
1
H
R
→
on
)
)
-
2
Re
(
α
2
*
X
→
2
H
R
→
on
)
+
2
Re
(
α
1
*
α
2
X
→
1
H
X
→
2
)
+
α
1
2
X
→
1
2
+
α
2
2
+
X
→
2
2
where,
{right arrow over (X)} 1 ={right arrow over (C)} 1 ·x{right arrow over (R)} off +{right arrow over (N)} 1 is the first sequence of the long preamble,
{right arrow over (X)} 2 ={right arrow over (C)} 2 ·x{right arrow over (R)} off +{right arrow over (N)} 2 is the second one,
{right arrow over (R)} on and {right arrow over (R)} off respectively designate on-baud and off-baud sampled reference sequence, α 1 and α 2 are the weighting coefficients,
C
→
1
=
ⅇ
j
φ
1
ⅇ
j
2
π
v
F
s
+
j
φ
1
⋮
ⅇ
j
2
π
v
F
s
63
+
j
φ
1
,
C
→
2
=
[
ⅇ
j
φ
2
ⅇ
j
2
π
v
F
s
+
j
φ
2
⋮
ⅇ
j
2
π
v
F
s
63
+
j
φ
2
,
and
ν designates a frequency offset
and, minimizing C(α 1 ,α 2 ) with respect to α 1 and α 2 yields,
∂
C
∂
α
1
=
-
R
→
on
H
X
→
1
+
α
2
*
X
→
2
H
X
→
1
+
α
1
*
X
→
1
2
=
∑
n
=
0
63
R
on
*
(
n
)
R
off
(
n
)
ⅇ
j
2
π
v
F
s
n
+
j
φ
1
+
α
2
*
ⅇ
j
(
φ
1
-
φ
2
)
∑
n
=
0
63
R
off
(
n
)
2
+
α
1
*
(
R
→
off
2
+
σ
N
2
)
=
-
ⅇ
j
φ
1
P
+
α
1
*
(
S
+
σ
N
2
)
+
α
2
*
ⅇ
j
(
φ
1
-
φ
2
)
S
∂
C
∂
α
2
=
-
R
→
on
H
X
→
2
+
α
1
*
X
→
1
H
X
→
2
+
α
2
*
X
→
2
2
=
-
∑
n
=
0
63
R
on
*
(
n
)
R
off
(
n
)
ⅇ
j
2
π
v
F
s
n
+
j
φ
2
+
α
1
*
ⅇ
j
(
φ
2
-
φ
1
)
∑
n
=
0
63
R
off
(
n
)
2
+
α
2
*
(
R
→
off
2
+
σ
N
2
)
=
-
ⅇ
j
φ
2
P
+
α
1
*
ⅇ
j
(
φ
2
-
φ
1
)
S
+
α
2
*
(
S
+
σ
N
2
)
with
:
σ
N
2
=
N
→
1
H
N
→
1
=
N
→
2
H
N
→
2
,
S
=
∑
n
=
0
63
R
off
(
n
)
2
and
P
=
∑
n
=
0
63
R
on
*
(
n
)
R
off
(
n
)
ⅇ
-
j
2
π
v
F
s
n
.
8. The method of claim 7 , wherein the steps of linearly combining and computing assume that {right arrow over (N)} 1 H {right arrow over (N)} 2 ={right arrow over (N)} 1 H {right arrow over (X)} 2 ={right arrow over (N)} 2 H {right arrow over (X)} 1 ={right arrow over (N)} 1 H {right arrow over (R)}={right arrow over (N)} 2 H {right arrow over (R)}=0, although such is not exactly true in reality, and thereby reduces computer processing required; and continuing with,
{
∂
C
∂
α
1
=
0
∂
C
∂
α
2
=
0
⇒
[
S
+
σ
N
2
ⅇ
j
(
φ
1
-
φ
2
)
S
ⅇ
j
(
φ
2
-
φ
1
)
S
S
+
σ
N
2
]
[
α
1
*
α
2
*
]
=
[
ⅇ
j
φ
1
P
ⅇ
j
φ
2
P
]
⇔
[
α
1
α
2
]
=
1
(
S
+
σ
N
2
)
2
-
S
2
[
S
+
σ
N
2
-
ⅇ
j
(
φ
2
-
φ
1
)
S
-
ⅇ
j
(
φ
1
-
φ
2
)
S
S
+
σ
N
2
]
[
ⅇ
-
j
φ
1
P
*
ⅇ
-
j
φ
2
P
*
]
⇔
[
α
1
α
2
]
=
P
*
2
S
+
σ
N
2
[
ⅇ
-
j
φ
1
ⅇ
-
j
φ
2
]
in the absence of any timing misalignment, frequency offset or Gaussian noise, the weighting coefficients are simply,
[
α
1
α
2
]
=
1
2
[
ⅇ
-
j
φ
1
ⅇ
-
j
φ
2
]
.