Method and apparatus for reducing impulse noise in a signal processing system
View Patent ↗An observed signal that is corrupted with impulse noise is recorded in a signal processing system of an image processing system or a digital subscriber line (xDSL). The observed signal that is recorded by the signal processing system includes a noise component and data component. The signal processing system estimates the parameters of an alpha-stable distribution using a modified iteratively reweighted least squares (IRLS) technique. The estimated parameters define a probability density function that is used to model the noise component of the observed signal. Once the parameters of the alpha-stable distribution are estimated, the signal processing system uses them to estimate model coefficients of a non-linear prediction filter such as a Volterra filter. Using the model coefficients, the non-linear prediction filter estimates the data component of the observed signal.
1. A method for reducing impulse noise in a signal processing system, comprising the steps of:
estimating parameters of an alpha-stable distribution to model impulse noise that corrupts data signals input into a transmission medium of the signal processing system;
sampling signals from the transmission medium; said sampling step storing the sampled signals in a memory; the sampled signals having a noise component and a data component; and
computing with a prediction filter an estimate of the data components of the sampled signals using the estimated parameters of the alpha-stable distribution.
2. The method according to claim 1 , wherein said sampling step is performed from a digital subscriber line (DSL).
3. The method according to claim 2 , further comprising the steps of:
recording samples of impulse noise signals {x 1 , x 2 , . . . , x n } transmitted over the transmission medium of the signal processing system; the samples of impulse noise signals recorded by said recording step having no data component; and
using the recorded samples of impulse noise to perform said estimating step.
4. The method according to claim 1 , wherein said sampling step is performed using an imaging device.
5. The method according to claim 4 , further comprising the steps of:
recording a first, a second, and a third image with the imaging device;
computing a difference between the first and the second image to define a centro-symmetrized difference image;
estimating a characteristic exponent of an alpha-stable distribution using the centro-symmetrized difference image; and
computing a sample of impulse noise signals by applying a centralizing transformation to the first, the second, and the third images.
6. The method according to claim 1 , further comprising the steps of:
generating a synthetic noise signal with a random number generator that uses the parameters of the alpha-stable distribution as input parameters to the random number generator; and
subtracting the synthetic noise signal from the sampled signal to produce a modified sampled signal; the modified sampled signal having a noise component with a symmetric distribution.
7. The method according to claim 1 , further comprising the steps of:
estimating a characteristic exponent of the alpha-stable distribution that models impulse noise corrupting the sampled signal; and
optimizing model coefficients of the prediction filter using the estimated characteristic exponent of the alpha-stable distribution.
8. The method according to claim 7 , wherein said optimizing step minimizes a p th -power error criterion to optimize the model coefficients of the prediction filter.
9. The method according to claim 8 , wherein said optimizing step defines the p th -power error criterion using the characteristic exponent of the alpha-stable distribution.
10. The method according to claim 8 , wherein said optimizing step minimizes the p th -power error criterion by performing the following computation iteratively:
C ( k )=( X ext T WX ext ) −1 X ext T W x t ,
where C are model coefficients, W is a diagonal weight matrix, and x t is an observed signal block of L samples, where:
C
=
[
a
_
b
_
c
_
]
,
such
that
a
_
=
[
a
1
⋮
a
N
]
,
b
_
=
[
b
1
⋮
b
N
(
N
+
1
)
2
]
,
and
c
_
=
[
c
1
⋮
c
N
(
N
+
1
)
(
N
+
2
)
6
]
,
x
_
t
=
[
x
[
t
×
L
]
⋮
x
[
t
×
L
+
L
-
1
]
]
,
where
t
=
0
,
1
,
2
,
3
,
…
X
ext
=
⌊
X
(
1
)
X
(
2
)
X
(
3
)
⌋
,
such
that
X
(
1
)
=
[
x
[
t
×
L
]
0
⋯
0
x
[
t
×
L
+
1
]
x
[
t
×
L
]
⋯
0
⋮
⋮
⋮
x
[
t
×
L
+
L
-
1
]
x
[
t
×
L
+
L
-
2
]
⋯
x
[
t
×
L
+
L
-
N
]
]
X
(
2
)
=
[
x
2
[
t
×
L
]
0
⋯
0
x
2
[
t
×
L
+
1
]
x
[
t
×
L
]
x
[
t
×
L
]
⋯
0
⋮
⋮
⋮
x
2
[
t
×
L
+
N
]
x
[
t
×
L
+
N
]
x
[
t
×
L
+
N
-
1
]
⋯
x
2
[
t
×
L
]
⋮
⋮
⋮
x
2
[
t
×
L
+
L
-
1
]
x
[
t
×
L
+
L
-
1
]
x
[
t
×
L
+
L
-
2
]
⋯
x
2
[
t
×
L
+
L
-
N
]
]
X
(
3
)
=
[
x
3
[
t
×
L
]
0
⋯
0
x
3
[
t
×
L
+
1
]
x
2
[
t
×
L
+
1
]
x
[
t
×
L
]
⋯
0
⋮
⋮
⋮
x
3
[
t
×
L
+
N
]
x
2
[
t
×
L
+
N
]
x
[
t
×
L
+
N
-
1
]
⋯
x
3
[
t
×
L
]
⋮
⋮
⋮
x
3
[
t
×
L
+
L
-
1
]
x
2
[
t
×
L
+
L
-
1
]
x
[
t
×
L
+
L
-
2
]
⋯
x
3
[
t
×
L
+
L
-
N
]
]
.
11. The method according to claim 8 , wherein said step of estimating the characteristic exponent comprises the steps of:
computing a moment of the alpha-stable distribution; and
estimating the characteristic exponent using the computed moment.
12. The method according to claim 11 , wherein said step of computing the moment of the alpha-stable distribution comprises the step of computing an absolute fractional lower order moment  p according to the following equation:
A
^
p
=
1
n
∑
k
=
1
n
X
k
p
,
where p is the order of the moment and X k is one sample from a sequence of noise samples {X 1 , X 2 , . . . , X n }.
13. The method according to claim 12 , wherein said step of estimating the characteristic exponent comprises the step of estimating the characteristic exponent by computing {circumflex over (Z)}=log  p x =log  p y using the computed absolute fractional lower order moment Ŝ p by performing the steps of:
partitioning the sequence of noise samples into two parts U and V , with each part containing data samples U 1 , U 2 , U 3 , . . . and V 1 , V 2 , V 3 , . . . , respectively;
computing the moment  p x by summing noise samples in the sequence of noise samples as:
X 1 =U 1 +V 1 , X 2 =U 2 +V 2 , X 3 =U 3 +V 3 , . . . ; and
computing the moment  p r by concatenating noise samples in the sequence of noise samples as:
Y 1 =U 1 , Y 2 =V 1 , Y 3 =U 2 , Y 4 =V 2 , Y 5 =U 3 , Y 6 =V 3 , . . . .
14. The method according to claim 13 , wherein said estimating step estimates the characteristic exponent to be equal to
p
log
2
Z
^
when
Z
^
<
p
log
2
α
min
and
Z
^
>
p
log
2
2
,
where p is the order of the moment.
15. The method according to claim 11 , wherein said step of computing the moment of the alpha-stable distribution comprises the step of computing an absolute logarithmic moment using the following equation:
L
^
I
=
1
n
∑
k
=
1
n
log
X
k
,
where X k is one sample from a sequence of noise samples (X 1 , X 2 , . . . , X n }.
16. The method according to claim 15 , wherein said estimating step estimates the characteristic exponent by computing {circumflex over (Z)}={circumflex over (L)} 1 x −{circumflex over (L)} 1 r using the computed absolute logarithmic moment {circumflex over (L)} 1 , by performing the steps of:
partitioning the sequence of noise samples into two parts U and V , with each part containing data samples U 1 , U 2 , U 3 , and V 1 , V 2 , V 3 , . . . , respectively;
computing the moment {circumflex over (L)} 1 x by summing noise samples in the sequence of noise samples as:
X 1 =U 1 +V 1 , X 2 =U 2 +V 2 , X 3 =U 3 +V 3 , . . . ; and
computing the moment {circumflex over (L)} 2 x by concatenating noise samples in the sequence of noise samples as:
Y 1 =U 1 , Y 2 =V 1 , Y 3 =U 2 , Y 4 =V 2 , Y 5 =U 3 , Y 6 =V 3 . . . .
17. The method according to claim 16 , wherein said estimating step estimates the characteristic exponent to be equal to
log
2
Z
^
when
Z
^
<
log
2
α
min
and
Z
^
>
log
2
2
.
18. The method according to claim 16 , wherein said computing step computes a signed logarithmic moment using the following equation:
Λ
^
=
1
n
∑
k
=
1
n
sign
(
X
k
)
log
X
k
.
19. The method according to claim 18 , further comprising the step of estimating a symmetry parameter of an alpha-stable distribution by computing
tan
(
(
Λ
^
/
L
^
1
)
α
π
/
2
)
)
tan
(
α
π
/
2
)
.
20. The method according to claim 1 , wherein said computing step extrapolates with the prediction filter an estimate of the data components of the sampled signals using the estimated parameters of the alpha-stable distribution.
21. The method according to claim 1 , wherein said computing step interpolates with the prediction filter an estimate of the data components of the sampled signals using the estimated parameters of the alpha-stable distribution.
22. The method according to claim 1 , further comprising the step of computing a moment of the alpha-stable distribution.
23. The method according to claim 22 , wherein said step of computing a moment comprises the step of computing one of:
an absolute fractional lower order moment
A
^
p
=
1
n
∑
k
=
1
n
X
k
p
,
a signed fractional lower order moment
S
^
p
=
1
n
∑
k
=
1
n
sign
(
X
k
)
X
k
p
,
a signed logarithmic moment
Λ
^
=
1
n
∑
k
=
1
n
sign
(
X
k
)
log
X
k
,
a first absolute logarithmic moment
L
^
1
=
1
n
∑
k
=
1
n
log
X
k
,
a second absolute logarithmic moment
L
^
2
=
1
n
-
1
(
∑
k
=
1
n
log
X
k
-
L
^
1
)
2
,
a third absolute logarithmic moment
L
^
3
=
1
n
-
1
(
∑
k
=
1
n
log
X
k
-
L
^
1
)
3
,
an empirical characteristic function moment
φ
^
(
p
)
=
1
n
∑
k
=
1
n
ⅇ
j
p
X
k
,
a first upper extreme value moment
Y
_
^
1
=
1
n
/
r
∑
k
=
1
n
/
r
K
_
k
,
a first lower extreme value moment
Y
1
_
^
=
1
n
/
r
∑
k
=
1
n
/
r
K
_
k
,
a second upper extreme value moment
Y
_
^
2
=
1
(
n
/
r
)
-
1
∑
k
=
1
n
/
r
(
K
_
k
-
Y
_
^
1
)
2
,
and
second lower extreme value moment
Y
^
2
_
=
1
(
n
/
r
)
-
1
∑
k
=
1
n
/
r
(
K
_
k
-
Y
^
1
_
)
2
,
where:
X k is one sample from a sequence of noise samples {X 1 , X 2 , . . . , X n },
p is the order of the moment,
{overscore (K)} k =max {log X r(k−1)+1 , log X r(k+1)+2 , . . . log X r(k−1)+ r− 1 },
K k =max {log−X r(k−1)+1 , log−X r(k−1)+2 , . . . log−X r(k−1)+r−1 {, and
r is a block length of data.
24. The method according to claim 23 , further comprising the step of computing a characteristic exponent α of the alpha-stable distribution.
25. The method according to claim 24 , wherein said step of computing the characteristic exponent a of the alpha-stable distribution comprises the step of computing one of:
a first auxiliary variable {circumflex over (Z)}=log  p x −log  p y ,
a second auxiliary variable {circumflex over (Z)}={circumflex over (L)} 1 x −{circumflex over (L)} 1 r , and
a third auxiliary variable
Z
^
=
(
1
-
L
^
3
1.2020569
)
-
1
/
3
,
and
a fourth auxiliary variable
Z
^
=
π
2
6
(
1
Y
_
2
+
1
Y
_
2
)
1
/
2
.
26. The method according to claim 24 , further comprising the step of computing a symmetry parameter β of the alpha-stable distribution.
27. The method according to claim 26 , wherein said step of computing the symmetry parameter β of the alpha-stable distribution comprises the step of computing one of:
a first estimate of the symmetry parameter
β
^
=
tan
(
(
Λ
^
/
L
^
1
)
α
π
/
2
)
)
tan
(
α
π
/
2
)
,
and
a second estimate of the symmetry parameter
β
^
=
1
-
2
exp
(
α
(
Y
_
1
-
Y
_
1
)
)
.
28. The method according to claim 24 , further comprising the step of computing a dispersion γ of the alpha-stable distribution.
29. The method according to claim 28 , wherein said step of computing the dispersion γ of the alpha-stable distribution comprises the step of computing
γ
^
=
(
Γ
(
1
-
p
)
cos
(
p
π
/
2
)
Γ
(
1
-
p
/
α
)
cos
(
p
θ
/
α
)
)
α
/
p
cos
θ
,
where
θ
=
arc
tan
(
β
tan
(
α
π
/
2
)
)
.
30. The method according to claim 24 , further comprising the step of computing a location parameter δ of the alpha-stable distribution.
31. The method according to claim 30 , wherein said step of computing the location parameter δ of the alpha-stable distribution comprises the step of computing a fractile f of the sequence of noise, where
f
=
1
2
-
θ
π
α
,
θ
=
arc
tan
(
β
tan
(
α
π
/
2
)
)
.
32. The method according to claim 1 , further comprising the steps of:
computing a characteristic exponent α of the alpha-stable distribution
computing a location parameter δ of the alpha-stable distribution,
computing a symmetry parameter β of the alpha-stable distribution, and
computing a dispersion γ of the alpha-stable distribution.
33. The method according to claim 32 , wherein said step of computing the characteristic exponent α and said step of computing the dispersion γ of the alpha-stable distribution comprises the step of computing the following matrix equation
[
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
log
(
t
j
)
log
(
t
k
)
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
log
(
t
k
)
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
log
(
t
k
)
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
C
j
k
]
[
α
log
γ
]
=
[
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
log
(
t
k
)
ψ
j
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
ψ
j
C
j
k
]
,
where
{t 1 , t 2 , . . . , t m } is some set of positive real numbers,
C equals W −1 where W is an m×m matrix with elements
W jk ={exp (γ + [t j α′ +t k α′ −t j α′ t k α′ ])−1)/{γ 2 t j α′ t k α′ }
and α′, γ′ are existing estimates of the characteristic exponent and dispersion, and
where Ψ k =log(−log(Re{circumflex over (φ)}(t k ))) for sequence Y for k=1, . . . , m, and
Y is a centro-symmetrized version of the sequence of noise samples.
34. The method according to claim 33 , wherein said stop of computing the location parameter δ and said step of computing the symmetry parameter β of the alpha-stable distribution comprises the step of computing the following matrix equation:
[
κ
2
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
(
t
j
t
k
)
α
-
1
C
j
k
κ
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
t
k
α
-
1
C
j
k
κ
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
t
k
α
-
1
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
C
j
k
]
[
β
δ
]
=
[
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
t
k
α
-
1
ω
j
C
j
k
∑
j
=
1
,
k
=
1
j
=
m
,
k
=
m
ω
j
C
j
k
]
,
where
κ
=
γ
tan
(
α
π
2
)
,
C equals W −1 where W is the m×matrix with elements
W
s
r
=
1
2
(
R
s
2
+
I
s
2
)
(
R
r
2
+
I
r
2
)
t
s
t
r
{
R
s
+
r
(
I
s
I
r
-
R
s
R
r
)
+
R
s
-
r
(
I
s
I
r
+
R
s
R
r
)
+
I
s
+
r
(
I
s
R
r
+
R
s
I
r
)
+
I
s
-
r
(
I
s
R
r
-
R
s
I
r
)
-
4
R
s
I
s
R
r
I
r
}
where
R s =Re (φ(t s )), I s =Im (φ( t s )), R s+r =Re (φ( t s +t r )),
where
φ
(
t
)
=
exp
(
-
γ
t
α
(
1
+
j
β
′
tan
α
π
2
)
+
j
t
δ
′
)
for existing estimates β′,δ′ of the skew and location parameters, and
ω k =−Im log({circumflex over (φ)}(t k ))/t k for the sequence of noise samples.
35. An apparatus for reducing impulse noise in a signal processing system, comprising:
a parameter estimation module for estimating parameters of an alpha-stable distribution; the alpha-stable distribution modeling impulse noise that corrupts data signals input into a transmission medium of the signal processing system;
a memory for accumulating sampled signals output from the transmission medium; the sampled signals having a noise component and a data component; and
a signal estimation module for computing an estimate of the data component of the sampled signals output from the transmission medium using the estimated parameters of the alpha-stable distribution.
36. The apparatus according to claim 35 , wherein said signal estimation module further comprises:
a prediction filter for estimating, using model coefficients, the data components of the sampled signals output from the transmission medium; the estimated data components corresponding to an estimation of the data signals input into the transmission medium; and
a coefficient optimization module for optimizing the model coefficients of the prediction filter using one of the estimated parameters of the alpha-stable distribution received from said parameter estimation module.
37. The apparatus according to claim 36 , wherein the dependence of the prediction filter output on the model coefficients is linear.
38. The apparatus according to claim 37 , wherein said prediction filter is a Volterra filter.
39. The apparatus according to claim 36 , wherein the dependence of the prediction filter output on the data components of the sampled signals input into said prediction filter is non-linear.
40. The apparatus according to claim 36 , wherein said parameter estimation module further comprises:
means for computing moments of the alpha-stable distribution; and
means for estimating the parameters of the alpha-stable distribution with the computed moments.
41. The apparatus according to claim 40 , wherein said parameters estimation module further comprises means for transforming the alpha-stable distribution to obtain deskewed alpha-stable random variables.
42. The apparatus according to claim 40 , wherein said parameters estimation module further comprises means for transforming the alpha-stable distribution to obtain centralized alpha-stable random variables.
43. The apparatus according to claim 36 , wherein said parameter estimation module adaptively estimates the parameters of the alpha-stable distribution.
44. The apparatus according to claim 36 , wherein said coefficient optimization module uses the parameters of the alpha-stable distribution to specify a minimum dispersion error criterion for determining the model coefficients of said prediction filter.
45. The apparatus according to claim 44 , wherein said coefficient optimization module minimizes a cost function defined by the minimum dispersion error criterion to optimize the model coefficients of said prediction filter.
46. The apparatus according to claim 45 , wherein said coefficient optimization module minimizes a p th -power error criterion to optimize the model coefficients of said prediction filter.
47. The apparatus according to claim 36 , wherein transmission medium into which the data signals are input is a twisted pair.
48. The apparatus according to claim 35 , wherein the signal processing system operates in a digital subscriber line (DSL).
49. The apparatus according to claim 35 , wherein the signal processing system operates in an imaging device.