Method for optimization of Q-Filter kernel parameters
View Patent ↗A Q-Filter is a reconfigurable technique that performs a continuum of linear and nonlinear filtering operations. It is modeled by unique mathematical structure, utilizing a function called the Q-Measure, defined using a set of adjustable kernel parameters to enable efficient hardware and software implementations of a variety of useful, new and conventional, filtering operations. The Q-Measure is based on an extension of the well-known Sugeno λ-Measure. In order to optimize the Q-Filter kernel parameters, the value of an error function is minimized. The error function is based on difference between the filtered signal and target signal, with the target signal being a desired result of filtering.
1. A method for filtering a signal using a Q-filter, the method comprising the steps of:
receiving with a microprocessor, values for λ, f, and t, wherein λ comprises a q-measure scalar parameter value, f comprises a q-measure vector of density generator parameter values, t comprises a vector of desired target signals; and
optimizing via the microprocessor filtering parameters by minimizing an error function R, wherein R is a function of input signal vector X, λ, f, t; and t*, where t* comprises a vector of filtered signals; and
filtering via the microprocessor the signal using the optimized filter parameters.
2. The method of claim 1 wherein R is an average squared error of (t-t*).
3. The method of claim 1 wherein the step of minimizing R(X, λ, f, t, t*) comprises the step of finding values for ∂R/∂λ and ∂R/∂f, where
R
=
1
2
M
∑
k
=
1
M
(
Q
(
x
k
,
λ
,
f
)
-
t
k
)
2
,
and M is a number of training samples.
4. The method of claim 3 wherein:
∂
R
∂
λ
=
1
M
∑
k
=
1
M
(
Q
k
-
t
k
)
∂
Q
k
∂
λ
∂
R
∂
f
i
=
1
M
∑
k
=
1
M
(
Q
k
-
t
k
)
∂
Q
k
∂
f
i
wherein, Q k =Q(x k , λ, f); and
∂
Q
k
∂
λ
,
∂
Q
k
∂
f
i
can be calculated numerically as,
∂
Q
k
∂
λ
=
Q
k
(
λ
+
Δλ
)
-
Q
k
(
λ
)
Δλ
∂
Q
k
∂
f
i
=
Q
k
(
f
i
+
Δ
f
i
)
-
Q
k
(
f
)
Δ
f
i
.
5. The method of claim 4 wherein:
Q
(
x
,
λ
,
f
)
=
∑
i
=
1
n
x
(
i
)
Δ
q
i
=
1
ψ
1
∑
i
=
1
n
x
(
i
)
f
(
i
)
(
1
+
λψ
i
+
1
)
,
and
wherein n is a window size of the Q-Filter; x (i) is a sorted sequence of x i , so that x (1) ≦x (2) . . . ≦x (n) ;and Ψ i are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψ n+1 =0
Ψ n =f (n)
Ψ i =f (i) +Ψ i+1 +λf (i) Ψ i+1 .
6. The method of claim 1 further comprising the step of:
receiving via the microprocessor linear-regression coefficients for α and β; and
wherein the step of optimizing comprises the step of minimizing R wherein R is a function of α, β, X, λ, f, t and t*.
7. The method of claim 6 wherein the step of minimizing R(α, β, X, λ, f, t)
comprises the step of finding values for ∂R/∂λ and ∂R/∂f, ∂R/∂α, and ∂R/∂β, where
R
=
1
2
M
∑
k
=
1
M
(
α
Q
(
x
k
,
λ
,
f
)
+
β
-
t
k
)
2
.
8. The method of claim 7 wherein:
∂
R
∂
λ
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
α
∂
Q
k
∂
λ
∂
R
∂
f
i
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
α
∂
Q
k
∂
f
i
∂
R
∂
α
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
Q
k
∂
R
∂
β
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
wherein, Q k =Q(x k , λ, f); and
∂
Q
k
∂
λ
,
∂
Q
k
∂
f
i
can be calculated numerically as,
∂
Q
k
∂
λ
=
Q
k
(
λ
+
Δλ
)
-
Q
k
(
λ
)
Δλ
∂
Q
k
∂
f
i
=
Q
k
(
f
i
+
Δ
f
i
)
-
Q
k
(
f
)
Δ
f
i
.
9. The method of claim 8 wherein:
Q
(
x
,
λ
,
f
)
=
∑
i
=
1
n
x
(
i
)
Δ
q
i
=
1
ψ
1
∑
t
=
1
n
x
(
i
)
f
(
i
)
(
1
+
λψ
i
+
1
)
wherein n is the window size of the Q-Filter; x (i) is a sorted sequence of x i , so that x (1) ≦x (2) . . .≦x (n) ; and Ψ i are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψ n+1 =0
Ψ n =f (n)
Ψ i =f (i) +Ψ i+1 +λf (i) Ψ i+1 .
10. An apparatus comprising:
a microprocessor configured for receiving values for λ, f, and t, wherein λ comprises a q-measure scalar parameter value, f comprises a q-measure vector of density generator parameter values, t comprises a vector of desired target signals, the microprocessor also configured for optimizing filter parameters by minimizing an error function R, wherein R is a function of input signal vector X, λ, f, t; and t*, where t* comprises a vector of filtered signals; and
a filter coupled to the filter parameter optimizer for filtering a signal using the optimized filter parameters.
11. The apparatus of claim 10 wherein R is an average squared error of (t-t*).
12. The apparatus of claim 10 wherein the minimizing of R(X, λ, f, t, t*) takes place by finding local minimum of ∂R/∂λ and ∂R/∂f, where
R
=
1
2
M
∑
k
=
1
M
(
Q
(
x
k
,
λ
,
f
)
-
t
k
)
2
,
and M is a number of training samples.
13. The apparatus of claim 12 wherein:
∂
R
∂
λ
=
1
M
∑
k
=
1
M
(
Q
k
-
t
k
)
∂
Q
k
∂
λ
∂
R
∂
f
i
=
1
M
∑
k
=
1
M
(
Q
k
-
t
k
)
∂
Q
k
∂
f
i
wherein, Q k =Q(x k , λ, f); and
∂
Q
k
∂
λ
,
∂
Q
k
∂
f
i
can be calculated numerically as,
∂
Q
k
∂
λ
=
Q
k
(
λ
+
Δλ
)
-
Q
k
(
λ
)
Δλ
∂
Q
k
∂
f
i
=
Q
k
(
f
i
+
Δ
f
i
)
-
Q
k
(
f
)
Δ
f
i
.
14. The apparatus of claim 13 wherein:
Q
(
x
,
λ
,
f
)
=
∑
i
=
1
n
x
(
i
)
Δ
q
i
=
1
ψ
1
∑
i
=
1
n
x
(
i
)
f
(
i
)
(
1
+
λψ
i
+
1
)
,
and
wherein n is a window size of the Q-Filter; x (i) is a sorted sequence of x i , so that x (1) ≦x (2) . . . ≦x (n) ; and Ψ i are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψ n+1 =0
Ψ n =f (n)
Ψ i =f (i) +Ψ i+1 +λf (i) Ψ i+1 .
15. The apparatus of claim 10 wherein the microprocessor configured for receives linear-regression coefficients for α and β and minimizes R wherein R is a function of α, β, X, λ, f, t and t*.
16. The apparatus of claim 15 wherein R(α, β, X, λ, f, t) is minimized by finding local minimum for ∂R/∂λ and ∂R/∂f, ∂R/∂α, and ∂R/∂β, where
R
=
1
2
M
∑
k
=
1
M
(
aQ
(
x
k
,
λ
,
f
)
+
β
-
t
k
)
2
.
17. The apparatus of claim 16 wherein:
∂
R
∂
λ
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
α
∂
Q
k
∂
λ
∂
R
∂
f
i
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
α
∂
Q
k
∂
f
i
∂
R
∂
α
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
Q
k
∂
R
∂
β
=
1
N
∑
k
=
1
N
(
α
Q
k
+
β
-
t
k
)
wherein, Q k =Q(x k , λ, f); and
∂
Q
k
∂
λ
,
∂
Q
k
∂
f
i
can be calculated numerically as,
∂
Q
k
∂
λ
=
Q
k
(
λ
+
Δ
λ
)
-
Q
k
(
λ
)
Δ
λ
∂
Q
k
∂
f
i
=
Q
k
(
f
i
+
Δ
f
i
)
-
Q
k
(
f
)
Δ
f
i
.
18. The apparatus of claim 17 wherein:
Q
(
x
,
λ
,
f
)
=
∑
i
=
1
n
x
(
i
)
Δ
q
i
=
1
ψ
1
∑
i
=
1
n
x
(
i
)
f
(
i
)
(
1
+
λ
ψ
i
+
1
)
wherein n is the window size of the Q-Filter; x (i) is a sorted sequence of x i , so that x (1) ≦x (2) . . . ≦x (n) ; and Ψ i are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψ n+1 =0
Ψ n =f (n)
Ψ i =f (i) +Ψ i+1 +λf (i) Ψ i+1 .