# US 7,613,758 B2 · Method for optimization of Q-Filter kernel parameters

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- Number: US 7,613,758 B2
- Document ID: US7613758B2
- Status: Granted
- Kind: B2
- Application number: 11/179,104
- Filed: 2005-07-12
- Published: 2006-08-31
- Granted: 2009-11-03
- Earliest claimed: 2005-02-16
- Assignees: Motorola, Inc.
- Inventors: Weimin Xiao; Magdi A. Mohamed
- Web page: https://patentgrep.com/patent/US7613758B2

## CPC

- H03H 17/0261: {Non linear filters}
- H03H 17/0294: {Variable filters; Programmable filters}
- H03H: IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS (waveguides, resonators, lines or other devices of the waveguide type H01P)

## Abstract

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.

## Claims

### Claim 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.

### Claim 2 (depends on 1)

The method of claim 1 wherein R is an average squared error of (t-t*).

### Claim 3 (depends on 1)

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.

### Claim 4 (depends on 3)

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, Qk=Q(xk, 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
.

### Claim 5 (depends on 4)

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 xi, so that x(1)≦x(2). . . ≦x(n);and Ψi are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψn+1=0
Ψn=f(n)
Ψi=f(i)+Ψi+1+λf(i)Ψi+1.

### Claim 6 (depends on 1)

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*.

### Claim 7 (depends on 6)

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
.

### Claim 8 (depends on 7)

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, Qk=Q(xk, 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
.

### Claim 9 (depends on 8)

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 xi, so that x(1)≦x(2). . .≦x(n); and Ψi are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψn+1=0
Ψn=f(n)
Ψi=f(i)+Ψi+1+λf(i)Ψi+1.

### Claim 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.

### Claim 11 (depends on 10)

The apparatus of claim 10 wherein R is an average squared error of (t-t*).

### Claim 12 (depends on 10)

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.

### Claim 13 (depends on 12)

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, Qk=Q(xk, 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
.

### Claim 14 (depends on 13)

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 xi, so that x(1)≦x(2). . . ≦x(n); and Ψi are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψn+1=0
Ψn=f(n)
Ψi=f(i)+Ψi+1+λf(i)Ψi+1.

### Claim 15 (depends on 10)

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*.

### Claim 16 (depends on 15)

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
.

### Claim 17 (depends on 16)

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, Qk=Q(xk, 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
.

### Claim 18 (depends on 17)

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 xi, so that x(1)≦x(2). . . ≦x(n); and Ψi are computed recursively for i=n+1, n, . . . , 1, as follows:
Ψn+1=0
Ψn=f(n)
Ψi=f(i)+Ψi+1+λf(i)Ψi+1.