Data processing: measuring – calibrating – or testing – Measurement system – Measured signal processing
Reexamination Certificate
2000-06-05
2002-08-06
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Measured signal processing
C702S191000
Reexamination Certificate
active
06430525
ABSTRACT:
FIELD OF THE INVENTION
The present invention is directed to the field of signal processing, and, more particularly, is directed to systems and methods for signal averaging.
BACKGROUND OF THE INVENTION
Digital signal processing techniques are frequently employed to enhance a desired signal in a wide variety of applications, such as health care, communications and avionics, to name a few. Signal enhancement includes smoothing, filtering and prediction. These processing techniques each operate on a block of input signal values in order to estimate the signal at a specific point in time.
FIG. 1
illustrates that smoothing, filtering and prediction can be distinguished by the time at which an output value is generated relative to input values. Shown in
FIG. 1
is a time axis
100
and a block
101
of input signal values depicted in this example as occurring within a time window between points t
min
and t
max
. Specifically, the block
101
includes a set of discrete input values {v
i
; i=1, 2, . . . n} occurring at a corresponding set of time points {t
i
; i=1, 2, . . . n}. A smoother operates on the block
101
of input values to estimate the signal at a time point, t
s
102
between t
min
and t
max
. That is, a smoother generates an output value based upon input values occurring before and after the output value. A filter operates on the block
101
of input values to estimate the signal at a time t
f
104
, corresponding to the most recently occurring input value in the block
101
. That is, a filter generates a forward filtered output value at the time t
f
based upon input values occurring at, and immediately before, the output value. A filter also operates on the block
101
to estimate the signal at a time t
b
105
at the beginning of the block
101
to generate a backward filtered value. A forward predictor operates on the block of input values
101
to estimate the signal at time t
pf
106
, which is beyond the most recently occurring value in the block
101
. That is, a forward predictor generates a forward predicted output value based upon input values occurring prior to the output value. A backward predictor operates on the block
101
of input values to estimate the signal at time t
pb
108
, which is before the earliest occurring value in the block
101
. That is, a backward predictor generates a backward predicted output value based upon input values occurring after the output value.
SUMMARY OF THE INVENTION
A common smoothing technique uses an average to fit a constant, v
A
, to a set of data values, {v
i
; i=1, 2, . . . , n}:
v
A
=
1
n
·
∑
i
=
1
n
v
i
(
1
)
A generalized form of equation (1) is the weighted average
v
WA
=
∑
i
=
1
n
w
i
·
v
i
∑
i
=
1
n
w
i
(
2
)
Here, each value, v
i
, is scaled by a weight, w
i
, before averaging. This allows data values to be emphasized and de-emphasized relative to each other. If the data relates to an input signal, for example, values occurring during periods of low signal confidence can be given a lower weight and values occurring during periods of high signal confidence can be given a higher weight.
FIG. 2A
illustrates the output of a constant mode averager, which utilizes the weighted average of equation (2) to process a discrete input signal, {v
i
; i an integer}
110
. The input signal
110
may be, for example, a desired signal corrupted by noise or a signal having superfluous features. The constant mode averager suppresses the noise and unwanted features, as described with respect to
FIG. 5
, below. A first time-window
132
defines a first set, {v
i
; i=1, 2, . . . , n}, of signal values, which are averaged together to produce a first output value, z
1
122
. A second time-window
134
, shifted from the previous window
132
, defines a second set {v
i
; i=2, 3, . . . , n+1}of signal values, which are also averaged together to produce a second output value z
2
124
. In this manner, a discrete output signal, {z
j
; j an integer}
120
is generated from a moving weighted average of a discrete input signal {v
i
; i an integer}
110
, where:
z
j
=
∑
i
=
j
n
+
j
-
1
w
i
v
i
/
∑
i
=
j
n
+
j
-
1
w
i
(
3
)
A common filtering technique computes a linear fit to a set of data values, {v
i
; i=1, 2, . . . , n}:
{circumflex over (v)}
i
=&agr;·t
i
+&bgr; (4)
where &agr; and &bgr; are constants and t
i
is the time of occurrence of the i
th
value.
FIG. 2B
illustrates the output of a linear mode averager, which uses the linear fit of equation (4) to process a discrete input signal, {v
i
; i an integer}
110
. The input signal
110
may be, for example, a desired signal with important features corrupted by noise. The linear mode averager reduces the noise but tracks the important features, as described with respect to
FIG. 6
below. A first time-window
132
defines a first set, {v
i
; i=1, 2, . . . , n}, of signal values. A linear fit to these n values is a first line
240
, and the value along this line at max {t
1
, t
2
, . . . , t
n
} is equal to a first output value, z
1
222
. A second time-window
134
shifted from the previous window
132
defines a second set, {v
i
; i=2, 3, . . . , n+1 }, of signal values. A linear fit to these n values is a second line
250
, and the value along this line at max {t
2
, t
3
, . . . , t
n
+1} is equal to a second output value, z
2
224
. In this manner, a discrete output signal, {z
j
; j an integer}
220
is generated from a moving linear fit of a discrete input signal {v
i
; i an integer}, where:
z
j
=
α
j
·
t
n
+
j
-
1
MAX
+
β
j
(5a)
t
n
+
j
-
1
MAX
=
max
{
t
j
,
t
j
+
1
,
…
,
t
n
+
j
-
1
}
(5b)
In general, the time windows shown in
FIGS. 2A-2B
may be shifted from each other by more than one input value, and values within each time window may be skipped, i.e., not included in the average. Further, the t
i
's may not be in increasing or decreasing order or uniformly distributed, and successive time windows may be of different sizes. Also, although the discussion herein refers to signal values as the dependent variable and to time as the independent variable to facilitate disclosure of the present invention, the concepts involved are equally applicable where the variables are other than signal values and time. For example, an independent variable could be a spatial dimension and a dependent variable could be an image value.
The linear mode averager described with respect to
FIG. 2B
can utilize a “best” linear fit to the input signal, calculated by minimizing the mean-squared error between the linear fit and the input signal. A weighted mean-squared error can be described utilizing equation (4) as:
ϵ
(
α
,
β
)
=
∑
i
=
1
n
w
i
(
v
i
-
v
^
i
)
2
/
∑
i
=
1
n
w
i
(6a)
ϵ
(
α
,
β
)
=
∑
i
=
1
n
w
i
[
v
i
-
(
α
·
t
i
+
β
)
]
2
/
∑
i
=
1
n
w
i
(6b)
Conventionally, the least-mean-squared (LMS) error is calculated by setting the partial derivatives of equation (6b) with respect to &agr; and &bgr; to zero:
∂
∂
α
ϵ
(
α
,
β
)
=
0
(7a)
∂
∂
β
ϵ
(
α
,
β
)
=
0
(7b)
Substituting equation (6b) into equation (7b) and taking the derivative yields:
-
2
∑
i
=
1
n
w
i
[
v
i
-
(
α
·
t
i
+
β
)
]
/
∑
i
=
1
n
w
i
=
0
(8)
Solving equation (8) for &bgr; and substituting the expression of equation (2) yields:
β
=
∑
i
=
1
n
w
i
·
v
i
∑
i
=
1
n
w
i
-
&al
Al-Ali Ammar
Weber Walter M.
Hoff Marc S.
Knobbe Martens Olson & Bear LLP
Masimo Corporation
Miller C Steven
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