# US 8,150,643 B1 · Method of detecting system function by measuring frequency response

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- Number: US 8,150,643 B1
- Document ID: US8150643B1
- Status: Granted
- Kind: B1
- Application number: 12/217,013
- Filed: 2008-06-30
- Granted: 2012-04-03
- Earliest claimed: 2004-12-20
- Assignees: Battelle Energy Alliance, LLC; Montana Tech of the University of Montana; Qualtech Systems, Inc.
- Inventors: John L. Morrison; William H. Morrison; Jon P. Christophersen
- Web page: https://patentgrep.com/patent/US8150643B1

## CPC

- G01R 31/389: Measuring internal impedance, internal conductance or related variables
- G01R: MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES (indicating correct tuning of resonant circuits H03J3/12)

## Abstract

Real-time battery impedance spectrum is acquired using a one-time record. Fast Summation Transformation (FST) is a parallel method of acquiring a real-time battery impedance spectrum using a one-time record that enables battery diagnostics. An excitation current to a battery is a sum of equal amplitude sine waves of frequencies that are octave harmonics spread over a range of interest. A sample frequency is also octave and harmonically related to all frequencies in the sum. The time profile of this signal has a duration that is a few periods of the lowest frequency. The voltage response of the battery, average deleted, is the impedance of the battery in the time domain. Since the excitation frequencies are known and octave and harmonically related, a simple algorithm, FST, processes the time record by rectifying relative to the sine and cosine of each frequency. Another algorithm yields real and imaginary components for each frequency.

## Claims

### Claim 1

A method for detecting function of a unit under test by measuring frequency response the method comprising; the acts of:
setting measurement conditions comprising choosing a Root Mean Square (RMS) level of current, identifying a start frequency, and choosing a period of a lowest frequency,
selecting a number of octave harmonic frequencies relative to the start frequency over which the function of the unit under test will be tested, wherein the number of octave harmonic frequencies is one or more;
assembling an excitation time record including a sum of sinusoids of the number of octave harmonic frequencies and a duration of the period of the lowest frequency;
conditioning the excitation time record to be compatible with the measurement conditions;
exciting the unit under test with the excitation time record and simultaneously capturing a resulting response time record with a data acquisition system having a sample frequency that is octave related and harmonically related to the frequencies within the sum of sinusoids and at least four times the highest frequency in the sum of sinusoids;
rectifying the response time record relative to a sine at each frequency of the octave harmonic frequencies with a first sine rectification sum m11, comprising:
m
11
=
sum
=
∑
n
=
0
N
1
2
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
1
2
N
1
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
Where: TR is a captured response time record with discrete time index n and duration of one period of a first frequency f1
Ni is a number of sample points for a period of the first frequency f1 wherein for each sum of sinusoid frequencies fi, Ni is constrained as log2 (Ni) and is an integer greater than 1 and wherein the frequency of interest is given as:
f
i
=
1
N
i
⁢
Δ
⁢
⁢
t
wherein Δt is a sample period
summing the sine rectification sum for each frequency of interest of the octave harmonic frequencies to create a sine-rectified response time record sum, normalizing the sine-rectified response time record sum to a number of periods known to be within the sine-rectified response time record sum for each frequency, and storing a normalized sine sum result, m1,i, where i is an ith frequency of interest;
rectifying the response time record relative to a cosine at each frequency of the octave harmonic frequencies with a first cosine rectification sum m21, comprising:
m
21
=
sum
=
∑
n
=
0
N
1
4
-
1
⁢
⁢
T
⁢
⁢
R
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
1
4
3
⁢
N
1
4
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
1
4
N
1
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
summing the cosine rectification sum for each frequency of interest of the octave harmonic frequencies to create a cosine-rectified response time record sum, normalizing the cosine-rectified response time record sum to the number of periods known to be within the cosine-rectified response time record sum for each frequency, and storing a normalized cosine sum result, m2,i, where i is the ith frequency of interest;
determining a real and an imaginary form of a magnitude and a phase of each of the octave harmonic frequencies:
[
V
i
⁢
cos
⁢
⁢
ϕ
i
V
i
⁢
sin
⁢
⁢
ϕ
i
]
=
[
K
4
K
1
⁢
K
4
-
K
2
⁢
K
3
,
K
2
K
2
⁢
K
3
-
K
1
⁢
K
4
K
3
K
2
⁢
K
3
-
K
1
⁢
K
4
,
K
1
K
1
⁢
K
4
-
K
2
⁢
K
3
]
⁡
[
m
1
,
i
m
2
,
i
]
Where:
Vi is a magnitude of the voltage for the frequency fi
φi is a phase of the voltage for the frequency fi
m1,i, m2,i are rectification sums for the frequency fi
K
1
=
∑
n
=
0
N
i
2
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
2
N
i
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
2
=
∑
n
=
0
N
i
2
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
2
N
i
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
3
=
∑
n
=
0
N
i
4
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
4
3
⁢
N
i
4
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
+
∑
n
=
3
⁢
N
i
4
N
i
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
4
=
∑
n
=
0
N
i
4
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
4
3
⁢
N
i
4
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
+
∑
n
=
3
⁢
N
i
4
N
i
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
and, obtaining the magnitude and the phase via a standard rectangular to polar conversion:
Let:
⁢
⁢
V
i
⁢
sin
⁢
⁢
ϕ
i
=
C
1
V
i
⁢
cos
⁢
⁢
ϕ
i
=
C
2
,
then:
⁢
⁢
V
i
=
C
1
2
+
C
2
2
ϕ
i
=
tan
-
1
⁡
(
C
1
C
2
)
⁢
assembling the magnitude and the phase at each of the octave harmonic frequencies to determine the frequency response.

### Claim 2 (depends on 1)

The method of claim 1, wherein rectifying the response time record relative to the sine at each of the octave harmonic frequencies is performed for signals that pass through zero at a discrete time step, wherein N is a number of samples over the period:
Rs
⁡
(
n
)
=
[
0
,
n
=
0
1
,
0
<
n
<
N
2
0
,
n
=
N
2
-
1
,
N
2
<
n
<
N
]
.

### Claim 3 (depends on 1)

The method of claim 1, wherein rectifying the response time record relative to the cosine at each of the octave harmonic frequencies is performed for signals that pass through zero at a discrete time step, wherein N is a number of samples over the period:
Rc
⁡
(
n
)
=
[
1
,
0
≤
n
<
N
4
0
,
n
=
N
4
-
1
,
N
4
<
n
<
3
⁢
N
4
0
,
n
=
3
⁢
N
4
1
3
⁢
N
4
<
n
<
N
]
.

### Claim 4 (depends on 1)

The method of claim 1, further comprising the acts of:
repeating the original acts of; selecting the number of octave harmonic frequencies, assembling the excitation time record, conditioning the excitation time record, exciting the unit under test, rectifying the response time record relative to the sine at each frequency, summing the sine rectification sum for each frequency of interest, rectifying the response time record relative to the cosine at each frequency, summing the cosine rectification sum for each frequency of interest, determining the real and the imaginary form, and assembling the magnitude and the phase, at least once, wherein the octave harmonic frequencies are shifted in the repeated acts relative to the original acts; and
interleaving the frequency response of the original acts with the frequency responses of the repeated acts to obtain a higher resolution frequency response.

### Claim 5 (depends on 1)

The method of claim 1, wherein:
rectifying the response time record relative to a sine at each frequency of the octave harmonic frequencies includes a second sine rectification sum m12, comprising;
sum
=
∑
n
=
0
N
2
2
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
2
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
N
2
+
N
2
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
+
N
2
2
2
⁢
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
m
12
=
sum
2
;
and
rectifying the response time record relative to a cosine at each frequency of the octave harmonic frequencies includes a second cosine rectification sum m22, comprising;
sum
=
∑
n
=
0
N
2
4
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
4
3
⁢
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
2
4
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
N
2
+
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
+
N
2
4
N
2
+
3
⁢
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
+
3
⁢
N
2
4
2
⁢
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
⁢
⁢
m
22
=
sum
2
;
where: N2 is a number of sample points in a second frequency at an octave harmonic of the first frequency and 2N2=N1.

### Claim 6 (depends on 5)

The method of claim 5, wherein:
rectifying the response time record relative to a sine at each frequency of the octave harmonic frequencies includes a third sine rectification sum m13, comprising;
sum
=
∑
n
=
0
N
3
2
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
2
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
+
N
3
2
2
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
2
⁢
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
2
⁢
N
3
+
N
3
2
3
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
3
⁢
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
3
⁢
N
3
+
N
3
2
4
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
m
13
=
sum
4
;
and
rectifying the response time record relative to a cosine at each frequency of the octave harmonic frequencies includes a third cosine rectification sum m23, comprising, comprising:
sum
=
∑
n
=
0
N
3
4
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
4
3
⁢
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
4
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
N
3
+
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
+
N
3
4
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
+
3
⁢
N
3
4
2
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
2
⁢
N
3
⁢
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
2
⁢
N
3
+
N
3
4
2
⁢
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
+
3
⁢
N
3
4
3
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
3
⁢
N
3
+
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
3
⁢
N
3
+
N
3
4
3
⁢
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
+
3
⁢
N
3
4
4
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
⁢
⁢
m
23
=
sum
4
;
where: N3 is a number of sample points in a third frequency at an octave harmonic of the first frequency and 4N3=N1.

### Claim 7 (depends on 6)

The method of claim 6, comprising:
rectifying the response time record relative to a sine at each frequency of the octave harmonic frequencies includes at least one additional sine rectification sum m1n; and
rectifying the response time record relative to a cosine at each frequency of the octave harmonic frequencies includes at least one additional cosine rectification sum m2n; wherein a last frequency includes 2n-1 periods within a period of the first frequency.

### Claim 8

A method of calibrating phase response of a captured time record for a unit under test by measuring frequency response of a reference standard the method comprising the acts of:
setting measurement conditions comprising choosing a Root Mean Square (RMS) level of current, identifying a start frequency, and choosing a period of a lowest frequency;
determining a theoretical frequency response for the reference standard;
selecting a number of frequencies and a frequency spread over which the reference standard will be tested, wherein the frequency spread is by octaves;
assembling a phase-shifted excitation time record including a sum of sinusoids of the number of frequencies with a pre-selected phase shift, wherein a duration of the excitation time record is greater than or equal to one period of a lowest of the number of frequencies;
conditioning the phase-shifted excitation time record to be compatible with the measurement conditions;
exciting the reference standard with the phase-shifted excitation time record and simultaneously capturing a response time record with a data acquisition system having an appropriate sample frequency;
processing the response time record to obtain estimated magnitude and estimated phase at each of the number of frequencies to determine the frequency response;
wherein, phase calibration is achieved through a selection of multiple known phase-shifts until enough points are obtained to determine phase calibration constants for each of the number of frequencies using a regression analysis.

### Claim 9 (depends on 8)

The method of claim 8, wherein the act of processing the response time record includes:
rectifying the response time record relative to a sine at each of the number of frequencies, with a first sine rectification sum m11, comprising:
m
11
=
sum
=
∑
n
=
0
N
1
2
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
1
2
N
1
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
Where: TR is a captured voltage time record with discrete time index n and duration of one period of a first frequency f1
N1 is a number of sample points for a first frequency f1 wherein for each sum of sinusoid frequencies fi, Ni is constrained as log2 (Ni) and is an integer greater than 1 and wherein the frequency of interest is given as:
f
i
=
1
N
i
⁢
Δ
⁢
⁢
t
wherein Δt is a sample period
summing the sine rectification sum for each frequency of interest of the number of frequencies to create a sine-rectified response time record sum, normalizing the sine-rectified response time record sum to a number of periods known to be within the sine-rectified response time record sum for each frequency, and storing a normalized sine sum result, m1,i, where i is the ith frequency of interest;
rectifying the response time record relative to the cosine at each of the number of frequencies, with a first cosine rectification sum m21, comprising:
m
21
=
sum
=
∑
n
=
0
N
1
4
-
1
⁢
⁢
T
⁢
⁢
R
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
1
4
3
⁢
N
1
4
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
1
4
N
1
-
1
⁢
⁢
T
⁢
⁢
R
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
summing the cosine rectification sum for each of the number of frequencies to create a cosine-rectified response time record sum, normalizing the cosine-rectified response time record sum to the number of periods known to be within the cosine-rectified time record sum for each frequency, and storing a normalized cosine sum result, m2,i, where i is an ith frequency of interest;
determining the real and imaginary form of the magnitude and phase of each octave harmonic frequency:
[
V
i
⁢
cos
⁢
⁢
ϕ
i
V
i
⁢
sin
⁢
⁢
ϕ
i
]
=
[
K
4
K
1
⁢
K
4
-
K
2
⁢
K
3
,
K
2
K
2
⁢
K
3
-
K
1
⁢
K
4
K
3
K
2
⁢
K
3
-
K
1
⁢
K
4
,
K
1
K
1
⁢
K
4
-
K
2
⁢
K
3
]
⁡
[
m
1
,
i
m
2
,
i
]
Where:
Vi is a magnitude of the voltage for the frequency fi
φi is a phase of the voltage for the frequency fi
m1,i, m2,i are rectification sums for the frequency fi
K
1
=
∑
n
=
0
N
i
2
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
2
N
i
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
2
=
∑
n
=
0
N
i
2
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
2
N
i
-
1
⁢
(
cos
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
3
=
∑
n
=
0
N
i
4
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
4
3
⁢
N
i
4
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
+
∑
n
=
3
⁢
N
i
4
N
i
-
1
⁢
(
sin
⁡
(
2
⁢
π
N
i
⁢
n
)
)
K
4
=
∑
n
=
0
N
i
4
-
1
⁢
⁢
(
cos
⁡
(
2
⁢
⁢
π
N
i
⁢
n
)
)
-
∑
n
=
N
i
4
3
⁢
N
i
4
-
1
⁢
(
cos
⁡
(
2
⁢
⁢
π
N
i
⁢
n
)
)
+
∑
n
=
3
⁢
N
i
4
N
i
-
1
⁢
(
cos
⁡
(
2
⁢
⁢
π
N
i
⁢
n
)
)
and, obtaining the magnitude and the phase via a standard rectangular to polar conversion:
Let
⁢
:
⁢
⁢
V
i
⁢
sin
⁢
⁢
ϕ
i
=
⁢
C
1
V
i
⁢
cos
⁢
⁢
ϕ
i
=
⁢
C
2
,
⁢
then
⁢
:
⁢
⁢
V
i
=
C
1
2
+
C
2
2
ϕ
i
=
tan
-
1
⁡
(
C
1
C
2
)
.
⁢
⁢

### Claim 10 (depends on 9)

The method of claim 9, wherein:
rectifying the response time record relative to a sine at each of the number of frequencies includes a second sine rectification sum m12, comprising:
sum
=
∑
n
=
0
N
2
2
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
2
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
N
2
+
N
2
2
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
+
N
2
2
2
⁢
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
m
12
=
sum
2
;
and
rectifying the response time record relative to a cosine at each of the number of frequencies includes a second cosine rectification sum m22, comprising:
sum
=
∑
n
=
0
N
2
4
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
4
3
⁢
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
2
4
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
N
2
+
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
2
+
N
2
4
N
2
+
3
⁢
N
2
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
2
+
3
⁢
N
2
4
2
⁢
N
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
m
22
=
sum
2
;
where: N2 is a number of sample points in a second frequency at an octave harmonic of the first frequency and 2N2=N1.

### Claim 11 (depends on 10)

The method of claim 10, wherein:
rectifying the response time record relative to a sine at each of the number of frequencies includes a third sine rectification sum m13, comprising:
sum
=
∑
n
=
0
N
3
2
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
2
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
+
N
3
2
2
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
2
⁢
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
2
⁢
N
3
+
N
3
2
3
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
3
⁢
N
3
+
N
3
2
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
3
⁢
N
3
+
N
3
2
4
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
m
13
=
sum
4
;
and
rectifying the response time record relative to a cosine at each of the number of frequencies includes a third cosine rectification sum m23, comprising, comprising:
sum
=
∑
n
=
0
N
3
4
-
1
⁢
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
4
3
⁢
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
4
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
N
3
+
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
N
3
+
N
3
4
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
N
3
+
3
⁢
N
3
4
2
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
2
⁢
N
3
+
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
2
⁢
N
3
+
N
3
4
2
⁢
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
2
⁢
N
3
+
3
⁢
N
3
4
3
⁢
N
3
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
3
⁢
N
3
+
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
-
∑
n
=
3
⁢
N
3
+
N
3
4
3
⁢
N
3
+
3
⁢
N
3
4
-
1
⁢
TR
⁢
(
n
⁢
⁢
Δ
⁢
⁢
t
)
+
∑
n
=
3
⁢
N
3
+
3
⁢
N
3
4
4
⁢
N
3
-
1
⁢
TR
⁡
(
n
⁢
⁢
Δ
⁢
⁢
t
)
⁢
⁢
m
23
=
sum
4
;
where: N3 is a number of sample points in a third frequency at an octave harmonic of the first frequency and 4N3=N1.

### Claim 12 (depends on 11)

The method of claim 11, wherein:
rectifying the response time record relative to a sine at each of the number of frequencies includes at least one additional sine rectification sum m1n; and
rectifying the response time record relative to a cosine at each of the number of frequencies includes at least one additional sine rectification sum m2n;
wherein a last frequency includes 2n-1 periods within a period of the first frequency.