Signal processing for accelerating moving targets
View Patent ↗A method for performing signal processing for accelerating moving targets, in one implementation, encompasses a method for performing coherent integration of pulses within a CPI for SMTI radar. In an embodiment, the method comprises the steps of determining the Fast Fourier Transform (FFT) for each pulse, and multiplying the FFT by a pulse compression reference function. The method then proceeds by shifting phase of the pulse-compressed FFT by applying a first factor derived from a ground reference point and a second factor derived from a velocity-acceleration hypothesis to provide phase-shifted data, shifting the envelope of the phase-shifted data by applying one factor derived from range history and a second factor derived from a velocity-acceleration hypothesis to provide aligned data, and determining the Inverse FFT for the aligned data to provide a set of target data of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index, the latter three indices each referenced to the starting time of the first CPI. In a further embodiment, the invention encompasses first performing coherent integration for each CPI within a dwell of CPIs, then performing non-coherent integration in an efficient way, taking full advantage of the alignment to the starting time of the first CPI.
1. A method for processing pulses within a CPI for SMTI radar, the method comprising the steps of:
(a) determining Fast Fourier Transform (FFT) for each pulse;
(b) multiplying the FFT by a pulse compression reference function;
(c) shifting phase of the pulse-compressed FFT by applying a first factor derived from a ground reference point and a second factor derived from a velocity-acceleration hypothesis to provide phase-shifted data;
(d) shifting envelope of the phase-shifted data by applying one factor derived from range history and a second factor derived from a velocity-acceleration hypothesis to provide aligned data; and
(e) determining Inverse FFT for the aligned data to provide a set of target data of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index.
2. The method in accordance with claim 1 , wherein the SMTI radar has a characteristic transmitted chirp waveform, and the step (b) of multiplying the FFT by a pulse compression reference function further comprises the steps of determining a transfer function of the transmitted chirp waveform, determining a conjugate of the transfer function, and multiplying the FFT by the conjugate of the transfer function of the transmitted chirp waveform.
3. The method in accordance with claim 1 , wherein pulse number is indexed by n, and wherein the step (c) of shifting phase of the pulse-compressed FFT by applying a first factor derived from a ground reference point and a second factor derived from a velocity-acceleration hypothesis to provide phase-shifted data further comprises the step of:
multiplying the n th pulse in the frequency domain, after multiplication by the pulse compression reference function, by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
]
}
exp
{
j
4
π
λ
(
[
V
0
(
i
2
)
+
A
(
i
3
)
t
p
]
n
Δ
t
+
1
2
A
(
i
3
)
(
n
Δ
t
)
2
)
}
where:
R G (t) represents range history of a GRP;
V 0 (i 2 ) represents a velocity hypothesis; and
A(i 3 ) represents a range acceleration hypothesis.
4. The method in accordance with claim 3 , wherein the step (d) of shifting envelope of the phase-shifted data by applying one factor derived from range history and a second factor derived from a velocity-acceleration hypothesis to provide aligned data further comprises the step of:
multiplying the phase-shifted data by:
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
]
}
·
exp
{
j
2
π
s
k
^
M
[
V
0
(
i
2
)
(
t
p
+
n
Δ
t
)
+
1
2
A
(
i
3
)
(
t
p
2
+
2
t
p
n
Δ
t
+
(
n
Δ
t
)
2
)
]
}
where:
s represents range sampling interval;
R G (t) represents range history of a GRP;
M represents number of samples;
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
;
V 0 (i 2 ) represents a velocity hypothesis;
A(i 3 ) represents a range acceleration hypothesis.
5. The method in accordance with claim 4 , further comprising the steps of:
selecting a set of velocity hypotheses given by
V
0
(
i
2
)
=
(
i
2
-
N
2
-
1
2
)
Δ
V
,
i
2
=
0
,
…
,
N
2
-
1
;
and
selecting a set of acceleration hypotheses given by
A
(
i
3
)
=
(
i
3
-
N
3
-
1
2
)
Δ
A
,
i
3
=
0
,
…
,
N
3
-
1.
6. The method in accordance with claim 5 , further comprising the step of:
multiplying the pulse-compressed FFT by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
-
N
2
-
1
2
Δ
V
·
n
Δ
t
]
}
.
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
-
N
2
-
1
2
Δ
V
(
t
p
+
n
Δ
t
)
]
}
where:
M represents number of samples; and
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
.
7. The method in accordance with claim 6 , further comprising the steps of:
determining
fact
(
n
,
k
)
=
2
π
[
1
λ
n
Δ
t
·
(
2
t
p
+
n
Δ
t
)
+
k
^
2
sM
(
t
p
+
n
Δ
t
)
2
]
for n=0, . . . ,N−1, and k=0, . . . ,M−1; and
determining
W
(
k
)
=
exp
[
j
2
πΔ
V
Δ
t
(
2
λ
+
k
^
sM
)
]
for k=0, . . . ,M−1, and
FFT{W(k) −n 2 /2 }.
8. The method in accordance with claim 7 , further comprising the steps of:
for i 3 =0, . . . ,N 3 −1, performing the following operations:
(a) multiply the result of the multiplication of claim 6 by
exp [jA(i 3 )fact(n,k)]
for n=0, . . . ,N−1, and k=0, . . . ,M−1, yielding a result of the form F(k,n,i 3 );
(b) perform a transformation from the pulse index n to the velocity hypothesis index i 2 by a Chirp-Z Transform computation comprising:
G ( k,i 2 ,i 3 )=Σ n=0 N−1 F ( k,n,i 3 ) W ( k ) i 2 n
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(c) multiply G(k,i 2 ,i 3 ) by
exp
(
j
2
πΔ
Vt
p
k
^
sM
i
2
)
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(d) perform an IFFT with respect to k to yield an output of the form H(p,i 1 ,i 2 ,i 3 ).
9. A method for processing pulses within a CPI for SMTI radar, wherein the SMTI radar has a characteristic transmitted chirp waveform, the method comprising the steps of:
(a) determining Fast Fourier Transform (FFT) for each pulse;
(b) multiplying the FFT by a pulse compression reference function to provide a pulse-compressed FFT by:
determining a transfer function of the transmitted chirp waveform;
determining a conjugate of the transfer function; and
multiplying the FFT by the conjugate of the transfer function of the transmitted chirp waveform;
(c) selecting a set of velocity hypotheses given by
V
0
(
i
2
)
=
(
i
2
-
N
2
-
1
2
)
Δ
V
,
i
2
=
0
,
…
,
N
2
-
1
and selecting a set of acceleration hypotheses given by
A
(
i
3
)
=
(
i
3
-
N
3
-
1
2
)
Δ
A
,
i
3
=
0
,
…
,
N
3
-
1
;
(d) multiplying the pulse-compressed FFT by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
-
N
2
-
1
2
Δ
V
·
n
Δ
t
]
}
.
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
-
N
2
-
1
2
Δ
V
(
t
p
+
n
Δ
t
)
]
}
where:
M represents number of samples; and
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
;
(e) determining
fact
(
n
,
k
)
=
2
π
[
1
λ
n
Δ
t
·
(
2
t
p
+
n
Δ
t
)
+
k
^
2
sM
(
t
p
+
n
Δ
t
)
2
]
for n=0, . . . ,N−1, and k=0, . . . ,M−1;
(f) determining
W
(
k
)
=
exp
[
j
2
πΔ
V
Δ
t
(
2
λ
+
k
^
sM
)
]
for k=0, . . . ,M−1, and
FFT{W(k) −n 2 /2 };
(g) for i 3 =0, . . . ,N 3 −1, performing the following operations:
(1) multiply the result of step (d) by
exp [jA(i 3 )fact(n,k)]
for n=0, . . . ,N−1, and k=0, . . . ,M−1, yielding a result of the form F(k,n,i 3 );
(2) perform a transformation from the pulse index n to the velocity hypothesis index i 2 by a Chirp-Z Transform computation comprising:
G ( k,i 2 ,i 3 )=Σ n=0 N−1 F ( k,n,i 3 ) W ( k ) i 2 n
for k=0, . . . ,M−1, and i 2 =0,. . . ,N 2 −1;
(3) multiply G(k,i 2 ,i 3 ) by
exp
(
j
2
πΔ
Vt
p
k
^
sM
i
2
)
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(4) perform an IFFT with respect to k to yield an output of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index.
10. A method for processing radar data over a dwell of CPIs for SMTI radar, the method comprising the steps of:
(a) performing coherent integration for each CPI within the dwell to provide an output of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index;
(b) establishing an increment value ΔA′ between acceleration hypotheses such that
Δ
A
′
≤
β
λ
T
C
T
D
where:
β denotes a fraction of a velocity resolution cell that an actual velocity is allowed to migrate through due to acceleration mismatch;
T D denotes dwell time; and
T C denotes length of a CPI;
(c) introducing an increased number of acceleration hypotheses corresponding to
A
′
(
i
3
′
)
=
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
,
i
3
′
=
0
,
…
,
N
3
′
-
1
where:
i 3 ′ denotes an index associated with the increased number of acceleration hypotheses; and
N 3 ′ denotes total number of increased acceleration hypotheses;
(d) choosing an acceleration index according to the relation
i
~
3
=
N
3
-
1
2
+
round
[
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
Δ
A
]
;
(e) choosing a velocity hypothesis index according to the relation
i
~
2
=
i
2
+
round
{
[
A
′
(
i
3
′
)
-
A
(
i
~
3
)
]
t
p
Δ
V
}
;
(f) summing power over all CPIs in the dwell to complete the NCI process through the relation
Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 ;
where:
P is the total number of CPIs in the dwell.
11. The method in accordance with claim 10 , further comprising reducing dimensionality of NCI output by computing a maximum with respect to the acceleration hypothesis according to the following relation:
max_power( i 1 ,i 2 )=max i 3 ′ Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 .
12. The method in accordance with claim 10 , wherein the step (a) of performing coherent integration for each CPI within the dwell further comprises the steps of:
(a) determining Fast Fourier Transform (FFT) for each pulse;
(b) multiplying the FFT by a pulse compression reference function;
(c) shifting phase of the pulse-compressed FFT by applying a first factor derived from a ground reference point and a second factor derived from a velocity-acceleration hypothesis to provide phase-shifted data;
(d) shifting envelope of the phase-shifted data by applying one factor derived from range history and a second factor derived from a velocity-acceleration hypothesis to provide aligned data; and
(e) determining Inverse FFT for the aligned data to provide a set of target data of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index.
13. The method in accordance with claim 12 , wherein the SMTI radar has a characteristic transmitted chirp waveform, and the step (b) of multiplying the FFT by a pulse compression reference function further comprises the steps of determining a transfer function of the transmitted chirp waveform, determining a conjugate of the transfer function, and multiplying the FFT by the conjugate of the transfer function of the transmitted chirp waveform.
14. The method in accordance with claim 12 , wherein performing coherent integration for each CPI within the dwell further comprises the steps of:
selecting a set of velocity hypotheses given by
V
0
(
i
2
)
=
(
i
2
-
N
2
-
1
2
)
Δ
V
,
i
2
=
0
,
…
,
N
2
-
1
;
and
selecting a set of acceleration hypotheses given by
A
(
i
3
)
=
(
i
3
-
N
3
-
1
2
)
Δ
A
,
i
3
=
0
,
…
,
N
3
-
1.
15. The method in accordance with claim 12 , further comprising the step of:
multiplying the pulse-compressed FFT by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
-
N
2
-
1
2
Δ
V
·
n
Δ
t
]
}
·
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
-
N
2
-
1
2
Δ
V
(
t
p
+
n
Δ
t
)
]
}
where:
M represents number of samples; and
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
.
16. The method in accordance with claim 15 , further comprising the steps of:
determining
fact
(
n
,
k
)
=
2
π
[
1
λ
n
Δ
t
·
(
2
t
p
+
n
Δ
t
)
+
k
^
2
sM
(
t
p
+
n
Δ
t
)
2
]
for n=0, . . . ,N−1, and k=0, . . . ,M−1; and
determining
W
(
k
)
=
exp
[
j2πΔ
V
Δ
t
(
2
λ
+
k
^
sM
)
]
for k=0, . . . ,M−1, and
FFT{W(k) −n 2 /2 }.
17. The method in accordance with claim 16 , further comprising the steps of:
for i 3 =0, . . . ,N 3 −1, performing the following operations:
(a) multiply the result of the multiplication of claim 15 by
exp [jA(i 3 )fact(n,k)]
for n=0, . . . ,N−1, and k=0, . . . ,M−1, yielding a result of the form F(k,n,i 3 );
(b) perform a transformation from the pulse index n to the velocity hypothesis index i 2 by a Chirp-Z Transform computation comprising:
G ( k,i 2 ,i 3 )=Σ n=0 N−1 F ( k,n,i 3 ) W ( k ) i 2 n
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(c) multiply G(k,i 2 ,i 3 ) by
exp
(
j2πΔ
Vt
p
k
^
sM
i
2
)
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(d) perform an IFFT with respect to k to yield an output of the form H(p,i 1 ,i 2 ,i 3 ).
18. An SMTI radar system for processing radar data over a dwell of CPIs comprising:
means for performing coherent integration for each CPI within the dwell to provide an output of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index;
means for establishing an increment value ΔA′ between acceleration hypotheses such that
Δ
A
′
≤
β
λ
T
C
T
D
where:
β denotes a fraction of a velocity resolution cell that an actual velocity is allowed to migrate through due to acceleration mismatch;
T D denotes dwell time; and
T C denotes length of a CPI;
means for introducing an increased number of acceleration hypotheses corresponding to
A
′
(
i
3
′
)
=
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
,
i
3
′
=
0
,
…
,
N
3
′
-
1
where:
i 3 ′ denotes an index associated with the increased number of acceleration hypotheses; and
N 3 ′ denotes total number of increased acceleration hypotheses;
means for choosing an acceleration index according to the relation
i
~
3
=
N
3
-
1
2
+
round
[
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
Δ
A
]
;
means for choosing a velocity hypothesis index according to the relation
i
~
2
=
i
2
+
round
{
[
A
′
(
i
3
′
)
-
A
(
i
~
3
)
]
t
p
Δ
V
}
;
means for summing power over all CPIs in the dwell to complete the NCI process through the relation
Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 ;
where:
P is the total number of CPIs in the dwell.
19. The SMTI radar system of claim 18 , further comprising means for reducing dimensionality of NCI output by means for computing a maximum with respect to the acceleration hypothesis according to the following relation:
max_power( i 1 ,i 2 )=max i 3 ′ Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 .
20. The SMTI radar system of claim 18 , wherein the means for performing coherent integration for each CPI within the dwell further comprises:
means for selecting a set of velocity hypotheses given by
V
0
(
i
2
)
=
(
i
2
-
N
2
-
1
2
)
Δ
V
,
i
2
=
0
,
…
,
N
2
-
1
;
and
means for selecting a set of acceleration hypotheses given by
A
(
i
3
)
=
(
i
3
-
N
3
-
1
2
)
Δ
A
,
i
3
=
0
,
…
,
N
3
-
1.
21. The SMTI radar system of claim 20 , further comprising:
means for multiplying the pulse-compressed FFT by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
-
N
2
-
1
2
Δ
V
·
n
Δ
t
]
}
·
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
-
N
2
-
1
2
Δ
V
(
t
p
+
n
Δ
t
)
]
}
where:
M represents number of samples; and
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
.
22. The SMTI radar system of claim 21 , further comprising:
means for determining
fact
(
n
,
k
)
=
2
π
[
1
λ
n
Δ
t
·
(
2
t
p
+
n
Δ
t
)
+
k
^
2
sM
(
t
p
+
n
Δ
t
)
2
]
for n=0, . . . ,N−1, and k=0, . . . ,M−1; and
means for determining
W
(
k
)
=
exp
[
j2πΔ
V
Δ
t
(
2
λ
+
k
^
sM
)
]
for k=0, . . . ,M−1, and
FFT{W(k) −n 2 /2 }.
23. The SMTI radar system of claim 22 , further comprising:
for i 3 =0, . . . ,N 3 −1:
means for multiplying the result of the multiplication of claim 21 by
exp [jA(i 3 )fact(n,k)]
for n=0, . . . ,N−1, and k=0, . . . ,M−1, yielding a result of the form F(k,n,i 3 );
means for performing a transformation from the pulse index n to the velocity hypothesis index i 2 by a Chirp-Z Transform computation comprising:
G ( k,i 2 ,i 3 )=Σ n=0 N−1 F ( k,n,i 3 ) W ( k ) i 2 n
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
means for multiplying G(k,i 2 ,i 3 ) by
exp
(
j2πΔ
Vt
p
k
^
sM
i
2
)
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
means for performing an IFFT with respect to k to yield an output of the form H(p,i 1 ,i 2 ,i 3 ).
24. A method for processing radar return pulse data over a dwell of CPIs for SMTI radar, the method comprising the steps of:
(a) determining Fast Fourier Transform (FFT) for each pulse;
(b) multiplying the FFT by a pulse compression reference function to provide a pulse-compressed FFT by:
determining a transfer function of the transmitted chirp waveform;
determining a conjugate of the transfer function; and
multiplying the FFT by the conjugate of the transfer function of the transmitted chirp waveform;
(c) selecting a set of velocity hypotheses given by
V
0
(
i
2
)
=
(
i
2
-
N
2
-
1
2
)
Δ
V
,
i
2
=
0
,
…
,
N
2
-
1
and selecting a set of acceleration hypotheses given by
A
(
i
3
)
=
(
i
3
-
N
3
-
1
2
)
Δ
A
,
i
3
=
0
,
…
,
N
3
-
1
;
(d) multiplying the pulse-compressed FFT by
exp
{
j
4
π
λ
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
t
p
)
-
N
2
-
1
2
Δ
V
·
n
Δ
t
]
}
·
exp
{
j
2
π
s
k
^
M
[
R
G
(
t
p
+
n
Δ
t
)
-
R
G
(
0
)
-
N
2
-
1
2
Δ
V
(
t
p
+
n
Δ
t
)
]
}
where:
M represents number of samples; and
k
^
=
{
k
,
k
=
0
,
…
,
M
/
2
-
1
k
-
M
,
k
=
M
/
2
,
…
M
-
1
}
;
(e) determining
fact
(
n
,
k
)
=
2
π
[
1
λ
n
Δ
t
·
(
2
t
p
+
n
Δ
t
)
+
k
^
2
sM
(
t
p
+
n
Δ
t
)
2
]
for n=0, . . . ,N−1, and k=0, . . . ,M−1;
(f) determining
W
(
k
)
=
exp
[
j2πΔ
V
Δ
t
(
2
λ
+
k
^
sM
)
]
for k=0, . . . ,M−1, and
FFT{W(k) −n 2 /2 };
(g) for i 3 =0, . . . ,N 3 −1, performing the following operations:
(1) multiply the result of step (d) by
exp [jA(i 3 )fact(n,k)]
for n=0, . . . ,N−1, and k=0, . . . ,M−1, yielding a result of the form F(k,n,i 3 );
(2) perform a transformation from the pulse index n to the velocity hypothesis index i 2 by a Chirp-Z Transform computation comprising:
G ( k,i 2 ,i 3 )=Σ n=0 N−1 F ( k,n,i 3 ) W ( k ) i 2 n
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1;
(3) multiply G(k,i 2 ,i 3 ) by
exp
(
j2πΔ
Vt
p
k
^
sM
i
2
)
for k=0, . . . ,M−1, and i 2 =0, . . . ,N 2 −1; and
(4) perform an IFFT with respect to k to yield an output of the form H(p,i 1 ,i 2 ,i 3 ), where p is CPI number, i 1 is range index, i 2 is velocity index, and i 3 is coarse acceleration index;
(h) establishing an increment value ΔA′ between acceleration hypotheses such that
Δ
A
′
≤
β
λ
T
C
T
D
where:
β denotes a fraction of a velocity resolution cell that an actual velocity is allowed to migrate through due to acceleration mismatch;
T D denotes dwell time; and
T C denotes length of a CPI;
(i) introducing an increased number of acceleration hypotheses corresponding to
A
′
(
i
3
′
)
=
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
,
i
3
′
=
0
,
…
,
N
3
′
-
1
where:
i 3 ′ denotes an index associated with the increased number of acceleration hypotheses; and
N 3 ′ denotes total number of increased acceleration hypotheses;
(j) choosing an acceleration index according to the relation
i
~
3
=
N
3
-
1
2
+
round
[
(
i
3
′
-
N
3
′
-
1
2
)
Δ
A
′
Δ
A
]
;
(e) choosing a velocity hypothesis index according to the relation
i
~
2
=
i
2
+
round
{
[
A
′
(
i
3
′
)
-
A
(
i
~
3
)
]
t
p
Δ
V
}
;
(k) summing power over all CPIs in the dwell to complete the NCI process through the relation
Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 ;
where:
P is the total number of CPIs in the dwell.
25. The method in accordance with claim 24 , further comprising reducing dimensionality of NCI output by computing a maximum with respect to the acceleration hypothesis according to the following relation:
max_power( i 1 ,i 2 )=max i 3 ′ Σ p=0 P−1 |H ( p,i 1 ,ĩ 2 ,ĩ 3 )| 2 .