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WJEEE World Journal of Electrical and Electronic Engineering 2770-5560 Trend
Research Publishing 10.31586/wjeee.2022.174 WJEEE-174 Research Article A Moving
Single-Station Doppler Ranging Solution by Means of Direction Finding Method Yu
Tao 1 * 1China Academy of Management Science, Beijing, China *Corresponding
author at: China Academy of Management Science, Beijing, China 10 12 2021 2 1 10
12 2021 10 12 2021 10 12 2021 10 12 2021 © Copyright 2021 by authors and Trend
Research Publishing Inc. 2021 This work is licensed under the Creative Commons
Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/ Abstract: The Doppler shift is an
angle dependent function. Based on the relationship between frequency shift and
angle, direction-finding method can be used directly to obtain the ranging
solution based on frequency shift measurement. The Doppler ranging solution
obtained by this method has excellent calculation accuracy and can keep the same
accuracy as the ranging solution based on frequency shift difference processing.
Single station location; Airborne passive positioning; Doppler shift; Doppler
frequency difference; Path difference equation; Direction-finding; Ranging
Introduction

Existing research results show that the location can be completed based on only
one detection by directly using the Doppler change rate[ 1,2,3]. Otherwise, if
based on Doppler frequency shift measurement, at least two detection are
generally required. One of the results of the author's early research on the
passive ranging method based on Doppler rate of change is that, based on the
mathematical definition, the Doppler rate of change is transformed into the
ratio of the Doppler frequency difference to the time difference by using the
difference method, and at the same time, by using the velocity vector
relationship, an airborne Doppler passive ranging method based on two
consecutive frequency shift measurement is derived[ 4,5]. The result of this
study is that there are both advantages and disadvantages. The advantages are
that direct measurement of Doppler rate of change is not required. The
disadvantages are that two measurements of the Doppler shift are required.

In fact, the Doppler shift is an angle dependent function. From the point of
view of pure mathematics, after measuring the frequency shift, the azimuth angle
between the movement direction of the detecting platform and the radial distance
of the target can be obtained directly according to the frequency shift if the
movement speed of the detecting platform and signal wavelength of the detecting
target are also known. Based on the relationship between frequency shift and
angle, this paper directly uses direction finding method based on trigonometric
function to give the ranging solution based on frequency shift measurement. In
order to compare the errors more clearly and make it easier for readers to read,
the author briefly describes the existing Doppler ranging solutions based on
difference processing after the derivation of Doppler ranging solutions by means
of direction finding method. The analysis results show that the Doppler ranging
solution given by direct direction finding method has excellent calculation
accuracy and can keep the same accuracy as the ranging solution based on
frequency shift difference processing.







Doppler ranging solution based on direction finding method

A Doppler receiver R is installed on the moving platform to detect the
stationary or slow-moving target T on the ground as shown inFigure 1, and the
received Doppler frequency shift is

(1) λ f d =vcosβ

where: f d is Doppler frequency shift; λ the wavelength; v the moving speed of
the moving platform; β leading angle.

Figure 1

Doppler frequency shift detection of moving single station.

Suppose there is a single station moving in a straight line, as shown inFigure
2, from position 1 to position 2. In this process, Doppler frequency shift
detection is performed twice in a row at a single moving station.

(2) λ f d1 =vcos β 1 (3) λ f d2 =vcos β 2

In the figure, d represents the moving distance of the sounding station, and r i
represents the radial distance between the sounding station and the target.

Figure 2

Detection of fixed target by moving single station.

According to the Doppler shift equation (1), the leading angle can be solved
directly

(4) β 1 = cos −1 ( λ v f d1 ) (5) β 2 = cos −1 ( λ v f d2 )

According to the geometric relation shown inFigure 2, the sine theorem is used

(6) d sin( β 2 − β 1 ) = r 1 sin β 2 = r 2 sin β 1

The radial distance r 2 from the detecting platform to the target at the current
position

(7) r 2 = dsin β 1 sin( β 2 − β 1 ) = dsin[ cos −1 ( λ v f d1 ) ] sin[ cos −1 (
λ v f d2 )− cos −1 ( λ v f d1 ) ] Ranging error

The relative ranging error is analyzed by total differential method. First set

(8) r 2 =d P Q

where:

P=sin β 1 Q=sinΔβ Δβ= β 2 − β 1 3.1. Ranging error resulting from frequency
shift measurement

Ranging error resulting from frequency shift f d1

(9) ∂ r 2 ∂ f d1 = d Q 2 ( Q ∂P ∂ f d1 −P ∂Q ∂ f d1 )

Range error resulting from frequency shift f d2

(10) ∂ r 2 ∂ f d2 = d Q 2 ( Q ∂P ∂ f d2 −P ∂Q ∂ f d2 )

where:

∂P ∂ f d1 =cos β 1 ∂ β 1 ∂ f d1 ∂P ∂ f d2 =0 ∂Q ∂ f d1 =cosΔβ ∂Δβ ∂ f d1 =cosΔβ(
∂ β 2 ∂ f d1 − ∂ β 1 ∂ f d1 ) ∂Q ∂ f d2 =cosΔβ ∂Δβ ∂ f d2 =cosΔβ( ∂ β 2 ∂ f d2 −
∂ β 1 ∂ f d2 )



thereinto

∂ β 1 ∂ f d1 =− λ vsin β 1 ∂ β 1 ∂ f d2 =0 ∂ β 2 ∂ f d1 =0 ∂ β 2 ∂ f d2 =− λ
vsin β 2

3.2. Ranging error caused by moving distance <math> <semantics> <mi>d</mi>
</semantics></math> (11) ∂ r 2 ∂d = sin β 1 sinΔβ 3.3. Ranging error caused by
flight speed <math> <semantics> <mi>v</mi> </semantics></math> (12) ∂ r 2 ∂v = d
Q 2 ( Q ∂P ∂v −P ∂Q ∂v )

where:

∂P ∂v =cos β 1 ∂ β 1 ∂v ∂Q ∂v =cosΔβ ∂Δβ ∂v =cosΔβ( ∂ β 2 ∂v − ∂ β 1 ∂v )

thereinto

∂ β 1 ∂v = λ f d1 v 2 1− ( λ v f d1 ) 2 = λ f d1 v 2 sin β 1 ∂ β 2 ∂v = λ f d2 v
2 1− ( λ v f d2 ) 2 = λ f d2 v 2 sin β 2



3.4. Computational formula

When the error of each observation is zero mean, independent of each other, the
relative ranging error can be calculated as follows

(13) σ r = 1 r 2 [ | ∂ r 2 ∂ f d1 | σ f +| ∂ r 2 ∂ f d2 | σ f +| ∂ r 2 ∂d | σ d
+| ∂ r 2 ∂v | σ v ]

where: σ θ , σ f , σ d , σ v are the root mean square errors of angle, frequency
shift, moving distance and flight velocity measurement errors respectively.

The preliminary calculation shows that the root mean square error of moving
distance and flight speed measurement error has little influence on the relative
ranging error. Therefore, in the following calculation process, only the
relative ranging error generated by the frequency shift measurement error is
considered

(14) σ r_DF = 1 r 2 [ | ∂ r 2 ∂ f d1 | σ f +| ∂ r 2 ∂ f d2 | σ f ] 4. Comparison
with the ranging solution based on frequency shift differential processing

The simulation results show that the Doppler ranging solution based on the
direction finding method has excellent calculation accuracy. At this point, it
seems meaningless to analyze the calculation accuracy of different moving
distances, different radial distances, different moving speeds and different
wavelengths separately. At the same time, the error calculation also shows that
the error characteristic of Doppler ranging solution based on direction finding
method is basically the same as that of the existing ranging solution based on
frequency shift difference processing. Therefore, the relative calculation
accuracy and ranging error analysis of Doppler ranging solutions based on
direction finding method are put in the section of comparison with the existing
ranging solutions based on frequency shift differential processing.

4.1. Ranging solution of difference based on frequency shift

This section first retells the mathematical derivation results of previous
studies[ 4,5]. Still based on the geometric model shown inFigure 2, it is
assumed that the Doppler change rate detected by the detection platform at
position 1 is

f d • = v t1 2 λ⋅ r 1

where, v t1 is the tangential velocity of the detection platform; r 1 the radial
distance between the target and the detection platform.

From the mathematical definition (15), the rate of Doppler change can be
approximated by the measured value of Doppler frequency difference between two
detection endpoints during a time period Δt obtained by time difference
measuring

f d • = Δ f d Δt = f d2 − f d1 Δt

Time difference Δt can also be expressed as the ratio of distance traveled to
speed traveled

Δt= d v

By integrating equations (15) and (16), and using the relationship v 2 = v r 2 +
v t 2 between velocity vector and its components, and the relationship v r =λ f
d between radial velocity and Doppler frequency shift, the ranging formula as
follows can be obtained

r 1 = ( v 2 − λ 2 f d1 2 )Δt λ| Δ f d | = d( v 2 − λ 2 f d1 2 ) vλ| Δ f d |

where: Δ f d = f d1 − f d2 .

4.2. Simulation modification

Parameter value used in calculation: r 1 =600km , d=100km , v=300m/s , λ=0.3m ,
σ f =50Hz .

The radial distance and Doppler frequency shift values at the two detection
positions are respectively selected to calculate.Figure 3 shows some of the
calculated curves selectively.

Figure 3

Relative calculation errors of ranging solution using frequency difference
directly.

The results show that, assuming that the detection platform moves the distance d
uniformly along a straight line, if the average frequency shift at two positions
is adopted, the ranging formula at position 1 has the minimum relative
calculation error, that is,

r 1 = d( v 2 −0.25 λ 2 ( f d1 + f d2 ) 2 ) vλ| Δ f d |

The dotted line inFigure 3 shows the relative calculation error of Equation
(19). This shows mathematically that there is a lag in the detection of target
distance. It is necessary to obtain the average value of Doppler frequency shift
on a moving distance, and then only the radial distance at the starting position
can be accurately obtained.

The dot line inFigure 3 shows the relative calculation error of equation (20),
which indicates that the Doppler frequency shift at the terminating position is
used to calculate the radial distance r 2 at position 2, and the relative
calculation error obtained is very poor

r 2 = d( v 2 − λ 2 f d2 2 ) vλ| Δ f d |

If equation (20) is simply modified to

r 2 = d( v 2 − λ 2 f d1 2 ) vλ| Δ f d |

The solid line inFigure 3 shows the relative calculation error of Equation (21).
The modified results show that, in the case of only two detection, in order to
directly use the Doppler shift measurement value at the current moving distance
to solve the radial distance r 2 at the end position more accurately, the
Doppler shift value at the starting position 1 should be used to replace the
frequency shift value at the second position 2.

4.3. Error calculation of frequency shift differential ranging solution

According to the results of simulation correction, formula (21) is adopted for
error analysis. Preset preestablish

r 2 = a 0 P 0 Q 0

where:

a 0 = d vλ P 0 = v 2 − λ 2 f d1 2 Q 0 =| Δ f d |

Ranging error resulting from frequency shift f d1

∂ r 2 ∂ f d1 = a 0 Q 0 2 ( Q 0 ∂ P 0 ∂ f d1 − P 0 ∂ Q 0 ∂ f d1 )

thereinto

∂ P 0 ∂ f d1 =−2 λ 2 f d1 ∂ Q 0 ∂ f d1 =1

Ranging error resulting from frequency shift f d2

∂ r 2 ∂ f d2 =− a 0 P 0 Q 0 2 ∂ Q 0 ∂ f d2

thereinto

∂ Q 0 ∂ f d2 =1

The relative ranging error caused by frequency shift measurement error

(25) σ r_FD = 1 r 2 [ | ∂ r 2 ∂ f d1 | σ f +| ∂ r 2 ∂ f d2 | σ f ]. 4.4.
Comparison of relative calculation accuracy

During the simulation calculation, the leading angle β 1 at starting point 1 of
the detection platform, the radial distance r 1 =600km from starting point 1 to
the target, the moving distance d=100km of the detection platform, the flight
speed v=300m/s and the wavelength λ=0.3m of the detection signal are preset.

Then the radial distance r 2 of the target at the second position 2 and other
geometric parameters are calculated by using trigonometric function relations.
According to the definition (1) of Doppler frequency shift, the Doppler
frequency shift value is calculated.

Figure 4 shows the relative calculation error curves of Doppler ranging
solutions based on direction finding method and existing ranging solutions based
on frequency shift measurement. Obviously, the relative calculation error of
Doppler ranging solution with direction finding method is almost zero, and the
most direct reason is that there is no approximate treatment in the process of
proving the ranging solution. In terms of mathematical expression, the leading
angle tends to be the same at both probe positions only when the leading angle
is small. Now, if you just look at the denominator and it tends to go to zero,
it seems to result in a singularity. But in fact, so the leading angle is going
to zero, so the numerator is going to zero. Therefore, according to the limit
rule of derivative, the whole ranging formula tends to a fixed value.

As far as the shape of the error curve is concerned, the author thinks that the
relative error distribution of the Doppler ranging solution by means of
direction finding method should be only some chaotic burrs, and has nothing to
do with physical characteristics. In other words, there is no functional
correlation between the distribution characteristics of the relative calculation
error curve and the leading angle in physical characteristics. However, the
relative calculation error of the ranging solution based on approximate
difference processing is related to the physical characteristics.

According to the mathematical definition (15) of Doppler rate of change, the
singularity of the equation does not appear when the tangential velocity and
distance are fixed. However, after the approximate difference treatment, the
Doppler change rate is directly determined by the difference between the two
frequency shifts. According to the definition (1) of Doppler, in the case of
fixed velocity, the magnitude of the Doppler shift is only related to the
leading angle. Therefore, when the leading angle is small, that is, when the
leading angle at the two detection positions tends to be the same, the magnitude
of the two frequency shifts is very close, and the difference between the two
frequency shifts tends to zero. At this point, the Doppler rate of change tends
to be singular, which leads to the relative calculation error of the approximate
ranging solution is relatively large when the leading angle is small.

Figure 4

Comparison of computational accuracy.

4.5. Comparison of ranging errors

The relative ranging error curves for the two ranging solutions have been shown
inFigure 5. When the leading angle is less than 40 degrees, there is a slight
difference between the two ranging methods. But when it is greater than 40
degrees and tends to 90 degrees, the relative ranging errors of the two
solutions tend to be the same. The author's existing and ongoing research
results show that[ 6,7,8], under the same frequency shift measurement method, no
matter what mathematical deduction method is adopted, the ranging error is
basically the same. In other words, the ranging error seems to be only related
to the frequency shift measurement method, and has nothing to do with the
mathematical expression of the ranging equation.

Figure 5

Comparison of relative ranging errors.

Conclusion

The existing ranging method based on frequency shift differential processing has
some approximation in the process of mathematical deduction and verification, so
the calculation accuracy of the mathematical formula obtained is relatively
poor. Based on the relationship between frequency shift and angle, the Doppler
ranging formula derived by direction finding method basically has no
approximation. Therefore, the calculation accuracy of the ranging formula
obtained is very good. Furthermore, the relative ranging error of the obtained
solutions is basically consistent with that of the existing solutions directly
based on frequency shift measurement.

Improving the accuracy of formula calculation is not only helpful for
engineering design, but also for theoretical research. The research in this
paper may provide a reference method for passive ranging technology of motion
detection platform.



References [1] Li Zong-hua, XiaoYu-qin, Zhou yi-yu, Sun Zhong-kang,
"Single-observer passive location and tracking algorithms using fre-quency and
spatial measurements", SYSTEMS ENGINEERING AND ELECTRONICS, vol. 26, no. 5, pp.
613-616, May 2004. [2] DIAO Ming, WANG Yue, "Research of passive location based
on the Doppler changing rate", SYSTEMS ENGINEERING AND ELECTRONICS, vol. 28, no.
5, pp. 696-698, June 2006. [3] TAN Xin-rong, GAO Xian-jun, LI Bao-zhu, WANG Yu,
"An improved tracking filter algorithm of single observer passive localization
based on Doppler changing rate", Electronic Design Engineering, vol. 22, no. 8,
pp. 77-80, April 2014. [4] YU Tao, "Airborne ranging principle based on Doppler
frequency", INFORMATION AND ELECTRONIC ENGINEERING, vol. 9, no. 1, pp. 22-25,
Feb. 2011. [5] Tao Yu, Technology of Passive detection location, Beijing:
National defense industry press, 2017 [6] Tao YU. "An Airborne Passive
Positioning Method Based on Angle and Frequency Difference Measurement", 2020
4th Inter-national Conference on Electronic Information Technology and Computer
Engineering Proceedings, 2020, pp.296-301. [7] Tao YU. "A Moving Single Station
Passive Ranging Method by Interchangeable Relationship between Frequency Shift
and Path Difference", To be published. [8] Tao Yu, "Recursive Equation of
Frequency Shift and Its Application in Passive Ranging", To be published.
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China Academy of Management Science, Beijing, China
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Corresponding author at: China Academy of Management Science, Beijing, China
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<abstract>
<bold>Abstract:</bold>
<bold> </bold>
The Doppler shift is an angle dependent function. Based on the relationship
between frequency shift and angle, direction-finding method can be used directly
to obtain the ranging solution based on frequency shift measurement. The Doppler
ranging solution obtained by this method has excellent calculation accuracy and
can keep the same accuracy as the ranging solution based on frequency shift
difference processing.
...
</abstract>
<kwd-group>
<kwd-group>
<kwd>Single station location; Airborne passive positioning; Doppler shift;
Doppler frequency difference; Path difference equation; Direction-finding;
Ranging</kwd>
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<title>Introduction</title>
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Existing research results show that the location can be completed based on only
one detection by directly using the Doppler change rate[
<xref ref-type="bibr" rid="R1">1</xref>
,
<xref ref-type="bibr" rid="R2">2</xref>
,
<xref ref-type="bibr" rid="R3">3</xref>
]. Otherwise, if based on Doppler frequency shift measurement, at least two
detection are generally required. One of the results of the author's early
research on the passive ranging method based on Doppler rate of change is that,
based on the mathematical definition, the Doppler rate of change is transformed
into the ratio of the Doppler frequency difference to the time difference by
using the difference method, and at the same time, by using the velocity vector
relationship, an airborne Doppler passive ranging method based on two
consecutive frequency shift measurement is derived[
<xref ref-type="bibr" rid="R4">4</xref>
,
<xref ref-type="bibr" rid="R5">5</xref>
]. The result of this study is that there are both advantages and disadvantages.
The advantages are that direct measurement of Doppler rate of change is not
required. The disadvantages are that two measurements of the Doppler shift are
required.
...
</p>
<p>In fact, the Doppler shift is an angle dependent function. From the point of
view of pure mathematics, after measuring the frequency shift, the azimuth angle
between the movement direction of the detecting platform and the radial distance
of the target can be obtained directly according to the frequency shift if the
movement speed of the detecting platform and signal wavelength of the detecting
target are also known. Based on the relationship between frequency shift and
angle, this paper directly uses direction finding method based on trigonometric
function to give the ranging solution based on frequency shift measurement. In
order to compare the errors more clearly and make it easier for readers to read,
the author briefly describes the existing Doppler ranging solutions based on
difference processing after the derivation of Doppler ranging solutions by means
of direction finding method. The analysis results show that the Doppler ranging
solution given by direct direction finding method has excellent calculation
accuracy and can keep the same accuracy as the ranging solution based on
frequency shift difference processing.</p>
<p/>
<p/>
<p/>
...
</sec>
<sec id="sec2">
<title>Doppler ranging solution based on direction finding method</title>
<p>
A Doppler receiver R is installed on the moving platform to detect the
stationary or slow-moving target T on the ground as shown inFigure
<xref ref-type="fig" rid="fig1"> 1</xref>
, and the received Doppler frequency shift is
...
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where:
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is Doppler frequency shift;
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...
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<mi>v</mi>
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the moving speed of the moving platform;
<math>
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<mi>β</mi>
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...
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leading angle.
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<label>Figure 1</label>
<caption>
<p>Doppler frequency shift detection of moving single station.</p>
...
</caption>
<graphic xlink:href="174.fig.001"/>
...
</fig>
<p>
Suppose there is a single station moving in a straight line, as shown inFigure
<xref ref-type="fig" rid="fig2"> 2</xref>
, from position 1 to position 2. In this process, Doppler frequency shift
detection is performed twice in a row at a single moving station.
...
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<mrow>
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<mo>=</mo>
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<mi>β</mi>
<mn>1</mn>
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...
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...
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...
</math>
...
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<semantics>
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<mi>λ</mi>
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<mi>f</mi>
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<mi>d</mi>
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...
</msub>
<mo>=</mo>
<mi>v</mi>
<mi>cos</mi>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
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<p>
In the figure,
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<mi>d</mi>
...
</semantics>
...
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represents the moving distance of the sounding station, and
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<msub>
<mi>r</mi>
<mi>i</mi>
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</msub>
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represents the radial distance between the sounding station and the target.
...
</p>
<fig id="fig2">
<label>Figure 2</label>
<caption>
<p>Detection of fixed target by moving single station.</p>
...
</caption>
<graphic xlink:href="174.fig.002"/>
...
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<p>According to the Doppler shift equation (1), the leading angle can be solved
directly</p>
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<mi>β</mi>
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</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<disp-formula id="FD5">
<label>(5)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<msup>
<mrow>
<mi>cos</mi>
...
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
...
</mrow>
...
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>
According to the geometric relation shown inFigure
<xref ref-type="fig" rid="fig2"> 2</xref>
, the sine theorem is used
...
</p>
<disp-formula id="FD6">
<label>(6)</label>
<math>
<semantics>
<mrow>
<mfrac>
<mi>d</mi>
<mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
<mo>−</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>
The radial distance
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
from the detecting platform to the target at the current position
...
</p>
<disp-formula id="FD7">
<label>(7)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
<mo>−</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mi>sin</mi>
<mrow>
<mo>[</mo>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
...
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
...
</mrow>
...
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mo>]</mo>
...
</mrow>
...
</mrow>
<mrow>
<mi>sin</mi>
<mrow>
<mo>[</mo>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
...
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
...
</mrow>
...
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
<mo>−</mo>
<msup>
<mrow>
<mi>cos</mi>
...
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
...
</mrow>
...
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mo>]</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
...
</sec>
<sec id="sec3">
<title>Ranging error</title>
<p>The relative ranging error is analyzed by total differential method. First
set</p>
<disp-formula id="FD8">
<label>(8)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<mi>d</mi>
<mfrac>
<mi>P</mi>
<mi>Q</mi>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>where: </p>
<inline-formula>
<math>
<semantics>
<mrow>
<mi>P</mi>
<mo>=</mo>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mi>Q</mi>
<mo>=</mo>
<mtext>sin</mtext>
<mi>Δ</mi>
<mi>β</mi>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mi>Δ</mi>
<mi>β</mi>
<mo>=</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
<mo>−</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<title>3.1. Ranging error resulting from frequency shift measurement</title>
<p>
Ranging error resulting from frequency shift
<math>
<semantics>
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</p>
<disp-formula id="FD9">
<label>(9)</label>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mrow>
<msup>
<mi>Q</mi>
<mn>2</mn>
...
</msup>
...
</mrow>
...
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>Q</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mi>P</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>
Range error resulting from frequency shift
<math>
<semantics>
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</p>
<disp-formula id="FD10">
<label>(10)</label>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mrow>
<msup>
<mi>Q</mi>
<mn>2</mn>
...
</msup>
...
</mrow>
...
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>Q</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mi>P</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>where:</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mn>0</mn>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mtext>cos</mtext>
<mi>Δ</mi>
<mi>β</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Δ</mi>
<mi>β</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mtext>cos</mtext>
<mi>Δ</mi>
<mi>β</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mtext>cos</mtext>
<mi>Δ</mi>
<mi>β</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Δ</mi>
<mi>β</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mtext>cos</mtext>
<mi>Δ</mi>
<mi>β</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p/>
<p>thereinto</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>λ</mi>
<mrow>
<mi>v</mi>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mn>0</mn>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mn>0</mn>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>λ</mi>
<mrow>
<mi>v</mi>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p/>
<title>
3.2. Ranging error caused by moving distance
<math>
<semantics>
<mi>d</mi>
...
</semantics>
...
</math>
...
</title>
<disp-formula id="FD11">
<label>(11)</label>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>d</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mi>sin</mi>
<mi>Δ</mi>
<mi>β</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<title>
3.3. Ranging error caused by flight speed
<math>
<semantics>
<mi>v</mi>
...
</semantics>
...
</math>
...
</title>
<disp-formula id="FD12">
<label>(12)</label>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mrow>
<msup>
<mi>Q</mi>
<mn>2</mn>
...
</msup>
...
</mrow>
...
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>Q</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mi>P</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>where:</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>P</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<mi>Δ</mi>
<mi>β</mi>
<mfrac>
<mrow>
<mo>∂</mo>
<mi>Δ</mi>
<mi>β</mi>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<mi>Δ</mi>
<mi>β</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p>thereinto</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>λ</mi>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<msqrt>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mn>2</mn>
...
</msup>
...
</mrow>
...
</msqrt>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>λ</mi>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mtext>sin</mtext>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>λ</mi>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<msqrt>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>λ</mi>
<mi>v</mi>
...
</mfrac>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mn>2</mn>
...
</msup>
...
</mrow>
...
</msqrt>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>λ</mi>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mi>sin</mi>
<msub>
<mi>β</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p/>
<p/>
<title>3.4. Computational formula</title>
<p>When the error of each observation is zero mean, independent of each other,
the relative ranging error can be calculated as follows</p>
<disp-formula id="FD13">
<label>(13)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>r</mi>
...
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
<mo>+</mo>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
<mo>+</mo>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>d</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>d</mi>
...
</msub>
<mo>+</mo>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<mi>v</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>v</mi>
...
</msub>
...
</mrow>
<mo>]</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<p>
where:
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>θ</mi>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>v</mi>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
are the root mean square errors of angle, frequency shift, moving distance and
flight velocity measurement errors respectively.
...
</p>
<p>The preliminary calculation shows that the root mean square error of moving
distance and flight speed measurement error has little influence on the relative
ranging error. Therefore, in the following calculation process, only the
relative ranging error generated by the frequency shift measurement error is
considered</p>
<disp-formula id="FD14">
<label>(14)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mrow>
<mi>r</mi>
<mo>_</mo>
<mi>D</mi>
<mi>F</mi>
...
</mrow>
...
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
<mo>+</mo>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
...
</mrow>
<mo>]</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
...
</sec>
<sec id="sec4">
<title>4. Comparison with the ranging solution based on frequency shift
differential processing</title>
<p>The simulation results show that the Doppler ranging solution based on the
direction finding method has excellent calculation accuracy. At this point, it
seems meaningless to analyze the calculation accuracy of different moving
distances, different radial distances, different moving speeds and different
wavelengths separately. At the same time, the error calculation also shows that
the error characteristic of Doppler ranging solution based on direction finding
method is basically the same as that of the existing ranging solution based on
frequency shift difference processing. Therefore, the relative calculation
accuracy and ranging error analysis of Doppler ranging solutions based on
direction finding method are put in the section of comparison with the existing
ranging solutions based on frequency shift differential processing.</p>
<title>4.1. Ranging solution of difference based on frequency shift</title>
<p>
This section first retells the mathematical derivation results of previous
studies[
<xref ref-type="bibr" rid="R4">4</xref>
,
<xref ref-type="bibr" rid="R5">5</xref>
]. Still based on the geometric model shown inFigure
<xref ref-type="fig" rid="fig2"> 2</xref>
, it is assumed that the Doppler change rate detected by the detection platform
at position 1 is
...
</p>
<math>
<semantics>
<mrow>
<mover>
<mrow>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>•</mo>
...
</mover>
<mo>=</mo>
<mfrac>
<mrow>
<msubsup>
<mi>v</mi>
<mrow>
<mi>t</mi>
<mn>1</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
<mrow>
<mi>λ</mi>
<mo>⋅</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
where,
<math>
<semantics>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>t</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
is the tangential velocity of the detection platform;
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
the radial distance between the target and the detection platform.
...
</p>
<p>
From the mathematical definition (15), the rate of Doppler change can be
approximated by the measured value of Doppler frequency difference between two
detection endpoints during a time period
<math>
<semantics>
<mrow>
<mi>Δ</mi>
<mi>t</mi>
...
</mrow>
...
</semantics>
...
</math>
obtained by time difference measuring
...
</p>
<math>
<semantics>
<mrow>
<mover>
<mrow>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>•</mo>
...
</mover>
<mo>=</mo>
<mfrac>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mrow>
<mi>Δ</mi>
<mi>t</mi>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
<mo>−</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
<mrow>
<mi>Δ</mi>
<mi>t</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
Time difference
<math>
<semantics>
<mrow>
<mi>Δ</mi>
<mi>t</mi>
...
</mrow>
...
</semantics>
...
</math>
can also be expressed as the ratio of distance traveled to speed traveled
...
</p>
<math>
<semantics>
<mrow>
<mi>Δ</mi>
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mi>v</mi>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
By integrating equations (15) and (16), and using the relationship
<math>
<semantics>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>=</mo>
<msubsup>
<mi>v</mi>
<mi>r</mi>
<mn>2</mn>
...
</msubsup>
<mo>+</mo>
<msubsup>
<mi>v</mi>
<mi>t</mi>
<mn>2</mn>
...
</msubsup>
...
</mrow>
...
</semantics>
...
</math>
between velocity vector and its components, and the relationship
<math>
<semantics>
<mrow>
<msub>
<mi>v</mi>
<mi>r</mi>
...
</msub>
<mo>=</mo>
<mi>λ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
between radial velocity and Doppler frequency shift, the ranging formula as
follows can be obtained
...
</p>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
<mo>)</mo>
...
</mrow>
<mi>Δ</mi>
<mi>t</mi>
...
</mrow>
<mrow>
<mi>λ</mi>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mrow>
<mi>v</mi>
<mi>λ</mi>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
where:
<math>
<semantics>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
<mo>−</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
.
...
</p>
<title>4.2. Simulation modification</title>
<p>
Parameter value used in calculation:
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
<mo>=</mo>
<mn>600</mn>
<mtext>km</mtext>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<mi>d</mi>
<mo>=</mo>
<mn>100</mn>
<mtext>km</mtext>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<mi>v</mi>
<mo>=</mo>
<mn>300</mn>
<mi>m</mi>
<mo>/</mo>
<mi>s</mi>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<mi>λ</mi>
<mo>=</mo>
<mn>0.3</mn>
<mi>m</mi>
...
</mrow>
...
</semantics>
...
</math>
,
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
<mo>=</mo>
<mn>50</mn>
<mi>H</mi>
<mi>z</mi>
...
</mrow>
...
</semantics>
...
</math>
.
...
</p>
<p>
The radial distance and Doppler frequency shift values at the two detection
positions are respectively selected to calculate.Figure
<xref ref-type="fig" rid="fig3"> 3</xref>
shows some of the calculated curves selectively.
...
</p>
<fig id="fig3">
<label>Figure 3</label>
<caption>
<p>Relative calculation errors of ranging solution using frequency difference
directly.</p>
...
</caption>
<graphic xlink:href="174.fig.003"/>
...
</fig>
<p>
The results show that, assuming that the detection platform moves the distance
<math>
<semantics>
<mi>d</mi>
...
</semantics>
...
</math>
uniformly along a straight line, if the average frequency shift at two positions
is adopted, the ranging formula at position 1 has the minimum relative
calculation error, that is,
...
</p>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<mn>0.25</mn>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mrow/>
...
</msubsup>
<mo>+</mo>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
<mrow/>
...
</msubsup>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mn>2</mn>
...
</msup>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mrow>
<mi>v</mi>
<mi>λ</mi>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
The dotted line inFigure
<xref ref-type="fig" rid="fig3"> 3</xref>
shows the relative calculation error of Equation (19). This shows mathematically
that there is a lag in the detection of target distance. It is necessary to
obtain the average value of Doppler frequency shift on a moving distance, and
then only the radial distance at the starting position can be accurately
obtained.
...
</p>
<p>
The dot line inFigure
<xref ref-type="fig" rid="fig3"> 3</xref>
shows the relative calculation error of equation (20), which indicates that the
Doppler frequency shift at the terminating position is used to calculate the
radial distance
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
at position 2, and the relative calculation error obtained is very poor
...
</p>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mrow>
<mi>v</mi>
<mi>λ</mi>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>If equation (20) is simply modified to</p>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
<mrow>
<mi>v</mi>
<mi>λ</mi>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>
The solid line inFigure
<xref ref-type="fig" rid="fig3"> 3</xref>
shows the relative calculation error of Equation (21). The modified results show
that, in the case of only two detection, in order to directly use the Doppler
shift measurement value at the current moving distance to solve the radial
distance
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
at the end position more accurately, the Doppler shift value at the starting
position 1 should be used to replace the frequency shift value at the second
position 2.
...
</p>
<title>4.3. Error calculation of frequency shift differential ranging solution
</title>
<p>According to the results of simulation correction, formula (21) is adopted
for error analysis. Preset preestablish</p>
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
<mo>=</mo>
<msub>
<mi>a</mi>
<mn>0</mn>
...
</msub>
<mfrac>
<mrow>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>where:</p>
<inline-formula>
<math>
<semantics>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
...
</msub>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mi>v</mi>
<mi>λ</mi>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
<mo>=</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
...
</msup>
<mo>−</mo>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mn>2</mn>
...
</msubsup>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mi>Δ</mi>
<msub>
<mi>f</mi>
<mi>d</mi>
...
</msub>
...
</mrow>
<mo>|</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p>
Ranging error resulting from frequency shift
<math>
<semantics>
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</p>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<msubsup>
<mi>Q</mi>
<mn>0</mn>
<mn>2</mn>
...
</msubsup>
...
</mrow>
...
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>−</mo>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>)</mo>
...
</mrow>
...
</mrow>
...
</semantics>
...
</math>
<p>thereinto</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mrow/>
...
</msubsup>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<msup>
<mi>λ</mi>
<mn>2</mn>
...
</msup>
<msubsup>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
<mrow/>
...
</msubsup>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mn>1</mn>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p>
Ranging error resulting from frequency shift
<math>
<semantics>
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
...
</p>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
...
</msub>
<msub>
<mi>P</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<msubsup>
<mi>Q</mi>
<mn>0</mn>
<mn>2</mn>
...
</msubsup>
...
</mrow>
...
</mfrac>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
...
</semantics>
...
</math>
<p>thereinto</p>
<inline-formula>
<math>
<semantics>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>Q</mi>
<mn>0</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
<mo>=</mo>
<mn>1</mn>
...
</mrow>
...
</semantics>
...
</math>
...
</inline-formula>
<p>The relative ranging error caused by frequency shift measurement error</p>
<disp-formula id="FD25">
<label>(25)</label>
<math>
<semantics>
<mrow>
<msub>
<mi>σ</mi>
<mrow>
<mi>r</mi>
<mo>_</mo>
<mi>F</mi>
<mi>D</mi>
...
</mrow>
...
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>1</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
<mo>+</mo>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mo>∂</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
<mrow>
<mo>∂</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>d</mi>
<mn>2</mn>
...
</mrow>
...
</msub>
...
</mrow>
...
</mfrac>
...
</mrow>
<mo>|</mo>
...
</mrow>
<msub>
<mi>σ</mi>
<mi>f</mi>
...
</msub>
...
</mrow>
<mo>]</mo>
...
</mrow>
<mo>.</mo>
...
</mrow>
...
</semantics>
...
</math>
...
</disp-formula>
<title>4.4. Comparison of relative calculation accuracy</title>
<p>
During the simulation calculation, the leading angle
<math>
<semantics>
<mrow>
<msub>
<mi>β</mi>
<mn>1</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
at starting point 1 of the detection platform, the radial distance
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>1</mn>
...
</msub>
<mo>=</mo>
<mn>600</mn>
<mtext>km</mtext>
...
</mrow>
...
</semantics>
...
</math>
from starting point 1 to the target, the moving distance
<math>
<semantics>
<mrow>
<mi>d</mi>
<mo>=</mo>
<mn>100</mn>
<mtext>km</mtext>
...
</mrow>
...
</semantics>
...
</math>
of the detection platform, the flight speed
<math>
<semantics>
<mrow>
<mi>v</mi>
<mo>=</mo>
<mn>300</mn>
<mi>m</mi>
<mo>/</mo>
<mi>s</mi>
...
</mrow>
...
</semantics>
...
</math>
and the wavelength
<math>
<semantics>
<mrow>
<mi>λ</mi>
<mo>=</mo>
<mn>0.3</mn>
<mi>m</mi>
...
</mrow>
...
</semantics>
...
</math>
of the detection signal are preset.
...
</p>
<p>
Then the radial distance
<math>
<semantics>
<mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
...
</msub>
...
</mrow>
...
</semantics>
...
</math>
of the target at the second position 2 and other geometric parameters are
calculated by using trigonometric function relations. According to the
definition (1) of Doppler frequency shift, the Doppler frequency shift value is
calculated.
...
</p>
<p>Figure 4 shows the relative calculation error curves of Doppler ranging
solutions based on direction finding method and existing ranging solutions based
on frequency shift measurement. Obviously, the relative calculation error of
Doppler ranging solution with direction finding method is almost zero, and the
most direct reason is that there is no approximate treatment in the process of
proving the ranging solution. In terms of mathematical expression, the leading
angle tends to be the same at both probe positions only when the leading angle
is small. Now, if you just look at the denominator and it tends to go to zero,
it seems to result in a singularity. But in fact, so the leading angle is going
to zero, so the numerator is going to zero. Therefore, according to the limit
rule of derivative, the whole ranging formula tends to a fixed value.</p>
<p>As far as the shape of the error curve is concerned, the author thinks that
the relative error distribution of the Doppler ranging solution by means of
direction finding method should be only some chaotic burrs, and has nothing to
do with physical characteristics. In other words, there is no functional
correlation between the distribution characteristics of the relative calculation
error curve and the leading angle in physical characteristics. However, the
relative calculation error of the ranging solution based on approximate
difference processing is related to the physical characteristics.</p>
<p>According to the mathematical definition (15) of Doppler rate of change, the
singularity of the equation does not appear when the tangential velocity and
distance are fixed. However, after the approximate difference treatment, the
Doppler change rate is directly determined by the difference between the two
frequency shifts. According to the definition (1) of Doppler, in the case of
fixed velocity, the magnitude of the Doppler shift is only related to the
leading angle. Therefore, when the leading angle is small, that is, when the
leading angle at the two detection positions tends to be the same, the magnitude
of the two frequency shifts is very close, and the difference between the two
frequency shifts tends to zero. At this point, the Doppler rate of change tends
to be singular, which leads to the relative calculation error of the approximate
ranging solution is relatively large when the leading angle is small.</p>
<fig id="fig4">
<label>Figure 4</label>
<caption>
<p>Comparison of computational accuracy.</p>
...
</caption>
<graphic xlink:href="174.fig.004"/>
...
</fig>
<title>4.5. Comparison of ranging errors</title>
<p>
The relative ranging error curves for the two ranging solutions have been shown
inFigure
<xref ref-type="fig" rid="fig5"> 5</xref>
. When the leading angle is less than 40 degrees, there is a slight difference
between the two ranging methods. But when it is greater than 40 degrees and
tends to 90 degrees, the relative ranging errors of the two solutions tend to be
the same. The author's existing and ongoing research results show that[
<xref ref-type="bibr" rid="R6">6</xref>
,
<xref ref-type="bibr" rid="R7">7</xref>
,
<xref ref-type="bibr" rid="R8">8</xref>
], under the same frequency shift measurement method, no matter what
mathematical deduction method is adopted, the ranging error is basically the
same. In other words, the ranging error seems to be only related to the
frequency shift measurement method, and has nothing to do with the mathematical
expression of the ranging equation.
...
</p>
<fig id="fig5">
<label>Figure 5</label>
<caption>
<p>Comparison of relative ranging errors.</p>
...
</caption>
<graphic xlink:href="174.fig.005"/>
...
</fig>
...
</sec>
<sec id="sec5">
<title>Conclusion</title>
<p>The existing ranging method based on frequency shift differential processing
has some approximation in the process of mathematical deduction and
verification, so the calculation accuracy of the mathematical formula obtained
is relatively poor. Based on the relationship between frequency shift and angle,
the Doppler ranging formula derived by direction finding method basically has no
approximation. Therefore, the calculation accuracy of the ranging formula
obtained is very good. Furthermore, the relative ranging error of the obtained
solutions is basically consistent with that of the existing solutions directly
based on frequency shift measurement. </p>
<p>Improving the accuracy of formula calculation is not only helpful for
engineering design, but also for theoretical research. The research in this
paper may provide a reference method for passive ranging technology of motion
detection platform.</p>
<p/>
...
</sec>
...
</body>
<back>
<ref-list>
<title>References</title>
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</article>