Direct Position Determination of Unknown Signals in the Presence of Multipath Propagation
<p>Multiple-path positioning problem with static transponders/receivers.</p> "> Figure 2
<p>Multiple-peak cost function of a frequency band signal and single peak cost function of a base band signal. (<b>a</b>) Cost function for a frequency band signal (<b>b</b>) Cost function for a base band signal.</p> "> Figure 3
<p>Layouts of the numerical examples. (<b>a</b>) Layout A (<b>b</b>) Layout B.</p> "> Figure 4
<p>Spatial spectrum of Signal Subspace Projection (SSP)-MUSIC and Noise Subspace Projection (NSP)-MUSIC in a single path scenario. (<b>a</b>) SSP-MUSIC (<b>b</b>) NSP-MUSIC.</p> "> Figure 5
<p>Spatial spectrum in baseband signal positioning. (<b>a</b>) Spatial spectrum of SSP-MUSIC and NSP-MUSIC (<b>b</b>) Spatial spectrum of Multi-path Propagation (MP)-MUSIC.</p> "> Figure 6
<p>Spatial spectrum when a transponder is close to the anther. (<b>a</b>) Spatial spectrum of SSP-MUSIC and NSP-MUSIC (<b>b</b>) Spatial spectrum of MP-MUSIC.</p> "> Figure 7
<p>Spatial spectrum for a single antenna of each receiving array. (<b>a</b>) Spatial spectrum of SSP-MUSIC and NSP-MUSIC (<b>b</b>) Spatial spectrum of MP-MUSIC.</p> "> Figure 8
<p>Spatial spectrum in a general scenario. (<b>a</b>) Spatial spectrum of SSP-MUSIC and NSP-MUSIC (<b>b</b>) Spatial spectrum of MP-MUSIC.</p> "> Figure 9
<p>Performance of MP-MUSIC-Active Set Algorithm (ASA) and MP-MUSIC-Interior Point Algorithm (IPA).</p> "> Figure 10
<p>Performances of MP-MUSIC and MP-ML (<math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>).</p> "> Figure 11
<p>Performance of MP-ML and MP-MUSIC with different <math display="inline"> <semantics> <mrow> <mi>J</mi> <mo>,</mo> <mi>K</mi> </mrow> </semantics> </math> combinations.</p> "> Figure 12
<p>Performances of MP-MUSIC and MP-ML with different numbers of snapshots.</p> "> Figure 13
<p>Time consumptions and RMSE of different numbers of emitters. (<b>a</b>) Time consumptions of MP-MUSIC and MP-ML (<b>b</b>) RMSE of MP-MUSIC and MP-ML.</p> "> Figure 14
<p>Positioning accuracies and time consumptions of MP-ML.</p> "> Figure 15
<p>MP-ML and CRLB of different numbers of receivers (<math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1024</mn> </mrow> </semantics> </math>).</p> "> Figure 16
<p>MP-ML and CRLB of different numbers of emitters (<math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>1024</mn> <mo>,</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>).</p> "> Figure A1
<p>Two paths with the same delay.</p> ">
Abstract
:1. Introduction
2. Problem Formulation
- MP-MUSIC method;
- MP-ML method.
3. MP-MUSIC Method
3.1. The Limitation of Existing MUSIC Methods
3.1.1. SSP-MUSIC
3.1.2. NSP-MUSIC
3.1.3. Singularity of the Manifold Matrix in the Presence of Multi-Path Propagation
3.1.4. Non-Negative Real Path Attenuation Constraints
3.2. Mathematical Model of MP-MUSIC
3.2.1. Remove the Imaginary Items in the Programming
3.2.2. Convexity of the Programming
3.2.3. Active Set Algorithm
3.2.4. MP-MUSIC Algorithm
Algorithm 1: MP-MUSIC algorithm. |
4. MP-ML Method
4.1. Mathematical Model of MP-ML
4.2. Remove Imaginary Items in the Programming
4.3. An Iterative Algorithm for Solving MP-ML
4.4. Getting the Initial Value
4.5. MP-ML Algorithm
Algorithm 2: MP-ML algorithm. |
5. Numerical Examples
5.1. Scenario Setting and Performance Index Definition
5.2. Performances of the MP-MUSIC Method
- SSP-MUSIC: Signal Subspace Projection MUSIC proposed in [18],
- NSP-MUSIC: Noise Subspace Projection MUSIC without non-negative and real constraints,
- MP-MUSIC-IPA: Noise subspace projection MUSIC with real and non-negative constraints in the multipath propagation scenario and solved by the Interior Point Algorithm,
- MP-MUSIC-ASA: Noise subspace projection MUSIC with real and non-negative constraints in the multipath propagation scenario and solved by the Active Set Algorithm.
5.2.1. SSP-MUSIC and NSP-MUSIC in a Single Path Propagation Positioning Scenario
5.2.2. SSP-MUSIC, NSP-MUSIC and MP-MUSIC in a Multi-Path Propagation Positioning
5.2.3. Performances of MP-MUSIC-ASA and MP-MUSIC-IPA
5.3. The Performance of MP-ML and MP-MUSIC
5.3.1. Insufficient Snapshots
5.3.2. Performances of Different K and J Combinations
5.3.3. The Performances of Different Numbers of Snapshots
5.3.4. Time Consumptions of MP-MUSIC and MP-ML
5.4. Performance of SGP and MP-ML
5.4.1. SPG and MP-ML with a Single Emitter
5.4.2. Performance of Positioning Multiple Emitters
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. Conditions of a Near Singular Manifold Matrix
Appendix B. Convergence of the Iterative Algorithm
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Description | Layout | R (m) | (m)/f (MHz) | M | B (kHz) | K | J | SNR (dB) |
---|---|---|---|---|---|---|---|---|
Baseband signal positioning | A | 30 | 1157.5/0.26 | 11 | 8 | 64 | 100 | 10 |
Transponders are close | B | 30 | 30/10 | 11 | 8 | 64 | 100 | 10 |
Single antenna of each receiver | A | 30 | 30/10 | 1 | 8 | 64 | 100 | 10 |
Standard scenario | A | 30 | 30/10 | 11 | 8 | 64 | 100 | 10 |
Description | Layout | R (m) | (m)/f (MHz) | M | B (kHz) | K | J | SNR (dB) |
---|---|---|---|---|---|---|---|---|
RMSE of MUSIC methods | A | 30 | 1157.5/0.26 | 11 | 8 | 128 | 100 |
Description | Layout | R (m) | (m)/f (MHz) | M | B (kHz) | K | J | SNR (dB) |
---|---|---|---|---|---|---|---|---|
MP-MUSIC | A | 30 | 1157.5/0.26 | 11 | 8 | 16 | 1 | |
MP-ML | A | 30 | 1157.5/0.26 | 11 | 8 | 16 | 1 | |
CRLB | A | 30 | 1157.5/0.26 | 11 | 8 | 16 | 1 |
Description | Layout | R (m) | (m)/f (MHz) | M | B (kHz) | K | J | SNR (dB) |
---|---|---|---|---|---|---|---|---|
MP-MUSIC Combination I | A | 30 | 1157.5/0.26 | 11 | 8 | 32 | 4 | |
MP-MUSIC Combination II | A | 30 | 1157.5/0.26 | 11 | 8 | 16 | 8 | |
MP-MUSIC Combination III | A | 30 | 1157.5/0.26 | 11 | 8 | 8 | 16 | |
MP-MUSIC Combination IV | A | 30 | 1157.5/0.26 | 11 | 8 | 4 | 32 | |
MP-ML | A | 30 | 1157.5/0.26 | 11 | 8 | 128 | 1 | |
CRLB | A | 30 | 1157.5/0.26 | 11 | 8 | 128 | 1 |
Description | Layout | R (m) | (m)/f (MHz) | M | B (kHz) | SNR (dB) | |
---|---|---|---|---|---|---|---|
MP-MUSIC | A | 30 | 1157.5/0.26 | 11 | 8 | , | 10 |
MP-ML | A | 30 | 1157.5/0.26 | 11 | 8 | , | 10 |
CRLB | A | 30 | 1157.5/0.26 | 11 | 8 | , | 10 |
Description | Layout | d | R (m) | (m)/f (MHz) | M | B (kHz) | SNR | K | J |
---|---|---|---|---|---|---|---|---|---|
MP-ML () | A | 30 | 1157.5/0.26 | 11 | 8 | 15 | 16 | 1 | |
MP-MUSIC () | A | 30 | 1157.5/0.26 | 11 | 8 | 15 | 16 | 1 | |
MP-ML () | A | 30 | 1157.5/0.26 | 11 | 8 | 15 | 128 | 1 | |
MP-MUSIC () | A | 30 | 1157.5/0.26 | 11 | 8 | 15 | 16 | 8 |
Description | Number of Receivers | R (m) | (m)/f (MHz) | M | B (kHz) | K | SNR (dB) |
---|---|---|---|---|---|---|---|
MP-ML | 2∼4 | 30 | 1157.5/0.26 | 11 | 8 | 128 | 10 |
CRLB | 1∼4 | 30 | 1157.5/0.26 | 11 | 8 | 128 | 10 |
SGP | 1 | 30 | 1157.5/0.26 | 11 | 8 | 128 | 10 |
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Du, J.; Wang, D.; Yu, W.; Yu, H. Direct Position Determination of Unknown Signals in the Presence of Multipath Propagation. Sensors 2018, 18, 892. https://doi.org/10.3390/s18030892
Du J, Wang D, Yu W, Yu H. Direct Position Determination of Unknown Signals in the Presence of Multipath Propagation. Sensors. 2018; 18(3):892. https://doi.org/10.3390/s18030892
Chicago/Turabian StyleDu, Jianping, Ding Wang, Wanting Yu, and Hongyi Yu. 2018. "Direct Position Determination of Unknown Signals in the Presence of Multipath Propagation" Sensors 18, no. 3: 892. https://doi.org/10.3390/s18030892