Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks
<p>The network topology of the TO LPWAN.</p> "> Figure 2
<p>The capture region defined by center and radius, based on SN pair (<math display="inline"><semantics> <msub> <mi>s</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math>).</p> "> Figure 3
<p>The blue regions outside of the circle are the ICD regions for <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math>, following IC of <math display="inline"><semantics> <msub> <mi>s</mi> <mn>1</mn> </msub> </semantics></math>, for <math display="inline"><semantics> <mrow> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>≥</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math></p> "> Figure 4
<p>The capture and ICD Region for a pair of SNs <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>s</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics></math> and: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>≤</mo> <mfrac> <mn>1</mn> <msup> <mi>τ</mi> <mn>2</mn> </msup> </mfrac> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mfrac> <mn>1</mn> <msup> <mi>τ</mi> <mn>2</mn> </msup> </mfrac> <mo><</mo> <mi>z</mi> <mo>≤</mo> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> </semantics></math>; and (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>></mo> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> </semantics></math>.</p> "> Figure 5
<p>The capture margin of the <math display="inline"><semantics> <mi>τ</mi> </semantics></math> with fixed <span class="html-italic">z</span>.</p> "> Figure 6
<p>The location margin of <math display="inline"><semantics> <mi>δ</mi> </semantics></math>. The blue region shows the location of GW: with location margin (<b>b</b>); and without location margin (<b>a</b>).</p> "> Figure 7
<p>Crescent of each pair of SNs and location of 3 SNs and 2 GWs. The crescent region where the GW can decode both collided packets sent by the pair of SNs with capture effect and IC method, are shown in (<b>a</b>). In (<b>b</b>–<b>d</b>), the OPs are generated by the crescents corresponding to groups of sensor pairs <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mrow> <mo>}</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>s</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mrow> <mo>}</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>s</mi> <mn>3</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>s</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mn>3</mn> </msub> <mrow> <mo>}</mo> </mrow> </mrow> </semantics></math>, respectively, such as <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mn>12</mn> </msub> <mo>}</mo> </mrow> </semantics></math> in (<b>b</b>).</p> "> Figure 8
<p>Example of the suggested WBG Algorithm with 3 SNs and GWs. (<b>a</b>) shows the complete bipartite graph, where <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. The weights of the edges joining <math display="inline"><semantics> <msub> <mi>P</mi> <mn>11</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> and joining <math display="inline"><semantics> <msub> <mi>P</mi> <mn>17</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> are 1 and 2, respectively, as shown in (<b>b</b>).</p> "> Figure 9
<p>PGL Algorithm with 3 SN and 2 GW.</p> "> Figure 10
<p>Contentions of each SN when a GW is placed at different locations. <math display="inline"><semantics> <msup> <mn>1</mn> <mo>♯</mo> </msup> </semantics></math> GW is at the location by WBG placement, <math display="inline"><semantics> <msup> <mn>2</mn> <mo>♯</mo> </msup> </semantics></math> GW is at the location of capture circle only and <math display="inline"><semantics> <msup> <mn>3</mn> <mo>♯</mo> </msup> </semantics></math> GW is at location of out of any crescentm as shown in (<b>a</b>,<b>b</b>) by zooming in. (<b>c</b>) The coordinate location of SNs and GWs in different strategy. (<b>d</b>–<b>f</b>) The contention of each SN with three cases of the GW placement, respectively.</p> "> Figure 11
<p>Average Contention for different number of SNs with one and two GWs, by WBG and PGL algorithms with capture only and capture with IC for different pixel densities.</p> "> Figure 12
<p>Simulation visualizations for three GWs placed by PGL (<b>a</b>) and by naive placement (<b>c</b>) along a sine wave composed by 100 SNs. (<b>b</b>) The average contentions comparing in different number of GWs by capture and IC to capture only separately; and (<b>d</b>) the average contentions for capture and IC comparing in different number of GWs placed by PGL and naive.</p> "> Figure 13
<p>Simulation visualizations for three GWs placed by PGL (<b>a</b>) and by naive placement (<b>c</b>) in the circumference of a circle composed by 100 SNs. (<b>b</b>) The average contentions comparing in different number of GWs by capture and IC to capture only separately; and (<b>d</b>) the average contentions comparing for capture and IC in different number of GWs placed by PGL and naive.</p> "> Figure 14
<p>Simulation visualizations for three GWs placed by PGL (<b>a</b>) and by naive placement (<b>c</b>) in the circumference composed by 100 SNs in a uniformly random distribution. (<b>b</b>) The average contentions comparing in different number of GWs by capture and IC to capture only separately; and (<b>d</b>) the average contentions for capture and IC comparing in different number of GWs placed by PGL and naive.</p> "> Figure 15
<p>The results with GWs placed by the PGL algorithm with capture and IC and capture only both show that a given number of GWs can give a predictable contentions reduction even with more SNs. However, with capture and IC, the reduction of contentions has much more than that with capture only.</p> "> Figure 16
<p>To maintain a desired minimum average contention, the number of GWs placed by the PGL algorithm, both with capture and IC and capture only, will grow with the increasing number of SNs. However, the required minimum number of GWs placed by PGL with capture and IC will be less than that with capture only.</p> ">
Abstract
:1. Introduction
2. System Model
2.1. Capture Effect
2.2. Interference Cancellation
3. Single GW Placement for Two SNs
3.1. Capture Circle
3.2. IC and the Decoding Circle
- (1)
- if , then ,In this case, g will be outside a circle with center and radius , as shown in Figure 3. This is the condition for to be decoded following IC of , when . We refer to this circle as the IC and decoding (ICD) circle.
- (2)
- if , then ,In this case, g is inside a circle whose center is and radius is . This area for g is illustrated as in Figure 2, with replaced by and , exchanged with each other, for to be decoded after IC of .
3.3. IC and the Decoding Crescent
3.4. Margins
4. Algorithms for Multiple GWs Placement
4.1. Algorithm of Weight Bipartite Graph (WBG)
- If an OP is strictly inside of the capture region and outside of ICD region (inside of the ICD circle) decided by an ordered pair of SNs , an edge with a weight of exists between the vertices OP and . In other words, an edge is weighted by if the OP is inside a capture circle but outside the decoding crescent.
- If an OP is inside or on the boundary of the capture region and inside or on the boundary of ICD region (outside of the ICD circle) decided by an ordered pair of SNs , which means in or touching a crescent, the edge between the OP and has a weight of .
- Otherwise, there is no edge between the OP and .
Algorithm 1 Algorithm of WBG |
|
4.2. Algorithm of PGL
Algorithm 2 Algorithm of PGL |
|
5. Numerical and Simulation Results
5.1. Sensor Contention
5.2. Simple Scenario
5.3. Comparison of the WBG and PGL Algorithm
5.4. Study of PGL for Larger and Different Network Topologies
5.5. Contentions versus GWs Number
5.6. Required GWs and Minimum Contentions
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Tian, H.; Weitnauer, M.A.; Nyengele, G. Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks. Sensors 2018, 18, 3884. https://doi.org/10.3390/s18113884
Tian H, Weitnauer MA, Nyengele G. Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks. Sensors. 2018; 18(11):3884. https://doi.org/10.3390/s18113884
Chicago/Turabian StyleTian, Hongxian, Mary Ann Weitnauer, and Gedeon Nyengele. 2018. "Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks" Sensors 18, no. 11: 3884. https://doi.org/10.3390/s18113884