Eric Monteiro

Steve Hranilovic

Contributions

2

• Differentiable non-convex formulation

solvable by interior point methods

– Can specify perceived color of light source

• Efficient design heuristic for large

constellations

– Can NOT specify perceived color of light source

Color-Shift Keying (CSK)

3

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Constraints:

• �� �� �� � • 0 ��,�,� ��,�,�• ∑���� �����

Advantages:• Zero flicker

• Reduced Inrush Currents

4

Transmittable Set

4

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Transmittable Set

5

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6

System Model

6

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Filtered Photo-detectorsRGB Light Emitting Diode

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System Model

7

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8

Basis Change

8

Basis Change

9

• Non-Convex

• Discontinuous

10

Objective

max !"#min�&' !" ( !) *

*

Optimization Methods

Stochastic

- Works with any objective

- Avoids poor local minima

- Brute force/inefficient

- Converge in probability

11

Deterministic

- Well defined stopping criteria

- Comparatively fast (per trial)

- Easy to implement

- Sensitive to starting point

Interior Point Methods

• Intended for Convex Optimization

– Finds local minima of non-convex problems

• Assumptions:

– Linear inequality constraints

– Affine Equality Constraints

– Continuous Objective

12

Continuous Approximation

• Minimum approximation:

13

min +� ,�

(-. /01234

�/6

max 74#(-. /012 !"1!) 8

8

�&'/6

• New Objective:

Results: Validity

14

Results: Arbitrary Regions

15

Results: Color Balance

16

Results: Color Balance

17

0 2 4 6 8 10 12 14 1610

-6

10-5

10-4

10-3

10-2

10-1

100

No Balance

White Balance

Edge Balance

SNR 10-:;<=>? @⁄ B

Sym

bo

l E

rro

r R

ate

Results: Large Constellations

18

Heuristic

19

Heuristic

20

5 10 15 20 25 300

.1

.2

.3

0.4

0.5

0.6

.7

.8

Constellation Size

No Balance

No Balance: Hexagonal

Comparison

21

No

rma

lize

d M

inim

um

Dis

tan

ce

Constellation Size

22

Comparison

22

No

rma

lize

d M

inim

um

Dis

tan

ce

Constellation Size

6 7 8 9 10 11

0.4455

0.4596

0.4738

0.4879

0.5020

0.5162

Conclusions

• A systematic optimization approach to CSK

design has been presented, which functions

under:

– Any constellation size

– Arbitrary constraint region and color balance

• Under no color balance, an efficient heuristic

based on hexagonal lattices has been

demonstrated

23

QUESTIONS?

24

Detailed Constraints

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

!" � 1 CD��E�

25

FGH � ����

I!" J

K �L 0 00 ⋱ 00 0 L

N � J⋮J

GH � vec>!")

Minimum example

26

-0S+< ≪+*U.+6 � 1

min +<, +* ,�

(-. 013V 0138

, (-. 013V � +<

Optimization Example

27

!" ← XU.+:YZSUXS[.;�:[.S6 ← 1\]�^_6 � 6`abcNd!efg ← argmax !"#

(-. /012 !"1!) 88

�&'/6

6 ← 26!" ← !efg_�N\]�^_

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

Results: Cross-talk

28

0

0.5

10

0.51

0

0.5

1

+k�l �0.2474

� � 1 0 00 0.8 0.10 0.1 0.8