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Electromagnetic Sensing for Space-borne Imaging

Electromagnetic Sensing for Space-borne Imaging. Lecture 7 Beam Steering, Optimal Beam Forming. Beam Steering: Consider the 1-D Array Example, But let the weights be more general. Beam Steering: Consider the 1-D Array Example, But let the weights be more general.

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Electromagnetic Sensing for Space-borne Imaging

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  1. Electromagnetic Sensing for Space-borne Imaging Lecture 7Beam Steering, Optimal Beam Forming

  2. Beam Steering: Consider the 1-D Array Example, But let the weights be more general

  3. Beam Steering: Consider the 1-D Array Example, But let the weights be more general

  4. Let’s construct a reasonably general 2-D Array Model yP m n D xP Square Array, N apertures on a side

  5. 2-D Flat Panel Array Model

  6. 2-D Flat Panel Array Model – Beam Steering

  7. yQ xQ What does the ground plane power distribution look like? • The small spots are potential power reception areas (covered by rectennas). We would like these to be small. • The big circle is the maximum coverage area. We would like this to be large enough to service ~ 10 reception stations.

  8. How to get the desired power distribution on the ground • Assumptions: • The phased array geometry and dimensions are given. • We want a specific field amplitude distribution on the ground. Denote this by • Question: What phased array gains will produce the best possible approximation to in the sense that we minimize: • Where and have the same total power

  9. “The old RF guys never died. They just phased array” - Anon. Gain Time delay 4 1 3 2 5 Assumption: The individual transmitter apertures all have immediate neighbors with spacing

  10. L2 Space of Lebesgue Integrable Functions • “L” for Henri Lebesgue, superscript “2” to denote a generalization of the vector 2-norm • Generally, a finite or infinite dimensional, linear vector space • An inner product is defined • The inner product induces the norm • The specific L2space we use is the space of all square-integrable, complex-valued functions of two variables. For an f(x,y) in this space:

  11. Inner Product and Norm – Dirac “Bra-Ket” Notation

  12. Operators, Norm Preserving Transformations

  13. Operators, Norm Preserving Transformations

  14. L2 Formulation of the optimization problem

  15. L2 Solution of the Optimization Problem

  16. L2 Solution of the Optimization Problem

  17. L2 Solution of the Optimization Problem

  18. L2 Solution of the Optimization Problem

  19. L2 Solution of the Optimization Problem

  20. Field Point Spread Function

  21. There is another way to suppress side beams- Randomize the transmitter locations! yP D xP

  22. Power Distribution for a Randomized Array

  23. Power Distribution for a Randomized Array

  24. Power Distribution for a Randomized Array

  25. Power Distribution for a Randomized Array

  26. Total power Delivered by a Randomized Array

  27. Optimal Average Distance Between Transmitters

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