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Group Representation Patterns in Digital Signal Processing

Nir Sochen Department of Applied Mathematics Tel-Aviv University, Israel. Group Representation Patterns in Digital Signal Processing. Joint work with Shamgar Gurevich (Berkeley) and Ronny Hadani (Chicago). Dagstuhl, December 4, 2008. Goal. To describe a way in which

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Group Representation Patterns in Digital Signal Processing

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  1. Nir Sochen Department of Applied Mathematics Tel-Aviv University, Israel Group Representation Patternsin Digital Signal Processing Joint work with Shamgar Gurevich (Berkeley) and Ronny Hadani (Chicago) Dagstuhl, December 4, 2008

  2. Goal To describe a way in which a large class of signals with good properties can be described in a deterministic way

  3. Outline • Applications: Sparsity, Radar and communication • Unrelated problem: DFT diagonalization • Representation theory • The Weil representation • The Oscillator system • Main result

  4. Outline • Application 1: Sparsity (See Bruckstein, Donoho & Elad) • Application 2: Radar • Application 3: Communication • Unrelated problem: DFT diagonalization • Representation theory • The Weil representation • The Oscillator system • Main result

  5. Application 1 - Sparsity:General Notions Signals are elements in the Hilbert space H = C( Fp ) We define “atoms” in the Hilbert space Each signal is a combination of atoms The collection of atoms is a “dictionary”

  6. = where L • L>=P, • Sigmais full rank, and • Columns are normalized P

  7. Mutual Incoherence Mutual Incoherence (cross-correlation): The largest correlation between atoms Note that Thm: If there exists a representation of such that then it is unique and recoverable

  8. Sparsity: Conclusion We need as large as possible dictionary with as small as possible mutual incoherence (or cross-correlation)

  9. Outline • Application 1: Sparsity (See Bruckstein, Donoho & Elad) • Application 2: Radar • Application 3: Communication • Unrelated problem: DFT diagonalization • Representation theory • The Weil representation • The Oscillator system • Main result

  10. Application 2: Radar

  11. Basic Principles

  12. Range Measurement and Time Shift

  13. Velocity Measurement and Phase Shift

  14. Combined effect • The emitted signal is S(t) • The echo is R(t) • The signal is time shifted by • The signal is modulated • The phase change depends on the velocity

  15. Time and frequency shifts Denote a time shift by And a frequency shift by

  16. Phase translation matrix

  17. ? Ambiguity function In order to determine the distance and the velocity of an object we need to find the Time and Frequency shifts This is done via the ambiguity function

  18. Outline • Application 1: Sparsity (See Bruckstein, Donoho & Elad) • Application 2: Radar • Application 3: Communication • Unrelated problem: DFT diagonalization • Representation theory • The Weil representation • The Oscillator system • Main result

  19. Application 3: Communication

  20. CDMA Basic Principles Problem: N people transmit messages at the same time Solution1: Build signals for p dimensional space The received signal is

  21. Wireless Communication We need as large as possible dictionary with as small as possible mutual incoherence And with robustness to time-frequency shifts

  22. Outline • Application 1: Sparsity (See Bruckstein, Donoho & Elad) • Application 2: Radar • Application 3: Communication • Unrelated problem: DFT diagonalization • Representation theory • The Weil representation • The Oscillator system • Main result

  23. Strategy of construction • The dictionary of signals is a disjoint union of bases. • First, we give a detail account of the construction of one special basis. • Then we indicate how to generalize it.

  24. First basisDiagonalization of the DFT  Large multiplicities of eigenvalues Goal:Find canonical basis of eigenfunctions for the DFT. Approach: Look for its symmetries.

  25. Intertwining time and phase The DFT intertwines time and phase translations:

  26. Combinning time and phase translation • The combined operation: (PxP matrix)‏ • Rewriting the two relations:

  27. Characterization of the DFT The following linear system for DFT: comprises constraints. Theorem (Stone-von Neumann):  DFT is characterized (up to a scalar) by the system.

  28. DFT characterization (cont.) Rewriting the linear system for DFT as: • Note: - the group of 2x2 matrices with elements in and det =1.‏

  29. Generalization Consider a linear system for a matrix It comprises constraints. Theorem (Stone-von Neumann):  The system characterizes a matrix (up to a scalar).

  30. The Weil representation Theorem (Schur): There exists a unique choice of matrices such that The homomorphism is called the “Weil representation”. Note that in this language:

  31. Summarizing • The Weil representation provides a homomorphism ‏

  32. Fourier SL(2)

  33. Solution: diagonalization of the DFT • The Weil representation provides a homomorphism • This homomorphism enables to change problems: Finding the symmetry group of DFT (difficult)‏  Finding the symmetry group of W in (easy).‏

  34. Symmetries of W in SL(2) • Let • T can be explicitly described Main property: is commutative, i.e.,

  35. Canonical symmetries of DFT • Let • Note • Main property: G is commutative! Because:

  36. Facts from linear algebra • Every is diagonalizable. • If with then they can be diagonalized simultaneously. • If with then they can be diagonalized simultaneously.

  37. In our situation • Apply to The collection can be diagonalized together.  The Hilbert spaces decomposes with

  38. The canonical basis for the DFT Theorem: We have  We resolved the degeneracies!  • Choose a representative from each invariant subspace and get an orthonormal basis • We call it “the canonical basis of eigenfunctions for the DFT”.

  39. Pictorially

  40. Generalization

  41. Properties of the oscillator dictionaries A deterministic collection of sequences • Small auto-correlation

  42. Main Result (cont.)‏ • Small and stable cross-correlation • Small supremum:

  43. P=401 Eigenvector 43

  44. P=401 random vector

  45. Efficiency‏ • Naively one should do p2 measurements • in order to evaluate the ambiguity function. • Here we need only O(p) measurements:

  46. Thank you Publications: ( Hompages: Gurevich, Hadani, Sochen)‏ • PNAS, “The finite harmonic oscillator and its associated sequences”. July 2008. • IEEE Trans. IT, “ The finite harmonic oscillator and its applications to sequences, communications, and radar”, Sep. 2008. • JFAA, Special Issue on Sparsity, “On some deterministic dictionaries that support sparsity”. To appear, 2008.

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