Sub- Nyquist Sampling of Wideband Signals

1. Sub-Nyquist Sampling of Wideband Signals Optimization of the choice of mixing sequences Itai Friedman Tal Miller Supervised by: Deborah Cohen Technion – Israel Institute of Technology

2. Presentation Outline • Brief System Description • Project Objective • System Simulation • Literature Review • Project Gantt

3. Spectrum Sparsity • Spectrum is underutilized • In a given place, at a given time, only a small number of PUs transmit concurrently Shared Spectrum Company (SSC) – 16-18 Nov 2005

4. Model ~ ~ ~ ~ • Input signal in Multiband model: • Signal support is but it is sparse. • N – max number of transmissions • B – max bandwidth of each transmission • Output: • Reconstructed signal • Blind detection of each transmission • Minimal achievable rate: 2NB << fNYQ Mishali & Eldar ‘09

5. The Modulated Wideband Converter (MWC) ~ ~ ~ ~ Mishali & Eldar ‘10

6. MWC – Recovery ~ ~ ~ ~ Now we can solve a linear set of equations for input signal:

7. MWC – Recovery System

8. MWC – Mixing & Aliasing • System requirement: are periodic functions with period called “Mixing functions” • Examples for : … 1 -1 Frequency domain

9. MWC – Mixing & Aliasing • In the sequences case:

10. Project Objective • Questions: • What are the best Mixing functions ? • Focusing on {+1,-1} functions, what properties should the sequences have? • Main Objective: • Finding optimal Mixing function sequences for effective reconstruction

11. What is our part in the system? • Analog signal generation • Mixing • Filtering • Sampling • Recovery • The code already exists, we modify the mixing functions generator

12. System Simulation • Simulation parameters:

20. MWC – Support Recovery (CTF) Problem:infinite number of linear systems (f is continuous) • Solve in the time domain for each n: • Time consuming • Not robust to noise • CTF (Continuous To Finite): Infinite problem (IMV)  One finite-dimensional problem

21. Spark • Definitions: • The spark of a given matrix A is the smallest number of columns that are linearly dependent • spark(A)≥k if every set of (k-1) columns are linearly independent

22. Spark • Definitions: • The spark of a given matrix A is the smallest number of columns that are linearly dependent • spark(A)≥k if every set of (k-1) columns are linearly independent • Theorem (reconstruction): • For any vector , there exists at most one k-sparse signal , such that if and only if Spark(A)>2k . In particular, for uniqueness we must have that m ≥2k

23. RIP: (Restricted Isometry Property) • Definitions: • A matrix A has RIP(k) if there exists a such that:

24. RIP: (Restricted Isometry Property) • Definitions: • A matrix A has RIP(k) if there exists a such that: • Properties: • If A satisfies RIP(2k) for any , then spark(A)>2k (reconstruction guarantee) • RIP based theorems give bounds on reconstruction error in the presence of noise (dependence on reconstruction algorithm and noise level)

25. Probabilistic Views Mishali & Eldar ‘10 • Problem: Calculating Spark/RIP is NP-hard • Solution: Take on a probabilistic worldview

26. Probabilistic Views Mishali & Eldar ‘10 • Problem: Calculating Spark/RIP is NP-hard • Solution: Take on a probabilistic worldview • Assume A matrix is random, very high chances ( ) that A has appropriate Spark/RIP qualities for reconstruction, for all k-sparse signals x Worldview 1

27. Probabilistic Views Mishali & Eldar ‘10 • Problem: Calculating Spark/RIP is NP-hard • Solution: Take on a probabilistic worldview • Assume A matrix is random, very high chances ( ) that A has appropriate Spark/RIP qualities for reconstruction, for all k-sparse signals x • Problems: • Lack of characterization of Astructure • In practice, implementing A on hardware is deterministic and not dynamic Worldview 1

28. Probabilistic Views Mishali & Eldar ‘10 • Problem: Calculating Spark/RIP is NP-hard • Solution: Take on a probabilistic worldview • Assume signal itself is random • Signal randomness is demandedin the properties: • StRIP– Statistical RIP • ExRIP– Expected RIP Worldview 2

29. StRIP Mishali & Eldar ‘10 • Definition: The A matrix has the if A has with probability at least p for all k-sparse signals x such that: • supp(x) is uniformly distributed

30. StRIP Mishali & Eldar ‘10 • Definition: The A matrix has the if A has with probability at least p for all k-sparse signals x such that: • supp(x) is uniformly distributed

31. ExRIP Mishali & Eldar ‘10 • Definition: The A matrix has the if A has with probability at least p for all k-sparse signals x such that: • supp(x) is uniformly distributed • Nonzero values of x are i.i.d

32. ExRIP Mishali & Eldar ‘10 • Theorem: Let be the MWC sensing matrix. If the nonzeros of x are drawn from a symmetric distribution, then has the ExRIP with probability:

33. ExRIP Mishali & Eldar ‘10

34. ExRIP Mishali & Eldar ‘10 • ExRIPguarantees for different families of binary sequences: • Further literature review is needed in the field of families of binary mixing sequences

35. Project Gantt

36. Thank you For listening And thanks Debby for the basis to our presentation