Sub- Nyquist Sampling of Wideband Signals

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

Presentation Outline • Brief System Description • Project Objective • Simulation Method • Common Communication Sequences • Lu Gan’s Sequences • Sequences Comparison • Expander Performance • Conclusions and Insights

Motivation: 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

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

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

MWC – Recovery System

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

Project Objective • Main objective: Finding optimal Mixing sequencesfor effective signal reconstruction • Finding the characteristics of those sequences.

Research Environment • Based on the basic version of the MWC simulation. • Expanded to support: • Various kinds of sequences • Calculating the correlation parameters • The Expander • Designed to calculate the recovery probability under various conditions

Simulation Method • Building a certain sensing matrix A. • Counting successful recoveries for different signals. • Successful Recovery = supp(original signal) supp(reconstructed signal)

Simulation Method • , with random carriers and energies. • White noise is added according to SNR level.

ExRIP: Conditioning of The Modulated Wideband Converter • The article discusses a few common communication sequences: Gold Kasami and Hadamard. • It also introduces the correlation parameters . Mishali & Eldar ‘10

ExRIP: Conditioning of The Modulated Wideband Converter Mishali & Eldar ‘10

ExRIP: MWC Conditioning • A formula for the recovery probability is obtained. • The theoretical results for the sequences are: Mishali & Eldar ‘10

ExRIP: MWC Conditioning • We simulated the sequences for SNR=10,100dB. • are similar to the article. Mishali & Eldar ‘10

Conclusion: the formula for p obtains a general estimation of the sequences performance, but SNR level is not considered. Mishali & Eldar ‘10

Deterministic Sequences for the MWC • This article offers new sequences for the MWC. • The simulation conditions use deterministic energies. This condition is easier: Gan & Wang ‘13

Deterministic Sequences for the MWC • From now on we will use the same conditions. Gan & Wang ‘13

Matrix from Single Sequence • The following matrix structure is offered: • is a circulant matrix. • Sequences proposed for the first row: Maximal and Legendre. • is a random subsampling operator, which chooses m rows out of M. Gan & Wang ‘13

Random Selection of Rows • We tested the necessity of rows random selection by using three different row selection methods: • Choosing first m rows • Choosing every 6th row, total of m rows • Random selection (MATLAB’s randpermfunction) Gan & Wang ‘13

Random Selection of Rows • The deterministic selection methods led to poor results. • Insight: the correlation parameters do not predict system’s performance: same parameters but dramatically different p. Gan & Wang ‘13

Examination of Article’s Conditions • The theorem in the article predicts high recovery probability for if the signal is ZERO in baseband: • We examined this condition for different sequences: Gan & Wang ‘13

gfhgcg • The condition is not necessary, same results (except for Wrong-Legendre). Gan & Wang ‘13

Matrix from Periodic Complementary Pair (PCP) • Another matrix structure is offered: • is a matrix constructed from a PCP. • is a permutation operator. • is defined in the same way as before. Gan & Wang ‘13

Various Sequences Performance • scscdscsdcdsc

Flatness in Freq. Domain • To understand the poor performance of the Wrong-Legendresequence, we observed the sequences in the frequency domain:

Flatness in Freq. Domain • Unlike the other sequences, Hadamardand Wrong-Legendre are not flatin the frequency domain, thus their poor performance. • HOWEVER, this is an FFT of a single row and it lacks information on the entire matrix. • Therefore, frequency flat sequences can still have poor results.

MWC Performance with Expander • We simulated the Expander in our system by adding additional digital processing, and expanding the sensing matrix A to . • The simulations results:

MWC Demo Performance • Simulation Parameters:

Conclusions and Insights • A few sequences have very good and similar performance: Random, Gold, LU-Maximal, LU-Legendre, LU-PCP. • p>0.9 for SNR>10. • The main difference between these sequences is in the level of randomness: from full randomness, through random cyclic shifts of a single row, to a completely deterministic matrix.

Conclusions and Insights • Lack of flatness in the frequency domain indicates poor performance of the sequence. The opposite is not necessarily true. • The correlation parameters do not predict well the performance of the sequences. • Using the Expander with q=3,5 does not effect the system’s performance.

Future Work • Implementation of the sequences for different systems that use sub-nyquist sampling principles. • Optimization of the mixing sequences for the specifications of a certain MWC system.

Future Work • Examination of different periodic mixing functions other than the {+1,-1} sequences. • Optimization of the mixing sequences for sparse wideband signals with known carriers, as suggested by Prof. Eldar (Huawei)

Thank you For listening Thanks to Debby For Everything For a broader review, see project book

Sub- Nyquist Sampling of Wideband Signals