Sub-Nyquist Sampling of Wideband Signals

Itai Friedman Tal Miller

Supervised by:

Deborah CohenProf. Yonina Eldar

Technion – Israel Institute of Technology

Optimization of the choice of mixing sequences

Final Presentatio

n

Presentation OutlineBrief System Description Project ObjectiveSimulation MethodCommon Communication SequencesLu Gan’s SequencesSequences ComparisonExpander PerformanceConclusions and Insights

Motivation: Spectrum Sparsity

Spectrum is underutilizedIn 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 transmissionsB – max bandwidth of each transmission

Output:

Reconstructed signalBlind detection of each transmission

Minimal achievable rate: 2NB << fNYQ

~ ~~~

Mishali & Eldar ‘09

NYQf

The Modulated Wideband Converter (MWC)

~ ~~~

ip t

iy n

Mishali & Eldar ‘10

1

2 sT

1

2 sT

1

2 sT

snT

snT

snT

MWC – Recovery System

MWC – Mixing & AliasingSystem requirement:

are periodic functions with period called “Mixing functions”

Examples for :…

ip t

1

-1

pT

Frequency domain

ip t

Project ObjectiveMain objective: Finding optimal Mixing sequences for effective signal reconstructionFinding the characteristics of those sequences.

Research EnvironmentBased on the basic version of the MWC simulation.Expanded to support:

Various kinds of sequencesCalculating the correlation parameters

The ExpanderDesigned 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.

sin ( ) cos(2 )i ii

Signal E c t f t

ExRIP: Conditioning of The Modulated

Wideband ConverterThe 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

2

2 3, 1

1( )

m

i ki k

S S Sm M

22, 1

1( ) ( )

( )

mTi k

i k

S S SmM

22, 1

1( ) ( )

( )

mTi k

i k

S S SmM

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

MWCThis 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

-20 -15 -10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Random vs. deterministic energies selection method

Input SNR (dB)

p

Gold 80X511 determinstic

Gold 80X511 random

LU: Maximal 80X511 determinstic

LU: Maximal 80X511 random

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

S R C

C M M

R

Random Selection of Rows

We tested the necessity of rows random selection by using three different row selection methods:

Choosing first m rowsChoosing every 6th row, total of m rowsRandom selection (MATLAB’s randperm function)

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

, ,

100SNR

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

S R C

( ) 0,2

BX f f

gfhgcg

The condition is not necessary, same results (except for Wrong-Legendre).

Gan & Wang ‘13

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

Gold 80X511

Gold 80X511 zero baseband

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

LU: m-sequence 80X511

LU: m-sequence 80X511 zero baseband

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

Wrong-Legendre 80X509

Wrong-Legendre 80X509 zero baseband

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

LU: Legendre 80X509

LU: Legendre 80X509 zero baseband

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

S R GPG M M

PR

Various Sequences Performance

scscdscsdcdsc

-20 -15 -10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input SNR (dB)

p

Random 80X511

Gold 80X511

LU: Maximal 80X511Wrong-Legendre 80X509

LU: Legendre 80X509

LU: PCP 80X511

Flatness in Freq. DomainTo understand the poor performance of the Wrong-Legendre sequence, we observed the sequences in the frequency domain:

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

550

FF

T

Random

MaximalHadamard

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

550

FF

T

WrongLegendre

LegendreGold

Flatness in Freq. DomainUnlike the other sequences, Hadamard and Wrong-Legendre are not flat in 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:

mq M

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

Random1 80X511

Random1 80X511 expander q=3Random1 80X511 expander q=5

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

Gold 80X511

Gold 80X511 expander q=3

Gold 80X511 expander q=5

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

Wrong-Legendre 80X509

Wrong-Legendre 80X509 expander q=3Wrong-Legendre 80X509 expander q=5

LU: Maximal 80X511

LU: Maximal 80X511 expander q=3LU: Maximal 80X511 expander q=5

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

LU: Legendre 80X509

LU: Legendre 80X509 expander q=3LU: Legendre 80X509 expander q=5

-20 -15 -10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Input SNR (dB)

p

LU: PCP 80X511

LU: PCP 80X511 expander q=3LU: PCP 80X511 expander q=5

MWC Demo Performance

Simulation Parameters:6, 20 , 24 , 6.44p nyqN B MHz f Mhz f Ghz

4, 5, 263m q M

-5 0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input SNR (dB)

p

Demo Recovery Rate

RandomGold

LU: Maximal

LU: Legendre

LU: Legendre Zero-BasebandLU: PCP

Conclusions and InsightsA 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 InsightsLack 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 WorkImplementation 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 WorkExamination 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 youFor listening

Thanks to DebbyFor Everything

For a broader review, see project book