Sub-Nyquist Sampling of Wideband Signals

Itai Friedman Tal Miller

Supervised by:

Deborah Cohen

Technion – Israel Institute of Technology

Optimization of the choice of mixing sequences

Presentation Outline

Brief System Description Project ObjectiveSystem SimulationLiterature Review Project Gantt

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

Sz f

~~~~

z f

SA

y f

A

S Sz f A y f †

1

2 sT

1

2 sT

1

2 sT

1

2 sT

Now we can solve a linear set of equations for input signal:

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

In the sequences case:

A

MWC – Mixing & Aliasing

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

ip t

What is our part in the system?

Analog signal generation

Mixing

Filtering

Sampling

Recovery

The code already exists, we modify the mixing functions generator

System Simulation• Simulation parameters:

10

10, 50

15

50

51.28

195

nyquist

channels number

s p

sequence length

f GHz

N B MHz

SNR dB

m

f f MHz

M

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5

x 104 Original signal

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5

x 104 Original noised signal

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5

x 104 Reconstructed signal

t

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 109

0

2

4

6

8

10

12

14

16

18x 10

5 Spectrum Amplitude of original signal

Frequency (Hz)

|X(f

)|

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 109

0

2

4

6

8

10

12

14

16

18x 10

5 Spectrum Amplitude of noised signal

Frequency (Hz)

|X(f

)|

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 109

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

6 Spectrum Amplitude of reconstructed signal

Frequency (Hz)

|X(f

)|

0 20 40 60 80 100 120 140 160 180 200

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sign matrix row 1

sign

val

ue

index in row 1

MWC – Support Recovery (CTF)

Solve in the time domain for each n:

Time consumingNot robust to noise

CTF (Continuous To Finite):

Problem: infinite number of linear systems (f is continuous)

Infinite problem (IMV) One finite-dimensional problem

SparkDefinitions:The spark of a given matrix A is the smallest number of columns that are linearly dependentspark(A)≥k if every set of (k-1) columns are linearly independent

SparkDefinitions:The spark of a given matrix A is the smallest number of columns that are linearly dependentspark(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

y Axmy

x

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

(0,1)k

2 2 2

2 2 2(1 ) (1 )k kx Ax x

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)

(0,1)k

2 2 2

2 2 2(1 ) (1 )k kx Ax x

2 (0,1)k

Probabilistic ViewsProblem: Calculating Spark/RIP is NP-hardSolution: Take on a probabilistic worldview

Mishali & Eldar ‘10

Probabilistic ViewsProblem: Calculating Spark/RIP is NP-hardSolution: 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

1p

Worldview 1

Mishali & Eldar ‘10

Probabilistic ViewsProblem: Calculating Spark/RIP is NP-hardSolution: 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 xProblems:

Lack of characterization of A structureIn practice, implementing A on hardware is deterministic and not dynamic

1p

Worldview 1

Mishali & Eldar ‘10

Probabilistic ViewsProblem: Calculating Spark/RIP is NP-hardSolution: Take on a probabilistic worldview

Assume signal itself is randomSignal randomness is demanded in the properties:

StRIP – Statistical RIPExRIP – Expected RIP

Worldview 2

Mishali & Eldar ‘10

StRIPDefinition: 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

( , , )kStRIP k p( , )kRIP k

Mishali & Eldar ‘10

StRIPDefinition: 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

( , , )kStRIP k p( , )kRIP k

Mishali & Eldar ‘10

ExRIPDefinition: 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 distributedNonzero values of x are i.i.d

( , , )kExRIP k p( , )kRIP k

Mishali & Eldar ‘10

ExRIPTheorem: Let be the MWC sensing matrix. If the nonzeros of x are drawn from a symmetric distribution, then has the ExRIP with probability:

2

2

(1 ) (1 ( ) 2 ( ))1

( ) ( ( ) ( )) ( ) 1

k M

k

k k M k

k

C S Sp

B C S S C M S

/SF mM

Mishali & Eldar ‘10

ExRIP2

2, 1

1( ) ( )

( )

mTi k

i k

S S SmM

22, 1

1( ) ( )

( )

mTi k

i k

S S SmM

2

2 3, 1

1( )

m

i ki k

S S Sm M

1MM

M

4 4

1

{ }K

K ii

C E u u

2

42

1

{ }K

K ii

B E u u

Mishali & Eldar ‘10

ExRIPExRIP guarantees for different families of binary sequences:

Further literature review is needed in the field of families of binary mixing sequences

Mishali & Eldar ‘10

Project GanttOctobe

rSeptemb

erAugus

tJuly June Achievements

exams

Further literature review of mixing functions field

Simulating sequences based on conclusions from literature

Determining optimal sequences

Summing up results into project book

Thank youFor listening

And thanks Debby for the basis to our presentation