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Sub- Nyquist Sampling of Wideband Signals. Optimization of the choice of mixing sequences. Final Presentation. Itai Friedman Tal Miller Supervised by: Deborah Cohen Prof. Yonina Eldar Technion – Israel Institute of Technology. Presentation Outline. Brief System Description - PowerPoint PPT Presentation
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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