Periodic Non-Uniform Sampling (PNS) for Satellite Communications Marie Chabert 1 , Bernard Lacaze 2 , Marie-Laure Boucheret 1 , Jean-Adrien Vernhes 1,2,3,4 1 Universit´ e de Toulouse, IRIT-ENSEEIHT 2 T´ eSA laboratory 3 CNES (French Spatial Agency) 4 Thales Alenia Space [email protected]
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Periodic Non-Uniform Sampling (PNS) for Satellite Communications · 2016-02-02 · Periodic Non-Uniform Sampling (PNS) for Satellite Communications ... 2 The PNS solution Signal model
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Periodic Non Uniform Sampling (PNS): Landau criterionfe = 2B.
Sx(f)
fB− B+
−fc fc−fmax fmax−fmin fmin
Figure: Passband model
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Periodic Non uniform Sampling (PNS) of orderL
Definition
PNSL: L interleaved uniform sampling sequencesXi = {X(n+ δi), n ∈ Z}, δi ∈]0, 1[, i ∈ {0, L}.
t
nTe (n+ 1)Te (n+ 2)Te (n+ 3)Te
PNSL:
tL−1:
···
t2:
t1:
t0:
∆0
Te
∆1
Te
∆2
Te
∆L−1
Te
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Periodic Non uniform Sampling (PNS) of order 2
Definition
PNS2: 2 interleaved uniform sampling sequencesX0 = {X(n), n ∈ Z} and X∆ = {X(n+ ∆), n ∈ Z}, ∆ ∈]0, 1[.
t
nTe (n+ 1)Te (n+ 2)Te (n+ 3)Te
PNS2:
t1:
t0:
∆0 = 0
Te
∆1
Te
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Periodic Non uniform Sampling (PNS) of order 2
X(n)
t
X(n+Δ)
Figure: Example
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
PNS2 reconstruction - Filter formulation1
µt
ψt
⊕µ∆
⊕+−
X0
X∆ D
X0
XK
X
(a) Orthogonal scheme
ηt
ψt
⊕X0
X∆
X
X′0
X′K
(b) Symmetrical scheme
General filter expressions
µt(f) = St(f)S0(f)e
2iπft
ηt(f) = µt(f)− µ∆(f)ψt(f)
ψt(f) = e2iπf(t−∆) S0(f)St−∆(f)−S∗∆(f)St(f)
S20(f)−|St−∆(f)|2
with: Sλ(f) =∑n∈Z sX(f + n)e2iπnλ, f ∈ (− 1
2 ,12 )
1B. Lacaze. “Filtering from PNS2 Sampling”. In: Sampling Theory in Signal andImage Processing (STSIP) 11.1 (2012), pp. 43–53.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
PNS2 reconstruction - Interpolation formulas2
Closed-form reconstruction formulas
Hypothesis: Bandpass signal composed of two sub-bands, nooversampling.
Simple exact PNS2 reconstruction formulas :
X(t) =A0(t) sin [2πk(∆− t)] +A∆(t) sin [2πkt]
sin [2πk∆]
with Aλ(t) =∑n∈Z
sin [π(t− n− λ)]
π(t− n− λ)X(n+ λ)
if 2k∆ /∈ Z
2B. Lacaze. “Equivalent circuits for the PNS2 sampling scheme”. In: IEEETransactions on Circuits and Systems I: Regular Papers 57.11 (2010), pp. 2904–2914.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Practical sampling device
Time Interleaved Analog to Digital Converters (TI-ADCs)
Periodic Non-Uniform Sampling (PNS) for Satellite Communications 20 / 44
Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Improved PNS
Additional functionalities
Convergence speed improvement for an increasing joint filtertransfer function regularitya.
Selective signal reconstruction with interference rejection for awell-chosen joint filter bandb.
Analytical signal reconstruction for analytic joint filtersc.
aM. Chabert and B. Lacaze. “Fast convergence reconstruction formulas for periodicnonuniform sampling of order 2”. In: IEEE ICASSP 2012.
bJ.-A. Vernhes, M. Chabert, and B. Lacaze. “Conversion Numerique-Analogiqueselective d’un signal passe-bande soumis a des interferences”. In: GRETSI 2013.
cJ.-A. Vernhes et al. “Selective Analytic Signal Construction From A Non-UniformlySampled Bandpass Signal”. In: IEEE ICASSP 2014.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Rectangular filter
HR(f)
f
1
fcfmin fmax-fc -fmin-fmax
B+N (k)B−N (k)
k- 12 k+ 12-k+ 1
2-k- 12
Figure: Rectangular filter
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Trapezoidal filter
HT (f)
f
1
fc
Btr Btr
fmin fmax-fc -fmin-fmax
B+N (k)B−N (k)
k- 12 k+ 12-k+ 1
2-k- 12
Figure: Trapezoidal filter
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Raised cosine filter
HCS(f)
f
1
fc
Btr Btr
fmin fmax-fc -fmin-fmax
B+N (k)B−N (k)
k- 12 k+ 12-k+ 1
2-k- 12
Figure: Raised Cosine Filter
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
2 The PNS solutionSignal modelSampling frequency requirementsPNS sampling scheme and reconstruction formulasPractical sampling device: the TI-ADCs
3 Improved PNSPrincipleConvergence speed improvementSelective reconstruction with interference cancelationAnalytic signal reconstruction
4 PNS delay estimationPNS delay estimation with a learning sequenceBlind PNS delay estimation
5 Conclusion
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
PNS delay estimation with a learning sequence3
Using a learning sequence
Principle:Learning sequence with a priori known spectrum:
cosine wave,bandlimited white noise.
Sampling using the unsynchronized TI-ADC.PNS reconstruction for varying delays.Criterion optimization w.r.t the delay.
Limitation: no superimposition with the signal of interest
part of the Built-In Self Test (BIST),online updates during silent periods.
Advantages:
low complexity and thus low consumption.
3J.-A. Vernhes et al. “Adaptive Estimation and Compensation of the Time Delay ina Periodic Non-uniform Sampling Scheme”. In: SampTA 2015.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Principle: orthogonality property
Known delay: orthogonal equivalent scheme
Orthogonality between D = {D(n), n ∈ Z} and X0 = {X(n), n ∈ Z}:
E[D(n)X∗0 (m)] = 0 , ∀(n,m) ∈ Z
with:D(n) = X(n+ ∆)− µ∆[X0](n)
µt
ψt
⊕µ∆
⊕+−
X0
X∆ D
X0
XK
X
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Principle: orthogonality property
Unknown delay: loss of orthogonality
Sampling sequences: X0,X∆.
Reconstruction using a wrong delay ∆ ∈]0, 1[, ∆ 6= ∆.
Loss of orthogonality criterion:
σ2∆
= E[|X(n+ ∆)− µ∆[X0](n)|2
]
=
∫ ∞
−∞
∣∣e2iπf∆ − µ∆(f)∣∣2 sX(f) df
For simplificity:
Baseband learning sequence: sX(f) = 0 for f /∈(− 1
2, 1
2
)Delay filter µ∆(f): µ∆[X0](n) = X(n+ ∆)
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Principle: orthogonality property
Unknown delay: loss of orthogonality
Simplified criterion closed-form expression:
σ2∆
= E[|X(n+ ∆)−X(n+ ∆)|2
]
=∫ 1
2
− 12
∣∣∣e2iπf(∆−∆) − 1∣∣∣2
sX(f) df
with:
µ∆[X0](n) = X(n+ ∆) =∑
k
sin[π(∆− k)]
π(∆− k)X(n+ k)
Comparison between closed-form expression and empiricalestimation for particular learning sequences ⇒ ∆ estimation.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Examples of learning sequences
Cosine wave
Learning sequence: Cosine wave at frequency f0 defined by
sX(f) =1
2(δ(f − f0) + δ(f + f0)) , − 1
2< f0 <
1
2
Criterion closed-form expression:
σ2∆
= 4 sin2[πf0(∆−∆)
]
Estimation of ∆:
∆ = ∆− 1
2πf0arccos
[1−
σ2∆
2
]
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Learning sequence example 2
Bandlimited white noise
Learning sequence: bandlimited white noise defined by
sX(f) =
{1 on (− 1
2 + ε, 12 − ε) , 0 < ε < 1
2
0 elsewhere
Criterion closed-form expression:
σ2∆≈
∫ 12−ε− 1
2 +ε
∣∣∣e2iπf(∆−∆) − 1∣∣∣2
df
≈ 2(1− 2ε)(
1− sinc[π(∆−∆)(1− 2ε)])
Estimation of ∆ from:
sinc[π(∆−∆)(1− 2ε)] = 1−σ2
∆
2(1− 2ε)
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Performance analysis
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
·104
10−8
10−7
10−6
10−5
N
E[ |∆−
∆|2]
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Blind PNS delay estimation4
Principle: stationarity property
Property: wide sense stationarity of the reconstructed signal
X(∆) = {X(∆)(t), t ∈ R} if and only if ∆ = ∆. In particular:
P (∆)(tm) = E
[∣∣∣X(∆) (tm)∣∣∣2], tm =
m
M + 1, m = 1, ...,M
independent of tm.
Strategy: estimation of the reconstructed signal power P (∆)(tm)for m = 1, ...,M for different values of ∆:
P (∆)(tm) =1
N
N2∑
n=−N2
∣∣∣X(∆) (n+ tm)∣∣∣2
, m = 1, ...,M.
4J.-A. Vernhes et al. “Estimation du retard en echantillonnage periodique nonuniforme - Application aux CAN entrelaces desynchronises”. In: GRETSI 2015.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
2 The PNS solutionSignal modelSampling frequency requirementsPNS sampling scheme and reconstruction formulasPractical sampling device: the TI-ADCs
3 Improved PNSPrincipleConvergence speed improvementSelective reconstruction with interference cancelationAnalytic signal reconstruction
4 PNS delay estimationPNS delay estimation with a learning sequenceBlind PNS delay estimation
5 Conclusion
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Conclusion
Contributions
PNS as an alternative sampling scheme proposed for TI-ADCs.
Additional functionalities for telecommunications:
improved convergence speeda,selective reconstruction with interference rejectionb,analytical signal reconstructionc.
Estimation of the desynchronisation:
from a learning sequenced, blindlye.
aM. Chabert and B. Lacaze. “Fast convergence reconstruction formulas for periodicnonuniform sampling of order 2”. In: IEEE ICASSP 2012.
bJ.-A. Vernhes, M. Chabert, and B. Lacaze. “Conversion Numerique-Analogiqueselective d’un signal passe-bande soumis a des interferences”. In: GRETSI 2013.
cJ.-A. Vernhes et al. “Selective Analytic Signal Construction From A Non-UniformlySampled Bandpass Signal”. In: IEEE ICASSP 2014.
dJ.-A. Vernhes et al. “Adaptive Estimation and Compensation of the Time Delay ina Periodic Non-uniform Sampling Scheme”. In: SampTA 2015.
eJ.-A. Vernhes et al. “Estimation du retard en echantillonnage periodique nonuniforme - Application aux CAN entrelaces desynchronises”. In: GRETSI 2015.
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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Thanks for your attention
Questions?
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