2015 12-10 chabert

Periodic Non-Uniform Sampling (PNS)for Satellite Communications

Marie Chabert1, Bernard Lacaze2, Marie-Laure Boucheret1,Jean-Adrien Vernhes1,2,3,4

1Universite de Toulouse, IRIT-ENSEEIHT 2TeSA laboratory3CNES (French Spatial Agency) 4Thales Alenia Space

[email protected]

Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

Outline

1 IntroductionProblem formulationProposed approach

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

Marie Chabert IRIT-ENSEEIHT – TeSA – CNES – TAS

Periodic Non-Uniform Sampling (PNS) for Satellite Communications 2 / 44


Problem formulation

Satellite Communication Context

Context: increasing frequency bandwidth in satellitecommunications.

Technical challenge: onboard high-rate analog-to-digitalconversion.

Economical and ecological constraints: cost, complexity, weightand power consumption of electronic devices.

Trend: migration of signal processing from analog to digital world.




Proposed approach

Periodic Non uniform Sampling (PNS)

Electronic device: unsynchronized Time Interleaved ADCs.

Requirement: desynchronization estimation.

Additional functionalities:

fast convergence reconstruction,selective reconstruction and interference rejection.analytic signal reconstruction.




Outline





5 Conclusion




Signal model

Signal model

Stationary random process: X = {X(t), t ∈ R} with zero mean,finite variance and power spectral density sX(f):

sX(f) =

∫ ∞

−∞e−2iπfτRX(τ) dτ

RX(τ) = E[X(t)X∗(t− τ)] correlation function of X

Bandpass process: sX(f) support included in the normalized kth

Nyquist band BN (k):

BN (k) =

(−(k +

1

2),−(k − 1

2)

)∪(k − 1

2, k +

1

2

)




Nyquist band

Sx(f)

f

k- 12 k k+ 12

fmin fmax-k+ 12

-k-k- 12

B+N (k) = 1B−N (k) = 1

Figure: Nyquist band




Sampling frequency requirements

Case of a high frequency pass-band signal

Uniform low-pass sampling: Shannon criterion fe = 2fmax.

Uniform band-pass sampling: constrained Landau criterionfe ≥ 2B.

Periodic Non Uniform Sampling (PNS): Landau criterionfe = 2B.

Sx(f)

fB− B+

−fc fc−fmax fmax−fmin fmin

Figure: Passband model




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




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




Periodic Non uniform Sampling (PNS) of order 2

X(n)

t

X(n+Δ)

Figure: Example




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.




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.




Practical sampling device

Time Interleaved Analog to Digital Converters (TI-ADCs)

Structure: L time-interleaved multiplexed low-rate (fs) ADCs sharethe high-rate (fe = Lfs) sampling operation.

Advantages: high sampling rates at low cost, low complexity, lowpower consumption.

Limitations: mismatch errors including desynchronization.

For uniform sampling: perfect synchronization required.

For Periodic Non Uniform Sampling (PNS): possiblyunsynchronized.




Synchronized Time Interleaved Analog toDigital Converters (TI-ADCs)

M

U

X

X[n]Lfs

X(t)

ADCL−1

nTs +L−1L

Ts

fs

ADCL−2

nTs +L−2L

Ts

fs

ADC2

nTs +2LTs

fs

ADC1

nTs +1LTs

fs

ADC0

nTs

fs

(c) Architecture

t∼ Lfs

Ts 2Ts 3Ts 4Ts

TI-ADC:

ADCL−1:

ADCL−2:

ADC2:

ADC1:

ADC0:

∼ fs

∼ fs

∼ fs

∼ fs

∼ fsTs

1LTs

2LTs

L−2L Ts

L−1L Ts

(d) Elementary and global sampling operations

Figure: Synchronized TI-ADCs




Synchronized TI-ADCs: an ideal model

Synchronization at all price

Associated sampling scheme: uniform sampling.

In practice: design imperfections and operating conditions ⇒desynchronization.

Common solution: calibration and hardware corrections.




Unsynchronized TI-ADC

X[n]

R

E

C

O

N

S

T

Lfs

δi, i = 0, ..., L− 1

X(t)

ADCL−1

nTs + δL−1

fs

ADCL−2

nTs + δL−2

fs

ADC2

nTs + δ2

fs

ADC1

nTs + δ1

fs

ADC0

nTs + δ0

fs

(a) Realistic/desynchronizedTI-ADC architecture

t∼ Lfs

Ts 2Ts 3Ts 4Ts

TI-ADC:

ADCL−1:

ADCL−2:

ADC2:

ADC1:

ADC0:

∼ fs

∼ fs

∼ fs

∼ fs

∼ fs

δ0

δ1

δ2

δL−2

δL−1

(b) Elementary and global (non uniform)sampling operations

Figure: Realistic/desynchronized TI-ADC model




Unsynchronized TI-ADCs: a realistic model

Contributions: ”desynchronization... so?”

Proposed sampling scheme: Periodic Non uniform Sampling.

No hardware correction of the desynchronization required.

Estimation of the desynchronization:

hypothesis: slow variations of the desynchronization,from a training sequence,blindly.

Complexity moved from analog to digital world:

Digital compensation of the desynchronization.Additional functionalities:

improved reconstruction speed,selective reconstruction with interference rejection,analytic signal reconstruction.

Dirty RF paradigm: how to cope with low-cost imperfect analogdevices thanks to subsequent digital processing.




Outline





5 Conclusion




Improved PNS

Principle

Integration of a filtering operation in the reconstruction step.

Condition: oversampling.

Joint filter H with transfer function H(f).

Reconstruction of U = H(X) from the filtering ofX0 = {X(n), n ∈ Z} and X∆ = {X(n+ ∆), n ∈ Z} by:

ηHt (f) = ie2iπftH(f + k)e2iπk(t−∆) −H(f − k)e−2iπk(t−∆)

2 sin 2πk∆,

ψHt (f) = ie2iπf(t−∆)H(f − k)e−2iπkt −H(f + k)e2iπkt

2 sin 2πk∆.




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.




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




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




Raised cosine filter

HCS(f)

f

1

fc

Btr Btr


B+N (k)B−N (k)

k- 12 k+ 12-k+ 1

2-k- 12

Figure: Raised Cosine Filter




Convergence speed improvement: performanceanalysis

20 40 60 80 100 120 140 160 180 200 220 240 260 280 30010−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

N

EQ

MN

Filtre rectangulaireFiltre trapezoıdalFiltre en cosinus sureleve




Selective reconstruction with interferencecancelation

f

k + 12k − 1

2

Btot

Sx1(f) Sx2

(f) Sx3(f)

B B BBtr Btr Btr Btr

H1(f) H2(f) H3(f)

Figure: Selective reconstruction with interference cancelation




Selective reconstruction with interferencecancelation: performance analysis

20 40 60 80 100 120 140 160 180 200 220 240 260 280 30010−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

N

EQ

MN

Filtre rectangulaireFiltre trapezoıdal

Filtre en cosinus sureleve




Analytic signal reconstruction

HR(f)

f

1

fcfmin fmax-fc -fmin-fmax

B+N (k)B−N (k)

k- 12 k+ 12-k+ 1

2-k- 12





HT (f)

f

1

fc

Btr Btr


B+N (k)B−N (k)

k- 12 k+ 12-k+ 1

2-k- 12





HCS(f)

f

1

fc

Btr Btr


B+N (k)B−N (k)

k- 12 k+ 12-k+ 1

2-k- 12




Analytic signal reconstruction: performanceanalysis

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 30010−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

N

EQ

MN

Filtre rectangulaireFiltre trapezoıdal

Filtre cosinus sureleve




Outline





5 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.




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





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+ ∆)





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.




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

]




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)




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]




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.




Performance analysis

0.17 0.18 0.19 0.2 0.21 ∆ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.3310−1

100

101

102

103

∆

P(∆

)m

(a) Estimated power at differenttimes tm

0.17 0.18 0.19 0.2 0.21 ∆ 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.3310−4

10−3

10−2

10−1

100

101

102

103

104

105

∆

(b) Variance of the estimatedpower

Figure: Blind estimation principle




Outline





5 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.




Thanks for your attention

Questions?



2015 12-10 chabert

Technology