Marie Chabert - Periodic Non-Uniform Sampling (PNS) for Satellite Communications

Periodic Non-Uniform Sampling (PNS)
for Satellite Communications
Marie Chabert1, Bernard Lacaze2, Marie-Laure Boucheret1,
Jean-Adrien Vernhes1,2,3,4
1Universit´
e de Toulouse, IRIT-ENSEEIHT 2T´
eSA laboratory
3CNES (French Spatial Agency) 4Thales Alenia Space
[email protected]

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Introduction The PNS solution Improved PNS PNS delay estimation Conclusion
Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
Sampling frequency requirements
PNS sampling scheme and reconstruction formulas
Practical sampling device: the TI-ADCs
3 Improved PNS
Principle
Convergence speed improvement
Selective reconstruction with interference cancelation
Analytic signal reconstruction
4 PNS delay estimation
PNS delay estimation with a learning sequence
Blind PNS delay estimation
5 Conclusion
Marie Chabert IRIT-ENSEEIHT – T´
eSA – CNES – TAS
Periodic Non-Uniform Sampling (PNS) for Satellite Communications 2 / 44

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Problem formulation
Satellite Communication Context
Context: increasing frequency bandwidth in satellite
communications.
Technical challenge: onboard high-rate analog-to-digital
conversion.
Economical and ecological constraints: cost, complexity, weight
and power consumption of electronic devices.
Trend: migration of signal processing from analog to digital world.

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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.

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Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
5 Conclusion

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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−2iπfτ RX
(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

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Nyquist band
Sx
(f)
f
k-1
2
k k+1
2
fmin
fmax
-k+1
2
-k
-k-1
2
B+
N
(k) = 1
B−
N
(k) = 1
Figure: Nyquist band

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Case of a high frequency pass-band signal
Uniform low-pass sampling: Shannon criterion fe
= 2fmax.
Uniform band-pass sampling: constrained Landau criterion
fe
≥ 2B.
Periodic Non Uniform Sampling (PNS): Landau criterion
fe
= 2B.
Sx
(f)
f
B− B+
−fc
fc
−fmax
fmax
−fmin
fmin
Figure: Passband model

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Periodic Non uniform Sampling (PNS) of order
L
Deﬁnition
PNSL: L interleaved uniform sampling sequences
Xi
= {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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Periodic Non uniform Sampling (PNS) of order 2
Deﬁnition
PNS2: 2 interleaved uniform sampling sequences
X0
= {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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Periodic Non uniform Sampling (PNS) of order 2
X(n)
t
X(n+Δ)

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PNS2 reconstruction - Filter formulation1
µt
ψt
⊕
µ∆
⊕
+
−
X0
X∆ D
˜
X0
˜
XK
˜
X
(a) Orthogonal scheme
ηt
ψt
⊕
X0
X∆
˜
X
˜
X0
˜
X
K
(b) Symmetrical scheme
General ﬁlter expressions
µt
(f) = St
(f)
S0(f)
e2iπft
ηt
(f) = µt
(f) − µ∆
(f)ψt
(f)
ψt
(f) = e2iπf(t−∆) S0(f)St−∆(f)−S∗
∆
(f)St
(f)
S2
0
(f)−|St−∆(f)|2
with: Sλ
(f) =
n∈Z
sX
(f + n)e2iπnλ, f ∈ (−1
2
, 1
2
)
1B. Lacaze. “Filtering from PNS2 Sampling”. In: Sampling Theory in Signal and
Image Processing (STSIP) 11.1 (2012), pp. 43–53.

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PNS2 reconstruction - Interpolation formulas2
Closed-form reconstruction formulas
Hypothesis: Bandpass signal composed of two sub-bands, no
oversampling.
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: IEEE
Transactions on Circuits and Systems I: Regular Papers 57.11 (2010), pp. 2904–2914.

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Practical sampling device
Time Interleaved Analog to Digital Converters (TI-ADCs)
Structure: L time-interleaved multiplexed low-rate (fs) ADCs share
the high-rate (fe
= Lfs) sampling operation.
Advantages: high sampling rates at low cost, low complexity, low
power consumption.
Limitations: mismatch errors including desynchronization.
For uniform sampling: perfect synchronization required.
For Periodic Non Uniform Sampling (PNS): possibly
unsynchronized.

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Synchronized Time Interleaved Analog to
Digital Converters (TI-ADCs)
M
U
X
X[n]
Lfs
X(t)
ADCL−1
nTs
+ L−1
L
Ts
fs
ADCL−2
nTs
+ L−2
L
Ts
fs
ADC2
nTs
+ 2
L
Ts
fs
ADC1
nTs
+ 1
L
Ts
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
∼ fs
Ts
1
L
Ts
2
L
Ts
L−2
L
Ts
L−1
L
Ts
(d) Elementary and global sampling operations
Figure: Synchronized TI-ADCs

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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.

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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/desynchronized
TI-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

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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 analog
devices thanks to subsequent digital processing.

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Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
5 Conclusion

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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 of
X0
= {X(n), n ∈ Z} and X∆
= {X(n + ∆), n ∈ Z} by:
ηH
t
(f) = ie2iπft
H(f + k)e2iπk(t−∆) − H(f − k)e−2iπk(t−∆)
2 sin 2πk∆
,
ψH
t
(f) = ie2iπf(t−∆)
H(f − k)e−2iπkt − H(f + k)e2iπkt
2 sin 2πk∆
.

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Improved PNS
Additional functionalities
Convergence speed improvement for an increasing joint filter
transfer function regularitya.
Selective signal reconstruction with interference rejection for a
well-chosen joint filter bandb.
Analytical signal reconstruction for analytic joint filtersc.
aM. Chabert and B. Lacaze. “Fast convergence reconstruction formulas for periodic
nonuniform sampling of order 2”. In: IEEE ICASSP 2012.
bJ.-A. Vernhes, M. Chabert, and B. Lacaze. “Conversion Num´
erique-Analogique
s´
elective d’un signal passe-bande soumis `
a des interf´
erences”. In: GRETSI 2013.
cJ.-A. Vernhes et al. “Selective Analytic Signal Construction From A Non-Uniformly
Sampled Bandpass Signal”. In: IEEE ICASSP 2014.

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Rectangular ﬁlter
HR(f)
f
1
fc
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2
Figure: Rectangular ﬁlter

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Trapezoidal ﬁlter
HT (f)
f
1
fc
Btr
Btr
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2
Figure: Trapezoidal ﬁlter

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Raised cosine ﬁlter
HCS(f)
f
1
fc
Btr
Btr
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2
Figure: Raised Cosine Filter

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Convergence speed improvement: performance
analysis
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
N
EQMN
Filtre rectangulaire
Filtre trap´
ezo¨
ıdal
Filtre en cosinus sur´
elev´
e

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Selective reconstruction with interference
cancelation
f
k + 1
2
k − 1
2
Btot
Sx1
(f) Sx2
(f) Sx3
(f)
B B B
Btr
Btr
Btr
Btr
H1
(f) H2
(f) H3
(f)
Figure: Selective reconstruction with interference cancelation

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Selective reconstruction with interference
cancelation: performance analysis
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
N
EQMN
Filtre trap´
ezo¨
ıdal
Filtre en cosinus sur´
elev´
e

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HR(f)
f
1
fc
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2

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HT (f)
f
1
fc
Btr
Btr
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2

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HCS(f)
f
1
fc
Btr
Btr
fmin
fmax
-fc
-fmin
-fmax
B+
N
(k)
B−
N
(k)
k-1
2
k+1
2
-k+1
2
-k-1
2

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Analytic signal reconstruction: performance
analysis
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
N
EQMN
Filtre trap´
ezo¨
ıdal
Filtre cosinus sur´
elev´
e

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Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
5 Conclusion

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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 in
a Periodic Non-uniform Sampling Scheme”. In: SampTA 2015.

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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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Unknown delay: loss of orthogonality
Sampling sequences: X0
, X∆.
Reconstruction using a wrong delay ∆ ∈]0, 1[, ∆ = ∆.
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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Unknown delay: loss of orthogonality
Simpliﬁed criterion closed-form expression:
σ2
∆
= E |X(n + ∆) − X(n + ∆)|2
= 1
2
− 1
2
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 empirical
estimation for particular learning sequences ⇒ ∆ estimation.

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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 ∆:
ˆ
∆ = ∆ −
1
2πf0
arccos 1 −
σ2
∆
2

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Learning sequence example 2
Bandlimited white noise
Learning sequence: bandlimited white noise deﬁned by
sX
(f) =
1 on (−1
2
+ ε, 1
2
− ε) , 0 < ε < 1
2
0 elsewhere
Criterion closed-form expression:
σ2
∆
≈ 1
2
−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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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 | ˆ
∆ − ∆|2

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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
N
2
n=− N
2
X( ˜
∆) (n + tm
)
2
, m = 1, ..., M.
4J.-A. Vernhes et al. “Estimation du retard en ´
echantillonnage p´
eriodique non
uniforme - Application aux CAN entrelac´
es d´
esynchronis´
es”. In: GRETSI 2015.

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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 0.33
10−1
100
101
102
103
˜
∆
P( ˜
∆)
m
(a) Estimated power at diﬀerent
times 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 0.33
10−4
10−3
10−2
10−1
100
101
102
103
104
105
˜
∆
(b) Variance of the estimated
power
Figure: Blind estimation principle

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Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
5 Conclusion

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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 periodic
nonuniform sampling of order 2”. In: IEEE ICASSP 2012.
bJ.-A. Vernhes, M. Chabert, and B. Lacaze. “Conversion Num´
erique-Analogique
s´
elective d’un signal passe-bande soumis `
a des interf´
erences”. In: GRETSI 2013.
cJ.-A. Vernhes et al. “Selective Analytic Signal Construction From A Non-Uniformly
Sampled Bandpass Signal”. In: IEEE ICASSP 2014.
dJ.-A. Vernhes et al. “Adaptive Estimation and Compensation of the Time Delay in
a Periodic Non-uniform Sampling Scheme”. In: SampTA 2015.
eJ.-A. Vernhes et al. “Estimation du retard en ´
echantillonnage p´
eriodique non
uniforme - Application aux CAN entrelac´
es d´
esynchronis´
es”. In: GRETSI 2015.

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Thanks for your attention
Questions?

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Marie Chabert - Periodic Non-Uniform Sampling (PNS) for Satellite Communications

Marie Chabert - Periodic Non-Uniform Sampling (PNS) for Satellite Communications

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