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
1
Universit´e de Toulouse, IRIT-ENSEEIHT 2
T´eSA laboratory
3
CNES (French Spatial Agency) 4
Thales Alenia Space
marie.chabert@enseeiht.fr

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

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.

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
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
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−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

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

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)
fB−
B+
−fc fc−fmax fmax−fmin fmin
Figure: Passband model

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

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

Periodic Non uniform Sampling (PNS) of order 2
X(n)
t
X(n+Δ)

PNS2 reconstruction - Filter formulation1
µt
ψt
⊕µ∆
⊕+
−
X0
X∆ D
˜X0
˜XK
˜X
(a) Orthogonal scheme
ηt
ψt
⊕
X0
X∆
˜X
˜X0
˜XK
(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.

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.

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.

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

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/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

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.

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

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 Numérique-Analogique
sélective d’un signal passe-bande soumis à des interférences”. In: GRETSI 2013.
cJ.-A. Vernhes et al. “Selective Analytic Signal Construction From A Non-Uniformly
Sampled Bandpass Signal”. In: IEEE ICASSP 2014.

Rectangular ﬁlter
HR
(f)
f
1
fcfmin 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

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

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

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 trapézo¨ıdal
Filtre en cosinus surélevé

Selective reconstruction with interference
cancelation
f
k + 1
2k − 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

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 en cosinus sur´elev´e

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

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

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

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 cosinus sur´elev´e

Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
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 in
a 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[, ∆ = ∆.
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
− 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.

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

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)

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

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 échantillonnage périodique non
uniforme - Application aux CAN entrelacés désynchronisés”. 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 0.33
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

Outline
1 Introduction
Problem formulation
Proposed approach
2 The PNS solution
Signal model
3 Improved PNS
Principle
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 periodic
nonuniform sampling of order 2”. In: IEEE ICASSP 2012.
bJ.-A. Vernhes, M. Chabert, and B. Lacaze. “Conversion Numérique-Analogique
sélective d’un signal passe-bande soumis à des interférences”. 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 échantillonnage périodique non
uniforme - Application aux CAN entrelacés désynchronisés”. In: GRETSI 2015.

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2015 12-10 chabert

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