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

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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 marie.chabert@enseeiht.fr

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 3 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 4 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 6 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 7 / 44

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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 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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 8 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 9 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 10 / 44

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Periodic Non uniform Sampling (PNS) of order 2 X(n) t X(n+Δ) Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 11 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 12 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 13 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 14 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 15 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 16 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 17 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 18 / 44

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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∆ . Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 20 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 21 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 22 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 23 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 24 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 25 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 26 / 44

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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 rectangulaire Filtre trap´ ezo¨ ıdal Filtre en cosinus sur´ elev´ e Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 27 / 44

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Analytic signal reconstruction 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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 28 / 44

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Analytic signal reconstruction 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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 29 / 44

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Analytic signal reconstruction 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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 30 / 44

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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 rectangulaire Filtre trap´ ezo¨ ıdal Filtre cosinus sur´ elev´ e Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 31 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 33 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 34 / 44

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Principle: orthogonality property 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 + ∆) Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 35 / 44

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Principle: orthogonality property 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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 36 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 37 / 44

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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) Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 38 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 39 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 40 / 44

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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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 41 / 44

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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. Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 43 / 44

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Thanks for your attention Questions? Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 44 / 44

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

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

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