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

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December 10, 2015

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

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SCEE Team

December 10, 2015
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  1. 1.

    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
  2. 2.

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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
  5. 5.

    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 5 / 44
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    Introduction The PNS solution Improved PNS PNS delay estimation 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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 6 / 44
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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    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 ˜ 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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    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, 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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    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 19 / 44
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    Introduction The PNS solution Improved PNS PNS delay estimation 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∆ . Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 20 / 44
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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    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-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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    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-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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    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 32 / 44
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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 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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    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 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 34 / 44
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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[, ∆ = ∆. 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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    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 − 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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    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 ∆: 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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    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, 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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    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 | ˆ ∆ − ∆|2 Marie Chabert IRIT-ENSEEIHT – T´ eSA – CNES – TAS Periodic Non-Uniform Sampling (PNS) for Satellite Communications 39 / 44
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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 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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

    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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    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 42 / 44
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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 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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    Introduction The PNS solution Improved PNS PNS delay estimation Conclusion

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