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