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Multichannel Sampling of Pulse Streams at the Rate of Innovation

Kfir Gedalyahu, Ronen Tur, Yonina C. Eldar

arXiv:1004.5070v5cs.IT

TL;DR

Infinite pulse streams require sampling below the Nyquist rate while preserving delays and amplitudes, but earlier methods either missed the innovation rate or handled only restricted pulse shapes. The paper proposes multichannel modulation and integration to recover Fourier coefficients and then pulse parameters at the rate of innovation. It reports stable recovery for arbitrary pulse shapes, improved noise robustness, high-innovation-rate operation, and practical resilience to channel failures.

  • Problem

    Earlier infinite-stream sampling methods either did not achieve the rate of innovation, restricted pulse shapes to Diracs, or imposed delay constraints.

  • Method

    The paper uses multichannel waveform modulation and integration to recover Fourier coefficients, then applies spectral estimation to obtain pulse delays and amplitudes.

  • Results

    The proposed scheme recovers arbitrary-shaped pulse streams at the rate of innovation and is reported to be more noise-robust than previous methods, including at high innovation rates.

  • Takeaways & Limitations

    Mixing Fourier coefficients supports practical waveform generation and recovery despite failures in one or more sampling channels.

Abstract

from arXiv · show

We consider minimal-rate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on sub-Nyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimal-rate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.

I. INTRODUCTION

The paper targets stable, minimal-rate recovery of infinite pulse streams with arbitrary known pulse shapes, addressing limitations in earlier methods. It proposes multichannel modulation and integration to recover pulse delays and amplitudes at the rate of innovation.

  • Motivation: Finite-rate-of-innovation pulse streams can be characterized by their pulse delays and amplitudes rather than sampled at the full Nyquist rate.For L pulses per interval T, the rate of innovation is 2L/T.
  • Limitations of prior work: Previous infinite-stream methods either failed to achieve the rate of innovation, restricted pulses to Diracs, constrained delays, or became unstable at high innovation rates.Earlier approaches also differed in noise robustness and sampling-kernel assumptions.
  • Proposed approach: The proposed multichannel system modulates the signal with selected waveforms, integrates over finite intervals, and recovers Fourier coefficients before estimating delays and amplitudes spectrally.The design uses oscillators, mixers, and integrators and derives conditions guaranteeing recovery.
  • Robustness and implementation: Mixing Fourier coefficients distributes information across channels, allowing recovery even when one or more sampling channels fail.The approach also supports practical waveform generation and hardware implementation.
  • Reported contributions: The infinite-stream extension achieves perfect reconstruction at the rate of innovation for general finite-support pulse shapes and shows improved noise robustness in simulations.The paper also discusses shift-invariant pulse streams and conditions for generating modulating waveforms.

B. Relation to Model-Based Complex Sinusoids Estimation

The paper converts pulse-delay recovery into model-based complex-sinusoid estimation using Fourier-series coefficients. It then designs multichannel time-domain sampling to obtain those coefficients from a minimal number of measurements.

  • Fourier representation: Known pulse streams are represented through Fourier-series coefficients whose frequency-domain modulations encode the unknown pulse delays.The pulse spectrum is separated from the delay-dependent exponential terms using an invertible diagonal matrix.
  • Parameter estimation: At least K ≥ 2L suitable Fourier coefficients allow standard spectral-estimation methods to recover distinct delays and then amplitudes.The cited methods include annihilating filters, matrix pencil, Kumaresan–Tufts, and ESPRIT.
  • Sampling architecture: The sampling problem is to obtain the Fourier-coefficient vector from time-domain samples rather than access it directly.The direct multichannel scheme addresses this by integrating modulated versions of the signal.
  • Direct multichannel sampling: In direct sampling, each channel uses a complex exponential and an integrator over [0,T) to produce one Fourier coefficient.This implementation requires many oscillators at exact multiples of a common base frequency.

D. Mixing the Fourier Coefficients

The paper generalizes direct sampling by mixing multiple Fourier coefficients in each channel. Full-column-rank mixing permits coefficient recovery while enabling simpler hardware and resilience to channel failures.

  • Context and robustness: The generalized scheme is motivated by sub-Nyquist multiband sampling and inherits the noise-robustness advantage associated with the earlier time-limited-filter method.The theorem also applies to periodic pulse streams represented by Fourier series.
  • Failure robustness: Mixing distributes each Fourier coefficient across several channels, so the signal can remain recoverable after channel failures when enough operating channels remain.The required Fourier coefficients can still be recovered if their number exceeds 2L.
  • Mixing architecture: Each channel modulates the signal with a weighted sum of cisoids, so its integrated sample becomes a linear combination of Fourier coefficients.The channel-dependent weights form the mixing matrix S.
  • Recovery condition: When S has full column rank and p ≥ K, the Fourier-coefficient vector is recovered from the samples by x = S†c.Direct sampling is the special case p = K and S = I.

1) Cosine and Sine waveforms: 

The proposed infinite-stream scheme uses periodic modulating waveforms and integrator samples to split the problem into independent finite pulse streams. With p = K = 2L, it enables perfect reconstruction at the rate of innovation.

  • Cosine and Sine waveforms: Sine, cosine, and constant waveforms provide an invertible real-valued mixing scheme, avoiding the complex exponentials required by direct multichannel sampling.The real-valued construction offers a practical implementation advantage.
  • Cosine and Sine waveforms: Periodic waveforms can be Fourier-expanded and shaped to retain only a finite set of coefficients, yielding the required modulation form.Proper waveform selection makes the mixing matrix S left invertible.
  • Cosine and Sine waveforms: A periodic ±1 rectangular-pulse stream can serve every channel through delayed copies, removing the need for multiple oscillators and accurately matched frequency multiples.The common-waveform construction simplifies hardware implementation.
  • Cosine and Sine waveforms: Resetting each integrator every T seconds makes each sample depend only on one interval, reducing the infinite-stream problem to a sequence of finite pulse streams.The interval-local samples are represented through Fourier-series coefficients and a left-invertible mixing matrix.
  • Cosine and Sine waveforms: With p = K = 2L, the cisoid recovery problem is solvable and the infinite pulse stream is perfectly reconstructed at the rate of innovation.The theorem requires p ≥ |K| ≥ 2L and a left-invertible coefficient matrix S.

B. Stream of Pulses with Shift-Invariant Structure

For shift-invariant pulse streams, repeated delays across periods provide shared information that can reduce the required number of sampling channels. When amplitudes vary sufficiently, the sampling rate becomes (L + 1)/T.

  • Stream of Pulses with Shift-Invariant Structure: The shift-invariant model keeps each pulse delay constant relative to the beginning of every period, while amplitudes may vary across periods.The amplitude vector a[m] contains the L pulse amplitudes in period m.
  • Stream of Pulses with Shift-Invariant Structure: The general condition p ≥ 2L remains sufficient, but the additional structure can reduce the number of channels below 2L depending on η.Here η is the dimension of the minimal subspace containing the amplitude vectors across periods.
  • Stream of Pulses with Shift-Invariant Structure: When η = L, ESPRIT or MUSIC recovers the delays using p ≥ L + 1 channels; when η < L, smoothing is required and p ≥ 2L channels are needed.The two regimes correspond to full and deficient amplitude-subspace dimension.
  • Stream of Pulses with Shift-Invariant Structure: When amplitudes vary sufficiently from period to period, shared delay information reduces the sampling rate to (L + 1)/T.The same cross-period information can improve delay estimation in noise relative to recovering each period separately.

C. Channel Synchronization

The multichannel architecture introduces hardware and synchronization burdens, but channel offsets can be modeled through an effective mixing matrix. Known offsets can then be compensated by matrix inversion.

  • Channel Synchronization: Compared with single-channel schemes, the architecture requires additional hardware in each channel and precise synchronization of sampling times.These are identified as the two main disadvantages of the proposed scheme.
  • Channel Synchronization: Channel synchronization can be implemented with a zero-delay device or through calibration that measures each channel’s relative delay.Calibration may occur during manufacturing or power-on using a known signal.
  • Channel Synchronization: The offset model assumes each channel’s timing error lies within [−∆max, ∆max] and imposes pulse-support constraints so intervals remain independently processable.The support condition prevents boundary effects between adjacent intervals.
  • Channel Synchronization: A channel offset changes the effective pulse delay from t_l to t_l − ∆_i in the channel’s Fourier coefficients.The timing error therefore enters the channel measurement model explicitly.
  • Channel Synchronization: When offsets are known, misalignment is compensated by forming and inverting the effective mixing matrix ˜S.Unknown-delay effects are treated separately from the known-offset correction.
  • Channel Synchronization: Periodic modulation waveforms are filtered to reject unwanted Fourier coefficients, leaving a finite set that can be selected to make S left invertible.The filter response is specified at discrete Fourier-series frequencies, allowing practical analog-filter design freedom.

B. Pulse Sequence Modulation

Pulse-sequence modulation realizes the required multichannel mixing through periodic sequences and shaping filters. Left invertibility follows from rank conditions on the sequence-derived matrices and the filter response.

  • Pulse Sequence Modulation: The practical modulation construction uses p periodic waveforms generated from sequences α_i[n], with p ≥ K required.The sequences are combined with a pulse shape and then filtered before sampling.
  • Pulse Sequence Modulation: The mixing matrix factors into A, W, and diagonal Φ, where A contains the sequences and W is a Vandermonde matrix.The factorization separates sequence design, Fourier structure, and pulse-shape weighting.
  • Pulse Sequence Modulation: The factorized mixing matrix is left invertible when Φ is invertible and AW is left invertible.These conditions are sufficient to guarantee recovery through the multichannel mixing system.
  • Pulse Sequence Modulation: Because W is Vandermonde, N ≥ K ensures full column rank, while appropriate sequence choices can ensure that AW is left invertible.The pulse spectrum must also vanish on K so that Φ remains invertible.
  • Pulse Sequence Modulation: The construction requires p ≥ K, N ≥ K, a shaping-filter response satisfying the specified condition, pulse-spectrum zeros on K, and full column rank of AW.Under these conditions, Proposition 1 guarantees that S is left invertible.
  • Pulse Sequence Modulation: A cyclic shift of one common sequence provides a concrete configuration satisfying the sequence-based design requirements.This configuration reduces the need to construct unrelated sequences for every channel.

1) Single Generator:

The design can use one pulse generator to create delayed modulating waveforms across channels, simplifying hardware while preserving the required Fourier-coefficient recovery structure.

  • 1) Single Generator:: A single pulse generator can produce each channel waveform as a delayed version of the generator output.The delay for channel i is (i − 1)T/N.
  • 1) Single Generator:: Choosing p = N = K makes the system matrix circulant, with invertibility determined by the nonzero DFT of the generating sequence.The DFT of α[n] must not contain zero values.
  • 1) Single Generator:: The selected pulse has nonzero frequency response on the required index set, enabling recovery of the relevant Fourier coefficients.The example uses p = N = K and sequences α_i[n] consisting of ±1 values generated by cyclic shifts.
  • 1) Single Generator:: The shaping filter transfers only Fourier coefficients indexed by K = {−3, . . ., 3} and suppresses the others.In the example, rectangular alternating-pulse waveforms are smoothed by lowpass filtering.
  • 1) Single Generator:: The implementation includes oscillators, mixers, and integrators, with the modulating waveform examined before and after filtering in the time domain.The supplied figure concerns the time-domain waveform and its filtered version.
  • 1) Single Generator:: Channel-failure handling assumes that malfunctioning channels are identified by external hardware before recovery proceeds.This assumption is stated as part of the failure-robust design setup.

2) Robustness to Sampling Channels Failure:

The multichannel design distributes Fourier-coefficient information across channels so recovery can remain possible when channels fail, while relating to single-channel SoS filtering.

  • 2) Robustness to Sampling Channels Failure:: With at most p_e failed channels, the system requires p ≥ N + p_e and every remaining set of p − p_e rows must retain rank K.This ensures the reduced matrix is left invertible for every possible failure pattern.
  • 2) Robustness to Sampling Channels Failure:: Two generators can support the failure-robust design when p ≥ 2p_e, with each half of the channels using delayed versions of one generator.Properly selected generator sequences can satisfy the required rank condition.
  • 2) Robustness to Sampling Channels Failure:: Up to 6 failed channels are tolerated in the example with L = 4, N = K = 9, and p = 18, while preserving perfect recovery of the pulses.The worst-case reduced-matrix condition number remains relatively low for p_e ≤ 6.
  • 2) Robustness to Sampling Channels Failure:: The direct scheme cannot always provide 9 consecutive Fourier coefficients after channel failures, so annihilating-filter or matrix-pencil recovery is not guaranteed.The proposed mixing distributes coefficient information across several channels instead.
  • 2) Robustness to Sampling Channels Failure:: For periodic pulse streams, the single-channel SoS-filter scheme produces samples represented through a Vandermonde-type matrix and diagonal coefficient matrix.The samples use Ts = T/p and coefficients b_k indexed by the chosen set K.
  • 2) Robustness to Sampling Channels Failure:: The proposed modulation interpretation expresses each channel as an inner product with a delayed, reflected periodic continuation of the SoS filter.This explains the equivalence between filtering followed by sampling and modulation followed by integration.

B. Multichannel Schemes for Shift-Invariant Pulse Streams

The shift-invariant comparison shows that the earlier method supports broader pulse support and single-channel operation, whereas the proposed scheme uses simpler correction and finite-interval integration.

  • B. Multichannel Schemes for Shift-Invariant Pulse Streams: The earlier shift-invariant scheme filters each channel, uniformly samples at 1/T, and applies a digital correction bank before ESPRIT delay recovery.The correction filter operates on the sampling sequences in the DTFT domain.
  • B. Multichannel Schemes for Shift-Invariant Pulse Streams: Its sampling rate is generally 2L/T and can fall to (L + 1)/T for signals satisfying a specified dimension condition.The reduction condition depends on the span of output sample vectors rather than directly on the vectors a[m].
  • B. Multichannel Schemes for Shift-Invariant Pulse Streams: The earlier method does not require condition (2), supports pulses with infinite time support, and can use a single channel with serial-to-parallel conversion.These are identified as its two main advantages over the proposed method.
  • B. Multichannel Schemes for Shift-Invariant Pulse Streams: The proposed correction stage reduces to matrix inversion equivalent to a one-tap digital correction filter bank, rather than the generally longer filter bank used previously.This gives the proposed stage lower computational complexity.
  • B. Multichannel Schemes for Shift-Invariant Pulse Streams: The earlier method requires infinitely many samples because its bandlimited kernels have infinite support, whereas the proposed scheme integrates finite intervals and can process relevant periods separately.This difference matters when only a finite signal interval is of interest.

C. Modulated Wideband Converter

The proposed architecture is conceptually related to the Modulated Wideband Converter but replaces lowpass sampling with integration to recover Fourier coefficients, and experiments evaluate noise, stability, and implementation trade-offs.

  • C. Modulated Wideband Converter: The Modulated Wideband Converter targets multiband signals by periodically modulating each channel, lowpass filtering, and uniformly sampling at a low rate.Its mixing stage moves portions of all unknown bands to baseband.
  • C. Modulated Wideband Converter: The proposed method uses integration after mixing because it measures Fourier coefficients rather than multiband spectral content.In the MWC, mixing reduces sampling relative to Nyquist; here it supports rate-of-innovation sampling.
  • C. Modulated Wideband Converter: Hardware for the MWC can be adapted to the proposed scheme by adding shaping filters to a four-channel prototype operating near 20 MHz with 108 rectangular pulses per period.The prototype uses p = 4 channels and N = 108 rectangular pulses in each period.
  • C. Modulated Wideband Converter: At the rate of innovation, experiments compare noise performance across tones, filtered rectangular alternating pulses, and SoS waveforms against integrator and exponential-filter methods.The simulations also examine shift-invariant recovery, synchronization errors, and practical shaping filters.
  • C. Modulated Wideband Converter: For L = 2 Diracs, the proposed technique outperforms integrator and exponential-filter methods in delay-estimation noise robustness across all tested configurations.Tone- and SoS-based schemes have a 2 dB advantage over alternating pulses.
  • C. Modulated Wideband Converter: For L = 10 Diracs with N = p = K = 21, the proposed method remains stable while the compared integrator and exponential-filter methods become unstable.This experiment demonstrates stability at high model orders.
  • C. Modulated Wideband Converter: Tone and SoS schemes gain about 3.5 dB over the single-generator pulse scheme, creating a choice between noise performance and lower hardware complexity.The pulse-sequence approach still has reasonable performance and can simplify implementation.
  • C. Modulated Wideband Converter: A further experiment compares the proposed tones configuration with B-spline and E-spline methods using p = 64 samples for L = 4 Diracs.The comparison uses oversampling and the Kumaresan–Tufts method for delay recovery.

B. Sampling of Pulses with SI Structure

The SI-structured pulse-stream experiment compares separate-period and joint-delay recovery, then evaluates synchronization errors and practical shaping filters. Joint recovery improves delay estimation, while the proposed approach remains effective under channel offsets and practical Chebyshev filtering.

  • SI recovery: Joint SI recovery uses ESPRIT to estimate common delays across periods, unlike standard recovery, which processes each period independently.The experiment compares these two recovery options for streams of Diracs with shift-invariant structure.
  • SI recovery: Above 15dB SNR, SI recovery clearly outperforms standard recovery because it uses mutual information between periods.The reported improvement is in time-delay estimation error.
  • SI recovery: Up to 20dB SNR, the proposed method and perform similarly, but at higher SNR suffers a finite-sample error floor.The comparison highlights the proposed scheme's operation over finite time intervals for finite pulse periods.
  • Synchronization errors: When synchronization error is below 10 percent of the estimation error, noise dominates; for larger offsets, delay error degrades linearly.For ∆max < 0.03T, the estimation error is bounded from below by the synchronization error.
  • Practical shaping filters: Chebyshev Type I filters are used because their steeper roll-off rejects undesired coefficients, while pass-band ripple is digitally corrected during matrix inversion.The filters have 3 dB ripple and are considered practical alternatives to ideal shaping filters.
  • Practical shaping filters: A tenth-order Chebyshev filter closely approaches ideal-LPF performance, while a sixth-order filter provides a good approximation below 50dB SNR.These results support implementing waveform generation with practical analog filters.
  • Conclusion: The proposed multichannel scheme recovers pulse delays and amplitudes at the innovation rate for general pulse shapes and remains robust at high innovation rates.The architecture uses waveform mixing followed by integration and supports simplified hardware and channel-failure robustness.
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