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Xampling: Analog to Digital at Sub-Nyquist Rates

Moshe Mishali, Yonina C. Eldar, Oleg Dounaevsky, Eli Shoshan

arXiv:0912.2495v1nlin.CGnlin.AOphysics.ins-det

TL;DR

Wideband sub-Nyquist conversion requires hardware that handles unknown carrier positions and nonordinary circuit operations. This paper realizes a modulated wideband converter, supporting 2 GHz Nyquist-rate inputs with 120 MHz occupancy and sampling as low as 280 MHz.

  • Problem

    The modulated wideband converter requires simultaneous mixing with many sinusoids and periodic waveforms alternating at 2.075 GHz, unlike conventional single-sinusoid RF mixing and standard FPGA clocking.

  • Method

    The paper implements a modular modulated wideband converter using wideband passive mixing, frequency equalization, tunable waveform power control, and specialized ECL shift-register circuitry.

  • Results

    280 MHz sampling supports inputs with an approximately 2 GHz Nyquist rate and 120 MHz spectrum occupancy, while experiments verify expected sub-Nyquist aliases and preserved signal shapes.

  • Takeaways & Limitations

    The modular circuit provides a hardware realization of sub-Nyquist sampling for wideband multiband inputs with unknown carrier positions and may also scale toward high-speed ADC operation.

Abstract

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We present a sub-Nyquist analog-to-digital converter of wideband inputs. Our circuit realizes the recently proposed modulated wideband converter, which is a flexible platform for sampling signals according to their actual bandwidth occupation. The theoretical work enables, for example, a sub-Nyquist wideband receiver, which has no prior information on the transmitter carrier positions. Our design supports input signals with 2 GHz Nyquist rate and 120 MHz spectrum occupancy, with arbitrary transmission frequencies. The sampling rate is as low as 280 MHz. To the best of our knowledge, this is the first reported wideband hardware for sub-Nyquist conversion. Furthermore, the modular design is proven to compete with state-of-the-art Nyquist ADCs in terms of resolution bits and full-scale range. We describe the various circuit design considerations, with an emphasis on the nonordinary challenges the converter introduces: mixing a signal with a multiple set of sinusoids, rather than a single local oscillator, and generation of highly-transient periodic waveforms, with transient intervals on the order of the Nyquist rate. A series of hardware experiments validates the design and demonstrate sub-Nyquist sampling.

I. INTRODUCTION

Existing ADC technology struggles to directly sample wideband signals whose carriers may lie far above ADC capabilities. The paper presents hardware for sub-Nyquist conversion based on actual spectrum occupancy, including 2 GHz Nyquist-rate inputs sampled as low as 280 MHz.

  • Wideband receivers may intercept several narrowband transmissions at carrier frequencies around tens of GHz, beyond existing ADC capabilities.The motivation is the gap between communication technology and ADC sampling-rate improvements.
  • Demodulation shifts a selected band to baseband but requires knowledge of its exact carrier frequency.This requirement can involve user assistance for tuning or carrier selection.
  • 2 GHz Nyquist-rate signals with up to 120 MHz spectrum occupancy can be sampled at rates as low as 280 MHz.The rate is 14% of the Nyquist rate and is slightly above the 240 MHz theoretical lower bound for unknown carrier positions.
  • The modulated wideband converter mixes with highly transient periodic waveforms, lowpass filters the result, and uses standard low-rate ADCs.The approach shifts complexity toward RF mixing rather than the ADC track-and-hold circuitry.
  • At Nyquist rates, the modular design reaches 2.075 GSamples/sec with 5.7 ENOB over a 10 mVptp full-scale range.MAX109 is cited as achieving 2.2 GSamples/sec with 6.68 ENOB over 500 mVptp.
  • The prototype addresses two nonordinary circuit challenges: simultaneous mixing with multiple sinusoids and generation of waveforms with transients near the 2 GHz Nyquist rate.The design uses commercial devices operated beyond their ordinary specifications, with auxiliary circuitry and timing techniques.
  • Hardware experiments verify mixing-waveform periodicity, analog-path gains, and sub-Nyquist sampling.The paper reports a series of laboratory tests validating the implemented converter.

II. THEORETICAL BACKGROUND

The MWC models wideband inputs as multiband signals with unknown carrier locations and uses periodic mixing, lowpass filtering, and uniform low-rate sampling. Designed mixing functions distribute information from all bands into recoverable baseband mixtures.

  • A. The modulated wideband converter: A multiband signal has spectrum concentrated in N intervals, each no wider than B, with carrier locations unrestricted below fmax.The highest possible frequency fmax determines the Nyquist rate.
  • A. The modulated wideband converter: Each MWC channel multiplies x(t) by a periodic function pi(t), lowpass filters the product, and samples it uniformly.The basic configuration uses m ≥ 4N channels and fs = fp ≥ fNYQ/B.
  • A. The modulated wideband converter: Periodic mixing produces weighted sums of fp-shifted copies of X(f), while the lowpass filter retains frequencies up to fs/2.The weights are the Fourier coefficients of the mixing functions, and the resulting mixtures are sampled as yi[n].
  • A. The modulated wideband converter: Deliberate aliasing places portions of every band in baseband so differently designed channels capture different spectrum combinations for reconstruction.The output spectrum consists of overlaid slices whose combinations encode the original multiband signal.
  • A. The modulated wideband converter: Low-rate ADCs can follow the RF front-end because preceding lowpass filters limit their inputs, shifting the Nyquist burden to RF mixers.This avoids requiring the low-rate ADCs to connect directly to the wideband input.
  • A. The modulated wideband converter: The Fourier coefficients of pi(t) determine the highest input frequency the MWC can capture, with nonnegligible coefficients required through the relevant harmonic index.Signals can lie anywhere below Lfp when coefficients remain sufficiently large for reconstruction.
  • A. The modulated wideband converter: The advanced MWC configuration sets fs = qfp, reducing the required number of channels by increasing the sampling rate per channel.Here q is an integer greater than one.

B. Xampling and related works

Xampling targets structured wideband signals with low-rate, efficient analog-to-digital conversion, and the MWC was identified as the only surveyed system satisfying all four criteria. The prototype translates that premise into a modular hardware design with specified multiband inputs, waveform generation, filtering, sampling, and channel choices.

  • Xampling methodology: Xampling targets broad analog inputs, low sampling rates, efficient implementation, and low-rate processing of information bands.
  • Related works: Pointwise strategies retain Nyquist-rate analog bandwidth, while random demodulation uses a sensitive discretized model, incurs severe computation, and prevents low-rate processing.
  • Related works: The MWC was found to be the only surveyed system satisfying all four Xampling criteria, motivating this paper's transition from theory to hardware.
  • Prototype specifications: The prototype targets three concurrent transmissions across N = 6 bands with maximal bandwidth B = 19 MHz and a 2.075 GHz Nyquist rate.
  • Prototype specifications: M = 108 sign-alternating waveform elements produce aliasing spacing fp = 19.212 MHz, while the advanced q = 3 configuration uses an approximately 33 MHz filter cutoff.
  • Prototype specifications: 280 MHz sampling approaches the 228 MHz multiband lower bound, and the modular implementation uses four acquired channels while selecting 16-bit resolution.The prototype uses a four-channel scope; theory specifies eight channels for the advanced configuration, with additional digital computations compensating the factor of two.

B. Circuit challenges

The hardware realization faces two nonordinary circuit challenges: multiplying by spectrally rich waveforms and generating periodic sign alternations at the 2.075 GHz Nyquist rate. These requirements create attenuation, nonlinear distortion, and severe timing constraints beyond ordinary mixer and waveform-generator operation.

  • The MWC requires simultaneous mixing with many sinusoids, unlike conventional RF mixers designed for a single sinusoid.The resulting attenuation and nonlinear distortion are addressed partly with frequency equalization and tunable power control.
  • Analog periodic waveforms lack enough high-frequency transients for sufficient aliasing, whereas digital waveforms face timing constraints set by the 480-picosecond clock interval.

IV. ANALOG BOARD

The analog board splits the input into four channels, mixes each with a periodic waveform, and lowpass-filters the results. Link-budget analysis addresses noise, nonlinear distortion, attenuation, and dynamic range.

  • Analog path: Four channels implement splitting, waveform mixing, and two-stage lowpass filtering in the analog path.Each channel equalizes and mixes with its corresponding p_i(t) before amplification and filtering.
  • Link budget: The link budget accounts for gain, noise figure, IP3, and nonordinary mixing, with experiments validating the manufactured board’s requirements.Low-power analysis is noise-dominated, whereas high-power analysis emphasizes spurious images and nonlinear distortion.
  • Link budget: 49 dB dynamic range is predicted for the system, while measurements report an actual 50 dB dynamic range.The calculation assumes equal powers across narrowband transmissions; attenuator settings are optimized for input power.
  • Link budget: 42 dB maximal SNDR is predicted, while experiments affirm 36.2 dB at -35 dBm input power.The architecture is evaluated as an alternative to Nyquist ADCs when m f_s ≥ f_NYQ.

C. Mixing with multiple sinusoids

Multiple-sinusoid mixing creates distinctive RF-path design problems, requiring a passive wideband mixer, calibrated LO power, and pre-mixer equalization.

  • Nonordinary mixing: The MWC mixes x(t) with multiple sinusoids rather than the single sinusoid assumed by standard switching-mixer operation.The selected passive mixer supports RF inputs from DC to 6 GHz, while its datasheet LO specification refers to a single source.
  • LO power: 17 dBm per harmonic risks excessive total power, whereas 17 dBm total power leaves each harmonic substantially weaker.This nonlinear-device trade-off motivates experimentally determining the appropriate LO setting.
  • Mixer characterization: Conversion loss of -16 dB, IP3 of 27 dBm, and LO power of 20 dBm replace datasheet values of -6 dB, 30 dBm, and 17 dBm.The revised parameters come from experiments characterizing multiple-sinusoid mixing.
  • Equalization: A wideband equalizer precedes the mixer because the mixed output overlays energy from the entire input spectrum.The equalizer attenuates frequencies up to 500 MHz by approximately 8 dB while leaving frequencies above 1.5 GHz unaltered.

D. Lowpass filtering of spectrum mixtures

Lowpass filtering emphasizes sharp cutoff transitions and high stopband attenuation because passband nonflatness can be corrected digitally.

  • Filter objectives: The filter design prioritizes a sharp cutoff and high attenuation beyond the cutoff rather than a flat passband.The passband response is digitally compensable, while stopband rejection is the analog design focus.
  • Filter implementation: A seventh-order elliptic filter is used twice with buffering to double stopband attenuation.The two-stage response is simulated and optimized for a smooth S21 transfer curve.
  • Flexibility: Cutoff frequencies up to 100 MHz are supported by re-optimizing component values.The design supports single-ended and double-balanced outputs; balanced outputs may provide increased noise immunity.

A. Description

The prototype combines a four-channel analog board with a programmable digital shift-register board that generates shifted periodic waveforms for the MWC.

  • Digital board: A 96-bit ECL shift register concatenates twelve 8-bit packages for periodic-waveform generation.The register can be loaded with a fixed hexadecimal pattern through switches or an onboard FPGA.
  • Analog board: The analog board has three stages: four-way input splitting, mixing with sign patterns p_i(t), and lowpass filtering.The block diagram describes the analog path end to end.
  • Digital board: Each analog channel receives a different shift-register tap, with configurable tap locations and bypass options.The four waveforms are shifted versions of one another, and each uses a 24-bit register segment.
  • Clock network: A 2.075 GHz sine clock is split into six paths, converted to balanced form, shaped toward ECL levels, and distributed to the register packages.The clock is synthesized by locking a VCO to a 25 MHz TCXO.

B. Design considerations

The design addresses the difficulty of implementing a 2.075 GHz discrete shift register and generating the periodic waveforms required by the MWC.

  • Shift-register implementation: Discrete devices replace FPGA logic because common FPGAs cannot operate at the required 2.075 GHz, while premium alternatives are expensive and power consuming.
  • Shift-register implementation: 2.075 GHz operation makes setup and hold timing difficult because the 480-ps clock period is comparable to inter-device routing delays.
  • Shift-register implementation: The design uses an MC10EP142MNG 8-bit ECL shift register rated for 2.8 GHz, exploiting its internally guaranteed timing and larger external clock-to-output delay.
  • Waveform generation: The selected sign pattern was chosen from options satisfying theoretical performance bounds, with practical emphasis on balancing coefficient powers.
  • Clock design: The clock design balances interference mitigation against VCO phase noise, while experiments at room temperature found clock deviation negligible.

VI. EXPERIMENTS

Experiments verify the clock and mixing-waveform periodicity required for MWC operation, including spectral stability despite nonideal time-domain transitions.

  • Experimental setup: The experiments use spectrum analysis, oscilloscopes, signal generation, network analysis, and power measurement to verify the design.
  • Waveform periodicity: 19.212 MHz spacing between equal-spaced Dirac spectral lines validates the selected waveform period and confirms steady periodicity.The spectral lines also show balanced power levels.
  • Time-domain waveform: The waveform transitions are not rectangular on the Nyquist grid, but this nonideal time-domain appearance does not affect practice because periodicity is the essential requirement.

B. Power level, SNDR and dynamic range

Power and distortion tests verify the analog path's output level and SNDR requirements, including 5.7 effective resolution bits at higher input power.

  • Power level: 5 dBm output power implies a minimum input power of -55 dBm for satisfying the required power level.The test used a 92.4 MHz sinusoid with -45 dBm input power and measured approximately 1 Vptp outputs.
  • SNDR: The low-power two-tone test measured -14.2 dBm output power per tone and a -78 dBm noise level at 1 kHz resolution bandwidth.
  • Filtering: The lowpass filter provides about 70 dB stopband attenuation, while passband nonflatness through 33 MHz is compensated digitally.
  • Measurement conditions: The spectrum-analyzer noise floor limits interpretation of the high-energy distortion panels, and panel (c) includes an external 20 dB attenuator.

D. Sub-Nyquist sampling

The X-ADC aliases wideband AM, FM, and PAM signals with arbitrary carrier positions into low frequencies while preserving their signal shapes for digital processing.

  • Aliasing procedure: The MWC aliases all spectrum slices into [−fp/2, fp/2], and the signal shape remains unaltered at the output.
  • Sub-Nyquist demonstration: AM, FM, and PAM carriers alias experimentally to 5.71 MHz, 4.82 MHz, and 6.73 MHz, respectively.The source carriers are 340.12 MHz, 629.2 MHz, and 1011.54 MHz.
  • System scope: The circuit realizes a sub-Nyquist sampler for wideband inputs under a multiband model with unknown carrier positions.
  • System scope: The implemented configuration handles a 2 GHz Nyquist-rate input with up to 120 MHz occupancy at sampling rates as low as 280 MHz.
  • Circuit challenges: The design emphasizes mixing with multiple sinusoids and generating periodic waveforms with Nyquist-rate transients using modified standard devices and auxiliary circuitry.
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