Andrew Harms

A Constrained Random Demodulator for Sub-Nyquist Sampling

This paper presents a significant modification to the Random Demodulator (RD) of Tropp for sub-Nyquist sampling of frequency-sparse signals. The modification, termed constrained random demodulator, involves replacing the random waveform, essential to the operation of the RD, with a constrained random waveform that has limits on its switching rate because fast switching waveforms may be hard to generate cleanly. The result is a relaxation on the hardware requirements with a slight, but manageable, decrease in the recovery guarantees. The paper also establishes the importance of properly choosing the statistics of the constrained random waveform. If the power spectrum of the random waveform matches the distribution on the tones of the input signal (i.e., the distribution is proportional to the power spectrum), then recovery of the input signal tones is improved. The theoretical guarantees provided in the paper are validated through extensive numerical simulations and phase transition plots.

Beating nyquist through correlations: A constrained random demodulator for sampling of sparse bandlimited signals

Technological constraints severely limit the rate at which analog-to-digital converters can reliably sample signals. Recently, Tropp et al. proposed an architecture, termed the random demodulator (RD), that attempts to overcome this obstacle for sparse bandlimited signals. One integral component of the RD architecture is a white noise-like, bipolar modulating waveform that changes polarity at a rate equal to the signal bandwidth. Since there is a hardware limitation to how fast analog waveforms can change polarity without undergoing shape distortion, this leads to the RD also having a constraint on the maximum allowable bandwidth. In this paper, an extension of the RD, termed the constrained random demodulator (CRD), is proposed that bypasses this bottleneck by replacing the original modulating waveform with a run-length limited (RLL) modulating waveform that changes polarity at a slower rate than the signal bandwidth. One of the main contributions of the paper is establishing that the CRD, despite employing a modulating waveform with correlations, enjoys some theoretical guarantees for certain RLL waveforms. In addition, for a given sampling rate and rate of change in the modulating waveform polarity, numerical simulations confirm that the CRD, using an appropriate RLL waveform, can sample a signal with an even wider bandwidth without a significant loss in performance.

Identification of Linear Time-Varying Systems Through Waveform Diversity

Linear, time-varying (LTV) systems composed of time shifts, frequency shifts, and complex amplitude scalings are operators that act on continuous finite-energy waveforms. This paper presents a novel, resource-efficient method for identifying the parametric description of such systems, i.e., the time shifts, frequency shifts, and scalings, from the sampled response to linear frequency modulated (LFM) waveforms, with emphasis on the application to radar processing. If the LTV operator is probed with a sufficiently diverse set of LFM waveforms, then the system can be identified with high accuracy. In the case of noiseless measurements, the identification is perfect, while in the case of noisy measurements, the accuracy is inversely proportional to the noise level. The use of parametric estimation techniques with recently proposed denoising algorithms allows the estimation of the parameters with high accuracy.

Predicting the array gain of partially coherent planar HF arrays

This paper concerns estimating the partially coherent signal gain of a large, fully-populated planar HF phasedarray using simulated data from a few widely distributed receive elements. Our approach involves estimating magnitude-squared coherence between sensor elements in order to estimate parameters of a 2-D exponential model for signal decoherence, which, in turn, is used to predict array gain. Via cross-validation, this approach is shown to correctly predict the array gain observed in simulated data using well-established models for ionospheric perturbations. Finally, it is used to predict the signal gains achievable for large fully-populated planar HF arrays.

Rapid sensing of underutilized, wideband spectrum using the Random Demodulator

Efficient spectrum sensing is an important problem given the large and increasing demand for wireless spectrum and the need to protect incumbent users. We can more efficiently use large swaths of underutilized spectrum by designing spectrum sensors that can quickly, and power-efficiently, find and opportunistically communicate over unused (or underutilized) pieces of spectrum, such as television bands. In this paper, we concentrate on a particular sensing architecture, the Random Demodulator (RD), and look at two aspects of the problem. First, we offer fundamental limits on how efficiently any algorithm can perform the sensing operation with the RD. Second, we analyze a very simple, low-complexity algorithm called one-step thresholding that has been shown to work near-optimally for certain measurement classes in a low SNR setting or when the non-zero input coefficients are nearly equal. We rigorously establish that the RD architecture is well-suited for near-optimal recovery of the locations of the non-zero frequency coefficients in similar settings using one-step thresholding and perform numerical experiments to offer some confirmation of our results.

Faster than Nyquist, slower than Tropp

The sampling rate of analog-to-digital converters is severely limited by underlying technological constraints. Recently, Tropp et al. proposed a new architecture, called a random demodulator (RD), that attempts to circumvent this limitation by sampling sparse, bandlimited signals at a rate much lower than the Nyquist rate. An integral part of this architecture is a random bi-polar modulating waveform (MW) that changes polarity at the Nyquist rate of the input signal. Technological constraints also limit how fast such a waveform can change polarity, so we propose an extension of the RD that uses a run-length limited MW which changes polarity at a slower rate. We call this extension a constrained random demodulator (CRD) and establish that it enjoys theoretical guarantees similar to the RD and that these guarantees are directly related to the power spectrum of the MW. Further, we put forth the notion of knowledge-enhanced CRD in the paper. Specifically, we show through simulations that matching the distribution of spectral energy of the input signal with the power spectrum of the MW results in the CRD performing better than the RD of Tropp et al.

Ocean acoustic waveguide invariant parameter estimation using tonal noise sources

The abundance of shipping noise sources in ocean littoral zones provides a great opportunity to estimate ocean environmental parameters. The waveguide invariant parameter β, defined as the ratio of inverse group and phase velocities between modes, has been used in a variety of applications including ranging of passive sources. Previous work utilizing the waveguide invariant in passive sonar has relied on processing the time-frequency intensity striations of broadband sources. In this paper, the reception of strong tonal components from transiting commercial ships of known location (e.g., from AIS data) are used for estimating β over the source-receiver path. A maximum likelihood estimate of β is derived by relating the fading characteristics of different tonal components over range. The method is verified on simulated data using a Pekeris waveguide model.

Efficient linear time-varying system identification using chirp waveforms

Linear, time-varying (LTV) systems are operators composed of time shifts, frequency shifts, and complex amplitude scalings that act on continuous finite-energy waveforms. This paper builds upon a novel, resource-efficient method previously proposed by the authors for identifying the parametric description of such systems from the sampled response to linear frequency modulated (LFM) waveforms. If the LTV operator is probed with a sufficiently diverse set of LFM pulses, more LFM pulses than reflectors, then the system can be identified with high accuracy. The accuracy is shown to be proportional to the uncertainty in the estimated frequencies and confirmed with numerical experiments.

Resource-efficient parametric recovery of linear time-varying systems

This paper presents a novel, resource-efficient method of identifying the parametric description of linear, time-varying (LTV) systems, consisting of time shifts, frequency shifts, and complex amplitude scalings. Linear Frequency Modulation (LFM) waveforms are used to probe the system, and the returns are used to identify the parametric description. The number of samples required for perfect recovery is shown to scale linearly in the number of descriptive parameters K, and the sampling rate required is sub-Nyquist when compared to the bandwidth of the LFM waveforms. The time-bandwidth product of the LFM waveforms scales quadratically in K. Numerical examples demonstrate perfect recovery of closely spaced targets in the delay-Doppler space.

Run-length limited codes for backscatter communication

In backscatter communications, ultra-low power devices signal by modulating the reflection of radio frequency signals emitted from an external source. Unlike conventional one-way communication, the backscatter channel experiences unique self-interference and spread Doppler clutter. Run-length limited (RLL) codes provide a method for spectrum shaping that requires no hardware changes to the communicating devices. The proposed coding framework is suitable for any arbitrarily-shaped pulse train or continuous wave reader waveform. It exploits the unique channel Doppler spread statistics to offer a trade-off between interference rejection and data rate. Analysis shows that code rates of 1 and 4/5 are achievable when dealing with low spread Doppler channels, which is an improvement over the current rate 1/2 with current mainstream backscatter communication techniques. Simulation results with realistic channel assumptions are analyzed and discussed to confirm the theoretical analysis.