T. Justin Shaw

Compressive sensing of sparse radio frequency signals using optical mixing.

We demonstrate an optical mixing system for measuring properties of sparse radio frequency (RF) signals using compressive sensing (CS). Two types of sparse RF signals are investigated: (1) a signal that consists of a few 0.4 ns pulses in a 26.8 ns window and (2) a signal that consists of a few sinusoids at different frequencies. The RF is modulated onto the intensity of a repetitively pulsed, wavelength-chirped optical field, and time-wavelength-space mapping is used to map the optical field onto a 118-pixel, one-dimensional spatial light modulator (SLM). The SLM pixels are programmed with a pseudo-random bit sequence (PRBS) to form one row of the CS measurement matrix, and the optical throughput is integrated with a photodiode to obtain one value of the CS measurement vector. Then the PRBS is changed to form the second row of the mixing matrix and a second value of the measurement vector is obtained. This process is performed 118 times so that we can vary the dimensions of the CS measurement matrix from 1×118 to 118×118 (square). We use the penalized ℓ(1) norm method with stopping parameter λ (also called basis pursuit denoising) to recover pulsed or sinusoidal RF signals as a function of the small dimension of the measurement matrix and stopping parameter. For a square matrix, we also find that penalized ℓ(1) norm recovery performs better than conventional recovery using matrix inversion.

Sensing RF signals with the optical wideband converter

The optical wideband converter (OWC) is a system for measuring properties of RF signals in the GHz band without use of high speed electronics. In the OWC the RF signal is modulated on a repetitively pulsed optical field with a large wavelength chirp, the optical field is diffracted onto a spatial light modulator (SLM) whose pixels are modulated with a pseudo-random bit sequences (PRBSs), and finally the optical field is directed to a photodiode and the resulting current integrated for each PRBS. When the number of PRBSs and measurements equals the number of SLM pixels, the RF signal can be obtained in principle by multiplying the measurement vector by the inverse of the square matrix given by the PRBSs and the properties of the optics. When the number of measurements is smaller than the number of pixels, a compressive sensing (CS) measurement can be performed, and sparse RF signals can be obtained using one of the standard CS recovery algorithms such as the penalized l1 norm (also known as basis pursuit) or one of the variants of matching pursuit. Accurate reconstruction of RF signals requires good calibration of the OWC. In this paper, we present results using the OWC for RF signals consisting of 2 sinusoids recovered using 3 techniques (matrix inversion, basis pursuit, and matching pursuit). We compare results obtained with orthogonal matching pursuit with nonlinear least squares to basis pursuit with an over-complete dictionary.

Multimode waveguide speckle patterns for compressive sensing.

Compressive sensing (CS) of sparse gigahertz-band RF signals using microwave photonics may achieve better performances with smaller size, weight, and power than electronic CS or conventional Nyquist rate sampling. The critical element in a CS system is the device that produces the CS measurement matrix (MM). We show that passive speckle patterns in multimode waveguides potentially provide excellent MMs for CS. We measure and calculate the MM for a multimode fiber and perform simulations using this MM in a CS system. We show that the speckle MM exhibits the sharp phase transition and coherence properties needed for CS and that these properties are similar to those of a sub-Gaussian MM with the same mean and standard deviation. We calculate the MM for a multimode planar waveguide and find dimensions of the planar guide that give a speckle MM with a performance similar to that of the multimode fiber. The CS simulations show that all measured and calculated speckle MMs exhibit a robust performance with equal amplitude signals that are sparse in time, in frequency, and in wavelets (Haar wavelet transform). The planar waveguide results indicate a path to a microwave photonic integrated circuit for measuring sparse gigahertz-band RF signals using CS.

Applications of the Orthogonal Matching Pursuit/ Nonlinear Least Squares Algorithm to Compressive Sensing Recovery

Compressive sensing (CS) has been widely investigated as a method to reduce the sampling rate needed to obtain accurate measurements of sparse signals (Donoho, 2006; Candes & Tao, 2006; Baraniuk, 2007; Candes & Wakin, 2008; Loris, 2008; Candes et al., 2011; Duarte & Baraniuk, 2011). CS depends on mixing a sparse input signal (or image) down in dimension, digitizing the reduced dimension signal, and recovering the input signal through optimization algorithms. Two classes of recovery algorithms have been extensively used. One class is based on finding the sparse target vector with the minimum ell-1 norm that satisfies the measurement constraint: that is, when the vector is transformed back to the input signal domain and multiplied by the mixing matrix, it satisfies the reduced dimension measurement. In the presence of noise, recovery proceeds by minimizing the ell-1 norm plus a term proportional to ell-2 norm of the measurement constraint (Candes and Wakin, 2008; Loris, 2008). The second class is based on „greedy“ algorithms such as orthogonal matching pursuit (Tropp and Gilbert, 2007) and iteratively, finds and removes elements of a discrete dictionary that are maximally correlated with the measurement. There is, however, a difficulty in applying these algorithms to CS recovery for a signal that consists of a few sinusoids of arbitrary frequency (Duarte & Baraniuk, 2010). The standard discrete Fourier transform (DFT), which one expects to sparsify a time series for the input signal, yields a sparse result only if the duration of the time series is an integer number of periods of each of the sinusoids. If there are N time steps in the time window, there are just N frequencies that are sparse under the DFT; we will refer to these frequencies as being on the frequency grid for the DFT just as the time samples are on the time grid. To recover signals that consist of frequencies off the grid, there are several alternative approaches: 1) decreasing the grid spacing so that more signal frequencies are on the grid by using an overcomplete dictionary, 2) windowing or apodization to improve sparsity by reducing the size of the sidelobes in the DFT of a time series for a frequency off the grid, and 3) scanning the DFT off integer values to find the frequency (Shaw & Valley, 2010). However, none of these approaches is really practical for obtaining high precision values of the frequency and amplitude of arbitrary sinusoids. As shown below in Section 6, calculations with time windows of more than 10,000 time samples become prohibatively slow; windowing distorts the signal and in many cases, does not improve sparsity enough for CS recovery algorithms

Calibration of a speckle-based compressive sensing receiver

Optical speckle in a multimode waveguide has been proposed to perform the function of a compressive sensing (CS) measurement matrix (MM) in a receiver for GHz-band radio frequency (RF) signals. Unlike other devices used for the CS MM, e.g. the digital micromirror device (DMD) used in the single pixel camera, the elements of the speckle MM are not known before use and must be measured and calibrated. In our system, the RF signal is modulated on a repetitively pulsed chirped wavelength laser source, generated from mode-locked laser pulses that have been dispersed in time or from an electrically addressed distributed Bragg reflector laser. Next, the optical beam with RF propagates through a multimode fiber or waveguide, which applies different weights in wavelength (or equivalently time) and space and performs the function of the CS MM. The output of the guide is directed to or imaged on a bank of photodiodes with integration time set to the pulse length of the chirp waveform. The output of each photodiode is digitized by an analog-to-digital converter (ADC), and the data from these ADCs are used to form the CS measurement vector. Accurate recovery of the RF signal from CS measurements depends critically on knowledge of the weights in the MM. Here we present results using a stable wavelength laser source to probe the guide.

Photonic systems for compressive sensing of high-speed RF signals

We review photonic systems for undersampling and compressive sensing of GHz-band RF signals. We focus on methods for performing the compressive measurement and on the properties of RF signals amenable to such sampling.

Use of optical speckle patterns for compressive sensing of RF signals in the GHz band

We demonstrate that speckle patterns at the output of multimode optical waveguides can be used for a compressive sensing (CS) measurement matrix (MM) to measure sparse RF signals in the GHz band (1-100 GHz). In our system mode-locked femtosecond laser pulses are stretched to a width on the order of the interpulse time, modulated by the RF, and injected into a multimode waveguide. The speckle pattern out of the guide is imaged onto an array of photodiodes whose output is digitized by a bank of ADCs. We have measured the CS MM for multimode fibers and used these MMs to demonstrate that sparse RF signals (sparsity K) modulated on a chirped optical carrier can be recovered from M measurements (the number of photodiodes) consistent with the CS relation M ~ K log(N/K) (N is the number of samples needed for Nyquist rate sampling). We demonstrate experimentally that speckle sampling gives comparable results to the photonic WDM sampling system used previously for periodic undersampling (multi-coset sampling) of RF chirp pulses. We have also calculated MMs for both multimode fibers and planar waveguides using their respective mode solutions to determine optimal waveguide parameters for a CS system. Our results suggest a path to a CS system for GHz band RF signals that can be completely constructed using photonic integrated circuit (PIC) technology.