Random Sampling for Analog-to-Information Conversion of Wideband Signals Jason Laska, Sami Kirolos, Yehia Massoud, Richard Baraniuk Department of Electrical and Computer Engineering Rice University Houston, TX Anna Gilbert, Mark Iwen, Martin Strauss Departments of Mathematics and EECS University of Michigan Ann Arbor, MI Abstract—We develop a framework for analog-to-information conversion that enables sub-Nyquist acquisition and processing of wideband signals that are sparse in a local Fourier representation. The first component of the framework is a random sampling system that can be implemented in practical hardware. The second is an efficient information recovery algorithm to compute the spectrogram of the signal, which we dub the sparsogram.A simulated acquisition of a frequency hopping signal operates at 33× sub-Nyquist average sampling rate with little degradation in signal quality. I. I NTRODUCTION Sensors, signal processing hardware, and algorithms are un- der increasing pressure to accommodate ever faster sampling and processing rates. In this paper we study the acquisition and analysis of wideband signals that are locally Fourier sparse (LFS) in the sense that at each point in time they are well-approximated by a few local sinusoids of constant fre- quency. Examples of LFS signals include frequency hopping communication signals, slowly varying chirps from radar and geophysics, and many acoustic and audio signals. LFS signals are sparse in a time-frequency representation like the short-time Fourier transform (STFT). In discrete time, the STFT corresponds to a Fourier analysis on a sliding window of the signal S(τ,ω)= ∞  ν=-∞ s(ν ) w(ν − τ ) e -jων ; (I.1) that is, S(τ,ω) is the Fourier spectrum of the signal s localized around time sample τ by the N -point window w. The spectrogram is the squared magnitude |S(τ,ω)| 2 . The defining property of an LFS signal is that S(τ,ω) ≈ 0 for most τ and ω. While LFS signals are simply described in time-frequency, they are wideband when there is no a priori restriction on the frequencies of the local sinusoids. Hence, the requirements of traditional Nyquist-rate sampling at two times the bandwidth can be excessive and difficult to meet. As a practical exam- ple, consider sampling a frequency-hopping communications signal that consists of a sequence of windowed sinusoids with frequencies distributed between f 1 and f 2 Hz. The bandwidth of this signal is f 2 − f 1 Hz, which dictates sampling above the Nyquist rate of 2(f 2 − f 1 ) Hz to avoid aliasing. However the description of the signal at any point in time is extremely simple: it consists of just a single sinusoid. Surely we should be able to acquire such a signal with fewer than 2(f 2 − f 1 ) samples per second. Fortunately, the past several years have seen several ad- vances in the theory of sampling and reconstruction that address these very questions. Leveraging the theory of stream- ing algorithms, we introduce in this paper a new framework for analog-to-information conversion that enables sub-Nyquist acquisition and processing of LFS signals. Our framework has two key components. The first is a random sampling system that can be implemented in practical hardware. The second is an efficient information recovery algorithm to compute the spectrogram of LFS signal, which we dub the sparsogram. This paper is organized as follows. We review the requisite theory and algorithms for random sampling in Section II and develop two implementations of random samplers in Section III. We conduct a number of experiments to validate our approach in Section IV and close with a discussion and conclusions in Section V. II. RANDOM SAMPLING AND I NFORMATION RECOVERY To set the stage for the random sampling and information recovery algorithm, we start with a description of the problem in the discrete setting. Let s be a discrete-time signal of length N (not discrete samples of an analog signal but simply a vector of length N ). We know that s is perfectly represented by its Fourier coefficients or spectrum. Suppose that we wish to use only m Fourier coefficients to represent the signal or spectrum. To minimize the MSE in our choice of m coefficients, the optimal choice of Fourier coefficients is the m largest Fourier coefficients in magnitude. Denote the optimal m-term Fourier representation of a signal s of length N by r opt . We assume that, for some M , we have (1/M ) ≤‖s − r opt ‖ 2 ≤‖s‖ 2 ≤ M. Gilbert et. al [1] have developed an algorithm that uses at most m · (log(1/δ), log N, log M, 1/ǫ) O(1) space and time and outputs a representation r such that ‖s − r‖ 2 2 ≤ (1 + ǫ)‖s − r opt ‖ 2 2 , with probability at least 1 − δ. The algorithm is randomized and the probability is over random choices made by the algorithm, not over the signal.