Compressive Sensing Off the Grid

Compressive sensing off the grid Abstract— We consider the problem of estimating the fre- Indeed, one significant drawback of the discretization quency components of a mixture of s complex sinusoids approach is the performance degradation when the true signal from a random subset of n regularly spaced samples. Unlike is not exactly supported on the grid points, the so called basis previous work in compressive sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the mismatch problem [13], [15], [16]. When basis mismatch normalized frequency domain [0; 1]. We propose an atomic occurs, the true signal cannot be sparsely represented by the norm minimization approach to exactly recover the unobserved assumed dictionary determined by the grid points. One might samples, which is then followed by any linear prediction method attempt to remedy this issue by using a finer discretization. to identify the frequency components. We reformulate the However, increasing the discretization level will also increase atomic norm minimization as an exact semidefinite program. By constructing a dual certificate polynomial using random the coherence of the dictionary. Common wisdom in com- kernels, we show that roughly s log s log n random samples pressive sensing suggests that high coherence would also are sufficient to guarantee the exact frequency estimation with degrade the performance. It remains unclear whether over- high probability, provided the frequencies are well separated. discretization is beneficial to solving the problems. Finer Extensive numerical experiments are performed to illustrate gridding also results in higher computational complexity and the effectiveness of the proposed method. Our approach avoids the basis mismatch issue arising from discretization by working numerical instability, overshadowing any advantage it might directly on the continuous parameter space. Potential impact have in sparse recovery. on both compressive sensing and line spectral estimation, in We overcome the issues arising from discretization by particular implications in sub-Nyquist sampling and super- working directly on the continuous parameter space for a resolution, are discussed. specific problem, where we estimate the continuous frequencies and amplitudes of a mixture of complex sinusoids from I. INTRODUCTION randomly observed time samples. This specific problem in In many modern signal processing systems, acquiring a fact captures all the essential ingredients of applying com- real-world signal by a sampling mechanism in an efficient pressive sensing to problems with continuous dictionaries. and cost-effective manner is a major challenge. For compu- In particular, the frequencies are not assumed to lie on tational, cost and storage reasons it is often desirable to not a grid, and can instead take arbitrary values in [−W; W ] only acquire, but also to subsequently compress the acquired where W is the bandwidth of the signal. With a time- signal. A fundamental idea that has the potential to overcome frequency exchange, our model is exactly the same as the the somewhat wasteful process of acquiring a signal using one in Candes, Romberg, and Tao’s foundational work on expensive hi-fidelity sensors, followed by compression and a compressive sensing [1], except that we do not assume the subsequent loss of fidelity, is the possibility of compressive frequencies lie on a equispaced grid. This major difference sensing: i.e. the realization that it is often possible to combine presents a significant technical challenge as the resulting the signal acquisition and the compression step by sampling dictionary is no longer an orthonormal Fourier basis, but the signal in a novel way [1]–[4]. Compressive sensing is an infinite dictionary with continuously many atoms and explores different mechanisms that allow one to acquire arbitrarily high coherence. a succinct representation of the underlying system while In this paper we consider signals whose spectra consist simultaneously achieving compression. of spike trains with unknown locations in a continuous Despite the tremendous impact of compressive sensing on normalized interval [0; 1] and whose amplitudes are random signal processing theory and practice, its development thus but unknown. Rather than sampling the signal at all times t = far has focused on signals with sparse representation in finite 0; : : : ; n−1 we randomly sample the signal at times t1; : : : tm discrete dictionaries. However, signals we encounter in the where each tj 2 f0; : : : ; n − 1g. Our main contributions in real world are usually specified by continuous parameters, this paper are the following: especially those in radar, sonar, sensor array, communication, 1) Provided the original signal has a resolvable spectrum seismology, remote sensing. Wideband analog signal with (in a sense that we make precise later), we show that sparse spectrum is another example that is closely tied to such a procedure is a viable means for sampling, and sampling theory [5]–[7]. In order to apply the theory of CS, that the original signal can be reconstructed exactly. researchers in these fields adopt a discretization procedure to 2) Our reconstruction algorithm is formulated as the so- reduce the continuous parameter space to a finite set of grid lution to an atomic norm [17] minimization problem. points [8]–[14]. While this seemingly simple strategy gives We show that this convex optimization problem can superior performance for many problems provided that the be exactly reformulated as a semidefinite program true parameters do fall into the grid set, the discretization (hence our methodology is computationally tractable) introduces recovery issues. by exploiting a well-known result in systems theory called the bounded real lemma. B. Atomic Norms 3) Our proof technique requires the demonstration of an jJj Define atoms a(f) 2 C ; f 2 [0; 1] via explicit dual certificate that satisfies certain interpola- tion conditions. The production of this dual certificate 1 i2πfj [a (f)]j = e ; j 2 J (2) requires us to consider certain random polynomial ker- pjJj nels, and proving concentration inequalities for these and rewrite the signal model (1) in matrix-vector form: kernels. These results may be of independent interest to the reader. s s ? X X iφ(ck) 4) We validate our theory by extensive numerical experi- x = cka(fk) = jckje a(fk) (3) ments that confirm random under-sampling as a means k=1 k=1 of compression, followed by atomic norm minimiza- where φ(·): C 7! [0; 2π) extracts the phase of a complex tion as a means of recovery are viable. number. The set of atoms A = feiφa(f); f 2 [0; 1]; φ 2 ? This paper is organized as follows. In Section II we intro- [0; 2π)g are building blocks of the signal x , the same duce the class of signals under consideration, the sampling way that canonical basis vectors are building blocks for procedure, and the recovery algorithm formally. Theorem sparse signals, and unit-norm rank one matrices are building II.2 is the main result of this paper. We outline connections blocks for low-rank matrices. In sparsity recovery and matrix to prior art and the foundations that we build upon in completion, the unit balls of the sparsity-enforcing norms, Section III. In Section IV we present the main proof idea e.g., the `1 norm and the nuclear norm, are exactly the convex (though we omit the detailed proofs due to space limitations). hulls of their corresponding building blocks. In a similar In Section V we present some supporting numerical experi- spirit, we define an atomic norm k · kA by identifying its ments. In Section VI we make some concluding remarks and unit ball with the convex hull of A: mention future directions. kxkA = inf ft > 0 : x 2 t conv (A)g n o X X iφk II. MODEL AND MAIN RESULTS = inf ck : x = cke a(fk) : ck≥0; φk2[0;2π) f2[0;1] k k We start with introducing the signal model and present the (4) main results. Roughly speaking, the atomic norm k · kA can enforce A. Problem Setup sparsity in A because low-dimensional faces of conv(A) correspond to signals involving only a few atoms. The idea Suppose we have a signal of using atomic norm to enforce sparsity for a general set of atoms was first proposed and analyzed in [17]. 1 s ? X i2πfkj Equation (4) indeed defines a norm if the set of atoms xj = p cke ; j 2 J; (1) jJj k=1 is bounded, centrally symmetric, and absorbent, which are satisfied by our choice of A. The dual norm of k · kA is composed of complex sinusoids with s distinct frequencies ∗ Ω = ff ; ··· ; f g ⊂ [0; 1] J jJj kzk = sup hz; xi = sup hz; eiφa(f)i (5) 1 s . Here is an index set and A R R kxk ≤1 φ2[0;2π);f2[0;1] is the size of J. In this paper, J is either f0; ··· ; n − 1g A or {−2M; ··· ; 2Mg for some positive integers n and M. In this paper, for complex column vectors x and z, the Any mixture of sinusoids with frequencies bandlimited to complex and real inner products are defined as [−W; W ], after appropriate normalization, can be assumed hz; xi = x∗z; hz; xi = Re(x∗z) (6) to have frequencies in [0; 1]. Consequently, a bandlimited R signal of such a form leads to samples of the form (1). respectively, where the superscript ∗ represents conjugate We emphasize that, unlike previous work in compressive transpose. sensing where the frequencies are assumed to lie on a finite set of discrete points [1], [13], [14], [18], the frequencies in C. Computational Method this work could lie anywhere in the normalized continuous With the atomic norm k·kA, we recover the missing entries domain [0; 1]. ? ? of x by solving the following convex optimization problem: Instead of observing xj for all j 2 J, we observe only entries in a subset T ⊂ J.

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