A Super-Resolution Algorithm for Multiband Signal Identification

Transcription

1 A Super-Resolution Algorithm for Multiband Signal Identification Zhihui Zhu, Dehui Yang, Michael B. Wakin, and Gongguo Tang Department of Electrical Engineering, Colorado School of Mines {zzhu, dyang, mwakin, Abstract Recent advances in convex optimization have led to super-resolution algorithms that provide exact frequency localization in multitone signals from limited time-domain samples. Such localization is accomplished by minimizing a certain atomic norm, which can be implemented in a semidefinite program. In this work, we consider the identification of multiband signals, which are comprised of multiple, unknown narrow bands of frequency content at multiple carrier frequencies. Integrating a basis of modulated discrete prolate spheroidal sequences (DPSS s) into the atomic norm minimization framework, we introduce a technique for estimating the unknown band positions based on limited time-domain samples of the signal. Index Terms Atomic norm, discrete prolate spheroidal sequences, multiband identification, super-resolution I. INTRODUCTION Super-resolution refers to the problem of identifying signal parameters with high resolution from limited measurements. The conventional super-resolution problem considers a multitone analog signal that can be expressed as a sum of J complex exponentials of various frequencies: x(t) = θ j e iπfjt. Suppose such a multitone signal x(t) is bandlimited with bandlimit Bnyq Hz, i.e., that max j F j Bnyq. Let x denote the length-n vector obtained by uniformly sampling x(t) over the time interval [0, NT s ) with sampling period T s B nyq : x[n] = θ j e iπfjn, n = 0,,..., N, where f j = F j T s. This model arises in problems such as radar signal processing with point targets [], [], line spectral estimation [3] [5], and spike detection for neural recordings [6]. We consider a more realistic model in which x(t) has a continuous-time Fourier transform (CTFT) supported on a union of several narrow bands J F = [F j B bandj /, F j + B bandj /], i.e., x(t) = X(F )e jπf t df. () F Here X(F ) denotes the CTFT of x(t). The band centers are given by the frequencies {F j } j [J] and the band widths are denoted by {B bandj } j [J], where [J] denotes the set {,,..., J}. Such multiband signal models arise in communications [7], radar signal processing with non-point targets [8], [9], and mitigation of narrowband interference [0], []. Our goal is to identify the band centers {F j } j [J] from the finite-length samples (see ()) of x(t). In [], the authors consider recovering the finite-length samples of a multiband signal x(t) from compressive measurements with a dictionary of multiband modulated discrete prolate spheroidal sequences (DPSS s) [3] dictionary. The representation ability of the multiband modulated DPSS dictionary for sampled multiband signals is further analyzed in [4]. However, the works [], [4] assume that either the band centers are known or the bands are located in certain predefined grids. Instead, we attempt to identify the band centers which are assumed to be located arbitrarily as long as they satisfy a certain separation condition. In this paper, we apply atomic norm techniques to invert the parameters in () from finite-length samples. The atomic norm is utilized to promote group sparsity and has an equivalent semidefinite programming (SDP) characterization. Thus the problem can be solved efficiently using an off-the-shelf solver [5]. Our work differs from previous works on line spectral estimation [3], [4], [6] in that we assume that the CTFT of the corresponding analog signal occupies certain bandwidths, rather than that it is composed of several spikes. The outline of this paper is as follows. The main problem is illustrated in Section II. Our approach based on atomic norm minimization is discussed in Section III. Section IV discusses the development of a recovery guarantee for the atomic norm minimization approach. Section V presents some simulations to support our proposed method. To begin, we define II. PROBLEM SETUP e f = [ e iπf e iπf(n )] T C N as a length-n sampled complex sinusoid of normalized frequency f [, ]. We assume each band has the same bandwidth, i.e., B bandj = B band for all j [J]. Consider a signal vector x that arises by sampling the analog signal x(t) over the time interval [0, NT s ) with sampling period T s. We

2 assume T s is chosen to satisfy the minimum Nyquist sampling rate, which means T s := B nyq max j [J] { F j ± B band / }. Under these assumptions, the sampled multiband signal x can be expressed as an integral of sampled pure tones (i.e., discrete-time sinusoids) x = x(f)e f df, () W where the digital frequency f is integrated over a union of intervals J [ W := T s F = [f j W, f j + W ], ]. Here, W = BbandTs, f j = T s F j and the weighting function x(f) = T s X(f/T s ) equals the scaled CTFT of the analog signal and corresponds to the discrete-time Fourier transform (DTFT) of its infinite sample sequence. However, we stress that our interest concerns the finite-length sample vector x and not the infinite sample sequence. Spectral analysis of x is complicated by its finite nature: conventional frequency analysis based on the discrete Fourier transform (DFT) will suffer from familiar leakage artifacts, making it difficult, for example, to reliably identify the band centers f j. However, taking each e fj to act as a modulator to the center frequency f j and viewing each g j as samples of a baseband signal formed by integrating over a collection of sinusoids e f with f [ W, W ], we can express the vector x: with x = ( f j+w f j W g j = W W ) x(f)e f df = e fj g j (3) x(f + f j )e f df. (4) Here represents the elementwise (Hadamard) product between two length-n vectors. On its surface, the observation model in (3) describes a collection of sinusoids with unknown frequencies f j modulated by different unknown waveforms g j [7], [8]. Our goal is to identify the frequencies Ω = {f,..., f J } and the unknowns waveforms g,..., g J. A. DPSS basis III. OUR APPROACH It is known [], [4] that vectors g j formed from integrating sinusoids over a narrow range of frequencies (as in (4)) can be approximated to a very high degree of accuracy using a basis constructed from the DPSS s [3] from time-frequency analysis. Given W (0, ), the DPSS vectors {s(l) N,W } l [N] are length-n vectors whose DTFT have a certain concentration in the digital frequency band [ W, W ]. Now define [ ] S := s () N,W s(l) (5) N,W to contain the first L DPSS vectors for some value of L {,,..., N} that we can choose as desired. The columns of S are orthonormal. Throughout the paper, for any matrix S with orthonormal columns, we use P S := I N SS H to denote an orthogonal projection from C N to the orthogonal complement of the subspace formed by the columns of S. Taking L NW, the dictionary S provides very accurate approximations (in a mean square error (MSE) sense) for all sampled sinusoids in the targeted band [4], [9], [0]. Theorem. [4], [9], [0] Fix W (0, ). Let S be an N L orthobasis defined in (5). (Asymptotic guarantee) For fixed η (0, ). choose L = NW ( + η). Then there exist positive constants C, C (where C, C may depend on W and ɛ) such that for all N N 0 P S e f C N 5/4 e CN, f [ W, W ]. (Nonasymptotic guarantee) For any ɛ (0, ), the orthobasis S satisfies P S e f e f ɛ for all f [ W, W ] with ( 60πCN L = C N log and with L = W W Here C N = 4 π log(8n) + 6. ɛ P S e f e f df ɛ ) +, ( ) 5CN C N log +. Nɛ It follows that, with L NW DPSS vectors (which can be much smaller than N), one obtains an efficient basis S R N L for representing most sampled baseband signals such as g j in (4). B. Atoms and Atomic Norm Utilizing the fact that each g j Sσ j for some σ j C L, we can rewrite (3) as x (Sσ j ) e fj = a σ j (S sign(σ j )) e fj, where sign(a) = a denotes the sign of a vector. Thus, x can be viewed as a sparse combination of elements from the atomic set A := { a(f, α) = (Sα) e f, α C L, α = }.

3 The atomic norm of x is then defined as x A = inf c c j 0, j : x = c j a(f j, α j ), (6) α j j j = f j [, ) which can be used as a regularizer for promoting a certain notion of simplicity or sparsity of x, in particular by representing x using as few items as possible from the atomic set A. The atomic norm (6) has the following equivalent form [7], [8] x A = inf u C N C,X { N trace(toep(u)) + trace(c) [ ] Toep(u) X x = S(X), X H C } 0. Here S(X) = l s(l) N,W x l with x l being the l-th column of X. C. Localizing the band centers As we explain below, the SDP formulation (7) can be used to recover the band centers {f j } j [J]. Suppose u is an optimal solution to (7). Then the Vandermonde decomposition of Toep(u) characterizes the band centers. The dual norm of p A is defined as p A = sup p, y R = sup S H (e f p). y A f [, ] The dual problem of computing the atomic norm (6) can be written as (7) maximize p C N x, p R, subject to p A (8) which also has an equivalent SDP formulation. The band locations can alternatively be identified from p, the optimal solution to (8). To be precise, consider the vector valued dual polynomial q(f) = S H (e f p). (9) The set of frequencies can be obtained by finding the peaks of q(f) : Ω = {f : q(f) = }. IV. RECOVERY GUARANTEE We can certify the optimality of computing the atomic norm defined in (6) using the following proposition. Let E f := diag(e f ) denote an N N diagonal matrix for any f [, ). Proposition. Suppose x = j (Sσ j) e fj with { σ j > 0, j =,,..., J. Also } suppose the columns E fj s (l) N,W, j =,..., J, l =,..., L are linearly independent. If there is a vector p C N such that the corresponding vector-valued dual polynomial q(f) = S H (e f p) satisfies q(f j ) = sign(σ j ), f j Ω, q(f) <, f / Ω, (0) then x = j (Sσ j) e fj is the unique atomic decomposition satisfying x A = j σ j. The above optimality conditions are derived from the facts that the atomic norm minimization is convex, strong duality holds, and thus both primal and dual optimal values are attainable. Inspired by [3], [4], where the dual polynomial is constructed with the square of the Fejér kernel, [7] and [8] construct a dual polynomial q(f) that satisfies (0) when the orthobasis S is populated from certain random distributions. However, here the DPSS basis S is not a random orthobasis and it has particular structural properties. The following result establishes that the DPSS vectors are approximately orthogonal to the modulated ones. Lemma. [, Lemma 5.] Fix ɛ (0, ). Let L = NW ( ɛ), and let S be the orthobasis as defined in (5). Then there exist constants C, C (where C, C may depend on W and ɛ) such that for all N N 0 and f W s (l) N,W, E f s (l ) N,W 3 C e C N, l, l [L] and S H E f S 3N C e C N. Rather than utilizing the Fejér kernel as in [3], [4], we construct a pre-certificate by simply solving the following problem: minimize p subject to q(f j ) = sign(σ j ), j [J]. () With this, we can rewrite e f p = E f p. Now the equality constraints in () can be written as a linear system of equations: S H E H f. p = S H E H f }{{ J } A sign(σ ). sign(σ J ) }{{} λ The optimality condition ensures that the optimal p has the form p = A H α = E fj Sα j for some α = [ α H α H J ] H with αj C L satisfying the normal equation AA H α = λ which gives (when AA H is invertible) α = (AA H ) λ, p = A H α = A H (AA H ) λ. With this form of p, we obtain the dual polynomial q(f) = S H E H f p = S H E H f A H (AA H ) λ. ()

4 To certify the optimality of computing the atomic norm (6), it remains to show q(f) <, f / {f, f,..., f J }. A. Simple case: S H E f S = 0 if f W Lemma () shows that S is nearly orthogonal to E f S for any f W. To simplify analysis, we could hypothetically assume S is exactly orthogonal to E f S, i.e., S H E f S = 0 for any f W. Under this case, the following result establishes conditions under which the dual polynomial obtained by () certifies the success of the atomic norm minimization. Proposition. Let x = j (Sσ j) e fj with σ j > 0. Additionally, assume sign(σ j ) are drawn independently and identically (i.i.d.) from the uniform distribution on the complex unit circle and f := min j j dist(f j f j ) 4W, where the distance dist(f j f j ) is understood as the warparound distance on the unit circle. Also suppose S H E f S = 0 when f W. Then with high probability, x = j (Sσ j) e fj is the unique atomic decomposition of x. Proof: Note that when S H E f S = 0 if f W and f 4W, we have AA H = I and the dual polynomial () becomes q(f) = S H E H f A H (AA H ) λ = j S H E H f f j Sλ j. We first consider the case where f is away from the bands, i.e., dist(f, f j ) W for all j [J]. Then q(f) = j S H E H f f j Sλ j = 0 since S H E H f f j S = 0. Now consider the case where f is within one band; that is dist(f f j ) W for some j {,,..., J}. In this case, we have q(f) = S H E H f f j S sign(σ j ) since dist(f, f j ) > W for all j j. Without loss of generality, let f j = 0. To show q(f) <, we first show that σ min (S H E H f S) (the smallest singular value of S H E H f S) is strictly less than for any f 0. We prove this by contradiction. Note that both S and E f S are orthonormal matrices with L columns. This implies that σ min (S H E f S) < as long as Range(S) Range(E f S). Now we suppose Range(S) = Range(E f S) for some f 0, which indicates that E f S = SR for some L L orthonormal matrix R. Then we have Range(E f (E f S)) = Range(E f SR) = Range(E f S) = Range(S), dual polynomial locations of band centers Figure. Dual polynomial q(f) (see (9)) for multiband signal identification. Here, the dashed red lines represent the true frequencies, which are correctly localized by examining the peaks of the dual polynomial. where the second equality follows because R is an orthonormal matrix. Similarly, we have Range(E f ( (E f }{{} W f S))) = = Range(S), which contradicts to the fact S H E H f W f completed by noting that q(f) = S H E H f f j S sign(σ j ) < S = 0. The proof is with high probability if sign(σ j ) is drawn i.i.d. from the uniform distribution on the complex unit circle {σ C L : σ = }. We note that Proposition is premised on the idealized assumption that S H E f S = 0 for any f W, which holds only approximately, not exactly. Thus, we save for future work the task of formalizing Proposition using the DPSS basis properties outlined in Lemma. V. NUMERICAL SIMULATIONS As an illustration, we test on a time-limited vector x obtained by setting N = 56 and sampling an analog signal that has a multiband spectrum. In this multiband spectrum, the range of active digital frequencies W is a finite union of J = 4 small intervals of width W = 3 within the interval [, ). We randomly generate each vector gj as a linear combination of the first L = 5 DPSS vectors. Figure displays the dual polynomial obtained by solving the atomic norm minimization problem. We observe that the band centers are correctly localized by the peaks of the dual polynomial. Figure shows the pre-certificate dual polynomial obtained by solving (). ACKNOWLEDGMENT This work was supported by NSF grants CCF-4096, CCF-46405, and CCF

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