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On the Asymptotic Equivalence of Circulant and Toeplitz Matrices Zhihui Zhu and Michael B

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On the Asymptotic Equivalence of Circulant and Toeplitz Matrices Zhihui Zhu and Michael B 1 On the Asymptotic Equivalence of Circulant and Toeplitz Matrices Zhihui Zhu and Michael B. Wakin Abstract—Any sequence of uniformly bounded N × N Her- Hermitian Toeplitz matrices fHN g by defining a function mitian Toeplitz matrices fHN g is asymptotically equivalent to a eh(f) 2 L2([0; 1]) with Fourier series2 certain sequence of N ×N circulant matrices fCN g derivedp from Z 1 the Toeplitz matrices in the sense that kHN − CN kF = o( N) −j2πkf as N ! 1. This implies that certain collective behaviors of the h[k] = eh(f)e df; k 2 Z; 0 eigenvalues of each Toeplitz matrix are reflected in those of the 1 X corresponding circulant matrix and supports the utilization of h(f) = h[k]ej2πkf ; f 2 [0; 1]: the computationally efficient fast Fourier transform (instead of e the Karhunen-Loeve` transform) in applications like coding and k=−∞ filtering. In this paper, we study the asymptotic performance of the individual eigenvalue estimates. We show that the asymptotic Usually eh(f) is referred to as the symbol or generating equivalence of the circulant and Toeplitz matrices implies the function for the N × N Toeplitz matrices fHN g. individual asymptotic convergence of the eigenvalues for certain Suppose eh 2 L1([0; 1]). Szego’s˝ theorem [1] states that types of Toeplitz matrices. We also show that these estimates N−1 asymptotically approximate the largest and smallest eigenvalues 1 X Z 1 lim #(λl(HN )) = #(eh(f))df; (1) for more general classes of Toeplitz matrices. N!1 N l=0 0 Keywords—Szego’s˝ theorem, Toeplitz matrices, circulant matri- ces, asymptotic equivalence, Fourier analysis, eigenvalue estimates where # is any function continuous on the range of eh. As one example, choosing #(x) = x yields N−1 1 X Z 1 I. INTRODUCTION lim λl(HN ) = eh(f)df: N!1 N 0 A. Szego’s˝ Theorem l=0 Toeplitz matrices are of considerable interest in statistical In words, this says that as N ! 1, the average eigenvalue of signal processing and information theory [1–5]. An N × N HN converges to the average value of the symbol eh(f) that 1 Toeplitz matrix HN has the form generates HN . As a second example, suppose eh(f) > 0 (and thus λl(HN ) > 0 for all l 2 [N] and N 2 N) and let # be the 2 h[0] h[−1] h[−2] : : : h[−(N − 1)] 3 log function. Then Szego’s˝ theorem indicates that h[1] h[0] h[−1] 6 7 1 . 1 Z 6 . 7 lim log (det (H )) = log h(f) df: HN = 6 h[2] h[1] h[0] . 7 N e 6 7 N!1 N 0 6 . .. 7 4 . 5 This relates the determinant of the Toeplitz matrix to its h[N − 1] ··· h[0] symbol. Szego’s˝ theorem has been widely used in the areas of or H [m; n] = h[m − n]; m; n 2 [N] := f0; 1;:::;N − N signal processing, communications, and information theory. 1g. The covariance matrix of a random vector obtained by A paper and review by Gray [2, 7] serve as a remarkable sampling a wide-sense stationary (WSS) random process is an elementary introduction in the engineering literature and offer example of such a matrix. arXiv:1608.04820v2 [cs.IT] 22 Feb 2017 a simplified proof of Szego’s˝ original theorem. The result has H Throughout this paper, we consider N that is Hermitian, also been extended in several ways. For example, the Avram- HH = H H i.e., N N , and we suppose that the eigenvalues of N Parter theorem [8, 9], a generalization of Szego’s˝ theorem, λ (H ) ≥ · · · ≥ λ (H ) are denoted and arranged as 0 N N−1 N . relates the collective asymptotic behavior of the singular values A AH Here the Hermitian transpose of a matrix is denoted by . of a general (non-Hermitian) Toeplitz matrix to the absolute Szego’s˝ theorem [1] describes the collective asymptotic value of its symbol, i.e., jh(f)j. Tyrtyshnikov [10] proved that behavior (as N ! 1) of the eigenvalues of a sequence of e Szego’s˝ theorem holds if eh(f) 2 R and eh(f) 2 L2([0; 1]), and This work was supported by NSF grant CCF-1409261. Zamarashkin and Tyrtyshnikov [11] further extended Szego’s˝ Z. Zhu and M. B. Wakin are with the Department of Electrical Engineering and Computer Science, Colorado School of Mines, Golden, CO 80401 USA. 2This can also be interpreted using the discrete-time Fourier transform P1 −j2πkf e-mail: fzzhu,[email protected]. (DTFT). That is, we can define bh(f) = k=−∞ h[k]e = eh(1 − f). 1Through the paper, finite-dimensional vectors and matrices are indicated However, it is more common to view h as the Fourier series of the symbol by bold characters and we index such vectors and matrices beginning at 0. eh; see [1, 6]. 2 theorem to the case when eh(f) 2 R and eh(f) 2 L1([0; 1]). in O(N log N) time3 using the fast Fourier transform (FFT). Sakrison [12] extended Szego’s˝ theorem to high dimensions. The individual eigenvalues of the circulant matrix approximate Gazzah et al. [13] and Gutierrez-Guti´ errez´ and Crespo [14] those of the Toeplitz matrix. We study the quality of this extended Gray’s results on Toeplitz and circulant matrices approximation. to block Toeplitz and block circulant matrices and derived An N × N circulant matrix CN is a special Toeplitz matrix Szego’s˝ theorem for block Toeplitz matrices. of the form Most relevant to our work, Bogoya et al. [15] studied the 2 c[0] c[1] c[2] : : : c[N − 1] 3 individual asymptotic behavior of the eigenvalues of Toeplitz c[N − 1] c[0] c[1] matrices by interpreting Szego’s˝ theorem in probabilistic lan- 6 7 6 . 7 guage. In the case that the range of h is connected, Bogoya et 6 . 7 e CN = 6 c[N − 2] c[N − 1] c[0] . 7 : al. related the eigenvalues to the values obtained by sampling 6 . 7 4 . .. 5 the symbol eh(f) uniformly in frequency on [0; 1]. c[1] ··· c[0] B. Motivation Circulant matrices arise naturally in applications involving Despite the power of Szego’s˝ theorem, in many scenarios the discrete Fourier transform (DFT) [3]; in particular, any (such as certain coding and filtering applications [2, 3]), one circulant matrix can be unitarily diagonalized using the DFT matrix. Circulant matrices offer a nontrivial but simple set may only have access to HN and not eh. In such cases, it is still desirable to have practical and efficiently computable of objects that can be used for problems involving Toeplitz matrices. For example, the product H x can be computed in estimates of individual eigenvalues of HN . We elaborate on N two example applications below. O(N log N) time by embedding HN into a (2N−1)×(2N−1) circulant matrix and using the FFT to perform matrix-vector i: Estimating the condition number of a positive-definite multiplication. Also Gray [2, 7] showed that Toeplitz and Toeplitz matrix. The linear system HN y = b arises naturally circulant matrices are asymptotically equivalent in a certain in many signal processing and estimation problems such as sense; this implies that their eigenvalues have similar collective linear prediction [4, 5]. The condition number κ(HN ) of the behavior. See Section II for formal definitions. Finally, we note Toeplitz matrix HN is important when solving such systems. that circulant matrices have been used as preconditioners [21, For example, the speed of solving such linear systems via 22] of Toeplitz matrices in iterative methods for solving linear the widely used conjugate gradient method is determined systems of the form HN y = b. by the condition number: the larger κ(HN ), the slower the We consider estimates for the eigenvalues of a Toeplitz convergence of the algorithm. In case of large κ(HN ), pre- matrix obtained from a well-constructed circulant matrix. The conditioning can be applied to ensure fast convergence. Thus eigenvalues of the circulant matrix can be computed efficiently estimating the smallest and largest eigenvalues of a symmetric without constructing the whole matrix; one merely applies the positive-definite Toeplitz matrix (such as the covariance matrix FFT to the first row of the matrix. We do not provide new of a random vector obtained by sampling a stationary random circulant approximations to Toeplitz matrices in this paper; process) is of considerable interest [16, 17]. rather we sharpen the analysis on the asymptotic equivalence of ii: Spectrum sensing algorithm for cognitive radio. Toeplitz and certain circulant matrices [2, 3, 7] by establishing Spectrum sensing is a fundamental task in cognitive ratio, results in terms of individual eigenvalues rather than collective which aims to best utilize the available spectrum by identifying behavior. To the best of our knowledge, this is the first work unoccupied bands [18–20]. Zeng and Ling [20] have proposed that provides guarantees for asymptotic equivalence in terms spectrum sensing methods for cognitive radio based on the of individual eigenvalues. eigenvalues of a Toeplitz covariance matrix. These eigenvalue- based algorithms overcome the noise uncertainty problem D. Circulant Approximations to H which exists in alternative methods based on energy detection. N We consider the following circulant approximations that Aside from the above applications, approximate and effi- have been widely used in information theory and applied ciently computable eigenvalue estimates can also be used as the mathematics. starting point for numerical algorithms that iteratively compute 1) CeN : Bogoya et al. [15] proved that the samples of the eigenvalues with high precision.
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