Compressibility of Deterministic and Random Infinite Sequences

Amini, Arash; Unser, Michaël; Marvasti, Farrokh

doi:10.1109/tsp.2011.2162952

Cited by 47 publications

(134 citation statements)

References 12 publications

(19 reference statements)

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“…Then, the Bayesian interpolator of the AR(1) process at the point , given the samples , is given by (29) Proof: We start by the definition of the Bayesian interpolator (posterior mean) (30) Since there is a bijection between the sets and , the condition in the expectation of (30) can be replaced according to (31) where the validity of the second equality comes from the fact that is statistically independent of and for (Lemma 1). Up to this point, we have simplified the general form of the Bayesian interpolator.…”

Section: Lemma 2: Letmentioning

confidence: 99%

See 1 more Smart Citation

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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Abstract-Bayesian estimation problems involving Gaussian distributions often result in linear estimation techniques. Nevertheless, there are no general statements as to whether the linearity of the Bayesian estimator is restricted to the Gaussian case. The two common strategies for non-Gaussian models are either finding the best linear estimator or numerically evaluating the Bayesian estimator by Monte Carlo methods. In this paper, we focus on Bayesian interpolation of non-Gaussian first-order autoregressive (AR) processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the Lévy-Khintchine representation theorem. We redefine the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. By providing analytic expressions, we show that the optimal interpolator is linear for all symmetric -stable distributions. The Bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines. We also show that the limiting case of Lévy-type AR(1) processes, the system of which has a pole at the origin, always corresponds to a linear Bayesian interpolator made of a piecewise linear spline, irrespective of the innovation distribution. Finally, we show the two mentioned cases to be the only ones within the family for which the Bayesian interpolator is linear.

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Section: Lemma 2: Letmentioning

confidence: 99%

“…It is shown in [31] that -stable priors become more compressible as decreases. This means that, in the realization of an -stable process, the intervals with large amplitudes are few and narrow.…”

Section: Simulationsmentioning

confidence: 99%

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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“…AR(1) systems and α-stable distributions are at the core of signal modeling and probability theory. Since stable processes have heavy-tailed statistics for α < 2, they are prototypical representatives for sparse signals [9], while one recovers the classical Gaussian processes for α = 2.…”

Section: Introductionmentioning

confidence: 99%

On the optimality of operator-like wavelets for sparse AR(1) processes

Pad

Unser

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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Sinusoidal transforms such as the DCT are known to be optimal-that is, asymptotically equivalent to the KarhunenLoève transform (KLT)-for the representation of Gaussian stationary processes, including the classical AR(1) processes. While the KLT remains applicable for non-Gaussian signals, it loses optimality and, is outperformed by the independentcomponent analysis (ICA), which aims at producing the most-decoupled representation. In this paper, we consider an extension of the classical AR(1) model that is driven by symmetric-alpha-stable (SαS) noise which is either Gaussian (α = 2) or sparse (0 < α < 2). For the sparse (non-Gaussian) regime, we prove that an expansion in a proper wavelet basis (including the Haar transform) is much closer to the optimal orthogonal ICA solution than the classical Fourier-type representations. Our criterion for optimality, which favors independence, is the Kullback-Leibler divergence between the joint pdf of the original signal and the product of the marginals in the transformed domain. We also observe that, for very sparse AR(1) processes (α ≤ 1), the operator-like wavelet transform is indistinguishable from the ICA solution that is determined through numerical optimization.

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“…Optimal sparse channel estimation often requires that its training signal satisfies restrictive isometry property (RIP) [16] in high probability. However, designing the RIP-satisfied training signal is a non-polynomial (NP) hard problem [17]. However, there exist some proposed methods which are stable with the cost of extra computational burden, especially in time-variant MIMO-OFDM systems.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Sparse Channel Estimation for Time-Variant MIMO Communication Systems

Gui

Mehbodniya

Adachi

2013

2013 IEEE 78th Vehicular Technology Conference (VTC Fall)

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Abstract-Channel estimation problem is one of the key technical issues in time-variant multiple-input multiple-output (MIMO) communication systems. To estimate the MIMO channel, least mean square (LMS) algorithm was applied to adaptive channel estimation (ACE). Since the MIMO channel is often described by sparse channel model, such sparsity can be exploited to improve the estimation performance by adaptive sparse channel estimation (ASCE) methods using sparse LMS algorithms. However, conventional ASCE methods have two main drawbacks: 1) sensitive to random scale of training signal and 2) unstable in low signal-to-noise ratio (SNR) regime. To overcome the two harmful factors, in this paper, we propose a novel ASCE method using normalized LMS (NLMS) algorithm (ASCE-NLMS). In addition, we also proposed an improved ASCE method using normalized least mean fourth (NLMF) algorithm (ASCE-NLMF). Two proposed methods can exploit the channel sparsity effectively. Also, stability of the proposed methods is confirmed by mathematical derivation. Computer simulation results show that the proposed sparse channel estimation methods can achieve better estimation performance than conventional methods.Keywords-least mean square (LMS), least mean fourth (LMF), normalized LMF (NLMF), adaptive sparse channel estimation (ASCE), multiple-input multiple-output (MIMO).

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Compressibility of Deterministic and Random Infinite Sequences

Cited by 47 publications

References 12 publications

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

On the optimality of operator-like wavelets for sparse AR(1) processes

Adaptive Sparse Channel Estimation for Time-Variant MIMO Communication Systems

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