A method for the determination of the eigenvalues and eigenfunctions of a certain class of linear integral equations

Pugachev, V.S.

doi:10.1016/0021-8928(59)90165-0

Cited by 2 publications

(3 citation statements)

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“…Note that this problem is discussed in detail on page 99 of [8]. Similar discussions can also be found in [6], [7], [18] and references therein.…”

Section: Characteristic Values and Functions Of A Continuous Randmentioning

confidence: 80%

“…In matrix form, are the rows of the transform matrix, and also called basis functions (3) with the matrix orthogonality property stated as (4) where indicates conjugated and transposed version of a matrix and is identity matrix. A signal vector (5) is mapped into the orthonormal space (subspace) through forward transform operator (6) where is transform coefficients vector as given (7) Similarly, the inverse transform yields the signal vector (8) We assume that the vector is populated by a wide-sense stationary (WSS) stochastic process. Then, we have (9) where is the expectation operator and is the autocorrelation sequence of the WSS process .…”

Section: Orthogonal Transformsmentioning

confidence: 99%

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An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

Torun

Akansu

2013

IEEE Trans. Signal Process.

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Signal dependent Karhunen-Loève transform (KLT), also called factor analysis or principal component analysis (PCA), has been of great interest in applied mathematics and various engineering disciplines due to optimal performance. However, implementation of KLT has always been the main concern. Therefore, fixed transforms like discrete Fourier (DFT) and discrete cosine (DCT) with efficient algorithms have been successfully used as good approximations to KLT for popular applications spanning from source coding to digital communications. In this paper, we propose a simple method to derive explicit KLT kernel, or to perform PCA, in closed-form for first-order autoregressive, AR (1), discrete process. It is a widely used approximation to many real world signals. The merit of the proposed technique is shown. The novel method introduced in this paper is expected to make real-time and data-intensive applications of KLT, and PCA, more feasible.Index Terms-Covariance analysis, eigenanalysis, explicit Karhunen-Loève transform (KLT) kernel, factor analysis, first-order autoregressive process, principal component analysis (PCA), signal dependent transform.

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“…Note that this problem is discussed in detail on page 99 of [8]. Similar discussions can also be found in [6], [7], [18] and references therein.…”

Section: Characteristic Values and Functions Of A Continuous Randmentioning

confidence: 80%

Section: Orthogonal Transformsmentioning

confidence: 99%

An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

Torun

Akansu

2013

IEEE Trans. Signal Process.

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“…There were prior studies to derive closed form kernel expressions for certain classes of stochastic processes reported in the literature. In particular, the derivation of such a kernel in its implicit form for processes with exponential correlation was reported [2][3][4][5][6]. Those methods require us to solve a transcendental tangent equation by using either numerical techniques with convergence concerns, or complex methods for explicit expression of KLT kernel.…”

Section: Introductionmentioning

confidence: 99%

A novel method to derive explicit KLT kernel for AR(1) process

Torun

Akansu

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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The derivation of explicit Karhunen-Loève transform (KLT) kernel for an auto-regressive order one, AR(1), process has been of a major implementation concern. The main reason is due to difficulties in finding the roots of a transcendental tangent equation as required in the derivation. We propose a novel method to derive explicit KLT kernel for discrete AR(1) process. The merit of the proposed technique is highlighted. The proposed method has a good potential to make real-time KLT applications more feasible in the coming years.

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A method for the determination of the eigenvalues and eigenfunctions of a certain class of linear integral equations

Cited by 2 publications

References 1 publication

An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

A novel method to derive explicit KLT kernel for AR(1) process

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