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### Open Access Journals

IPI

We propose a novel framework for studying radar pulse compression with continuous waveforms.
Our methodology is based on the recent developments of the mathematical
theory of comparison of measurements. First we show that a radar
measurement of a time-independent but spatially distributed radar target
is rigorously more informative than another one if the modulus of
the Fourier transform of the
radar code is greater than or equal to the modulus of
the Fourier transform of the second radar
code.
We then motivate the study by spreading a Gaussian pulse into a longer
pulse with smaller peak power and re-compressing the spread pulse into
its original form.
We then review the basic concepts of the theory and pose the conditions for statistically equivalent radar experiments.
We show that such experiments can be constructed by spreading the radar pulses via multiplication of their Fourier transforms by unimodular functions.
Finally, we show by analytical and numerical methods some examples of the spreading and re-compression of certain simple pulses.

IPI

We propose a method to construct perfect pulse-compression codes with autoregressive moving average algorithms. We first show the relation between the study of coding and decoding techniques in radar engineering and the study of unimodular polynomials with constrained coefficients. Then we extend the study to unimodular Fourier series and unimodular rational functions. We use the Fourier series and rational functions as transfer functions in the autoregressive moving average algorithms. We show that by a suitable choice of the coefficients, the autoregressive moving average algorithms are realisable, stable and causal. We show examples of some almost perfect codes, i.e. numerically truncated perfect codes. We end by proposing perfect code design principles for practical radar engineering purposes.

IPI

We propose a new class of Gaussian priors,

The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and different-order difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closed-form expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.

A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.

*correlation priors*. In contrast to some well-known smoothness priors, they have stationary covariances. The correlation priors are given in a parametric form with two parameters: correlation power and correlation length. The first parameter is connected with our prior information on the variance of the unknown. The second parameter is our prior belief on how fast the correlation of the unknown approaches zero. Roughly speaking, the correlation length is the distance beyond which two points of the unknown may be considered independent.The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and different-order difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closed-form expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.

A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.

IPI

We study flexible and proper smoothness priors for Bayesian
statistical inverse problems by using Whittle-Matérn Gaussian random fields.
We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the
discrete approximations can be expressed as solutions of sparse
stochastic matrix equations.
Such equations are known to be computationally efficient and
useful in inverse problems with a large number of unknowns.

The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.

As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.

The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.

As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.

IPI

In Bayesian statistical inverse problems the

*a priori*probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.## Year of publication

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