Perfect radar pulse compression via unimodular fourier multipliers
Lassi Roininen Markku S. Lehtinen Petteri Piiroinen Ilkka I. Virtanen
Inverse Problems & Imaging 2014, 8(3): 831-844 doi: 10.3934/ipi.2014.8.831
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.
keywords: radar experiments comparison of measurements. Pulse compression
Probabilistic interpretation of the Calderón problem
Petteri Piiroinen Martin Simon
Inverse Problems & Imaging 2017, 11(3): 553-575 doi: 10.3934/ipi.2017026

In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes. This probabilistic interpretation comes in three equivalent formulations which open up novel perspectives on the classical question of unique determinability of conductivities from boundary data. We aim to make this work accessible to both readers with a background in stochastic process theory as well as researchers working on deterministic methods in inverse problems.

keywords: Inverse conductivity problem Dirichlet form theory reflecting diffusion process boundary process Lévy system
Perfect and almost perfect pulse compression codes for range spread radar targets
Markku Lehtinen Baylie Damtie Petteri Piiroinen Mikko Orispää
Inverse Problems & Imaging 2009, 3(3): 465-486 doi: 10.3934/ipi.2009.3.465
It is well known that a matched filter gives the maximum possible output signal-to-noise ratio (SNR) when the input is a scattering signal from a point like radar target in the presence of white noise. However, a matched filter produces unwanted sidelobes that can mask vital information. Several researchers have presented various methods of dealing with this problem. They have employed different kinds of less optimal filters in terms of the output SNR from a point-like target than that of the matched filter. In this paper we present a method of designing codes, called perfect and almost perfect pulse compression codes, that do not create unwanted sidelobes when they are convolved with the corresponding matched filter. We present a method of deriving these types of codes from any binary phase radar codes that do not contain zeros in the frequency domain. Also, we introduce a heuristic algorithm that can be used to design almost perfect codes, which are more suitable for practical implementation in a radar system. The method is demonstrated by deriving some perfect and almost perfect pulse compression codes from some binary codes. A rigorous method of comparing the performance of almost perfect codes (truncated) with that of perfect codes is presented.
keywords: comparison of experiments Radar waveform Ambiguity function.
Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
Lassi Roininen Petteri Piiroinen Markku Lehtinen
Inverse Problems & Imaging 2013, 7(2): 611-647 doi: 10.3934/ipi.2013.7.611
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.
keywords: covariance convergence Stochastic difference equation statistical inversion.

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