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
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.
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.
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.