Title:

Theory and realization of novel algorithms for random sampling in digital signal processing

Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N(^2). The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the antialias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the autocorrelation sequence of a randomly sampled sequence as an alternative means to confirm its antialias property : The antialias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the autocorrelation of such sample sequences. A technique to evaluate the autocorrelation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudorandom sampling : To fully exploit its advantages, random sampling has been adopted in measurement Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N"^. The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the antialias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% , of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the autocorrelation sequence of a randomly sampled sequence as an alternative means to confirm its antialias property : The antialias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the autocorrelation of such sample sequences. A technique to evaluate the autocorrelation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudorandom sampling : To fully exploit its advantages, random sampling has been adopted in measurement instruments where computing a spectrum is either minimal or not required. Such applications in instrumentation are easily found in the literature. In this thesis, two applications in digital signal processing are introduced. (5) To suggest an inverse transformation for random sampling so as to complete a twoway process and to broaden its scope of application. Apart from the above, a case study of realizing in a transputer network the prime factor algorithm with regular sampling is given in Chapter 2 and a rough estimation of the signaltonoise ratio for a spectrum obtained from random sampling is found in Chapter 3. Although random sampling is aliasfree, problems in computational complexity and noise prevent it from being adopted widely in engineering applications. In the conclusions, the criteria for adopting random sampling are put forward and the directions for its development are discussed.
