Sampling Theorem: A Unified Outlook on Information Theory, Block and Convolutional Codes

Farokh MARVASTI  Mohammed NAFIE  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E76-A   No.9   pp.1383-1391
Publication Date: 1993/09/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
sampling theorem,  information theory,  real fields block and convolutional codes,  

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Redundancy is introduced by sampling a bandlimited signal at a higher rate than the Nyquist rate. In the cases of erasures due to fading or jamming, the samples are discarded. Therefore, what we get at the output of the receiver is a set if nonuniform samples obtained from a uniform sampling process with missing samples. As long as the rate of nonuniform samples is higher than the Nyquist rate, the original signal can be recovered with no errors. The sampling theorem can be shown to be equivalent to the fundamental theorem of information theory. This oversampling technique is also equivalent to a convolutional code of infinite constraint length is the Field of real numbers. A DSP implementation of this technique is through the use of a Discrete Fourier Transform (DFT), which happens to be equivalent to block codes in the field of real numbers. An iterative decoder has been proposed for erasure and impulsive noise, which also works with moderate amount of additive random noise. The iterative method is very simple and efficient consisting of modules of Fast Fourier Transforms (FFT) and Inverse FFT's. We also suggest a non-linear iterative method which converges faster than the successive approximation. This iterative decoder can be implemented in a feedback configuration. Besides FFT, other discrete transforms such as Discrete Cosine Transform, Discrete Sine Transform, Discrete Hartley Transform, and Discrete Wavelet Transform are used. The results are comparable to FFT with the advantage of working in the field of real numbers.