UBC Theses and Dissertations
Sigma-delta quantization and Sturmian words Ayaz, Ulaş
In this thesis, our main focus is Sigma-Delta quantization schemes. These are commonly used in state-of-art Analog-to-digital conversion technology. Their main advantage is the ease of implementation and more importantly their insensitivity to certain circuit imperfections. When we compare sigma-delta scheme with pulse-code modulation (PCM), sigma-delta is inferior in terms of rate distortion because an N-bit kth order sigma-delta quantizer produces an approximation with the error of order O(N-k) whereas the corresponding N-bit PCM scheme has accuracy of O(2−N)). However, this is a raw estimate of the actual rate-distortion characteristic of sigma-delta as one can further compress the bitstreams obtained via sigma-delta quantization. Even though this observation was made earlier in  under certain assumptions, to our knowledge, it was not investigated fully. In this thesis, such an investigation is made for first-order sigma-delta quantizers by using some results from symbolic dynamics literature on “Sturmian words”. Surprisingly, it turns out that the approximation error is a function of the “actual bit-rate”, i.e., the bit-rate after compressing an N-bit first-order sigma-delta encoding. In addition, in this thesis, we will introduce a new setup for sampling a bandlimited function and then quantizing these samples via first-order sigma-delta scheme. This simple but surprisingly efficient technique will allow us to get a better bound for the approximation rate of sigma-delta schemes and it will allow us to apply the derived results for compression of the bitstreams.
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