Sigma-Delta Quantization and Sturmian Words

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 sigmadelta 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 [10] 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.