Mean Squared Error Analysis of Quantizers With Error Feedback
Quantization is a fundamental process in digital signal processing. <inline-formula><tex-math notation="LaTeX"> $\Delta \Sigma$</tex-math></inline-formula> modulators are often utilized for quantization, which can be easily implemented with static uniform quantizers and error feedback filters. In this paper, we analyze the mean squared quantization error of the quantizer with error feedback including the <inline-formula><tex-math notation="LaTeX"> $\Delta \Sigma$</tex-math></inline-formula> modulators. First, we study the quantizer with an ideal optimal error feedback filter that minimizes the mean squared error (MSE) of quantization. We show that the amplitude response of the optimal error feedback filter can be parameterized by one parameter. This parameterization enables us to find the optimal error feedback filter numerically. Second, the relationship between the number of bits used for the quantizer and the achievable MSE is clarified by using the optimal error feedback filter. This makes it possible to investigate the efficiency of the quantizer with the optimal error feedback filter in terms of MSE. Then, ideal optimal error feedback filters are approximated by practically implementable filters using the Yule-Walker method and the linear matrix inequality-based method. Numerical examples are provided for demonstrating our analysis and synthesis.