# THE EXPANDED STATE MODEL AND FINITE WORD LENGTH EFFECTS IN 2-D DIGITAL FILTERS

#### Abstract

Digital filters process input data sequences and produce output data sequences which in some sense are better than the original. Digital filters are mathematically represented by transfer functions, impulse responses or state variable matrices. These representations imply that the filter computations are performed using infinite precision arithmetic. However, when implementing the mathematical filter model, one is required to use either hardware with finite word length or software with finite word machines. This finite word length implementation introduces various phenomena in filter behavior which are not present in the ideal filter. The main effect of finite word length implementation is the introduction of nonlinear error sources in the filter model. These result in small signal quantization errors as well as limit cycle and overflows. The analysis of these errors is performed using either a statistical approach where nonlinear characteristics are used. Extensive research has been done on the finite word length problems of one dimensional (1-D) digital filters. On the contrary, very little work has been done on the error analysis of two dimensional (2-D) digital filters. The research reported here for the first time takes a comprehensive look at the error analysis of 2-D digital filters. The analysis is performed in the framework of the 2-D expanded state model. This new model is an extension of the 1-D expanded state model introduced recently. The new model is used to perform the quantization error analysis of 2-D digital filters. The new model provides a straightforward approach for defining the "error system". Three error systems are introduced for the three sources of quantization errors (input quantization, multiplier quantization and summer quantization). For a given error system, the output noise variance is expressed in terms of the input noise variance and the error system matrices. The quantization noise analysis using the 2-D expanded state model is applicable in general to multiple input-multiple output systems. The 1-D and 2-D digital filters exhibit limit cycle phenomena due to finite word length implementations. The expanded state model offers a convenient and general approach to perform the limit cycle analysis of digital filters. The same error system used in the quantization noise analysis is also used in the limit cycle analysis. The expanded state model enables one to express the filter in terms of the expanded state matrices and the period of the limit cycle. This, then is used to establish bounds on the amplitude of the limit cycles. Asymptotic behavior of the digital filters is also studied. The overflow problem of 1-D and 2-D digital filters is analyzed by the 1-D and 2-D expanded state models. These models for the first time allow a convenient means of obtaining a transfer function and impulse response from the input to the summer outputs. Using this impulse response, the filter can be scaled to avoid summer overflow. In addition, the expanded state model is used to provide composite state variable and summer variable scaling. This technique allows the most flexible scaling strategy that can be applied to the 1-D and 2-D digital filters. Various examples are used to demonstrate these theoretical results.

#### Recommended Citation

Ambegaonkar, Prakash M., "THE EXPANDED STATE MODEL AND FINITE WORD LENGTH EFFECTS IN 2-D DIGITAL FILTERS" (1980). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI8111848.

https://epublications.marquette.edu/dissertations/AAI8111848