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# Multiresolution modeling of cyclostationary processes using the wavelet transform

#### Abstract

In this dissertation, we investigate two different multiresolution (MR) models for discrete-time second-order wide-sense cyclostationary processes of period P (CS(P) processes). These models utilize the discrete wavelet transform (DWT) to find the best possible statistical representation of such processes at different resolutions.^ The first model finds the best approximation $\ s\sb{0l},$ of a CS(P) process $s\sb0,$ from resolution 2$\sp{l}$ by minimizing the average variance of the error in this approximation on a wavelet packet tree. The basis which determines this best estimate is called the optimum wavelet packet basis of $s\sb0.$ The optimum basis selection algorithm determines the structure of the tree and the DWT filters at different nodes of this tree. We show that such an approach yields projections which significantly improve the quality of the low resolution estimates of $s\sb0.$^ In the second model, based on the modified binary wavelet tree, we develop a hierarchical family of sets of CS(P) processes ${\cal CS}(P,{\cal W}\sb{N})=\{{\cal CS}\sb{l}(P,{\cal W}\sb{N}),\ l=0, 1,\cdots, L\},$ where $P=2\sp{L}$ for some $L>0.$ Each $s\sb0\in {\cal CS}\sb{l}(P,{\cal W}\sb{N})$ can be parameterized by (i) l nominal DWT filter pairs $\{h\sbsp{k}{\*},g\sbsp{k}{\*}\},$ (ii) l mutually uncorrelated WSS residuals $r\sbsp{k}{\*}$ and (iii) a $CS(P\sb{l})$ projection process $s\sbsp{l}{\*},$ uncorrelated with the residuals $r\sbsp{k}{\*},\ k=1,2,\cdots,l,$ where $P\sb{l}=P/2\sp{l}.$ The sets ${\cal CS}\sb{l}(P,{\cal W}\sb{N})$ are nested within each other:$${\cal CS}\sb{l}(P,{\cal W}\sb{N})\supseteq{\cal CS}\sb{l+1}(P,{\cal W}\sb{N})$$The projection $s\sbsp{k}{\*}$ at resolution 2$\sp{k}$ is shown to obey similar hierarchical properties: $s\sbsp{k}{\*}\in{\cal CS}\sb{1-k}(P\sb{k},{\cal W}\sb{N}),$ where $P\sb{k}=P/2\sp{k},$ for $k=1,2,\cdots,l.$ Such a multiscale representation provides a significant reduction in the number of second-order moments required for the exact statistical representation of $s\sb0,$ enabling independent analysis and processing of its low resolution signal components $\{r\sbsp{1}{\*},r\sbsp{2}{\*},\cdots,r\sbsp{l}{\*}, s\sbsp{l}{\*}\}.$^ Finally, we present a model fitting algorithm to find the closest process $\ s\sb{0,l}\in{\cal CS}\sb{l}(P,{\cal W}\sb{N})$ to a $CS(P)$ process $s\sb0$ by minimizing the distance between $s\sb0$ and all $\tilde s\sb0\in{\cal CS}\sb{l}(P,{\cal W}\sb{N}).$ The effectiveness of this algorithm in fitting the hierarchical model to any $CS(P)$ process is demonstrated with experiments using synthesized $CS(P)$ processes and voiced speech. ^

#### Subject Area

Engineering, Electronics and Electrical

#### Recommended Citation

Moushumi Sen,
"Multiresolution modeling of cyclostationary processes using the wavelet transform"
(January 1, 1996).
*Doctoral Dissertations Available from Proquest.*
Paper AAI9709651.

http://scholarworks.umass.edu/dissertations/AAI9709651