Z Transforms

Signals and Systems Using Mathcad by Derose and Veronis.
Chapter 3 Frequency Domain Analysis - Z Transforms.
At end of this file summaries of all 3 transforms are provided from Linear Systems and Signals 2nd edition by B.P. Lathi.
Entered by: Karl S Bogha Dhaliwal.

When do we know which transform to use and is there a comparison?
It goes like this you surely may have a better answer.
1. Laplace - for systems analysis and design but less for signals.
2. Fourier - for signals analysis and design, and less for systems.
3. Z - for digital (discrete) signal analysis and design, and systems.

We describe a method to extract single trial Visual Evoked Potential (VEP) buried in ongoing Electroencephalogram (EEG) activity. The common method for separating VEP from EEG is to use signal averaging. But we use digital filters to extract VEP assuming that VEP spectra are in the gamma band. As an application, a Fuzzy ARTMAP (FA) neural network classifier with voting strategy is used with this extracted VEP to discriminate alcoholics from normal subjects. The VEP is extracted from subjects while seeing visuals of Snodgrass and Vanderwart picture set. The high FA classification of 96.5% shows the validity of the proposed method to successfully remove EEG contamination.

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=955269

We look at the design of oversampled filter banks and the resulting framelets. The framelets we design can improve shift invariant properties over decimated wavelet transform. Shift invariance has applications in many areas particularly denoising and coding and compression. Our contribution here is on filter bank completion. We develop factorization methods to find wavelet filters from given scaling filters. We look at a special class of framelets from a filter bank perspective, in that we design double density filter banks (DDFB's). We denote the z-transform of a sequence h(.) as H(z) and its Fourier transform as Hf(ω). Now, for the perfect reconstruction, i.e. Y(z) = X(z), it must be necessary that (1) H0(z) H˜0(z) + H1(z) H˜1(z) + H2(z) H˜2(z) = 2, (2) H0(z) H˜0(-z) + H1(z) H˜-1(z) + H2(z) H˜-2(z) = 0. Alternatively we can write the above perfect reconstruction conditions in the polyphase domain. The two polyphase matrices are given, where H˜(z) is a type 1 analysis polyphase matrix, and H(z) is the type 2 synthesis polyphase matrix, we can write the perfect reconstruction conditions as [H(z)]T H˜(z) = I.

Abstract—This paper deals with continued fraction expansion (CFE) based indirect discretization scheme for finding the rational approximation of fractional order differentiators and integrators and their discretized transfer functions. Indirect discretization approach is used for Al-...

We look at the design of oversampled filter banks and the resulting framelets. The framelets we design can improve shift invariant properties over decimated wavelet transform. Shift invariance has applications in many areas particularly denoising and coding and compression. Our contribution here is on filter bank completion. We develop factorization methods to find wavelet filters from given scaling filters. We look at a special class of framelets from a filter bank perspective, in that we design double density filter banks (DDFB's). We denote the z-transform of a sequence h(.) as H(z) and its Fourier transform as Hf(ω). Now, for the perfect reconstruction, i.e. Y(z) = X(z), it must be necessary that (1) H0(z) H˜0(z) + H1(z) H˜1(z) + H2(z) H˜2(z) = 2, (2) H0(z) H˜0(-z) + H1(z) H˜-1(z) + H2(z) H˜-2(z) = 0. Alternatively we can write the above perfect reconstruction conditions in the polyphase domain. The two polyphase matrices are given, where H˜(z) is a type 1 analysis polyphas...