Mathematics Yoshio Sep 8th, 2023 at 8:00 PM 8 0
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation.
The frequency response of a filter is determined by the interaction of a unit vector rotating around the unit circle with the poles and zeros of the filter.
The unit vector at rotation ω=0 corresponds to DC (0Hz).
The unit vector at rotation ω=π (180°) corresponds to Fs/2 or the Nyquist frequency.
When the tip of the unit vector gets close to a zero, the filter magnitude response is pushed downwards because zero is a root of the numerator polynomial. When the tip of the unit vector gets close to a pole, the filter magnitude response is pushed upwards because a pole is a root of the denominator polynomial.
![](https://vru.vibrationresearch.com/wp-content/uploads/2021/06/zdomain-frequency-response.png)
Pole-zero locations are important for:
Wavelets
Symlets
B-splineis
![](https://d23s79tivgl8me.cloudfront.net/user/14446/fig.%202_65944.jpg)
![](https://www.oreilly.com/api/v2/epubs/9780133038446/files/graphics/0271_fig01.jpg)
![](https://vru.vibrationresearch.com/wp-content/uploads/2021/06/zdomain-poles-zeros.png)
![](http://lcs-vc-marcy.syr.edu:8080/_images/image_4_2_4.png)
![](https://www.researchgate.net/profile/Larry-Reed/publication/364332657/figure/fig1/AS:11431281089928761@1665795442358/4-Example-pole-zero-map-of-pole-locations-and-discrete-time-domain-impulse-frequency.png)
![](https://www.oldfriend.url.tw/images/Designer/FFT/18.gif)
Nyquist frequency
The Nyquist frequency (also called the folding frequency), named after Harry Nyquist, is a characteristic of a sampler. It converts a continuous function or signal into a discrete sequence, it is the frequency you need to sample an analog signal at in order to reconstruct it adequately. The Nyquist frequency is defined as 2*(freq. of original signal).
Usually in practical cases, 5 to 10 times frequency of original signal is selected.
Laplace transform
Hilbert Transform
Gabor Transform
Riesz transform
wavelet transform
Markov chain
Z-transform
Advanced z-transform
Matched Z-transform method
Bilinear transform
Constant-Q transform
Impulse invariance Integral transform
Post's inversion formula Starred transform
Zak transform
Kirchhoff's Law, Junction & Loop Rule, Ohm's Law