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.

Pole-zero locations are important for:

Wavelets

Symlets

B-splineis

##### 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