Z轉換(Z Transform)

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

Poles and Zeros




Advanced z-transform

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Region of convergence ROC

Z轉換

The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges (i.e. doesn't blow up in magnitude to infinity):



ROC of z-transform is indicated with circle in z-plane.
ROC does not contain any poles.
If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0.
If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-plane except at z = ∞.
If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. |z| > a.
If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e. |z| < a.
If x(n) is a finite duration two sided sequence, then the ROC is entire z-plane except at z = 0 & z = ∞.


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