The Z transformation is used whenever you have to design a digital filter or analyze a discrete-type system. It is also used to analyze the output of a system by looking at the system response in terms of exponential and sinusoidal components. It is the discrete-time equivalent of the Laplace Transform. It converts a discrete time domain signal into a discrete frequency domain signal. A Z -transform requires you to know the sample time and integration method for your simulation. This method does not require you to explicitly code any integrators but you do need to use a different Z transform if you change the sample time of your simulation. In the Z-transform method you replace all of the inverse Z terms with a unit delay. The equation for the Z-transform of a sampled time-domain signal is
$$X(z)=\sum_{n=0}^\infty x[n]z^{-n}$$
This is identical to the Discrete-Time Fourier transform signal but with the additional real term \(r^{-n}\)
$$X(\omega)=\sum^\infty_{n=0}x[n]\cdot e^{-i\omega n}\cdot r^{-n}$$
If \(r<1\) the curve increases exponentially, and when \(r>1\) the curve decreases exponentially. This means the correlation looks for sinusoids and exponentials to correlate in the signal. Therefore we can pull out the \(-n\) exponent and get our equation for \(z\)
$$z^{-n}=(r\cdot e^{i\omega})^{-n}$$
$$z=re^{i\omega}$$
The continuous time equation
$$y[n-k]
$$
becomes
$$
Y\cdot z^{-k}
$$
The exponent of z is negative if there is a time delay, and it is positive if there is a time advance.
in the numerator: \(\frac{1}{z}\) becomes \(u_{i-1}\) \(\frac{1}{z^2}\) becomes \(u_{i-2}\) and coefficients without a z become \(u_i\). For the denominator the above rules become \(y\) instead of \(u\). Therefore the Z domain converts into the time domain.
Z domain:
$$\frac{Y(z)}{U(z)} = 1 + \frac{1}{z}$$
converts to time domain:
$$y_i=u_i+u_{i-1}$$
It can be thought of as corelating each time-domain sample with with every possible value of \(z^{n}\) with z being a complex number. The locations of the poles and zeros are sufficient to determine the magnitude and phase of every other point. It can be thought of as mapping the S plane onto a unit circle.
The Z-Transform can be used to analyze the frequency response as well as the stability of a system using LTI difference equations. The Z-Transform needs to make assumptions about infinite data, otherwise there are convergence issues to to a finite set of samples.
IIR Filters – the Z transform can be used to transform an analog filter to an IIR filter.
[[Finding Transfer Functions Using The Z-transform]]
[[Bilinear Transform]] – another type of discretizing transformation.
[[LTI Systems]] – The Z transform is used to analyze the frequency and stability of LTI systems
[[First-Order IIR Filter Stability Analysis]] – take the Z transform of the difference equation
[[Z-Transform Frequency Response Analysis]]
Discrete Impulse Signal
[[Discrete-Time Fourier Transform]] – similar to the z-transform but without the $r^{-n}$ term
Poles of a System – z-transform used to find poles
[[Impulse Invariance Method]] – another z-transformation method
[[Parameter Estimation of Aircraft Dynamic Models]] – uses recursive z-transform
[[Discrete Time Controller]] – desired eigenvalues must be converted into discrete time
Advantages of IIR filters – you can use the Z-transform method to mimic analog filter performances
Sources
- [1] MATLAB, 4 Ways to Implement a Transfer Function in Code | Control Systems in Practice, (Jan. 08, 2020). Accessed: Nov. 01, 2022. [Online Video]. Available: https://www.youtube.com/watch?v=nkq4WkX7CFU
- [2] D. Boschen, “Answer to ‘How/why are the $\mathcal Z$-transform and unit delays related?,’” Signal Processing Stack Exchange. Accessed: Feb. 19, 2023. [Online]. Available: https://dsp.stackexchange.com/a/31841
- [3] Phil’s Lab, Z-Transform – Practical Applications – Phil’s Lab #27, (Jul. 24, 2021). Accessed: Apr. 15, 2023. [Online Video]. Available: https://www.youtube.com/watch?v=5Z3KAKs-EZs
- [4] “Understanding the Z-Transform – YouTube.” Accessed: May 13, 2023. [Online]. Available: https://www.youtube.com/
Backlinks
Discrete Systems
[[Discrete-Time Fourier Transform]]
[[Implementing Transfer Functions]]
[[Laplace Transform]]
[[LTI Systems]]
Poles of a System
[[Signal Correlation]]
[[Unit Circle]]
[[Z-Domain]]
[[Zeros of a System]]