- Z Transform Table Discrete Time
- Z Transform Table For Normal Distribution
- Z Transform Tables
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z-Transform
Transform pair Table. The inverse z-transform equation is complicated. The easier way is to use the -transform pair table Time-domain signal z-transform ROC 1) πΏα½ α½ 1 All 2) π’α½ α½ 1 1β β1 1 3) βπ’α½β β1α½ 1 1β β1 0. Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) Kronecker delta Ξ΄0(k) 1. β β 1 k=0 1 0 kβ 0 Ξ΄0(n-k) 2. β β 1 n=k z-k 0 nβ k 1 1 3. Find inverse z-transform of We get, Using table, Example 7 Example 8. Ghulam Muhammad King Saud University 12 Inverse z- Transform: Examples Find inverse z-transform of Since, By coefficient matching, Therefore, Find inverse z-transform of Example 9 Example 10.
Sometimes one has the problem to make two samples comparable, i.e. to compare measured values of a sample with respect to their (relative) position in the distribution. An often used aid is the z-transform which converts the values of a sample into z-scores:
with
zi ... z-transformed sample observations
xi ... original values of the sample
... sample mean
s ... standard deviation of the sample
The z-transform is also called standardization or auto-scaling. z-Scores become comparable by measuring the observations in multiples of the standard deviation of that sample. The mean of a z-transformed sample is always zero. If the original distribution is a normal one, the z-transformed data belong to a standard normal distribution (ΞΌ=0, s=1).
The following example demonstrates the effect of the standardization of the data. Assume we have two normal distributions, one with mean of 10.0 and a standard deviation of 30.0 (top left), the other with a mean of 200 and a standard deviation of 20.0 (top right). The standardization of both data sets results in comparable distributions since both z-transformed distributions have a mean of 0.0 and a standard deviation of 1.0 (bottom row).
| Hint: | In some published papers you can read that the z-scores are normally distributed. This is wrong - the z-transform does not change the form of the distribution, it only adjusts the mean and the standard deviation. Pictorially speaking, the distribution is simply shifted along the x axis and expanded or compressed to achieve a zero mean and standard deviation of 1.0. |
Z Transform Table Discrete Time
Using this table for Z Transforms with Discrete Indices
Shortened 2-page pdf of Laplace Transforms and Properties
Shortened 2-page pdf of Z Transforms and Properties
All time domain functions are implicitly=0 for t<0 (i.e. they are multiplied by unit step).
Z Transform Table For Normal Distribution
| Entry | Laplace Domain | Time Domain (Note) All time domain functions are implicitly=0 for t<0 (i.e. they are multiplied by unit step). | Z Domain (t=kT) |
|---|---|---|---|
| unit impulse | unit impulse | ||
| unit step | (Note) u(t) is more commonly used to represent the step function, but u(t) is also used to represent other things. We choose gamma (Ξ³(t)) to avoid confusion (and because in the Laplace domain (Ξ(s)) it looks a little like a step input). | ||
| ramp | |||
| parabola | |||
| tn (n is integer) | |||
| exponential | |||
| power | |||
| time multiplied exponential | |||
| Asymptotic exponential | |||
| double exponential | |||
| asymptotic double exponential | |||
| asymptotic critically damped | |||
| differentiated critically damped | |||
| sine | |||
| cosine | |||
| decaying sine | |||
| decaying cosine | |||
| generic decaying oscillatory | |||
| generic decaying oscillatory (alternate) | (Note) atan is the arctangent (tan-1) function. The atan function can give incorrect results (typically the function is written so that the result is always in quadrants I or IV). To ensure accuracy, use a function that corrects for this. In most programming languages the function is atan2. Also be careful about using degrees and radians as appropriate. | ||
| Z-domain generic decaying oscillatory | (Note) | ||
| Prototype Second Order System (ΞΆ<1, underdampded) | |||
| Prototype 2nd order lowpass step response | |||
| Prototype 2nd order lowpass impulse response | |||
| Prototype 2nd order bandpass impulse response | |||
Using this table for Z Transforms with discrete indices
Commonly the 'time domain' function is given in terms of a discrete index, k, rather than time. This is easily accommodated by the table. For example if you are given a function:
Since t=kT, simply replace k in the function definition by k=t/T. So, in this case,
Z Transform Tables
and we can use the table entry for the ramp
The answer is then easily obtained