z-Transform

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:

Z Transform Tables Pdf

with

z_i ... z-transformed sample observations
x_i ... 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 (μ=0, s=1).

Z Transform Table. Some of the more commonly occuring Z transforms are shown below. Delta function, step, ramp, parabola, power, exponent, sine, cosine and damped sine and cosine functions. Return to main page Return to page index.

1. Z 1 d T 1t2 a n 0 d1t nT2 1 1 e Ts z z 1 1 s APPENDIX J z-Transform Table. Title: APPJ.qxd Author: SSEN 16 Created Date: 12/8/2008 10:41:20 AM.
2. Proofs for Z-transform properties, pairs, initial and final value. Includes derivative, binomial scaled, sine and other functions.
3. Z Transform Pairs Time Domain. Z Domain z z-1 k (unit impulse) 1 1 γk † (unit step) z (z) z1 1 1 (z) 1z ak z za 1 1 1 z a e-bTk bT z ze 1 bT 1 1 z e k 2 z z1 1 1 2 z 1z sin(bk) 2 zsin(b) z 2zcos(b) 1 1 12 z sin(b) 1 2z cos(b) z cos(bk) 2 z z cos(b) z 2zcos(b) 1 1 12 1 z cos(b) 1 2z cos(b) z aksin(bk) 22 azsin(b) z 2azcos(b) a 1 1 2 2 az.

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.

Inverse Z Transform Calculator

Z Transform Table With Roc