z-Transform

• Z Transform Table Pdf
• Inverse Z Transform Table
• Z Transform Table For Normal Distribution

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 ’re &jT EQUATION 33-1 The z-transform. The z-transform defines the relationship between the time domain signal, x n, and the z-domain signal, X (z). X (z) ’ j 4 n ’&4 x n z &n in the Laplace transform by introducing a new complex variable, s, defined to be: s ’F%jT. In this same way, we will define a new variable for the z-transform.

• This is the first part of a very concise and quite detailed explanation of the z-transform and not recommended for those dealing with the z-transform for the.
• Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1. – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. – – δ0(n-k) 1 n = k 0 n ≠ k z-k 3. S 1 1(t) 1(k) 1 1 1 −z−.

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).

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).

Z Transform Table Pdf

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 Table

Z Transform Table For Normal Distribution