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. Replace each term in the difference equation by its z-transform and insert the initial condi-tion(s). Solve the resulting algebraic equation. (Thus gives the z-transform Y(z) of the solution sequence.) 3. Find the inverse z-transform of Y(z). The third step is usually the most difficult. We will consider the problem of finding inverse z.

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:

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).

Inverse Z Transform Table

Z Transform Table Properties