• Z Transform Table
• Inverse Z Transform Calculator
• Find Z Transform Calculator
• Inverse Z Transform Calculator Online

Found this: A more elegant way is to go back to the time domain and compute the Z-transform sum. Or just use the tables that have become very popular Table of Laplace and Z Transforms. 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.

Calculator Use

The z-score is the number of standard deviations a data point is from the population mean. You can calculate a z-score for any raw data value on a normal distribution.

When you calculate a z-score you are converting a raw data value to a standardized score on a standardized normal distribution. The z-score allows you to compare data from different samples because z-scores are in terms of standard deviations.

A positive z-score means the data value is higher than average. A negative z-score means it's lower than average.

You can also determine the percentage of the population that lies above or below any z-score using a z-score table.

Using the Z-Score Calculator

This calculator can find the z-score given:

• A raw data point, population mean and population standard deviation
• Sample mean, sample size, population mean and population standard deviation
• A sample that is used to calculate sample mean and sample size; population mean and population standard deviation

With the first method above, enter one or more data points separated by commas or spaces and the calculator will calculate the z-score for each data point provided from the same population.

With the last method above enter a sample set of values. Enter values separated by commas or spaces.

You can also copy and paste lines of data from spreadsheets or text documents. See all allowable formats below.

Z-Score Formula

When calculating the z-score of a single data point x:

[ z = dfrac{x - mu}{sigma} ]

When calculating the z-score of a sample with known population standard deviation:

[ z = dfrac{overline{x} - mu}{dfrac{sigma}{sqrt{n}}} ]

In these z-score formulas:

• x is a raw data point
• x̄ is the sample mean
• n is the sample size
• μ is the population mean
• σ is the population standard deviation.

Given a Z domain function, there are several ways to perform an inverse Z Transform:

The only two of these that we will regularly use are direct computation and partial fraction expansion.

Inverse Z Transform by Long Division

Z Transform Table

To understand how an inverse Z Transform can be obtained by long division, consider the function

If we perform long division

we can see that

.

So the sequence f[k] is given by

Upon inspection

Note: We already knew this because the form of F(z) is one that we have worked with previously (i.e., the exponential function).

This technique is laborious to do by hand, but can be reduced to an algorithm that can be easily solved by computer.

Inverse Z Transform by Direct Computation

The need for this technique, as well as its implementation, will be made clear when we consider transfer functions in the Z domain. We will present this method at that time.

Inverse Z Transform by Partial Fraction Expansion

This technique uses Partial Fraction Expansion to split up a complicated fraction into forms that are in the Z Transform table. If you are unfamiliar with partial fractions, here is an explanation.

As an example consider the function

For reasons that will become obvious soon, we rewrite the fraction before expanding it by dividing the left side of the equation by 'z.'

Inverse Z Transform Calculator

Now we can perform a partial fraction expansion

These fractions are not in our table of Z Transforms. However if we bring the 'z' from the denominator of the left side of the equation into the numerator of the right side, we get forms that are in the table of Z Transforms; this is why we performed the first step of dividing the equation by 'z.'

So

or

Example

Verify the previous example by long division.

So

and the sequence f[k] is given by

Find Z Transform Calculator

Inverse Z Transform by Direct Inversion

This method requires the techniques of contour integration over a complex plane. In particular

.

The contour, G, must be in the functions region of convergence. This technique makes use of Residue Theory and Complex Analysis and is beyond the scope of this document. Reference.

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik CheeverDepartment of EngineeringSwarthmore College

Inverse Z Transform Calculator Online