# ZTransform

ZTransform[expr,n,z]

gives the Z transform of expr.

ZTransform[expr,{n1,n2,},{z1,z2,}]

gives the multidimensional Z transform of expr.

# Details and Options • The Z transform for a discrete function is given by .
• The multidimensional Z transform is given by .
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate answers that involve conditions on parameters Method Automatic method to use VerifyConvergence True whether to verify convergence
• In TraditionalForm, ZTransform is output using .

# Examples

open allclose all

## Basic Examples(3)

Transform a sequence:

Transform a multivariate sequence:

Transform a symbolic sequence:

## Scope(25)

### Basic Uses(7)

Transform a univariate sequence:

Transform a multivariate sequence:

Compute a typical transform:

Plot the magnitude using Plot3D, ContourPlot, or DensityPlot:

Plot the complex phase:

Generate conditions for the region of convergence:

Plot the region for :

Evaluate the transform at a point:

Plot the spectrum:

The phase:

Plot both the spectrum and the plot phase using color:

Plot the spectrum in the complex plane using ParametricPlot3D:

ZTransform will use several properties including linearity:

Shifts:

Multiplication by exponentials:

Multiplication by polynomials:

Conjugate:

Equations:

Rules:

### Special Sequences(13)

Discrete impulses:

Discrete unit steps:

Discrete ramps:

Polynomials result in rational transforms:

Factorial polynomials:

Exponential functions:

Exponential polynomials:

Factorial exponential polynomials:

Trigonometric functions:

Trigonometric, exponential and polynomial:

Combinations of the previous input will also generate rational transforms:

Different ways of expressing piecewise defined signals:

Rational functions:

Rational exponential functions:

Hypergeometric term sequences:

The DiscreteRatio is rational for all hypergeometric term sequences:

Many functions give hypergeometric terms:

Any products are hypergeometric terms:

Transforms of hypergeometric terms:

Holonomic sequences:

A holonomic sequence is defined by a linear difference equation:

Many special function are holonomic sequences in their index:

Special sequences:

Periodic sequences:

Multivariate transforms:

Multivariate periodic sequences:

### Special Operators(5)

Linearity:

There are several relations to the InverseZTransform:

Shifts:

Polynomial multiplication:

Exponential multiplication:

Differences and shifts:

Sums:

Integrals:

## Options(4)

### Assumptions(1)

Without assumptions, typically a general formula will be produced:

Use Assumptions to obtain the expression on a given range:

### GenerateConditions(1)

Set GenerateConditions to True to get the region of convergence:

### Method(1)

Different methods may produce different results:

### VerifyConvergence(1)

By default, convergence testing is performed: Setting will avoid the verification step:

## Applications(1)

Solving difference equations:

## Properties & Relations(6)

ZTransform is closely related to GeneratingFunction:

Use InverseZTransform to get the sequence from its transform:

ZTransform effectively computes an infinite sum:

Linearity:

Shifting:

Convolution:

Derivative:

Initial value property:

Final value property:

## Possible Issues(1)

A ZTransform may not converge for all values of parameters: Use GenerateConditions to get the region of convergence:

## Neat Examples(1)

Create a gallery of Z transforms: