gives the Z transform of expr.


gives the multidimensional Z transform of expr.

Details and Options


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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:


Multiplication by exponentials:

Multiplication by polynomials:


ZTransform automatically threads over lists:



TraditionalForm typesetting:

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)


There are several relations to the InverseZTransform:


Polynomial multiplication:

Exponential multiplication:

Differences and shifts:



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 VerifyConvergence->False 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:





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:

Introduced in 1999
Updated in 2008