Built-in Mathematica Symbol

ZTransform

ZTransform[expr, n, z]
gives the Z transform of expr.

ZTransform[expr, {n₁, n₂, ...}, {z₁, z₂, ...}]
gives the multidimensional Z transform of expr.

MORE INFORMATION

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

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Basic Examples (3)

Transform a sequence:

Transform a multivariate sequence:

Transform a symbolic sequence:

Transform a sequence:

Transform a multivariate sequence:

Transform a symbolic sequence:

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Scope (23)

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:

ZTransform automatically threads over lists:

Equations:

Rules:

TraditionalForm typesetting:

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:

Multivariate transforms:

Linearity:

Several relation exist to the InverseZTransform:

Shifts:

Polynomial multiplication:

Exponential multiplication:

Differences and shifts:

Sums:

Integrals:

Options (4)

Without assumptions typically a general formula will be produced:

Use Assumptions to obtain the expression on a given range:

Set GenerateConditions to True to get the region of convergence:

Different methods may produce different results:

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:

ExponentialGeneratingFunction:

FourierSequenceTransform:

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: