This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ZTransform

 ZTransformgives the Z transform of expr. ZTransformgives the multidimensional Z transform of expr.
• 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
Transform a sequence:
Transform a multivariate sequence:
Transform a symbolic sequence:
Transform a sequence:
 Out[1]=
 Out[2]=

Transform a multivariate sequence:
 Out[1]=
 Out[2]=

Transform a symbolic sequence:
 Out[1]=
 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:
Equations:
Rules:
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:
There are several relations 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:
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:
A ZTransform may not converge for all values of parameters:
Use GenerateConditions to get the region of convergence: