# BilateralZTransform

BilateralZTransform[expr,n,z]

gives the bilateral Z transform of expr.

BilateralZTransform[expr,{n1,,nk},{z1,,zk}]

gives the multidimensional bilateral Z transform of expr.

# Details and Options • The bilateral Z transform is the discrete analog of the bilateral Laplace transform and plays an important role in digital signal processing and other fields.
• The bilateral Z transform for a discrete function is given by .
• The multidimensional bilateral Z transform is given by .
• The sum is computed using numerical methods if the third argument, z, is given a numerical value.
• The bilateral Z transform of exists only for complex values of in an annulus given by . In some cases, the annulus of definition may extend to the exterior or the interior of a disk.
• • The following options can be given:
•  AccuracyGoal Automatic digits of absolute accuracy sought Assumptions \$Assumptions assumptions to make about parameters GenerateConditions True whether to generate answers that involve conditions on parameters Method Automatic method to use PerformanceGoal \$PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations

# Examples

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## Basic Examples(2)

Define an exponentially decaying sequence:

Compute its bilateral Z transform:

Complex plot of the bilateral Z transform:

Compute the transform at a single point:

Compute the bilateral Z transform of a multivariate function:

## Scope(8)

Bilateral Z transform of the UnitStep function:

Discrete power function:

Combination of power functions:

Discrete-time, finite support function:

Trigonometric sequence:

Calculate the bilateral Z transform at a single point:

Alternatively, calculate the transform symbolically:

Then evaluate it for a specific value of :

For some functions, the bilateral Z transform can be evaluated only numerically:

Plot the bilateral Z transform using numerical values only:

## Options(3)

### Assumptions(1)

Specify the range for a parameter using Assumptions:

### GenerateConditions(1)

Set GenerateConditions to False to obtain a result without conditions:

### WorkingPrecision(1)

Use WorkingPrecision to obtain a result with arbitrary precision:

## Applications(2)

Define finite duration signals:

Plot the signals in the time domain:

To find the convolution, first calculate the product of the transforms:

Then, perform inversion back to the time domain:

Plot the convolution in the time domain:

Alternatively, find the convolution using DiscreteConvolve:

Define infinite duration signals:

Plot the signals in the time domain:

To find the convolution, first calculate product of the transforms:

Perform the inversion back to the time domain:

Plot the convolution in the time domain:

Alternatively, find the convolution using DiscreteConvolve:

## Properties & Relations(7)

BilateralZTransform and InverseBilateralZTransform are mutual inverses:

BilateralZTransform is closely related to FourierSequenceTransform:

Linearity:

Time shifting:

Scaling in the -domain:

Convolution:

Differentiation in the -domain:

## Neat Examples(1)

Create a table of basic bilateral Z transforms: