BilateralZTransform

BilateralZTransform[expr,n,z]

gives the bilateral Z transform of expr.

BilateralZTransform[expr,{n₁,…,n_k},{z₁,…,z_k}]

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:

DiscreteDelta:

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:

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BilateralZTransform

Details and Options

Examples

Basic Examples (2)

Scope (8)